Research ArticleEnergy Effective Congestion Control for Multicast withNetwork Coding in Wireless Ad Hoc Network

Chuanxin Zhao1 Yonglong Luo1 Fulong Chen1 Ji Zhang2 and Ruchuan Wang3

1 Department of Computer and Technology Anhui Normal University Wuhu 241002 China2Department of Mathematics and Computing University of Southern Queensland Toowoomba QLD 4350 Australia3 College of Computer Nanjing University of Posts and Telecommunications Nanjing 210003 China

Correspondence should be addressed to Chuanxin Zhao zcxonline126com

Received 16 February 2014 Accepted 18 April 2014 Published 22 May 2014

Academic Editor Changzhi Wu

Copyright copy 2014 Chuanxin Zhao et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to improve network throughput and reduce energy consumption we propose in this paper a cross-layer optimizationdesign that is able to achieve multicast utility maximization and energy consumption minimization The joint optimizationof congestion control and power allocation is formulated to be a nonlinear nonconvex problem Using dual decomposition adistributed optimization algorithm is proposed to avoid the congestion by control flow rate at the source node and eliminate thebottleneck by allocating the power at the intermediate node Simulation results show that the cross-layer algorithm can increasenetwork performance reduce the energy consumption of wireless nodes and prolong the network lifetime while keeping networkthroughput basically unchanged

1 Introduction

As a practical solution for the broadband wireless Internetwireless multihop networks can provide good geographiccoverage with low cost Nodes at different locations commu-nicate with each other by relaying information over wirelesslinks However wireless links suffer limited capacity due tolimited resource fading channel and mutual interferenceand so forth Multiuser resource assignment in interference-limited wireless network is a complex yet critical issue Themutual interference among users is a major factor in limitingthe performance of communication systems However itcan be efficiently mitigated by carefully allocating wirelessresources such as transmission power and frequency bandsAn important consideration in the design of a multihopnetwork is the networkrsquos ability to efficiently support highthroughput multicast applications over wireless links One ofthemain challenges for designing high throughput inwirelessmultihop networks is the interference of multiple wirelesslinks Recently some techniques have been developed forenhancing the performance of wireless multihop networks

which include cross-layer design that considers networkutility maximization problem [1] and network coding whichallows intermediate nodes to perform coding operations inaddition to pure packet forwarding [2]

The network utility maximization (NUM) framework hasbeen applied widely in network for rate allocation throughcongestion control protocols [1] Congestion control regu-lates the source rates to avoid overwhelming link capacityOn the other hand feasible power allocation can efficientlyenhance link capacity Multicast flow causes more congestionthan unicast traffic due to the fact that multicast flow canbe distributed in large multicast trees With the multimediaapplications becoming more popular in wireless networkhow to enhance network performance for multicast applica-tions is an urgent issue This paper addresses the challengestogether by considering a joint optimization of multicastcongestion and power allocation for a wireless multihopnetwork We focus on optimal congestion control and powerallocation use the network utility maximization frameworkto design a cross-layer optimizationmodel and then proposean efficient distributed algorithm to solve the problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 135945 12 pageshttpdxdoiorg1011552014135945

2 Mathematical Problems in Engineering

The remainder of this paper is organized as followsThe related works are described in Section 2 and energyefficient utilitymodel is introduced in Section 3 A distributedalgorithm will be presented in Section 4 Finally a numericalsimulation is conducted to evaluate the performance of theproposed distributed algorithm

2 Related Work

Most resource allocation problems in wireless network canbe formulated as network utility maximization (NUM) prob-lems which optimize the resource allocation as a wholethrough crossing different layers in a communication net-work The network utility maximization problem is dividedinto multiple subproblems through dual decompositionwhere each decomposed subproblemcorresponds to resourceallocation on one layer In network utility maximizationthe utility function represents an objective to be maximizedwhile the constraints represent different underlying networkcharacteristics The primal variables correspond to resourcesand the Lagrange dual variables correspond to the interfacesamong the layers NUM substantially expands the scope ofthe classical network flow problem that relies on nonlinearconcave utility objective functions Moreover there is anelegant economic interpretation of the dual-based distributedalgorithm where the Lagrange dual variables can be inter-preted as shadow prices for resource allocation and each enduser in the network tries to maximize their net utilities andnet revenue respectively [3]

It has been widely recognized that cross-layer designcan potentially lead to substantial performance of wirelessnetworks [4] Price and Javidi investigated the distributedrate assignment in CDMA-based wireless networks andpresented a distributed rate assignment algorithm based onthe congestion and interference constraints at the MAC andtransport layer to enhance network capacity [5] Ghasemi andFaez [6] presented a new algorithm for jointly optimal controlof multicast session rate link attempt rate and link powerin contention based multihop wireless networks where thecross-layer resource allocation was formulated as a nonlinearoptimization The optimization variables were coordinatedthrough two shadow prices and a distributed algorithm forupdating those variables was proposed It was shown thatit is possible to achieve increased network throughput anddecrease the sessionrsquos end-to-end delay A jointly optimalcongestion control and power control algorithm for a generalad hoc network was presented in [7] to enhance the overallnetwork performance and it proposed a framework to adaptphysical layer resource allocation to enhance the end-to-end utilities In fact the mutual interference links could beactivated synchronously if the transmission power of thedifferent senders were properly adjusted [8] The work in [8]showed that it was of a great importance to choose a propertransmitting power in the interference-limited environmentThe increasing of power might result in high interferencewhich resulted in the total throughput utility reduction and awaste of energy Zhang and Lee studied energy efficient utilitymaximization for wireless networks withwithout multipath

by formulating them into convex program [9] The problemwas solved by a distributed dual decomposition algorithmvan Nguyen et al studied efficient and fair power allocationassociated with congestion control in orthogonal frequencydivision multiplexing (OFDM) based multihop cognitiveradio networks [10] They considered mutual relationshipthrough a cross-layer optimization design that addressedboth aggregate utility maximization and energy consump-tion minimization with outage constraint of primary userYuan et al proposed a cross-layer optimization frameworkfor throughputmaximization jointmulticast routing problemand the wireless medium contention problem in wirelessmesh networks [11] however they did not consider the issueof multicast congestion control

On the other hand network coding has extended thefunctionality of network nodes from storingforwardingpackets to performing algebraic operations on received dataStarting with the work of [12] employing network coding atintermediate nodes is advantageous to maximize multicastthroughput Various potential benefits of network codinghave been shown which include the improvement of multi-cast sessionrsquos throughput and the reduction of the overheadof probabilistic routing [13] Among the numerous networkcoding schemes distributed random linear network codingreceives more attention as it independently and randomlyselects linear mappings from inputs onto output links Thisencoding scheme has been implemented in practice whichis widely applied to P2P systems network security andnetworkmonitoring andmanagement [14 15] Chen et al [16]considered a flow control for network coding-basedmulticastflows inwired network [16] In their work optimizationmod-els were formulated for network resource allocation whichincluded two sets of decentralized controllers at sources andlinksnodes for congestion control and they are developedfor wired networks with given coding subgraphs and withoutgiven coding subgraphs respectively Based on their workwe will consider both flow and power control for networkcoding-based multicast flows in interference-limited wirelessmultihop network

3 Problem Formulation

31 Network Model We consider a wireless multihop net-work with 119871 = 1 |119871| logical links shared by 119873 =

1 |119873| nodes which are equipped with multiradiosworking on the orthogonal channels It is also assumed thatthere are119872 = 1 |119872|multicast sessions in the networkwhere each multicast session 119898 isin 119872 has a source node119904119898

isin 119873 and a destination node set 119863119898

sub 119873 In order toimprove the utilization of the network the random linearnetwork coding is assumed to be used in the multicast flowsand the network coding allows the flows of amulticast sessionto share network capacity by coding them together For thecase ofmultiple sessions sharing a network achieving optimalthroughput requires coding across sessions However it isdifficult to design such network codes Thus we limit ourconsideration to codes within each session

Mathematical Problems in Engineering 3

Traditionally resource allocations are optimized sepa-rately which is difficult to support high quality communi-cation In this paper we focus on the joint optimization ofmulticast flow rate control and power allocation where themulticast tree is established and the channels are assigned inadvance such as in [17] In the wireless network the multipleinformation flows share the network links and it is mandatedthat all the information flows on the links do not exceed thephysical channel capacity The data received by intermediatenode should be forwarded (the source nodes send data whilethe destination nodes receive data only) The relationship ofeach information flow and physical flow can be expressed as

sum

119895(119894119895)isin119871

119892119898119889

119894119895minus sum

119895(119895119894)isin119871

119892119898119889

119895119894=

119909119898 if 119894 = 119904

119898

minus119909119898 if 119894 = 119889

0 otherwiseforall119889 isin 119863

119898

119892119898119889

119894119895le 119891119898

119894119895 forall119889 isin 119863

119898

(1)

Here 119909119898 is the source rate and 119892119898119889

119894119895represents the flow of

session 119898 on link (119894 119895) sent to the destination node 119889 Linkcapacity can be shared by the flows on the links when therandom linear network coding is adopted and the limit ofnetwork capacity can be expressed as

119891119898

119894119895= max119889

119892119898119889

119894119895 119889 isin 119863

119898 (2)

Now let us explain with the help of Figure 1 multicastsession model which is similar to the model in [16] which isa typical butterfly diagram 119904 is the source node of the sessionwhile 119889

1and 119889

2are the destination nodes Network coding

can be performed on the wireless links (119908 V) to share linkcapacity for improving network capacity

Figure 1 shows a multicast session where the flow is sentfrom the source node 119904 to left and right trees which are furtherdepicted in Figure 2 The encoding can be performed onthe wireless links shared by the different multicast trees Asshown in Figure 1 the link (119908 V) is the shared part of twotrees the multicast trees can share link capacity by networkcoding themax rate of information flow for leftmulticast treeis 2 and the right multicast tree is 1

Figure 2 shows the multicast subtrees partitioned byFigure 1 In the figure the source node separates the flowsent to the destination node into two subtrees each of whichcan change the transmission rate through adjusting at thesource node As shown in Figure 2 in a multicast sessionthe link using random linear network coding can achievemaximum upper bound in the ideal case In order to expressthe corresponding relationship of the link 119897 and multicast 119898we define a multicast matrix 119867

119898

(|119867119898

| = |119871| lowast |119878|) and theelements of the matrix are defined as follows [16]

119867119898

119897119904=

1 119897 isin 119879119898119904

0 others(3)

Here 119879119898119904

represents the link set of multicast tree The sourcenode in a multicast tree 119879

119898119904sends the same information

s

t u

w

d1 d2

b1

b1

a1 a

2

a 1a

2

a 1 a 2

a 1+b1

a2

a1 + b1

a1 +

b1 a

2

Figure 1 Typical butterfly diagram

s s

t u

w w

d1 d1d2 d2

b1

b1b1

b1

b1

a1 a

2

a1 a

2

a 1a

2

a 1 a

2

a 1 a 2

Figure 2 Network coding at the shared link (119908 V) corresponded tothe multicast tree decomposition

flow to the destination nodes For the sake of simplicity weassume that network coding occurs within session and theconstraints of the physical flow rate for each multicast stream119898 on link 119897 can be expressed as

max119904

119867119898

119897119904119909119898119904

le 119888119897

forall119897 isin 119871 (4)

32 Network Interference Model In an interference-limitedwireless network the information theoretic capacity 119888

119897of

each link 119897 is not fixed It can be considered as a functionof the transmit power the interference of adjacent linkbandwidth and modulation and so forth Assuming that themodulation scheme has been determined the informationtheory capacity 119888

119897of link 119897 is considered as the function of the

transmit power and channel conditions as follows

119888119897(119875) = 119861 sdot log (1 + 119870SINR

119897(119875)) (5)

4 Mathematical Problems in Engineering

where 119861 is bandwidth 119875 is the vector of transmission powerand 119870 is a constant which depends on the modulation andbit error rate Without loss of generality we assume that119870 = 1 SINR

119897represents the SINR of the link 119897 where SINR

119897

is defined as

SINR119897(119875) =

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

(6)

where 119899119897represents white Gaussian noise of link 119897 at the

receiving node 119866119897119897

is the fading coefficient of the link 119897

from the transmitting node to the receiving node sum119896 = 119897

119866119896119897

is the interference of other links on the receiving node 119875119897is

transmit power of source node and 119875119896is interference power

of other nodes Assuming that the CDMA system is adoptedin the network with reasonable spreading gain and undernormal circumstances 119866

119897119897≫ 119866119896119897 119896 = 119897 (eg SINR gt 5) [9]

due to the fact that the interference is much smaller than thesignal power Equation (5) can be approximated instead as[18]

119888119897(119875) = log(

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

) (7)

In wireless multihop networks the sum of interferenceterms over 119896 = 119897 can be conducted in practice only over theactive links in the two-hop neighborhood Interference infor-mation can be measured or obtained by mutual broadcastingof neighbor nodes

33 Energy Efficient Multicast Rate Control Model The wire-less nodes are generally powered by a battery so powercontrol may conserve energy and reduce the interferenceamong nodes In this paper we establish the model forenergy efficient multicast rate control and power allocationoptimization The purpose of such optimization is to max-imize the fair allocation of link bandwidth and minimizepower for the network multicast session flow and reduceenergy consumption Assuming that 119880

119898(119909119898) is the utility of

each multicast 119898 in the network and the utility function isseparable where 119909

119898= sum119904119909119898119904

is the sum of data rates whichare distributed on the respective trees then the problem canbe formulated as P1 Consider

P1 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897) (8a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897 (8b)

119875119897ge 0 sum

119897isin119900(119894)

119875119897le 119875

max119894

119894 isin 119873 (8c)

where 119880119898is a continuously differentiable strictly concave

and incremental function 120573 is the equilibrium coefficientand 119881

119897is the cost function For simplicity the cost function

can be expressed as the weights of power value as

119881119897(119875119897) = 119908119897119875119897 119908119897ge 0 (9)

Network performance includes not only the networkthroughput but also the fairness of user which can be

balanced through selecting the appropriate utility functionWedesign the utility functionwhich can be flexibly adjustableby parameters as follows with reference to [19]

119880120572

119904(119909119904) =

119901119904(1 minus 120572)

minus1

1199091minus120572

119904 120572 = 1 120572 ge 0

119901119904log119909119904 120572 = 1

(10)

where 119901119904represents the weight of different flows and 120572 is the

fairness parameter While 120572 = 1 corresponds to proportionalfairness 120572 = 2 corresponds to the harmonic mean fairnessand 120572 rarr infin corresponds to the max-min fairness Theweight of eachmulticast session flow is set to be 1 in the paperwhich means that 119880

119898(119909119898) = log(119909

119898)

4 Optimization Approachand Distributed Algorithm

P1 is a nonconvex optimization problem since the linkconstraint (8b) is a nonconvex region However a simplevariable transformation119875

119897= log(119875

119897) can be used to transform

the problem into an equivalent convex optimization problemP2 Consider

P2 max sum

119898

119880119898

(sum

119904

119909119898

119904) minus 120573sum

119897

119881119897(119875119897)

st sum

119898

119867119898

119897119904119909119898

119904le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873

(11)

Furthermore the objective function of P2 is not strictlyconcave since some sources have multiple alternative mul-ticast trees that may cause an instability problem in theconvergence of an iterative algorithmmdasha persistent oscilla-tion of the flow rate around the optimal value It impliesthat although the dual variables may converge the primalvariables flow rates and transmit powers may not To dealwith this instability problem we use ideas from proximaloptimization algorithms [13 page 232] The basic idea isthat instead of P3 we try to solve an equivalent problem byintroducing a quadratic term of some auxiliary variables 119910

119898119904

so that the optimization problem becomes strictly concavewith respect to 119909

119898119904 Consider

P3 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

(12a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

(12b)

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873 (12c)

Mathematical Problems in Engineering 5

Due to the fact that 119888119897(119875) = log(119866

119897119897) + 119875

119897minus

log(sum119896 = 119897

119866119896119897exp(119875

119896) + 119899119897) is sum of a linear function and a

log-sum-exp function problem (12a) is convex according tothe theory of convex optimization

Theorem 1 The transformed problemP2 is a convex optimiza-tion problem

Proof The constraints (12b) and (12c) are convex in (119909 119875)

since the log-sum-exp is convex in its domain [20]Moreoverthe utilities in (12a) are assumed to be strictly concaveTherefore P2 is convex in (119909 119875)

Hence the maximized value of (12a) is unique ProblemP2 can be solved by Lagrange dual decomposition whichis the most important optimization method and has beenwidely used in constrained optimization problem The orig-inal problem P3 is decomposed into two suboptimizationproblems

We define the Lagrangian as

119871 (119909 119910 119875 120582 120583) = sum

119898

119880119898

(119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

minus sum

119897

120582119897(sum

119898

119867119898

119897119904119909119898119904

minus 119888119897(119875))

minus sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898

119867119898

119897119904119909119898119904

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

+ sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

(13)

where 120582 is the vector of Lagrange multiplier which isassociated with capacity constraints (12b) while 120583 is thevector of Lagrangemultipliers which is associatedwith powerconstraints (12c) and 119875

119897and 119909

119898119904are primal variables The

dual problem P3 can be expressed as an unconstrained min-max problem

min120582120583ge0

119863(120582 120583) (14)

where the dual function

119863(120582 120583) = max 119871 (119909 119910 119875 120582 120583)

= max 1198711(119909 119910 120582) + max 119871

2(119875 120582 120583)

(15)

From the prospective of economics to understand 1198711

each user is selfish and wants to maximize its own util-ity Yet the user increases bandwidth will also reduce theavailable bandwidth for other users 119871

2corresponds to the

balance maximizing link capacity and the minimizing powerconsumption Such optimization process takes the followingseveral steps

Firstly the price of the congestion on each link is updatedaccording to the flow based on the projection gradientmethod [20] the step of adjusting the direction for each linkis calculated by 120582

119897 the update algorithm can be expressed as

follows

120582119897(119905 + 1) = [120582

119897(119905) + 119886 (119905) (sum

119898119904

119867119898

119897119904119909119898119904

minus 119888119897(119875))]

+

(16)

Here 119886(119905) is the small positive step size and ldquo+rdquo rep-resents the projected onto the real number of positive realnumber space If congestion has arisen on the multicasttree 119879

119898 then congestion control prices will rise accordingly

indicating that the source node should reduce the data rateBased on the gradient projection algorithm smaller step

iterative update in each time slot is chosen for the prices asfollows

120583119897(119905 + 1) = [120583

119897(119905) + 120574

1(119905) (119875

max119894

minus sum

119897isin119900(119894)

119875119897)]

+

(17)

Second by considering the second optimization term in (15)

max 1198712(119875 120582 120583) = max(sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897))

(18)

Notice that 1198712aims to maximize the sum of weighted

capacities and minimize the power cost Thus (18) servesas a tool for power control at each link Similar to [16] wesubstitute formula (7) into 119871

2 Consider

1198712= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus 120573sum

119897

119908119890119875119897 + sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus sum

119894

(120573119908 + 120583119894) sum

119897isin119900(119894)

119890119875119897 + sum

119894

120583119894119875max119894

(19)

Theorem 2 Problem max 1198712(119875 120582 120583) is a convex optimiza-

tion

6 Mathematical Problems in Engineering

Proof We know from [7] that 119861sum119897[120582119897log(119866

119897119897119890119875119897) minus

120582119897log(sum

119896exp(119875

119896+ log119866

119897119896) + 119899119897)] is strict concave function

and another termminussum119894(120573119908119897+120583119894) sum119897isin119900(119894)

119890119875119897 is concave function

for 119875 obviously problem (18) is a convex optimizationTaking the derivative of 119871

2with respect to 119875

1205971198712(119875 120582 120583)

120597119875119897

= 120582119897119861 minus (120573119908 + 120583

119894) 119890119875119897

minus 119861sum

119895 = 119897

120582119895119866119895119897119890119875119897

sum119896 = 119895

119866119895119896119890119875119896 + 120590

119897

(20)

Coming back to 119875 instead of 119875

1205971198712(119875 120582 120583)

120597119875119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

120582119895119866119895119897

sum119896 = 119895

119866119895119896119875119896+ 120590119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

119866119895119897119898119895

(21)

where119898119895is a message calculated based on locally measurable

quantities for transmitter 119895 Consider

119898119895=

120582119895

sum119896 = 119895

119866119895119896119875119896+ 120590119897

(22)

Then we can write the gradient steps as the followingdistributed power control algorithm with message passing

119875119897(119905 + 1)

= [

[

119875119897(119905) + 120574(

119861120582119897(119905)

119875119897(119905)

minus (120573119908 + 120583119894(119905))

minus119861sum

119895 = 119897

119866119895119897119898119895(119905))]

]

+

119897isin119900(119894)

(23)

Next we consider the first optimization in (15)

max119909ge0119910

1198711(119909 119910 120582)

= max[sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898119904

119867119898

119897119904119909119898119904

minussum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)

2

]

= sum

119898

max119909ge0119910

[

[

119880119898

(sum

119904

119909119898119904

)

minus sum

119905isin119879119898

(119867119898

119897119904119909119898119904

sum

119897

120582119897minus

119888119898

2(119909119898119904

minus 119910119898119904

)2

)]

]

(24)

The first optimization in (15) can be used to regulate theflow rate at each source If source 119904 only has a single treethen we can get the optimal value through derivation of thesubproblem 119871

1

1205971198711(119909 119910 120582)

120597119909119898

= 1198801015840

119898(119909119898) minus sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904 (25)

According to the optimization theory the rate of sourcenode can obtain the optimal valuewhen the derivation is zeroso

119909119898

(119905) = (sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904)

minus1

(26)

Otherwise the optimization 1198711can be solved by a non-

linear Gauss-Seidel method which alternately (i) maximizesthe objective function over 119909

119898while keeping 119910

119898fixed and (ii)

maximizes 1198711over 119910

119898while keeping 119909

119898fixedThe algorithm

repeats the following steps

Step 1 Fix 119910119898

= 119910119898(119905) and maximize the problem P3 With

the addition of the quadratic term the problem P3 is nowstrictly convex with respect to 119909

119898119904 Hence the maximizer of

P3 is unique existenceTo be precise taking the derivative of 119871

1with respect to

119909119898119904

1205971198711(119909 119910 120582)

120597119909119898119904

=1

sum119904119909119898119904

minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

minus 119888119898119909119898119904

(27a)

Thus according to the first-order necessary optimalitycondition we have

119909lowast

119898119904=

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27b)

where 119911 = sum119904isin119879119898

119909lowast

119898119904can be calculated by summing both

sides of (27b) as formula (27c) Consider

119911 = sum

119905isin119879119898

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27c)

Assuming that 119891(119911) = 119911 minus sum119905isin119879119898

(1119888119898)[(1119911) minus sum

119897119867119898

119897119903120582119897+

119888119898119910119898119904

]+ it is a strictly increasing function we can easily

achieve unique solution for the equation 119891(119911) = 0 119911 =

(minus119887 + radic1198872 + 4|119879119898|)2 where 119887 = sum

119904isin119879119898(sum119897119867119898

119897119904120582119897minus 119888119898119910119898119904

)The iterative procedure follows as (27b) and (27c)

Step 2The algorithm sets 119910(119905 + 1) = 119909(119905 + 1) The advantageof network utility maximization is that the optimizationmodel can be implemented in a distributed networkThroughbroadcasting the information of the adjacent nodes as wellas the feedback price for each link on the path the networkis able to share the global information among the nodesAfter exchanging cross-layer information the information of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Traditionally resource allocations are optimized sepa-rately which is difficult to support high quality communi-cation In this paper we focus on the joint optimization ofmulticast flow rate control and power allocation where themulticast tree is established and the channels are assigned inadvance such as in [17] In the wireless network the multipleinformation flows share the network links and it is mandatedthat all the information flows on the links do not exceed thephysical channel capacity The data received by intermediatenode should be forwarded (the source nodes send data whilethe destination nodes receive data only) The relationship ofeach information flow and physical flow can be expressed as

sum

119895(119894119895)isin119871

119892119898119889

119894119895minus sum

119895(119895119894)isin119871

119892119898119889

119895119894=

119909119898 if 119894 = 119904

119898

minus119909119898 if 119894 = 119889

0 otherwiseforall119889 isin 119863

119898

119892119898119889

119894119895le 119891119898

119894119895 forall119889 isin 119863

119898

(1)

Here 119909119898 is the source rate and 119892119898119889

119894119895represents the flow of

session 119898 on link (119894 119895) sent to the destination node 119889 Linkcapacity can be shared by the flows on the links when therandom linear network coding is adopted and the limit ofnetwork capacity can be expressed as

119891119898

119894119895= max119889

119892119898119889

119894119895 119889 isin 119863

119898 (2)

Now let us explain with the help of Figure 1 multicastsession model which is similar to the model in [16] which isa typical butterfly diagram 119904 is the source node of the sessionwhile 119889

1and 119889

2are the destination nodes Network coding

can be performed on the wireless links (119908 V) to share linkcapacity for improving network capacity

Figure 1 shows a multicast session where the flow is sentfrom the source node 119904 to left and right trees which are furtherdepicted in Figure 2 The encoding can be performed onthe wireless links shared by the different multicast trees Asshown in Figure 1 the link (119908 V) is the shared part of twotrees the multicast trees can share link capacity by networkcoding themax rate of information flow for leftmulticast treeis 2 and the right multicast tree is 1

Figure 2 shows the multicast subtrees partitioned byFigure 1 In the figure the source node separates the flowsent to the destination node into two subtrees each of whichcan change the transmission rate through adjusting at thesource node As shown in Figure 2 in a multicast sessionthe link using random linear network coding can achievemaximum upper bound in the ideal case In order to expressthe corresponding relationship of the link 119897 and multicast 119898we define a multicast matrix 119867

119898

(|119867119898

| = |119871| lowast |119878|) and theelements of the matrix are defined as follows [16]

119867119898

119897119904=

1 119897 isin 119879119898119904

0 others(3)

Here 119879119898119904

represents the link set of multicast tree The sourcenode in a multicast tree 119879

119898119904sends the same information

s

t u

w

d1 d2

b1

b1

a1 a

2

a 1a

2

a 1 a 2

a 1+b1

a2

a1 + b1

a1 +

b1 a

2

Figure 1 Typical butterfly diagram

s s

t u

w w

d1 d1d2 d2

b1

b1b1

b1

b1

a1 a

2

a1 a

2

a 1a

2

a 1 a

2

a 1 a 2

Figure 2 Network coding at the shared link (119908 V) corresponded tothe multicast tree decomposition

flow to the destination nodes For the sake of simplicity weassume that network coding occurs within session and theconstraints of the physical flow rate for each multicast stream119898 on link 119897 can be expressed as

max119904

119867119898

119897119904119909119898119904

le 119888119897

forall119897 isin 119871 (4)

32 Network Interference Model In an interference-limitedwireless network the information theoretic capacity 119888

119897of

each link 119897 is not fixed It can be considered as a functionof the transmit power the interference of adjacent linkbandwidth and modulation and so forth Assuming that themodulation scheme has been determined the informationtheory capacity 119888

119897of link 119897 is considered as the function of the

transmit power and channel conditions as follows

119888119897(119875) = 119861 sdot log (1 + 119870SINR

119897(119875)) (5)

4 Mathematical Problems in Engineering

where 119861 is bandwidth 119875 is the vector of transmission powerand 119870 is a constant which depends on the modulation andbit error rate Without loss of generality we assume that119870 = 1 SINR

119897represents the SINR of the link 119897 where SINR

119897

is defined as

SINR119897(119875) =

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

(6)

where 119899119897represents white Gaussian noise of link 119897 at the

receiving node 119866119897119897

is the fading coefficient of the link 119897

from the transmitting node to the receiving node sum119896 = 119897

119866119896119897

is the interference of other links on the receiving node 119875119897is

transmit power of source node and 119875119896is interference power

of other nodes Assuming that the CDMA system is adoptedin the network with reasonable spreading gain and undernormal circumstances 119866

119897119897≫ 119866119896119897 119896 = 119897 (eg SINR gt 5) [9]

due to the fact that the interference is much smaller than thesignal power Equation (5) can be approximated instead as[18]

119888119897(119875) = log(

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

) (7)

In wireless multihop networks the sum of interferenceterms over 119896 = 119897 can be conducted in practice only over theactive links in the two-hop neighborhood Interference infor-mation can be measured or obtained by mutual broadcastingof neighbor nodes

33 Energy Efficient Multicast Rate Control Model The wire-less nodes are generally powered by a battery so powercontrol may conserve energy and reduce the interferenceamong nodes In this paper we establish the model forenergy efficient multicast rate control and power allocationoptimization The purpose of such optimization is to max-imize the fair allocation of link bandwidth and minimizepower for the network multicast session flow and reduceenergy consumption Assuming that 119880

119898(119909119898) is the utility of

each multicast 119898 in the network and the utility function isseparable where 119909

119898= sum119904119909119898119904

is the sum of data rates whichare distributed on the respective trees then the problem canbe formulated as P1 Consider

P1 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897) (8a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897 (8b)

119875119897ge 0 sum

119897isin119900(119894)

119875119897le 119875

max119894

119894 isin 119873 (8c)

where 119880119898is a continuously differentiable strictly concave

and incremental function 120573 is the equilibrium coefficientand 119881

119897is the cost function For simplicity the cost function

can be expressed as the weights of power value as

119881119897(119875119897) = 119908119897119875119897 119908119897ge 0 (9)

Network performance includes not only the networkthroughput but also the fairness of user which can be

balanced through selecting the appropriate utility functionWedesign the utility functionwhich can be flexibly adjustableby parameters as follows with reference to [19]

119880120572

119904(119909119904) =

119901119904(1 minus 120572)

minus1

1199091minus120572

119904 120572 = 1 120572 ge 0

119901119904log119909119904 120572 = 1

(10)

where 119901119904represents the weight of different flows and 120572 is the

fairness parameter While 120572 = 1 corresponds to proportionalfairness 120572 = 2 corresponds to the harmonic mean fairnessand 120572 rarr infin corresponds to the max-min fairness Theweight of eachmulticast session flow is set to be 1 in the paperwhich means that 119880

119898(119909119898) = log(119909

119898)

4 Optimization Approachand Distributed Algorithm

P1 is a nonconvex optimization problem since the linkconstraint (8b) is a nonconvex region However a simplevariable transformation119875

119897= log(119875

119897) can be used to transform

the problem into an equivalent convex optimization problemP2 Consider

P2 max sum

119898

119880119898

(sum

119904

119909119898

119904) minus 120573sum

119897

119881119897(119875119897)

st sum

119898

119867119898

119897119904119909119898

119904le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873

(11)

Furthermore the objective function of P2 is not strictlyconcave since some sources have multiple alternative mul-ticast trees that may cause an instability problem in theconvergence of an iterative algorithmmdasha persistent oscilla-tion of the flow rate around the optimal value It impliesthat although the dual variables may converge the primalvariables flow rates and transmit powers may not To dealwith this instability problem we use ideas from proximaloptimization algorithms [13 page 232] The basic idea isthat instead of P3 we try to solve an equivalent problem byintroducing a quadratic term of some auxiliary variables 119910

119898119904

so that the optimization problem becomes strictly concavewith respect to 119909

119898119904 Consider

P3 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

(12a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

(12b)

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873 (12c)

Mathematical Problems in Engineering 5

Due to the fact that 119888119897(119875) = log(119866

119897119897) + 119875

119897minus

log(sum119896 = 119897

119866119896119897exp(119875

119896) + 119899119897) is sum of a linear function and a

log-sum-exp function problem (12a) is convex according tothe theory of convex optimization

Theorem 1 The transformed problemP2 is a convex optimiza-tion problem

Proof The constraints (12b) and (12c) are convex in (119909 119875)

since the log-sum-exp is convex in its domain [20]Moreoverthe utilities in (12a) are assumed to be strictly concaveTherefore P2 is convex in (119909 119875)

Hence the maximized value of (12a) is unique ProblemP2 can be solved by Lagrange dual decomposition whichis the most important optimization method and has beenwidely used in constrained optimization problem The orig-inal problem P3 is decomposed into two suboptimizationproblems

We define the Lagrangian as

119871 (119909 119910 119875 120582 120583) = sum

119898

119880119898

(119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

minus sum

119897

120582119897(sum

119898

119867119898

119897119904119909119898119904

minus 119888119897(119875))

minus sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898

119867119898

119897119904119909119898119904

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

+ sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

(13)

where 120582 is the vector of Lagrange multiplier which isassociated with capacity constraints (12b) while 120583 is thevector of Lagrangemultipliers which is associatedwith powerconstraints (12c) and 119875

119897and 119909

119898119904are primal variables The

dual problem P3 can be expressed as an unconstrained min-max problem

min120582120583ge0

119863(120582 120583) (14)

where the dual function

119863(120582 120583) = max 119871 (119909 119910 119875 120582 120583)

= max 1198711(119909 119910 120582) + max 119871

2(119875 120582 120583)

(15)

From the prospective of economics to understand 1198711

each user is selfish and wants to maximize its own util-ity Yet the user increases bandwidth will also reduce theavailable bandwidth for other users 119871

2corresponds to the

balance maximizing link capacity and the minimizing powerconsumption Such optimization process takes the followingseveral steps

Firstly the price of the congestion on each link is updatedaccording to the flow based on the projection gradientmethod [20] the step of adjusting the direction for each linkis calculated by 120582

119897 the update algorithm can be expressed as

follows

120582119897(119905 + 1) = [120582

119897(119905) + 119886 (119905) (sum

119898119904

119867119898

119897119904119909119898119904

minus 119888119897(119875))]

+

(16)

Here 119886(119905) is the small positive step size and ldquo+rdquo rep-resents the projected onto the real number of positive realnumber space If congestion has arisen on the multicasttree 119879

119898 then congestion control prices will rise accordingly

indicating that the source node should reduce the data rateBased on the gradient projection algorithm smaller step

iterative update in each time slot is chosen for the prices asfollows

120583119897(119905 + 1) = [120583

119897(119905) + 120574

1(119905) (119875

max119894

minus sum

119897isin119900(119894)

119875119897)]

+

(17)

Second by considering the second optimization term in (15)

max 1198712(119875 120582 120583) = max(sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897))

(18)

Notice that 1198712aims to maximize the sum of weighted

capacities and minimize the power cost Thus (18) servesas a tool for power control at each link Similar to [16] wesubstitute formula (7) into 119871

2 Consider

1198712= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus 120573sum

119897

119908119890119875119897 + sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus sum

119894

(120573119908 + 120583119894) sum

119897isin119900(119894)

119890119875119897 + sum

119894

120583119894119875max119894

(19)

Theorem 2 Problem max 1198712(119875 120582 120583) is a convex optimiza-

tion

6 Mathematical Problems in Engineering

Proof We know from [7] that 119861sum119897[120582119897log(119866

119897119897119890119875119897) minus

120582119897log(sum

119896exp(119875

119896+ log119866

119897119896) + 119899119897)] is strict concave function

and another termminussum119894(120573119908119897+120583119894) sum119897isin119900(119894)

119890119875119897 is concave function

for 119875 obviously problem (18) is a convex optimizationTaking the derivative of 119871

2with respect to 119875

1205971198712(119875 120582 120583)

120597119875119897

= 120582119897119861 minus (120573119908 + 120583

119894) 119890119875119897

minus 119861sum

119895 = 119897

120582119895119866119895119897119890119875119897

sum119896 = 119895

119866119895119896119890119875119896 + 120590

119897

(20)

Coming back to 119875 instead of 119875

1205971198712(119875 120582 120583)

120597119875119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

120582119895119866119895119897

sum119896 = 119895

119866119895119896119875119896+ 120590119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

119866119895119897119898119895

(21)

where119898119895is a message calculated based on locally measurable

quantities for transmitter 119895 Consider

119898119895=

120582119895

sum119896 = 119895

119866119895119896119875119896+ 120590119897

(22)

Then we can write the gradient steps as the followingdistributed power control algorithm with message passing

119875119897(119905 + 1)

= [

[

119875119897(119905) + 120574(

119861120582119897(119905)

119875119897(119905)

minus (120573119908 + 120583119894(119905))

minus119861sum

119895 = 119897

119866119895119897119898119895(119905))]

]

+

119897isin119900(119894)

(23)

Next we consider the first optimization in (15)

max119909ge0119910

1198711(119909 119910 120582)

= max[sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898119904

119867119898

119897119904119909119898119904

minussum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)

2

]

= sum

119898

max119909ge0119910

[

[

119880119898

(sum

119904

119909119898119904

)

minus sum

119905isin119879119898

(119867119898

119897119904119909119898119904

sum

119897

120582119897minus

119888119898

2(119909119898119904

minus 119910119898119904

)2

)]

]

(24)

The first optimization in (15) can be used to regulate theflow rate at each source If source 119904 only has a single treethen we can get the optimal value through derivation of thesubproblem 119871

1

1205971198711(119909 119910 120582)

120597119909119898

= 1198801015840

119898(119909119898) minus sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904 (25)

According to the optimization theory the rate of sourcenode can obtain the optimal valuewhen the derivation is zeroso

119909119898

(119905) = (sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904)

minus1

(26)

Otherwise the optimization 1198711can be solved by a non-

linear Gauss-Seidel method which alternately (i) maximizesthe objective function over 119909

119898while keeping 119910

119898fixed and (ii)

maximizes 1198711over 119910

119898while keeping 119909

119898fixedThe algorithm

repeats the following steps

Step 1 Fix 119910119898

= 119910119898(119905) and maximize the problem P3 With

the addition of the quadratic term the problem P3 is nowstrictly convex with respect to 119909

119898119904 Hence the maximizer of

P3 is unique existenceTo be precise taking the derivative of 119871

1with respect to

119909119898119904

1205971198711(119909 119910 120582)

120597119909119898119904

=1

sum119904119909119898119904

minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

minus 119888119898119909119898119904

(27a)

Thus according to the first-order necessary optimalitycondition we have

119909lowast

119898119904=

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27b)

where 119911 = sum119904isin119879119898

119909lowast

119898119904can be calculated by summing both

sides of (27b) as formula (27c) Consider

119911 = sum

119905isin119879119898

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27c)

Assuming that 119891(119911) = 119911 minus sum119905isin119879119898

(1119888119898)[(1119911) minus sum

119897119867119898

119897119903120582119897+

119888119898119910119898119904

]+ it is a strictly increasing function we can easily

achieve unique solution for the equation 119891(119911) = 0 119911 =

(minus119887 + radic1198872 + 4|119879119898|)2 where 119887 = sum

119904isin119879119898(sum119897119867119898

119897119904120582119897minus 119888119898119910119898119904

)The iterative procedure follows as (27b) and (27c)

Step 2The algorithm sets 119910(119905 + 1) = 119909(119905 + 1) The advantageof network utility maximization is that the optimizationmodel can be implemented in a distributed networkThroughbroadcasting the information of the adjacent nodes as wellas the feedback price for each link on the path the networkis able to share the global information among the nodesAfter exchanging cross-layer information the information of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 119861 is bandwidth 119875 is the vector of transmission powerand 119870 is a constant which depends on the modulation andbit error rate Without loss of generality we assume that119870 = 1 SINR

119897represents the SINR of the link 119897 where SINR

119897

is defined as

SINR119897(119875) =

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

(6)

where 119899119897represents white Gaussian noise of link 119897 at the

receiving node 119866119897119897

is the fading coefficient of the link 119897

from the transmitting node to the receiving node sum119896 = 119897

119866119896119897

is the interference of other links on the receiving node 119875119897is

transmit power of source node and 119875119896is interference power

of other nodes Assuming that the CDMA system is adoptedin the network with reasonable spreading gain and undernormal circumstances 119866

119897119897≫ 119866119896119897 119896 = 119897 (eg SINR gt 5) [9]

due to the fact that the interference is much smaller than thesignal power Equation (5) can be approximated instead as[18]

119888119897(119875) = log(

119866119897119897119875119897

sum119896 = 119897

119866119896119897119875119896+ 119899119897

) (7)

In wireless multihop networks the sum of interferenceterms over 119896 = 119897 can be conducted in practice only over theactive links in the two-hop neighborhood Interference infor-mation can be measured or obtained by mutual broadcastingof neighbor nodes

33 Energy Efficient Multicast Rate Control Model The wire-less nodes are generally powered by a battery so powercontrol may conserve energy and reduce the interferenceamong nodes In this paper we establish the model forenergy efficient multicast rate control and power allocationoptimization The purpose of such optimization is to max-imize the fair allocation of link bandwidth and minimizepower for the network multicast session flow and reduceenergy consumption Assuming that 119880

119898(119909119898) is the utility of

each multicast 119898 in the network and the utility function isseparable where 119909

119898= sum119904119909119898119904

is the sum of data rates whichare distributed on the respective trees then the problem canbe formulated as P1 Consider

P1 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897) (8a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897 (8b)

119875119897ge 0 sum

119897isin119900(119894)

119875119897le 119875

max119894

119894 isin 119873 (8c)

where 119880119898is a continuously differentiable strictly concave

and incremental function 120573 is the equilibrium coefficientand 119881

119897is the cost function For simplicity the cost function

can be expressed as the weights of power value as

119881119897(119875119897) = 119908119897119875119897 119908119897ge 0 (9)

Network performance includes not only the networkthroughput but also the fairness of user which can be

balanced through selecting the appropriate utility functionWedesign the utility functionwhich can be flexibly adjustableby parameters as follows with reference to [19]

119880120572

119904(119909119904) =

119901119904(1 minus 120572)

minus1

1199091minus120572

119904 120572 = 1 120572 ge 0

119901119904log119909119904 120572 = 1

(10)

where 119901119904represents the weight of different flows and 120572 is the

fairness parameter While 120572 = 1 corresponds to proportionalfairness 120572 = 2 corresponds to the harmonic mean fairnessand 120572 rarr infin corresponds to the max-min fairness Theweight of eachmulticast session flow is set to be 1 in the paperwhich means that 119880

119898(119909119898) = log(119909

119898)

4 Optimization Approachand Distributed Algorithm

P1 is a nonconvex optimization problem since the linkconstraint (8b) is a nonconvex region However a simplevariable transformation119875

119897= log(119875

119897) can be used to transform

the problem into an equivalent convex optimization problemP2 Consider

P2 max sum

119898

119880119898

(sum

119904

119909119898

119904) minus 120573sum

119897

119881119897(119875119897)

st sum

119898

119867119898

119897119904119909119898

119904le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873

(11)

Furthermore the objective function of P2 is not strictlyconcave since some sources have multiple alternative mul-ticast trees that may cause an instability problem in theconvergence of an iterative algorithmmdasha persistent oscilla-tion of the flow rate around the optimal value It impliesthat although the dual variables may converge the primalvariables flow rates and transmit powers may not To dealwith this instability problem we use ideas from proximaloptimization algorithms [13 page 232] The basic idea isthat instead of P3 we try to solve an equivalent problem byintroducing a quadratic term of some auxiliary variables 119910

119898119904

so that the optimization problem becomes strictly concavewith respect to 119909

119898119904 Consider

P3 max sum

119898

119880119898

(sum

119904

119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

(12a)

st sum

119898

119867119898

119897119904119909119898119904

le 119888119897(119875) forall119904 forall119898 isin 119872 forall119897

(12b)

119875119897ge 0 sum

119897isin119900(119894)

119890119875119897 le 119875

max119894

119894 isin 119873 (12c)

Mathematical Problems in Engineering 5

Due to the fact that 119888119897(119875) = log(119866

119897119897) + 119875

119897minus

log(sum119896 = 119897

119866119896119897exp(119875

119896) + 119899119897) is sum of a linear function and a

log-sum-exp function problem (12a) is convex according tothe theory of convex optimization

Theorem 1 The transformed problemP2 is a convex optimiza-tion problem

Proof The constraints (12b) and (12c) are convex in (119909 119875)

since the log-sum-exp is convex in its domain [20]Moreoverthe utilities in (12a) are assumed to be strictly concaveTherefore P2 is convex in (119909 119875)

Hence the maximized value of (12a) is unique ProblemP2 can be solved by Lagrange dual decomposition whichis the most important optimization method and has beenwidely used in constrained optimization problem The orig-inal problem P3 is decomposed into two suboptimizationproblems

We define the Lagrangian as

119871 (119909 119910 119875 120582 120583) = sum

119898

119880119898

(119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

minus sum

119897

120582119897(sum

119898

119867119898

119897119904119909119898119904

minus 119888119897(119875))

minus sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898

119867119898

119897119904119909119898119904

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

+ sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

(13)

where 120582 is the vector of Lagrange multiplier which isassociated with capacity constraints (12b) while 120583 is thevector of Lagrangemultipliers which is associatedwith powerconstraints (12c) and 119875

119897and 119909

119898119904are primal variables The

dual problem P3 can be expressed as an unconstrained min-max problem

min120582120583ge0

119863(120582 120583) (14)

where the dual function

119863(120582 120583) = max 119871 (119909 119910 119875 120582 120583)

= max 1198711(119909 119910 120582) + max 119871

2(119875 120582 120583)

(15)

From the prospective of economics to understand 1198711

each user is selfish and wants to maximize its own util-ity Yet the user increases bandwidth will also reduce theavailable bandwidth for other users 119871

2corresponds to the

balance maximizing link capacity and the minimizing powerconsumption Such optimization process takes the followingseveral steps

Firstly the price of the congestion on each link is updatedaccording to the flow based on the projection gradientmethod [20] the step of adjusting the direction for each linkis calculated by 120582

119897 the update algorithm can be expressed as

follows

120582119897(119905 + 1) = [120582

119897(119905) + 119886 (119905) (sum

119898119904

119867119898

119897119904119909119898119904

minus 119888119897(119875))]

+

(16)

Here 119886(119905) is the small positive step size and ldquo+rdquo rep-resents the projected onto the real number of positive realnumber space If congestion has arisen on the multicasttree 119879

119898 then congestion control prices will rise accordingly

indicating that the source node should reduce the data rateBased on the gradient projection algorithm smaller step

iterative update in each time slot is chosen for the prices asfollows

120583119897(119905 + 1) = [120583

119897(119905) + 120574

1(119905) (119875

max119894

minus sum

119897isin119900(119894)

119875119897)]

+

(17)

Second by considering the second optimization term in (15)

max 1198712(119875 120582 120583) = max(sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897))

(18)

Notice that 1198712aims to maximize the sum of weighted

capacities and minimize the power cost Thus (18) servesas a tool for power control at each link Similar to [16] wesubstitute formula (7) into 119871

2 Consider

1198712= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus 120573sum

119897

119908119890119875119897 + sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus sum

119894

(120573119908 + 120583119894) sum

119897isin119900(119894)

119890119875119897 + sum

119894

120583119894119875max119894

(19)

Theorem 2 Problem max 1198712(119875 120582 120583) is a convex optimiza-

tion

6 Mathematical Problems in Engineering

Proof We know from [7] that 119861sum119897[120582119897log(119866

119897119897119890119875119897) minus

120582119897log(sum

119896exp(119875

119896+ log119866

119897119896) + 119899119897)] is strict concave function

and another termminussum119894(120573119908119897+120583119894) sum119897isin119900(119894)

119890119875119897 is concave function

for 119875 obviously problem (18) is a convex optimizationTaking the derivative of 119871

2with respect to 119875

1205971198712(119875 120582 120583)

120597119875119897

= 120582119897119861 minus (120573119908 + 120583

119894) 119890119875119897

minus 119861sum

119895 = 119897

120582119895119866119895119897119890119875119897

sum119896 = 119895

119866119895119896119890119875119896 + 120590

119897

(20)

Coming back to 119875 instead of 119875

1205971198712(119875 120582 120583)

120597119875119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

120582119895119866119895119897

sum119896 = 119895

119866119895119896119875119896+ 120590119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

119866119895119897119898119895

(21)

where119898119895is a message calculated based on locally measurable

quantities for transmitter 119895 Consider

119898119895=

120582119895

sum119896 = 119895

119866119895119896119875119896+ 120590119897

(22)

Then we can write the gradient steps as the followingdistributed power control algorithm with message passing

119875119897(119905 + 1)

= [

[

119875119897(119905) + 120574(

119861120582119897(119905)

119875119897(119905)

minus (120573119908 + 120583119894(119905))

minus119861sum

119895 = 119897

119866119895119897119898119895(119905))]

]

+

119897isin119900(119894)

(23)

Next we consider the first optimization in (15)

max119909ge0119910

1198711(119909 119910 120582)

= max[sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898119904

119867119898

119897119904119909119898119904

minussum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)

2

]

= sum

119898

max119909ge0119910

[

[

119880119898

(sum

119904

119909119898119904

)

minus sum

119905isin119879119898

(119867119898

119897119904119909119898119904

sum

119897

120582119897minus

119888119898

2(119909119898119904

minus 119910119898119904

)2

)]

]

(24)

The first optimization in (15) can be used to regulate theflow rate at each source If source 119904 only has a single treethen we can get the optimal value through derivation of thesubproblem 119871

1

1205971198711(119909 119910 120582)

120597119909119898

= 1198801015840

119898(119909119898) minus sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904 (25)

According to the optimization theory the rate of sourcenode can obtain the optimal valuewhen the derivation is zeroso

119909119898

(119905) = (sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904)

minus1

(26)

Otherwise the optimization 1198711can be solved by a non-

linear Gauss-Seidel method which alternately (i) maximizesthe objective function over 119909

119898while keeping 119910

119898fixed and (ii)

maximizes 1198711over 119910

119898while keeping 119909

119898fixedThe algorithm

repeats the following steps

Step 1 Fix 119910119898

= 119910119898(119905) and maximize the problem P3 With

the addition of the quadratic term the problem P3 is nowstrictly convex with respect to 119909

119898119904 Hence the maximizer of

P3 is unique existenceTo be precise taking the derivative of 119871

1with respect to

119909119898119904

1205971198711(119909 119910 120582)

120597119909119898119904

=1

sum119904119909119898119904

minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

minus 119888119898119909119898119904

(27a)

Thus according to the first-order necessary optimalitycondition we have

119909lowast

119898119904=

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27b)

where 119911 = sum119904isin119879119898

119909lowast

119898119904can be calculated by summing both

sides of (27b) as formula (27c) Consider

119911 = sum

119905isin119879119898

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27c)

Assuming that 119891(119911) = 119911 minus sum119905isin119879119898

(1119888119898)[(1119911) minus sum

119897119867119898

119897119903120582119897+

119888119898119910119898119904

]+ it is a strictly increasing function we can easily

achieve unique solution for the equation 119891(119911) = 0 119911 =

(minus119887 + radic1198872 + 4|119879119898|)2 where 119887 = sum

119904isin119879119898(sum119897119867119898

119897119904120582119897minus 119888119898119910119898119904

)The iterative procedure follows as (27b) and (27c)

Step 2The algorithm sets 119910(119905 + 1) = 119909(119905 + 1) The advantageof network utility maximization is that the optimizationmodel can be implemented in a distributed networkThroughbroadcasting the information of the adjacent nodes as wellas the feedback price for each link on the path the networkis able to share the global information among the nodesAfter exchanging cross-layer information the information of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Due to the fact that 119888119897(119875) = log(119866

119897119897) + 119875

119897minus

log(sum119896 = 119897

119866119896119897exp(119875

119896) + 119899119897) is sum of a linear function and a

log-sum-exp function problem (12a) is convex according tothe theory of convex optimization

Theorem 1 The transformed problemP2 is a convex optimiza-tion problem

Proof The constraints (12b) and (12c) are convex in (119909 119875)

since the log-sum-exp is convex in its domain [20]Moreoverthe utilities in (12a) are assumed to be strictly concaveTherefore P2 is convex in (119909 119875)

Hence the maximized value of (12a) is unique ProblemP2 can be solved by Lagrange dual decomposition whichis the most important optimization method and has beenwidely used in constrained optimization problem The orig-inal problem P3 is decomposed into two suboptimizationproblems

We define the Lagrangian as

119871 (119909 119910 119875 120582 120583) = sum

119898

119880119898

(119909119898119904

) minus 120573sum

119897

119881119897(119875119897)

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

minus sum

119897

120582119897(sum

119898

119867119898

119897119904119909119898119904

minus 119888119897(119875))

minus sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898

119867119898

119897119904119909119898119904

minus sum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)2

+ sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

(13)

where 120582 is the vector of Lagrange multiplier which isassociated with capacity constraints (12b) while 120583 is thevector of Lagrangemultipliers which is associatedwith powerconstraints (12c) and 119875

119897and 119909

119898119904are primal variables The

dual problem P3 can be expressed as an unconstrained min-max problem

min120582120583ge0

119863(120582 120583) (14)

where the dual function

119863(120582 120583) = max 119871 (119909 119910 119875 120582 120583)

= max 1198711(119909 119910 120582) + max 119871

2(119875 120582 120583)

(15)

From the prospective of economics to understand 1198711

each user is selfish and wants to maximize its own util-ity Yet the user increases bandwidth will also reduce theavailable bandwidth for other users 119871

2corresponds to the

balance maximizing link capacity and the minimizing powerconsumption Such optimization process takes the followingseveral steps

Firstly the price of the congestion on each link is updatedaccording to the flow based on the projection gradientmethod [20] the step of adjusting the direction for each linkis calculated by 120582

119897 the update algorithm can be expressed as

follows

120582119897(119905 + 1) = [120582

119897(119905) + 119886 (119905) (sum

119898119904

119867119898

119897119904119909119898119904

minus 119888119897(119875))]

+

(16)

Here 119886(119905) is the small positive step size and ldquo+rdquo rep-resents the projected onto the real number of positive realnumber space If congestion has arisen on the multicasttree 119879

119898 then congestion control prices will rise accordingly

indicating that the source node should reduce the data rateBased on the gradient projection algorithm smaller step

iterative update in each time slot is chosen for the prices asfollows

120583119897(119905 + 1) = [120583

119897(119905) + 120574

1(119905) (119875

max119894

minus sum

119897isin119900(119894)

119875119897)]

+

(17)

Second by considering the second optimization term in (15)

max 1198712(119875 120582 120583) = max(sum

119897

120582119897119888119897(119875) minus 120573sum

119897

119881119897(119875119897)

+ sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897))

(18)

Notice that 1198712aims to maximize the sum of weighted

capacities and minimize the power cost Thus (18) servesas a tool for power control at each link Similar to [16] wesubstitute formula (7) into 119871

2 Consider

1198712= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus 120573sum

119897

119908119890119875119897 + sum

119894

120583119894(119875

max119894

minus sum

119897isin119900(119894)

119890119875119897)

= sum

119897

120582119897119861[

[

log (119866119897119897119890119875119897) minus log(120590

119897+ sum

119895 = 119897

119890119875119895+ln119866119894119895)]

]

minus sum

119894

(120573119908 + 120583119894) sum

119897isin119900(119894)

119890119875119897 + sum

119894

120583119894119875max119894

(19)

Theorem 2 Problem max 1198712(119875 120582 120583) is a convex optimiza-

tion

6 Mathematical Problems in Engineering

Proof We know from [7] that 119861sum119897[120582119897log(119866

119897119897119890119875119897) minus

120582119897log(sum

119896exp(119875

119896+ log119866

119897119896) + 119899119897)] is strict concave function

and another termminussum119894(120573119908119897+120583119894) sum119897isin119900(119894)

119890119875119897 is concave function

for 119875 obviously problem (18) is a convex optimizationTaking the derivative of 119871

2with respect to 119875

1205971198712(119875 120582 120583)

120597119875119897

= 120582119897119861 minus (120573119908 + 120583

119894) 119890119875119897

minus 119861sum

119895 = 119897

120582119895119866119895119897119890119875119897

sum119896 = 119895

119866119895119896119890119875119896 + 120590

119897

(20)

Coming back to 119875 instead of 119875

1205971198712(119875 120582 120583)

120597119875119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

120582119895119866119895119897

sum119896 = 119895

119866119895119896119875119896+ 120590119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

119866119895119897119898119895

(21)

where119898119895is a message calculated based on locally measurable

quantities for transmitter 119895 Consider

119898119895=

120582119895

sum119896 = 119895

119866119895119896119875119896+ 120590119897

(22)

Then we can write the gradient steps as the followingdistributed power control algorithm with message passing

119875119897(119905 + 1)

= [

[

119875119897(119905) + 120574(

119861120582119897(119905)

119875119897(119905)

minus (120573119908 + 120583119894(119905))

minus119861sum

119895 = 119897

119866119895119897119898119895(119905))]

]

+

119897isin119900(119894)

(23)

Next we consider the first optimization in (15)

max119909ge0119910

1198711(119909 119910 120582)

= max[sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898119904

119867119898

119897119904119909119898119904

minussum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)

2

]

= sum

119898

max119909ge0119910

[

[

119880119898

(sum

119904

119909119898119904

)

minus sum

119905isin119879119898

(119867119898

119897119904119909119898119904

sum

119897

120582119897minus

119888119898

2(119909119898119904

minus 119910119898119904

)2

)]

]

(24)

The first optimization in (15) can be used to regulate theflow rate at each source If source 119904 only has a single treethen we can get the optimal value through derivation of thesubproblem 119871

1

1205971198711(119909 119910 120582)

120597119909119898

= 1198801015840

119898(119909119898) minus sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904 (25)

According to the optimization theory the rate of sourcenode can obtain the optimal valuewhen the derivation is zeroso

119909119898

(119905) = (sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904)

minus1

(26)

Otherwise the optimization 1198711can be solved by a non-

linear Gauss-Seidel method which alternately (i) maximizesthe objective function over 119909

119898while keeping 119910

119898fixed and (ii)

maximizes 1198711over 119910

119898while keeping 119909

119898fixedThe algorithm

repeats the following steps

Step 1 Fix 119910119898

= 119910119898(119905) and maximize the problem P3 With

the addition of the quadratic term the problem P3 is nowstrictly convex with respect to 119909

119898119904 Hence the maximizer of

P3 is unique existenceTo be precise taking the derivative of 119871

1with respect to

119909119898119904

1205971198711(119909 119910 120582)

120597119909119898119904

=1

sum119904119909119898119904

minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

minus 119888119898119909119898119904

(27a)

Thus according to the first-order necessary optimalitycondition we have

119909lowast

119898119904=

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27b)

where 119911 = sum119904isin119879119898

119909lowast

119898119904can be calculated by summing both

sides of (27b) as formula (27c) Consider

119911 = sum

119905isin119879119898

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27c)

Assuming that 119891(119911) = 119911 minus sum119905isin119879119898

(1119888119898)[(1119911) minus sum

119897119867119898

119897119903120582119897+

119888119898119910119898119904

]+ it is a strictly increasing function we can easily

achieve unique solution for the equation 119891(119911) = 0 119911 =

(minus119887 + radic1198872 + 4|119879119898|)2 where 119887 = sum

119904isin119879119898(sum119897119867119898

119897119904120582119897minus 119888119898119910119898119904

)The iterative procedure follows as (27b) and (27c)

Step 2The algorithm sets 119910(119905 + 1) = 119909(119905 + 1) The advantageof network utility maximization is that the optimizationmodel can be implemented in a distributed networkThroughbroadcasting the information of the adjacent nodes as wellas the feedback price for each link on the path the networkis able to share the global information among the nodesAfter exchanging cross-layer information the information of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Proof We know from [7] that 119861sum119897[120582119897log(119866

119897119897119890119875119897) minus

120582119897log(sum

119896exp(119875

119896+ log119866

119897119896) + 119899119897)] is strict concave function

and another termminussum119894(120573119908119897+120583119894) sum119897isin119900(119894)

119890119875119897 is concave function

for 119875 obviously problem (18) is a convex optimizationTaking the derivative of 119871

2with respect to 119875

1205971198712(119875 120582 120583)

120597119875119897

= 120582119897119861 minus (120573119908 + 120583

119894) 119890119875119897

minus 119861sum

119895 = 119897

120582119895119866119895119897119890119875119897

sum119896 = 119895

119866119895119896119890119875119896 + 120590

119897

(20)

Coming back to 119875 instead of 119875

1205971198712(119875 120582 120583)

120597119875119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

120582119895119866119895119897

sum119896 = 119895

119866119895119896119875119896+ 120590119897

=120582119897119861

119875119897

minus (120573119908 + 120583119894) minus 119861sum

119895 = 119897

119866119895119897119898119895

(21)

where119898119895is a message calculated based on locally measurable

quantities for transmitter 119895 Consider

119898119895=

120582119895

sum119896 = 119895

119866119895119896119875119896+ 120590119897

(22)

Then we can write the gradient steps as the followingdistributed power control algorithm with message passing

119875119897(119905 + 1)

= [

[

119875119897(119905) + 120574(

119861120582119897(119905)

119875119897(119905)

minus (120573119908 + 120583119894(119905))

minus119861sum

119895 = 119897

119866119895119897119898119895(119905))]

]

+

119897isin119900(119894)

(23)

Next we consider the first optimization in (15)

max119909ge0119910

1198711(119909 119910 120582)

= max[sum

119898

119880119898

(sum

119904

119909119898119904

) minus sum

119897

120582119897sum

119898119904

119867119898

119897119904119909119898119904

minussum

119898

119888119898

2sum

119904

(119909119898119904

minus 119910119898119904

)

2

]

= sum

119898

max119909ge0119910

[

[

119880119898

(sum

119904

119909119898119904

)

minus sum

119905isin119879119898

(119867119898

119897119904119909119898119904

sum

119897

120582119897minus

119888119898

2(119909119898119904

minus 119910119898119904

)2

)]

]

(24)

The first optimization in (15) can be used to regulate theflow rate at each source If source 119904 only has a single treethen we can get the optimal value through derivation of thesubproblem 119871

1

1205971198711(119909 119910 120582)

120597119909119898

= 1198801015840

119898(119909119898) minus sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904 (25)

According to the optimization theory the rate of sourcenode can obtain the optimal valuewhen the derivation is zeroso

119909119898

(119905) = (sum

119897

120582119897(119905)sum

119898119904

119867119898

119897119904)

minus1

(26)

Otherwise the optimization 1198711can be solved by a non-

linear Gauss-Seidel method which alternately (i) maximizesthe objective function over 119909

119898while keeping 119910

119898fixed and (ii)

maximizes 1198711over 119910

119898while keeping 119909

119898fixedThe algorithm

repeats the following steps

Step 1 Fix 119910119898

= 119910119898(119905) and maximize the problem P3 With

the addition of the quadratic term the problem P3 is nowstrictly convex with respect to 119909

119898119904 Hence the maximizer of

P3 is unique existenceTo be precise taking the derivative of 119871

1with respect to

119909119898119904

1205971198711(119909 119910 120582)

120597119909119898119904

=1

sum119904119909119898119904

minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

minus 119888119898119909119898119904

(27a)

Thus according to the first-order necessary optimalitycondition we have

119909lowast

119898119904=

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27b)

where 119911 = sum119904isin119879119898

119909lowast

119898119904can be calculated by summing both

sides of (27b) as formula (27c) Consider

119911 = sum

119905isin119879119898

1

119888119898

[1

119911minus sum

119897

119867119898

119897119904120582119897+ 119888119898119910119898119904

]

+

(27c)

Assuming that 119891(119911) = 119911 minus sum119905isin119879119898

(1119888119898)[(1119911) minus sum

119897119867119898

119897119903120582119897+

119888119898119910119898119904

]+ it is a strictly increasing function we can easily

achieve unique solution for the equation 119891(119911) = 0 119911 =

(minus119887 + radic1198872 + 4|119879119898|)2 where 119887 = sum

119904isin119879119898(sum119897119867119898

119897119904120582119897minus 119888119898119910119898119904

)The iterative procedure follows as (27b) and (27c)

Step 2The algorithm sets 119910(119905 + 1) = 119909(119905 + 1) The advantageof network utility maximization is that the optimizationmodel can be implemented in a distributed networkThroughbroadcasting the information of the adjacent nodes as wellas the feedback price for each link on the path the networkis able to share the global information among the nodesAfter exchanging cross-layer information the information of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

1 Inner power allocation algorithm(1) Initialize parameter 120583

119894(0) 1199051= 0

(2) Update 119875119897(1199051+ 1) 119906

119894(1199051+ 1)

(3) 1199051= 1199051+ 1 repeat (2) until iterations end

2 For each 119897 isin 119900(119894) executes update at each node(1) Receive messages 119898

119895(1199052) from all interfering nodes 119895 in the neighborhood and execute inner power allocation

(2) Update power using formula (23)(3) Compute 119898

119897(1199052) broadcast 119898

119897(1199052) to all interfering nodes in the neighborhood

(4) Compute link capacity and update link price using formula (17)(5) 1199052= 1199052+ 1 if 119905

2lt 119870 then goto (1) else end

3 At each source node(1) Receive from the reverse path sum

119904isin119879119898119867119898

1198971199041205821(119905) of link prices

(2) Update the multicast flow rate 119909119898119904

using (26) or (27b) and (27c)(3) Communicate 119909

119898119904(119905 + 1) to all links on the tree 119879

119898

(4) if 1199052gt 119870 then set y(119905 + 1) = x(119905) 1199052 = 1 and repeat to (1) until iterations end

Algorithm 1 Cross-layer multicast rate and power allocation optimization

physical layer and transport layer can be shared The controlstructure is showed in Figure 3

The power allocation is executed at the physical layer ofthe wireless network the information can be sent to upperlayer rate control can be implemented by adjusting the sizeof windows for TCP to avoid congestion Joint optimizationproblem can be decomposed into layers of suboptimiza-tion problem by dual decomposition each suboptimizationproblem uses gradient projection method to solve the opti-mization problem and then we design cross-layer multicastrate and power allocation optimization algorithm as followsThe parameters 119905 119905

1 and 119905

2represent the iteration of inner

power allocation intermediate nodes and source nodesrespectively

The above algorithm can be executed on the sourcenode and the intermediate node through the sharing ofinformation Link price updates for each intermediate noderequires the information from its nearest two-hop neighborswhich can be obtained through the broadcast of the two-hopneighbors The price of each link is provided by the feedbackreturned to the source node and the price information canbe added into the underlying link ACK data packets so it canbe sent hop by hop back to the source node in the process ofsending data along the path

Algorithm 1 can achieve a global optimal solution of theiterative algorithm under certain conditions

Definition 3 (see [20]) Assuming 120582lowast is optimal value of dual

variable There exists fix step size 120572 for every 120575 gt 0

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

119863(120582 (120591) 120583) minus 119863 (120582lowast

120583) le 120575 (28)

Algorithm 1 converges to 120582lowast and 120583

lowast

Theorem 4 Assuming 120582lowast is optimal value of dual variable

and 119886 is small step size if the Euclidean norm of subgradientis bounded nabla

120582119863(120582 120583)

2le 119866 is true for every 119905 Algorithm 1

converges to the range of the optimum radius 11988611986622

Proof Consider

1003817100381710038171003817120582 (119905 + 1) minus 120582lowast1003817100381710038171003817

2

2le

1003817100381710038171003817120582(119905) minus 119886nabla120582119863(120582(119905) 120583) minus 120582

lowast1003817100381710038171003817

2

2

le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2

minus 2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583))

+ 1198862

119905

sum

120591=1

1003817100381710038171003817nabla120582119863(120582 120583)1003817100381710038171003817

2

2

(29)

so

2119886

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1003817100381710038171003817120582 (1) minus 120582

lowast1003817100381710038171003817

2

2+ 1199051198862

119866

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le

1003817100381710038171003817120582 (1) minus 120582lowast1003817100381710038171003817

2

2+ 1199051198862

119866

2119905119886

(30)

From Definition 3

lim sup119905rarrinfin

1

119905

119905

sum

120591=1

(119863 (120582 (120591) 120583) minus 119863 (120582lowast

120583)) le1198861198662

2 (31)

For nabla120582119863(120582 120583) = (119862

1(1198751) minus sum

1198981198671

119898119904119909119898119904

119862119871(119875119871) minus

sum119898

119867119871

119898119904119909119898119904

)119879 obviously

1003817100381710038171003817nabla120582119863(120582 120583)10038171003817100381710038172

le radicsum

119897isin119871

(119862119897(119875

max119894

)) + sum

119897isin119871

119875max (32)

So formula (17) converges to 120582lowast

Similarly formula (18) converges to 120583lowast Therefore Algo-

rithm 1 will converge to global optimumwhile the step size issmall enough

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

f

p120583(t)

120582(t)

x(t)

Transport layer(rate control)

Network layer(multicast routing)

Network capacity

Physical layer(power allocation)

Figure 3The open loop control structure for layered protocol stackin wireless system

s1 s2

m1 m2m3

w1w2

12

x y z

a

a

a

b

b b

c

c

c

d

de

e ef

f

f

Figure 4 A simple network with two multicast sessions

5 Numerical Experiments and Analysis

In this section we conduct numerical simulation to evaluatethe performance of Algorithm 1 Without loss of generalitynetwork multicast tree has been established and channelshave been assigned at the stage of network initialization Weconsider a simple topology as shown in Figure 4 where eachnode has multiradios each link has been assigned channellabel as ldquo119886rdquo ldquo119887rdquo ldquo119888rdquo and so forth and the capacity of link isdetermined by transmitting power the path fading of the linkand the external interference

There are twomulticast sessions in this network scenariothe source node of session 1 is 119878

1 while the destination nodes

are 119909 and 119910 the source node of session 2 is 1198782 while the

destination nodes are 119910 and 119911 For the sake of simplicitythe same utility function 119880(119909) = log(119909) is adopted for allsessions The left multicast tree is represented as a solid linewhile the right multicast tree is identified as dotted line Theencoded link of session 1 is (119908

1 V1) while the encoded link

of session 2 is (1199082 V2) The channel assignment of each link

meets the condition where there is no strong interference inthe surrounding area

The system is initialized by the following configurationthe maximal transmit power is 25mW for all the nodesthe interference noise of network is 119899

119897= 00001mW the

bandwidth of each channel is 1 kHz and the step size ofiterative algorithm is unified set to be 00035 channel gaincoefficient is set according to the distance 119889 between any twonodes in a uniformly distributed random selection within[02119889minus4

minus 2119889minus4

] The initial power of all links is set to be01mW and the initial prices value is set to be 1 for congestionand power

51 Network Utility Figures 5(a) and 5(b) show respectivelythe evolution of rate for the multicast session with 120573 = 0

and 120573 = 1 both of which show that the flow rate convergesgradually along with the algorithmrsquos iteration On the otherhand the speed of growth for the rate slows down gradually inthis processThe rate for two sessions behaves similarly whichillustrates that the algorithm has a good fairness Also thesession rate when 120573 = 1 is lower than that when 120573 = 0 whichindicates that the rate is slightly reduced when increasing thevalue of balance coefficients

In order to measure the difference of the tree for the samemulticast session the process of rate change of the left and theright multicast trees for session 119878

1is shown in Figure 6 The

rates of left and right multicast trees improve rapidly in theearly stage but their patterns are different The price of thetree is subject to interference and bandwidth in which thetree featuring a lower price should be allocated more flow

Figure 7 shows the evolution of the congestion prices overdifferent trees The left and right tree price tends to be stableafter 200 generations which is similar to the flow evolutionprocess in Figure 6

52 Power Consumption In wireless network the link own-ing lower channel gain needs to increase the power for elim-inating the linkrsquos bottleneck Enhancing power will increaselink capacity but it will increase the interference to theneighbour nodes consequently reducing the capacity of thenearby links Therefore the allocation of power is not thebigger the better in practice A comparison of the nodesrsquotransmit power evolution process is shown in Figure 8 wherethe balance coefficient is set to be 0

From the top to the bottom in Figure 8 the power curvesrespond to nodes 119898

2 V2 1198983 1199042 1199041 1198981 V1 1199081 and 119908

2

The node power is different due to the mutual interferenceof the neighboring nodes The growth speed is not smoothhowever the power update of the most nodes is upward asthe cost of power consumption is not considered Some nodesboost power in order to improve link capacity in the iterativeprocess while other nodes do not do so as they have met thebandwidth requirements

Given wireless nodersquos limited energy realizing a highthroughput will shorten the life of nodes in the networkIn the experiment the balance coefficient value is set be 01for reducing energy consumption The results of power areshowed in Figure 9 which correspond to nodes 119898

2 1198983 1198981

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 200 400 600 800 10000

5

10

15M

ultic

ast s

essio

n ra

te (k

bps)

Iteration

Multicast session 1Multicast session 2

(a)

0 200 400 600 800 10000

5

10

15

Mul

ticas

t ses

sion

rate

(bps

)

Iteration

Session 1Session 2

(b)

Figure 5 (a) Session rate with 120573 = 0 (b) Session rate with 120573 = 1

0 200 400 600 800 10000

5

10

15

Tree

rate

(kbp

s)

Iteration

Left treeRight tree

Figure 6 Flow rate of left tree and right tree of session 1

V2 1199042 V11199081 1199041 and119908

2from top to bottomWith the iteration

of the algorithm the power of some nodes is significantlydecreased when balance parameter 120573 = 0

53 The Effect of Tradeoff between the Total Energy Cost andTotal Utility The parameter 120573 plays a balancing role betweenthe energy efficiency and network performanceThe results ofcomparing different values of balance coefficient 120573 are shownin Figure 10(a) In this experiment the balance coefficientis increased from 0 to 1 and the session rate is reducedto approximately 172 as a result The experiment resultsfor node transmission power are showed in Figure 10(b)which shows that the sum of node transmission power con-sumption is significantly decreased from 13mW to 08mW

0 200 400 600 800 100000

01

02

03

04

05

Pric

e of t

ree

Iteration

Left treeRight tree

Figure 7 Evolution of congestion prices over different multicasttrees

when the balance coefficient increases The total powerconsumption of the power control is decreased noticeablywhen network traffic is declined very slightly and the valueof the balance coefficient is set as 01 The results show thatthe network energy consumption is very sensitive to thebalance coefficient Nevertheless the balance coefficient haslittle effect on the rate of nodes which represents a majoradvantage for battery-powered wireless networks

54 Comparison with the Equal and Fixed Power AllocationAlgorithms We compare our algorithm with the equal andfixed power allocation for the session rate evolution Theequal power allocation assumes that the nodes are allocated

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 200 400 600 800 1000

06

08

10

12

14

16

18

20

22

24

26

Nod

e pow

er (m

W)

Iteration

Figure 8 Evolution of transmit power 120573 = 0

0 200 400 600 800 10000002040608101214161820222426

Nod

e pow

er (m

W)

Iteration

Figure 9 Evolution of transmit power 120573 = 01

00 02 04 06 08 1010

15

20

25

30

Sum

of s

essio

n ra

te (b

ps)

120573

(a)

00 02 04 06 08 1002468

101214

Sum

pow

er co

st (m

W)

120573

(b)

Figure 10 (a) The effect of the tradeoff factor on total flow rate (b) The effect of the tradeoff factor on total cost of power

a maximum power of 25mW to affiliated link and thefixed power allocation sets the link power to be 05mWIn this experiment we randomly vary channel gain andindependently run the simulation 10 times to measure theflow rate As can be seen from Figure 11 the session rate

based on our joint rate and power allocation is higher thanthat of the equal allocation The session rate of the fixedpower allocation is significantly lower than the other twoalternatives but due to its fewer optimized variables thefixed power allocation enjoys faster convergence Although

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 100 200 300 400 500 600 700 800 900 100015

20

25

Sum

of s

essio

n ra

te (k

bps)

Iteration

Equal allocation

Utility allocationFix 05mW

Figure 11 Session rate for three power allocations

utility power optimization converges slowly it needs feweriterations to reach stable approximation giving it morepractical engineering advantages

On the other hand the equal power allocation costs a totalpower of 225mW while the fixed power needs 85mW andthe utility-based power allocation consumes 13mW whichare significantly lower than the equal power allocation Theresults reveal that the joint optimization can improve networkperformance while reducing node power consumption andprolonging the life of network

6 Conclusions

In this paper we study the joint optimization of powerallocation and rate control for multicast flow with networkcoding for enhancing wireless network throughput Usingthe network utility maximization theory we propose amultisession flow congestion control and power allocationoptimizationmodel under the condition that network codingsubgraph is determined We also use the dual decompositionto decompose the problem into source flow control and theintermediate node power allocation optimization problemswhere each subproblem corresponds to resource allocationof the transport layer and the physical layer For dealingwith the instability problem by multiple subtrees a proximaloptimization algorithm is adopted Analysis demonstratesthat the distributed algorithm can converge under the con-dition that the step length is small enough Finally numericalsimulation shows that our optimization algorithm can fairlyallocate resources for network traffic Due to the powercontrolling the network nodersquos energy efficiency has beenimproved significantly with a small impact on the networkcapacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The subject is sponsored by the National Natural Sci-ence Foundation of China (nos 61370050 and 61373017)the Anhui Provincial Natural Science Foundation (no1308085QF118) and the Anhui Normal University DoctoralStartup Foundation (151305)

References

[1] S Samat M Patrick and R Catherine ldquoCross-layer optimiza-tion using advanced physical layer techniques in wireless meshnetworksrdquo IEEE Transactions on Wireless Communications vol11 no 4 pp 1622ndash1631 2012

[2] A Khreishah I M Khalil and P Ostovari ldquoFlow-based xornetwork coding for lossy wireless networksrdquo IEEE Transactionson Wireless Communications vol 11 no 6 pp 2321ndash2329 2012

[3] S Jagadeesan and V Parthasarathy ldquoCross-layer design inwireless sensor networksrdquo in Advances in Computer ScienceEngineering amp Applications vol 166 pp 283ndash295 2012

[4] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[5] J Price and T Javidi ldquoDistributed rate assignments for simul-taneous interference and congestion control in CDMA-basedwireless networksrdquo IEEE Transactions on Vehicular Technologyvol 57 no 3 pp 1980ndash1985 2008

[6] A Ghasemi and K Faez ldquoJointly rate and power controlin contention based MultiHop Wireless Networksrdquo ComputerCommunications vol 30 no 9 pp 2021ndash2031 2007

[7] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[8] C Long B Li Q Zhang B Zhao B Yang and X GuanldquoThe end-to-end rate control inmultiple-hopwireless networkscross-layer formulation and optimal allocationrdquo IEEE Journalon Selected Areas in Communications vol 26 no 4 pp 719ndash7312008

[9] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo Interna-tional Journal of Electronics and Communications vol 64 no 2pp 99ndash111 2010

[10] M van Nguyen S H Choong and S Lee ldquoCross-layeroptimization for congestion and power control inOFDM-basedmulti-Hop cognitive radio networksrdquo IEEE Transactions onCommunications vol 60 no 8 pp 2101ndash2112 2012

[11] J Yuan Z Li W Yu et al ldquoA cross-layer optimization frame-work for multihop multicast in wireless mesh networksrdquo IEEEJournal on Selected Areas in Communications vol 24 no 11 pp2092ndash2103 2006

[12] R Ahlswede N Cai S-Y R Li and R W Yeung ldquoNetworkinformation flowrdquo IEEE Transactions on Information Theoryvol 46 no 4 pp 1204ndash1216 2000

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[13] C Fragouli and E Soljanin ldquoNetwork coding applicationsrdquoFoundations and Trends in Networking vol 2 no 2 pp 135ndash2692007

[14] T Matsuda T Noguchi and T Takine ldquoSurvey of networkcoding and its applicationsrdquo IEICE Transactions on Communi-cations vol E94-B no 3 pp 698ndash717 2011

[15] B Li and D Niu ldquoRandom network coding in peer-to-peernetworks from theory to practicerdquo Proceedings of the IEEE vol99 no 3 pp 513ndash523 2011

[16] L Chen T Ho M Chiang S H Low and J C DoyleldquoCongestion control for multicast flows with network codingrdquoIEEE Transactions on Information Theory vol 58 no 9 pp5908ndash5921 2012

[17] C Z Kwong Zhu Fang ldquoCognitive wireless mesh network QoSconstraints ofmulticast routing algorithmsrdquo Journal of Softwarevol 23 no 11 pp 3029ndash3044 2012

[18] Y Xi and E M Yeh ldquoNode-based optimal power controlrouting and congestion control in wireless networksrdquo IEEETransactions on Information Theory vol 54 no 9 pp 4081ndash4106 2008

[19] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[20] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of