978-1-4673-4714-3/13/$31.00 ©2013 IEEE 1526

2013 Ninth International Conference on Natural Computation (ICNC)

A Slotted Two-dimensional Probability Multi-channel and Random Multi-access Protocol Based on the

Markov Chain Yifan Zhao, Hongwei Ding, Lin Lin, Haiying Deng, Yinbo Yi

Information College, Yunnan University, Kunming, China

Abstract— our paper analyzes a slotted two-dimensional probability multi-channel and random multi-access protocol based on two kinds of system models: the symmetry access model and the load equilibrium model on the Markov Chain. It analyzes the systemic throughput of the STPMR protocol along with the result of throughput based on priority under the two system models. The computer simulations show that our theoretical analysis is in concordance with the simulations.

Keywords- Symmetry Access Control (SAC), STPMR Protocol, Load Equilibrium Model (LEM)

I. INTRODUCTION To improve the QoS of wireless communication networks,

Ningyu Zhou presented a Slotted Two-Dimensional Probability Multi-Channel and Random Multi-Access Protocol (STPMR), which ensures the demands for different service QoS in Wireless Communication Networks (WCN) by assigning different priorities, with the application of multi-channels and the choice of parameters p1 and p2 to ensure the high channel utilization in different WCN. The system time is divided into time slots (the maximum system propagation delay equals a slot width), with the packet arriving during the idle time slot must wait until the next time slot to transmit. That avoids arbitrary of sending data in terminal and reduces the probability of collision in order to ensure the higher channel-utilization rate.

Our paper builds the way of the Symmetry Access Control (SAC) based on the STPMR protocol in the WCN. In a SAC mode, the system through the control channel assignment to realize load balance. Our paper analyzes two kinds of mathematical model which are SAC and Load Equilibrium Model (LEM), based on the Markov chain method, and calculates two key parameters (the system throughput and the priority throughput) in two control modes, and obtains the probability of working process in each state. The simulations show that the theoretical analysis and the experiment results are unanimous and reasonable, and explain the theory of Markov chain is used to analyze the throughput of the STPMR protocol is reasonable, and the derivation system throughput process also shows the simplicity of this method. Thus, the Markov chain method is suitable for analysis of the state space which is certain in network protocol model.

II. STPMR PROTOCOL WITH SAC

A. System model The wireless communication system has N channels. All

terminal users can occupy channel randomly. In order to improve the system access efficiency, we provide the terminal use slotted two time dimensional probability CSMA to occupy channel randomly. Due to the system have the voice streaming, the MPEG streaming, the data streaming, and the other business streaming, so according to the different characteristics of the business set up different priority, and at the same time, we regulate different terminal user occupancy multi-channel. In the ‘Research on Two Multiple Channels and Multiple Services ALOHA Systems with Priority Control’, Liangping Yuan proposed the rule of occupying multi-channel fixed, however, the kind of control strategy increased the load channel, and the balance load characteristic of channel load is poor. Our paper proposes a new multi-channel control mode can reach load balance.

In the MCN of the STPMR protocol, the system has N channels and N priorities. The priority sequence is arranged from low to high as priority 1, priority 2,.... priority N as shown in figure 1, i packet will arrive in priority i each time, priority 1 occupies channel 1, priority 2 occupies channel N and N-1, priority 3 occupies channel 1, 2 and 3, priority 4 occupies channel N-1 and N-2, N-3…, By analogy, priority for odd group occupancy the channel is from positive direction of the system, and priority for even group occupancy channel is from the reverse direction.

Fig.1 System model with SAC

So the system load is balance when the channel number N is odd, arrival rate for each channel:

λ⎟⎟⎠

⎞⎜⎜⎝

⎛ += 12N

G (1)

The system load is half balance when the channel number N is even, arrival rate for each channel:

1527

⎪⎪⎩

⎪⎪⎨

⎧

=

=⎟⎠⎞

⎜⎝⎛ +

=evenjN

oddjN

G,

2

,12

λ

λ (2)

The two-dimensional probability CSMA described here is a new MAC protocol. It is different from detection of CSMA. For the two-dimensional probability CSMA, the site must sense the channel before sending packet:

• If the channel is in idle, transmits the packet with the probability 1P , and gives up transmitting with the probability 11 P− .

• If the channel is sensed busy, it detects the channel with the probability 2P , and gives up detecting the channel with the probability 21 P− until the channel goes idle, and then transmits the packet with the probability 1P , and gives up transmitting with the probability 11 P− .

The p-persistent CSMA protocol transmits the packet with probability p when the channel is idle. It will give up transmitting the packet when the arrival rate is small, and lead to waste of resources. The p-detection CSMA protocol detects the channel persistently with the probability p when the channel is busy. It improves the weakness of smaller arrival rate in p-persistent CSMA, but if p is much bigger, it will lead to conflict by persistently detecting channel. The two-dimensional probability CSMA transmits the packet with probability 1P . If the channel is in idle and detect the channel persistently on the probability 2P . If the channel is busy, so it improves the throughput performance of the system by choosing the probability 1P or 2P .

B. Performance Analysis of the STPMR With the SAC We make assumptions for the WCN before analyzing

system performance.

(1) The access model of channel j (j=1, 2,…N) is the slotted two-dimensional probability CSMA;

(2) Assume the system propagation delay of the packet transmission time is a, and all packets are unit length and the length is an integer multiple a.

(3) The system time is divided into time slots by unit time. Assume the length of time slots is maximum transmission delay a. Packets arriving during the idle time slot must wait for the next time slot to transmit.

(4) In the first slot of the transmission period TP (delay is a), packets can detect the channel status.

(5) The number of users for each priority is infinite, and the arrival rate of the priority i on the channel j is the Poisson process with λ.

(6) Assume that the channels are ideal without noise or interference.

(7) Collision packets will be transmitted again at a later time.

When transmitted again, these packets do not influence the arrived process.

In the circumstances of the System terminal users is unlimited, the transmitting packet process in N channel of the WCN system can divided into three random events:

(1) Packet transmission successful (U);

(2) Packet transmission conflict (B);

(3) There is no packet to transmit, the channel is in idle (I);So there are three events appear random in the time, divided these three random events like this:

(4) An idle random event I, the channel is idle;

(5) A composite random event BU, the channel is busy.

So, there are two events appear in the time axis with the random process. According to the protocol, certain state space, with the characteristics of the Markov chain and analysis method of state transition, we can get five states in N channels of the WCN with transmitting packet:

(1) The channel is idle (A1);

(2) The packet collides when the channel is busy (A2);

(3) The packet transmits when the channel is busy (A3);

(4) The packet collides when the channel is idle (A4);

(5) The packet transmits successfully when the channel is idle (A5).

Fig.2 Sending packets situation for channel j

Consider the following diagram in Fig. 2 for channel j (j = 1, 2, …, N), packets arriving during the idle time slot will be transmitted in the following time slot with the probability 1P , while those arriving during the transmission period TP will detect that the channel is busy and the channel is available until idle with the probability 2P . Then the packets which were not sent will be transmitted with probability 1P in the following time slot. If the total number of packets which will be transmitted equals or exceeds two, the packets will conflict in the next TP with the probability 1. If it equals one, then the packet will be transmitted successfully. If it equals zero, the busy period ends. Furthermore, the length of each TP is 1+a in each busy period BU. Arrived sequence is a random Poisson process with an independent stationary increment, so the number of arrivals in any TP is independent of the arrivals in every other TP.

Theorem 1: For the STPMR protocol with SAC in the wireless communication network, the systemic throughput is given by:

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∑=

−+−

−+−

−++

−−++=

N

japGappG

apGappGj

jj

jj

eaae

eappaapepGS

1)1(

222)1(

1

)1)(1(

])()1([121

121

(3)

If N is odd, the packet arrival rate for the service is given by equal (1) and the channel Load is equilibrium; If N is even, the packet arrival rate for the service is given by equal (2) and the channel Load is half equilibrium.

Proof: We make the following definitions before analysis: )(tZ : The event that there are n packets arrived in interval t; )(tX : The event that there are )0( nmm ≤≤ packets

transmitted with probability 1P in interval t; )(tY : In the TP of (1+a) time slot, the random event that k

arrivals detect the channel persistently with the probability 2P . When the channel is idle, the random event that r arrivals will be transmitted with the probability 1P ;

For the j channel of the STPMAR control protocol in the WCN:

)(tX denotes the number of packets to be sent in interval t,

tpGm

jmnmmn

tG

mn

nj

mn

mnmmn

jj em

tpGppCe

ntG

ppCntZPmtXP

1

!)(

)1(!)(

)1())(())((

111

11

−−−∞

=

∞

=

−

=−=

−===

∑

∑ (4)

The probability of no packets will be transmitted during a and the probability that only one packet will be transmitted during a are given by:

apG jeaXP 1)0)(( −== (5) apG

jjaepGaXP 1

1)1)(( −== (6)

The probability in the (1+a) time slot is the random event that k arrivals detect the channel persistently with the probability 1P . When the channel is idle, the random event that j arrivals will be transmitted with the probability 2P is:

)1(21 21

!)]1([

))1(( appGj

j jej

appGjaYP +−+

==+ (7)

We analyze the throughput of the channel j (j=1, 2,….N) according to the definition and hypothesis. In the wireless communication network system of the STPMR control protocol, channel j has five states in any time slot as Fig.3. The Markov chain marks the state of mutually transfer situation. From the above analysis, we get the state transition probability matrix T with rows, columns are 1A , 2A , 3A , 4A , 5A in sequence.

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

+−+

+−+−

+−−

+−

+−+

+−+−

+−−

+−

+−+

+−+−

+−−

+−

+−+

+−+−

+−−

+−

−−−

−−

−

00)1(21)1(21

)1(21)1(21)1(211

)1(21

00)1(21)1(21

)1(21)1(21)1(211

)1(21

00)1(21)1(21

)1(21)1(21)1(211

)1(21

00)1(21)1(21

)1(21)1(21)1(211

)1(21

11

11

11001

appjGeappjG

appjGeappjG

appjGe

appjGe

appjGeappjG

appjGeappjG

appjGe

appjGe

appjGeappjG

appjGeappjG

appjGe

appjGe

appjGeappjG

appjGeappjG

appjGe

appjGe

apjGaepjG

apjGaepjG

apjGe

apjGe

T (8)

Fig.3 Transfer situation

1 2 3 4 5 1 2 3 4 5A A A A A A A A A AP P P P P T P P P P P⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ (9)

1 2 3 4 51A A A A AP P P P P+ + + + = (10)

By the assumptions of the system: the length of packet 1=L , the average length of state 1A is aL =1 ， the average

length of other four state are a+1 and aLLLL +==== 15432 ,

the throughput of channel j is：

∑=

−+−

−+−

−++

−−++=

N

japGappG

apGappGj

jj

jj

eaae

eappaapepGS

1)1(

222)1(

1

)1)(1(

])()1([121

121

(11)

When N is odd the throughput is given by equal (1) and load is equilibrium; when N is even the throughput is given by equal (2) and load is half equilibrium.

Theorem 2: For the STPMR protocol with the SAC, the throughput based on priority i in the system is given by:

When N is odd:

∑=

+=−+−

−+−

−++

−−++=

i

jNGapGappG

apGappG

pjjj

jj

eaae

eappaapepS

1)1

2()1(

222)1(

1

)1)(1(

])()1([121

121

λλ (12)

When N is even；

;,

,22

21

21

2

)(

)12

(

)(

)2

(

)(

)12

(

)(

1oddi

eveniSiSi

SiSi

S

NG

pijNG

pij

NG

pijNG

pij

P =

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=+

−++

=

=+=

=+=

λλ

λλ (13)

Proof: When the channel load is half equilibrium (N is

even), for priority i when i is odd and the arrival rate of 2

1+i

channels is given by equal (1) and 2

1−i channels is given by

equal (2) .When i is odd, the arrive rate of 2i channels is given

by equal (1) and 2i channels is given by equal (2). But when

the channel load is equilibrium (N is odd), i channels on priority i are all given by equal (1), so the throughput of priority i is proofed.

1529

C. Simulations and Analysis for the STPMR protocol with the symmetric access Simulations of the STPMR protocol with the symmetric

access are based on the analytical results of the previous section. In the simulations, assume the channel is ideal. The system time axis is divided into time slots, the length of time slot is the maximum delay 1.0=a , packet length is unit 1, and the arrival of priority i is iλ (i=1, 2,…..N), λλλλ ==== N...21 ,

let N=4, average arrival rate λ5.21 =G , the priority from high to low is: voice business streaming, MPEG video streaming, data business streaming and other business streaming. Let N=5, average arrival rate λ31 =G , the priority from high to low is: voice business streaming, MPEG video streaming, data business streaming and other business streaming 1 and streaming 2.

Figure 4 and figure 5 compare the theoretical and simulation value for the STPMR protocol systems with symmetric access respectively, corresponding value for N is 4 and 5. The results based on priority considered four channels with four priorities having 06.01 =P , 02 =P and 06.01 =P ,

08.02 =P and 06.01 =P , 1.02 =P and 06.01 =P , 12 =P . And let N=5, the S-G-P varieties for 5.01 =P and 5.02 =P respectively are shown in figure 6 and figure 7. And let N=5, the S max -P varieties for 5.01 =P and 5.02 =P respectively are shown in figure 8 and figure 9. And figure 10 shows the G-P1-P2 varieties when the system throughput is max.

G1 (p1=0.06，p2=0) G1 (p1=0.06，p2=0.08)

G1 (p1=0.06，p2=0.1) G1 (p1=0.06，p2=1)

Fig 4 Compare the theoretical and simulation value for N is 4

G1 (p1=0.06，p2=0) G1 (p1=0.06，p2=0.08)

G1 (p1=0.06，p2=0.1) G1 (p1=0.06，p2=1)

Fig 5 Compare the theoretical and simulation value for N is 5

Fig.6 the S-G-P varieties for p2=0.5 Fig.7 the S-G-P varieties for p1=0.5

Fig.8 the S max -P varieties for p2=0.5 Fig.9 the S max -P varieties for p1=0.5

The throughput plotted by the simulations agrees with the theoretical analysis. Explain the correctness of the Markov chain for throughput analysis, the derivation process also shows the simplicity of this method, thus, the analytical method can be used to analyze networks protocol model with certain state space.

The simulations also show the following results:

• Comparison of figure 4 and figure 5, for the same 1P and 2P , the higher priority data uses more system resources,

each priority throughput reached maximum as the network load from light to heavy, and with the further increase of the load each priority throughput gradually decreased, but no matter how the load changes, each priority occupy usable system resources in certain proportion. Also ensures the high QoS demands of high priority while taking into account the fair.

• Comparison of figure 6 and figure 7, for the same probability 1P , for light loads, the larger 2P is the higher throughput is; for heavy loads, the smaller 2P is the higher throughput is. It also shows that for the same probability 2P, in the situations of different loads, the change of throughput with 1P is similar to the situation that 1P is certain. So comprehensive two figures, for the change of 2P, there is more influence to throughput than 1P .

• Comparison of figure 7 and figure 8, let N=5, the system has a maximum throughput( 933.2max =S ), when 5.01 =P and 03.02 =P , if 03.01 ≺P , maxS will increases along with the increase of 1P and when 03.01 =P the maxS is

1530

maximum, then maxS will keep the maximum along with the change of 1P ; The system has a maximum throughput( 2479.3max =S ) when 5.01 =P and 12.02 =P , after that, maxS will decrease along with the change of 2P , maxS

is minimum when 12 =P .

• As showed in figure 10, When 1P is smaller, the combination of 1P and 2P corresponding to bigger G, the system throughput to the tolerance of load is much larger in the combination range of 1P and 2P .

III. STPMR PROTOCOL WITH LEM The system balances the loads through control of the

assignment of channel and the system balances the channel loads through control of the arrival rate based on priority.

A. System model As showed in figure 11, consider the system with the

STPMR protocol having N channels and N priorities. The service with priority i occupies channel 1 to channel i (i =1, 2,… N), the service with priority 1 occupies channel 1, the service with priority 2 occupies channel 1 and channel 2, and the service with priority N occupies channels 1 to N. The packet arrival rate for service with priority i on channel j is

)(,1

)( ijjN

G jpij ≤

+−=λ and the packet arrival rate of each

channel is G.

Fig.11 System model

B. System Analysis of the STPMR with the LEM Theorem 3: For the STPMR protocol with the LEM, the

systemic throughput is given by:

1 2 1

1 2 1

(1 )1 2 2 2

(1 )

N [ (1 ) ( ) ](1 )(1 )

Gp p a Gp a

Gp p a Gp a

Gp e p a a p p a eSae a e

− + −

− + −

+ + − −=+ + −

(14)

where the packet arrival rate for the service with priority i on

channel j is )(,1

)( ijjN

G jpij ≤

+−=λ , i=1, 2,… N.

Proof: For all N channels of the system are balanced, )1( NjGG j ≤≤= and Theorem 3 can be proved.

Theorem 4: For the STPMR protocol with the LEM, the throughput based on priority i in the system is given by:

∑= −

−+++−

−−−++

+−

+−=

N

j apjGea

appjGae

apjGeappaap

appjGepjG

jnRS1

})11)(1(

)1(21

]1)22()1(2[)1(21

1){

11

( (15)

Proof: For all the channels of the system are balanced, the packet arrival rate for service with priority i on channel j is

)(,1

)( ijjN

G jpij ≤

+−=λ and Theorem 4 can be proved.

C. Simulations and Analysis The simulation environment of the STPMR protocol with

the LEM is identical with the SAC. Figure 12 compares the theoretical and simulations value

for the STPMR protocol system with the LEM, the results based on priority considered five channels with five priorities having 06.01 =P , 02 =P and 06.01 =P , 08.02 =P and 06.01 =P ,

1.02 =P and 06.01 =P , 12 =P . And let N=5, the S-G-P varieties for 5.01 =P and 5.02 =P respectively are shown in figure 13 and figure 14. And let N=5, the S max -P varieties for 5.01 =P and

5.02 =P respectively are shown in figure 15 and figure 16. And figure 17 shows the G-P1-P2 varieties when the system throughput is max.

From figures 12-17 the theoretical analysis method can describe the system reasonably, the theoretical value agrees with the simulations results.

The simulations also show the following results:

• From figure 12, for the same 1P and 2P , the higher priority data uses more system resources, each priority throughput reached maximum as the network load from light to heavy, and with the further increase of the load each priority throughput gradually decreased, but no matter how the load changes, each priority occupy usable system resources in certain proportion. Also ensures the high QoS demands of high priority while taking into account the fair.

• Comparison of figure 13 and figure 14, for the same probability 1P , for light loads, the larger 2P is the higher throughput is; for heavy loads, the smaller 2P is the higher throughput is. It also shows that for the same probability 2P, in the situations of different loads, the change of throughput with 1P is similar to the situation that 1P is certain. So comprehensive two figures, for the change of 2P, there is more influence to throughput than 1P .

• Comparison of figure 15 and 16, let N=5, the system has a maximum throughput( 933.2max =S ),when 5.01 =P and

03.02 =P , if 03.01 ≺P , maxS will increases along with the increase of 1P and when 03.01 =P the maxS is maximum, then

maxS will keep the maximum along with the change of 1P ; The system has a maximum throughput( 2479.3max =S ) when 5.01 =P and 12.02 =P ,after that, maxS will decrease along with the change of 2P , maxS is minimum when 12 =P

.

• As shown in figure 17, when 1P is smaller, the combination of 1P and 2P corresponding to bigger G. So the system throughput to the tolerance of load is much larger in the combination range of 1P and 2P .

1531

G1(p1=0.06，p2=0) G1(p1=0.06，p2=0.08)

G1(p1=0.06，p2=0.1) G1(p1=0.06，p2=1)

Fig 12 Compare the theoretical and simulation value for N is 5

Fig.13 the S-G-P varieties for p2=0.5 Fig.14 the S-G-P varieties for p1=0.5

Fig.15 the S max -P varieties for p2=0.5 Fig.16 the S max for p1=0.5

Fig.10 the G-P1-P2 varieties when throughput max for SAC Fig.17 for LEM

Fig.18 the S max –P varieties for p1=0.06 Fig.19 the S max for p2=0.9

I. COMPARISON FOR SAC AND LEM The S max -P varieties for 06.01 =P and 9.02 =P are

respective shown in figure 18 (N=4) and figure 19(N=5).

II. CONCLUSIONS The paper is based on the Markov chain method to analyze

two kinds of mathematical model: the SAC and the LEM, and calculates two key parameters: the system throughput and priority throughput in the two control modes. The simulations show that the correctness of theoretical derivation, and explain the theory of the Markov chain used for throughput analysis of the STPMR protocol is correct, the derivation system throughput process also showed the simplicity of this method, so the Markov chain method is suitable for analysis of the state space that is certain in the network protocol model.

REFERENCES [1]. AHMAD I, LUO J C. On using game theory to optimize the rate

control in video coding. IEEE Transactions on Circuits and Systems for Video Technology, 2006, 16(2): 209 - 219.

[2]. Ghasem ， Sousa ES. Collaborative spectrum sensing for opportunistic accessing fading environment. in Proc. IEE DySPAN2005, 2005: 131-136.

[3]. Mitola J, Maguire G, Jr.Cognitive radio: Making software radios more personal. IEEE Personal Communications Magazing, 1999, 6(4): 13-18.

[4]. Mitola J. Cognitive radio for flexible mobile multimedia communications. Mobile Multimedia Communications 1999(MoMuC99), 1999 IEEE International Work-shop on , Nov. 1999, 15-17: 3-10.

[5]. Haykin S. Cognitive radio: Brain-empowered wireless communications, IEEE Journal on Selected Areas in Communication s, 2005, 23(2): 201-220.

[6]. FOSCHINI G J, MILJANIC Z. A simple distributed autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology, 1993, 42(4): 641-646.

[7]. ALPCAN T, BASAR T, SRIKANT R. CDMA uplink power control as a non-cooperative game. Proceedings of the 40th IEEE Conference on Decision and Control. 2001. 197-202.

[8]. KOSKIE S, GAJIC Z. A nash game algorithm for SIR-based power control in 3G wireless CDMA networks. IEEE/ACM Transactions on Networking, 2005, 13(5): 1017-1026.

[9]. MESHKATI F, CHIANG M, SCHWARTZ S C, et al. A non-cooperative power control game for multi-carrier CDMA systems. 2005 IEEE Wireless Communications and Networking Conference. 2005. 606-611.

[10]. SUNG C W, LEUNG K. A generalized framework for distributed power control in wireless networks. IEEE Transactions on Information Theory, 2005, 51(7):2625-2635.

[11]. BERGGREN F, KIM S L. Energy-efficient control of rate and power in DS-CDMA systems. IEEE Transactions on Wireless Communications, 2004, 3(3): 725-733.

[12]. YATES R D. A framework for uplink power control in cellular radio systems. IEEE Journal on Selected Areas in Communications, 1995, 13(7):1341-1347.

[13]. HUANG J W, BERRY R A, HONIG M L. Distributed interference compensation for wireless networks. IEEE Journal on Selected Areasin Communications, 2006,24(5):1074 - 1084.

[14]. SULIMAN I M, OPPERMANN I. A cooperative multihop radio resource allocation in next generation networks. 2005 IEEE 61st Vehicular Technology Conference. 2005. 2400-2404.

[15]. ZHU H, ZHU J, LIU K J R. Fair multiuser channel allocation for OFDMA networks using Nash bargaining solutions and coalitions. IEEE Transactions on Communications, 2005, 53 (8): 1366 - 1376.