Effects of Delay Constraints on Multihop Networksusing Rateless Codes

Ashish James†, A.S. Madhukumar†, Ernest Kurniawan§ and Surya Dharma Tio†

†CeMNeT, School of Computer Engineering, NTU, Singapore 639798§Institute for Infocomm Research, A*STAR, Singapore 138632

Email: {†ashi0010,†ASMadhukumar,†sdtio}@ntu.edu.sg and §ekurniawan@i2r.a-star.edu.sg

Abstract—When a message is forwarded through a network, anode will have multiple opportunities to reliably receive the mes-sage and cooperate in transmission. However, the transmit powerof each node in a network is limited and so is the transmissionrange. Therefore, multihop transmission is a potential techniqueto deliver the data over large distances. Rateless code propertiesideally suit decode and forward based relaying strategies as itimproves the outage and bandwidth efficiency. In decode andforward based multihop networks, the messages are decoded andrelayed over sequential point-to-point communication links. Ina conventional multihop network, the intermediate nodes willhave to wait indefinitely to correctly decode the information. Toquantify the impact of this unknown waiting time, this paperanalyses the performance of a multihop network with strictdelay constraints. The average packet loss for such a network isobtained as a function of the total delay. By optimizing the totaldelay based on the performance across each links, the averagepacket loss can be minimized.

I. INTRODUCTION

In ad-hoc wireless networks, the self-configured nodes areconnected with each other by wireless links. The transmitpower as well as the transmission range of each nodes insuch networks is limited. Multihop transmission is a potentialtechnique to reliably deliver the data over large distances. Insuch networks, messages are relayed over sequential point-to-point links. The intermediate nodes have multiple opportu-nities to reliably receive the packets transmitted by a sourcenode. It then forwards the received information towards thefinal destination. Routing protocols determine the interme-diate nodes which are selected as the forwarding nodes insuch multihop networks. Several advantages can be cited foremploying multiple hop transmission, such as interferencereduction, improved connectivity and allowing distant nodesto communicate with each other more efficiently [1]- [2].

Of the forwarding strategies employed at the intermediatenodes, decode and forward based relaying strategies mitigatethe effect of noise amplification and thereby offer an improvedperformance. Rateless codes or fountain codes introduced in[3] ideally suit decode and forward based relaying strategy,where the intermediate node monitor the transmitting nodevirtually indefinitely until it has enough mutual informationto decode. This improves the outage as well as the bandwidthefficiency. Rateless codes are erasure coding schemes wherethe source oblivious of the channel state information (CSI)generates as many encoding symbols as needed for correct

decoding at the destination node (accumulates enough mutualinformation for proper decoding) [4]. However, rateless codeshave been shown to offer good performance in other channelssuch as binary symmetric channel, additive white Gaussiannoise (AWGN) and fading channels [5]- [7]. In addition, a verylow encoding/decoding complexity favors the use of ratelesscodes. This motivates the analysis on the impact of multihoptransmission with rateless codes.

Rateless coding is advantageous for a network with multiplenodes, as the source data can be reconstructed from any subsetof the encoded packets. Successful relays in such schemesassist the source in transmission. The performance of suchschemes has been analyzed in [4] and [8]. In [9], a throughputoptimal rateless coding scheme is proposed to relay LubyTransform (LT) codes across multiple nodes. However theperformance of such networks depends on the worst linkand it becomes the bottleneck of the entire system as itaffects the reliability of information delivered across multiplehops. In addition, such nodes may cause excessive delayswhich are not desirable for delay sensitive communications.In this paper the analysis is performed in a delay constrainednetwork with multiple hops which seek to deliver latency-sensitive information. Their benefits and drawbacks are clearlyillustrated.

For conventional multihop transmission based on ratelesscodes, the source node forwards the information to theneighboring nodes. Each of the intermediate nodes act asindependent fountain decoder/encoder where it decodes, thenre-encodes and transmits to the next neighboring node. Thisprocess continues until the message reaches the destinationnode. Even though such a scheme assures the reliable deliveryof information, the total delay encountered is boundless. Mostreal-time communication systems are delay intolerant wherethe decoding and re-encoding needs to be completed within agiven duration. In such a delay constrained multihop networkall the packets cannot be received within the interval, asthe packet delays for each hop are dependent on the totaldelay. Those packets which are received correctly within theinterval are transmitted in the subsequent hops to enable real-time communication over multihop networks. The number ofpackets lost will be a function of the elapsed time, which inturn is a factor of the quality of service (QoS) requirementof the system. We highlight the trade-off existing between

978-1-4244-8331-0/11/$26.00 ©2011 IEEE

average throughput and the delay.The rest of the paper is organized as follows. The system

and channel model considered for the analysis is explained inSection II. Section III analyzes the performance of the con-ventional multihop network and delay constrained multihopnetwork based on rateless codes. The numerical results arepresented in Section IV. Finally, concluding remarks and futureextensions of the current work are presented in Section V.

II. ANALYSIS MODEL

In this paper a wireless multihop network which consistsof wireless nodes connected by wireless links as shown inFig. 1 is considered. Each node is equipped with a singleantenna and the terminals are constrained to employ half-duplex transmission.

Relay

RN11

Source

SN

Destination

DN

Relay

RN21

Relay

RNi1

Relay

RNn1

.

.

.

Relay

RN12

Relay

RN1m

.

.

.

Relay

RN22

Relay

RN2m

.

.

.

Relay

RNi2

Relay

RNim

.

.

.

Relay

RNn2

Relay

RNnm

Fig. 1. Multihop relay system model

A. Received Signal & Channel Model

The network consists of a source node (SN), and a desti-nation node (DN) separated by a large distance, which makesdirect transmission ineffective. In such an ad-hoc scenario amessage is delivered to the DN by cooperating with m×n relaynodes (RN) placed in between SN and DN. The objective ofthe system is to reliably deliver bits generated at the SN toDN through n hops using rateless coding. The nodes operatein a time-division manner, i.e. only one node is allowed totransmit during a time slot. The transmitting node forwardsinformation to the m neighboring nodes. Out of the m relaysavailable during each hop, the best relay (RNi j, i ∈ {1 to n}and j ∈ {1 to m}) is selected as the forwarding node withproper coordination among the RNs.

The channel between the nodes is modeled as quasi-staticflat fading channel as in [10]. All the links are also corruptedby additive white Gaussian noise (AWGN) with zero meanand variance N0. In order to mark low reliability symbols aserasures, an erasure technique presented in [11] is considered.In this technique, a threshold value, γt is defined for the fading.The received signal at node q during time instant t is given as

yq(t) = hpq(t)xp(t)+ηq (1)

where p∈ {SN,RNi j} and q∈ {RNi j,DN}, ηq is the AWGN atnode q and xp is the transmitted symbol. Each node is assumedto transmit under equal power constraint, εs∀p and hence

E[|xp|2] = εs. If the received signal level, γq = |hpq|2εsN0

is less

than the threshold value, γt(threshold is dependent on the codeemployed and on the outage requirement), then the receivedsymbols are declared as erased. Equalization is performed onthe received signal in (1) using technique such as zero forcing(ZF). After equalization, the stream of recovered bits is fedinto the decoder.

All the nodes are assumed to make use of rateless coding,whereby they can encode and transmit as many independentfountain coded symbols as needed, constrained by the totaldelay. The data packets at SN are grouped into blocks ofk packets, each composed of b information bits. In fountaincoding, a number δ is randomly selected (generated from somedegree distribution, ρ(δ)). Each time, δ out of the k input pack-ets are randomly selected, linearly combined and transmitted.This process continues until one of the relay nodes (RNi j)in the next hop has enough mutual information to decode.This continues for several hops until the message reachesthe DN. It is not practical to attempt decoding after eachreceived block; this paper assumes the decoding is initiatedonly when the received generator matrix becomes full rank.Rateless coding can be considered as a form of continuousincremental redundancy and feedback channels are requiredonly to signal the successful decoding at the destination.

B. Multihop Transmission Schemes

The present paper analyzes the conventional multihop trans-mission using rateless coding and effects of delay constraintson multihop transmission. The routing and medium accesscontrol for these transmissions are not considered within thescope of the present work. The best relay is selected duringeach hop for transmission with proper coordination among thenodes and there is no contention.

For the conventional multihop transmission, each relaynode on the route receives the message from the node inthe previous hop. The previous node can be the SN itselfor another RN in the route. Consider the case where nodeNi j (i ∈ {0 to (n + 1)}, j ∈ {0 to (m + 1)}) communicateswith node N(i+1) j (where i = 0, j = 0 corresponds to theSN communicating with the RN across the first hop andi = (n + 1), j = (m + 1) corresponds to the RN in the nth

hop communicating with the DN). The node Ni j (SN or RNi j)forwards fountain coded packets until node N(i+1) j (RN(i+1) jor DN) is able to decode properly (the rank of the generatormatrix at node N(i+1) j is full). Then node N(i+1) j fountainencodes the decoded information and forwards to the nextnode N(i+2) j in the route. It goes on until the message isdecoded at the DN and one transmission cycle ends.

For delay constrained multihop transmission, the messageis transmitted from SN to DN via n hops with a hard delaydeadline. The goal is to optimize the number of transmissionssubject to the maximization of quality of service (QoS) withminimum delay. The total maximum delay of the system isfixed at D seconds and the maximum delay per hop is D/nseconds (delay is assumed to be uniformly distributed on theroute). Though the transmission scheme remains similar to theconventional multihop transmission, the number of encoded

packets transmitted depends on the delay. The maximumnumber of channel uses that can happen within the specifieddelay per hop is taken as L. Of the ki packets transmittedby node Ni j, node N(i+1) j is able to decode only (ki − Δi)(where Δi is the number of packets lost during ith hop) packetsbecause of strict delay constraint. Node N(i+1) j then re-encodesthe correctly decoded packets and forwards to the next nodeN(i+2) j. This goes on until the message is decoded at the DN.There will be a loss of Δ (Δ = Δ1 + Δ2 + · · ·+ Δn) packets atthe DN owing to the strict delay deadline. The average packetloss in such a scenario is derived in the next section.

III. PERFORMANCE ANALYSIS

The average number of transmissions required for conven-tional multihop transmission and the average packet loss incase of delay constrained multihop transmission are analyzedin this section. The details of the analysis is given below.

A. Conventional Multihop Transmission

When no delay constraint is imposed on the system, theforwarding node transmits fountain coded packets until theneighboring node decodes correctly. For the system config-uration in Fig. 1, the intermediate relay nodes receive theinformation from the previous node in the route. It re-encodesand transmits when decoding is successful. Throughout thesequel it is assumed that the decoding is deemed to besuccessful once the received generator matrix is full ranki.e Rank (GR) = k, where GR is the generator matrix at thereceiving node. Denoting the erasure probability across nodesp and q as Ppq

e = Pr[γq < γt ], the probability of receivingexactly k encoded packets after T transmissions can be givenas

PTk (Ppq

e ) = (1−Pe)(

T −1k−1

)(Ppq

e )T−k (1−Ppqe )(k−1) (2)

Let Tr denote the earliest number of transmissions (Tr ≥ k)when the received generator matrix is full rank. Tr basicallycaptures the time when the intermediate node is ready forforwarding the information. If the binary generator matrixis considered to be equiprobable (the same analysis can beperformed for other cases), then the probability of having agenerator matrix of rank k after receiving r unerased encodedpackets can be expressed as [12]

Pkrank(r) =

{0 r < k

∏k−1i=0

(1−2i−r

)r ≥ k

(3)

If the intermediate node decodes after Tr transmissions; thenthe probability of having a full rank generator matrix (GR)at intermediate node after Tr transmissions can be computedusing (2) as

Pr[Rank(GR) = k,Tr] =Tr

∑i=k

Pkrank(i).P

Tri (Ppq

e ) (4)

By equating (4) for a specific value of Ppqe , the average

number of transmissions required for the generator matrixat the receiving node to be full ranked for a single hop

can be found. For the case when the probability of packeterasures are uniform across all the n hops, the total numberof transmissions (Ttot) required to have a full ranked generatormatrix at DN is given as

Ttot = n×Tr (5)

This gives an estimate of the maximum total delay that canbe encountered by the information packets to reach the DN.

B. Delay Constrained Multihop Transmission

In a delay constrained scenario, the transmission at eachhop is limited to a specified value ‘L’. This will imposethe maximum delay across n hops to be D seconds. So themaximum number of transmissions can be given as

Ttot = n×L (6)

Such a scenario does not guarantee the delivery of all the kpackets across the n hops. However, in this case more priorityis given to the overall delay and thereby only a subset of kpackets, (k−Δ) packets are successfully delivered to the DN(Δ is the total packet loss across n hops). Consider ki packetsare correctly delivered across i hops from SN to node Ni j. AfterL channel uses, node N(i+1) j transmits only a subset of the ki

packets transmitted by node Ni j. This is a factor of the numberof packets decoded correctly during the ith hop. It is assumedthat node N(i+1) j is able to decode correctly the number ofpackets equal to the rank of the generator matrix receivedat node N(i+1) j (practically in a Rayleigh fading channel thenumber of packets decoded correctly will be less than or equalto the rank of the generator matrix).

In order to find the number of packets lost during the ith

hop, there are two cases to be considered. 1) Node N(i+1) j isable to decode correctly the ki packets transmitted by nodeNi j i.e the generator matrix at node N(i+1) j is full rank with(L−1) or less number of transmissions. This is given by theprobability

Pr[Rank(GN(i+1) j)= ki,ki · · ·(L−1)]=

(L−1)

∑u=ki

u

∑j=ki

Pkirank( j).Pu

j

(P

Ni jN(i+1) je

)(7)

2) For the second case, with L transmissions node N(i+1) j isable to decode only pc packets correctly, i.e.

Pr[Rank(GN(i+1) j) = pc,L)] =

L

∑j=pc

Ppcrank( j).PL

j

(P

Ni jN(i+1) je

)(8)

where pc ∈{0 to ki}. Taking the total probability theorem overall the pc disjoint sets of the sample space, the number ofpackets decoded correctly in the ith hop is given as

ki+1=ki.Pr[Rank(GN(i+1) j)=ki,ki · · ·(L−1)]+

ki

∑pc=1

pc.Pr[Rank(GN(i+1) j)=pc,L] (9)

Then the packet loss across node Ni j to node N(i+1) j is

Δi = ki − ki+1 (10)

Then the total packet loss across n hops is given as

Δ ≤n

∑i=1

Δi

= k−k1+(k−k1)−k2 +(k−k1−k2)−k3+· · ·+(k−k1−k2−·· ·−kn−1)−kn

= nk−nk1−(n−1)k2−·· ·− kn (11)

where ki is the number of packets decoded correctly at the ith

hop. The average packet loss is a function of the channel usesper hop, which depends on the overall delay permitted for thenetwork. In order to limit the packet loss to a tolerable value,the maximum number of channel uses should be determinedbased on the current channel characteristics, transmit powerand the delay tolerance of the network. This optimization canbe achieved when

Δopt = arg minL

Δ (12)

The optimum packet loss is selected based on the allowednumber of channel uses for the current propagation character-istics. It is observed that there is a trade-off existing betweenthe overall delay and the average throughput of the network.

IV. NUMERICAL RESULTS & DISCUSSION

In this section, we evaluate the performance of a conven-tional multihop network and a delay constrained multihopnetwork. The information packet generated at the SN isencoded with a 1/4 rate convolutional code before ratelessencoding. All the packets are of equal length. For computa-tional tractability, we consider a multihop network as shown inFig. 1 with n hops. Only one RN in each hop forwards the in-formation to the nodes in the next hop. As explained in sectionII, each of the wireless links are assumed to be statisticallyindependent and identically Rayleigh distributed. In order todetermine whether packets are erased, an SNR threshold isset. It is based on the maximum capacity achievable acrosseach links. If the instantaneous SNR falls below the thresholdSNR, the rateless coded packets are assumed to be erased.

A. Conventional Multihop Transmission

For a conventional multihop transmission the links alongeach hop are assumed to be uniformly distributed. The averagenumber of transmissions per hop is evaluated in such ascenario and is shown in Fig. 2. It is observed that when theSNR is low or when the links are least reliable, the number oftransmissions required is very large. Thus the weakest link in amultihop network becomes the bottleneck of the whole system.As the link reliability increases (at high SNRs) the number oftransmissions required decreases. With rateless decoding, thenumber of linear packet combinations required is only slightlylarger than k when the channel is reliable. For eg. only 14linear packet combination is required to correctly decode 12(k) packets at an SNR of 10 dB. Based on these results, theoverall performance of a multihop system employing ratelesscodes will be dragged down by the presence of a hop withpoor propagation characteristics. In other words, the weakestlink becomes the bottleneck of the whole system.

-10 -8 -6 -4 -2 0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

Ave

rage

num

ber

of t

rans

mis

sion

s pe

r ho

p

SNR

Fig. 2. Variation of average number of transmissions required per hop withSNR for a packet size (k) = 12

B. Delay Constrained Multihop Transmission

For delay constrained multihop transmission, the number ofchannel uses per hop is limited to a specific value (L). Thistakes care of the infinite delay if the link is unreliable but doesnot guarantee the delivery of the entire set of k packets. Sothere is a trade-off with delay against reliability as obtainedin (11). Only those packets which are decoded correctly aretransmitted in the next hop. Figure 3 shows the variation ofpacket loss with received SNR. It is visible that the numberof packets lost decreases as the reliability of link improves(SNR increases). As only the correctly decoded packets aretransmitted during each hop, packet loss increases with numberof hops. But as the number of channel uses per hop remainsconstant across the number of hops, the code-rate effectivelydecreases with number of hops. Thereby the performancedegradation with number of hops is limited. It is visible thatthe reliability of such networks improves with the link qualityand reduces the total delay when the link is unreliable fortransmission.

Figure 4 shows the variation of packet loss with numberof channel uses per hop at an SNR of 0 dB. It is observedthat the number of packets that can be correctly decoded is afactor of the reliability of the links irrespective of the numberof channel uses. In order to constrain the overall delay forpacket reception at DN, the number of channel uses per hopis limited to a specific value. Thus as given by (12), the numberof channel uses (L) should be optimized based on the packetloss which is a function of probability of erasure along thelinks. Optimizing the number of channel uses mitigates theinfinite delay and delivers the number of packets based on thereliability of the links.

Figure 5 illustrates the variation of packet error rate (PER)with number of hops for multihop transmission based onconventional and delay constrained scenarios. Hard decisiondecoding for both rateless coding and convolutional codingis performed at the receiver. For delay constrained scenario

-10 -8 -6 -4 -2 0 2 4 6 8 100

10

20

30

40

50

60

70

80

90P

acke

t Lo

ss (

%)

SNR

No. of hops = 5No. of hops = 2No. of hops = 1

Fig. 3. Comparison of percentage of packet loss with SNR for varyingnumber of hops with a packet size (k) = 12 and number of channel uses perhop (L) = 20

15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Pac

kets

Los

s (%

)

Number of channel uses/hop (L)

No. of hops = 1No. of hops = 2No. of hops = 5

Fig. 4. Variation of packet loss with number of channel uses per hop for anaverage SNR of 0 dB

0 2 4 6 8 10 12 14 16 18 2010

-3

10-2

10-1

100

PE

R

SNR

NH = 1, CMTNH = 2, CMTNH = 5, CMTNH = 1, DMT, L = 20NH = 2, DMT, L = 20

Fig. 5. PER performance comparison for conventional multihop transmissionand transmission constrained by delay. NH = No. of hops, CMT = Conven-tional multihop transmission, DMT = delay constrained multihop transmissionand L = No. of channel uses/hop

CRC is performed to detect decoding errors. For conventionalmultihop transmission it is evident that there is an increase inPER with number of hops as the error propagates from one hopto the next hop till it reaches the DN. In a delay constrainedscenario, only the packets decoded correctly are transmittedin the subsequent hops. This results in better reliability withlower number of packet decoding at the DN and a lower PER.

V. CONCLUSION

In this paper, the performance of rateless codes in amultihop scenario is analyzed. In a conventional multihopnetwork, the total time for communication is dependent onthe performance of the worst link of the network. Thusthe more reliable links are not utilized, which restrict theoverall throughput of the network. The performance of real-time communications in wireless multihop networks dependson the overall allocated time and the packet lost duringtransmission across the network. The average packet loss isobtained as a function of the total number of transmissions,where the transmissions across hops are limited based on theQoS requirement of the network. Moreover, if the number oftransmissions across hops is selected appropriately, the totaldelay and the packet loss can be minimized.

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