Suoping Li1,,*,Yongqiang Zhou,Xijuan Yang,and Zufang Dou

1.School of Science,Lanzhou University of Technology,Lanzhou 730050,China;2.School of Electrical&Information Engineering,Lanzhou University of Technology,Lanzhou 730050,China

Performance evaluation of multiple relays cooperative GBn-ARQ with limited retransmission

Suoping Li1,2,*,Yongqiang Zhou2,Xijuan Yang2,and Zufang Dou2

1.School of Science,Lanzhou University of Technology,Lanzhou 730050,China;
2.School of Electrical&Information Engineering,Lanzhou University of Technology,Lanzhou 730050,China

We investigate the performance of multiple relays cooperative go backnautomatic repeat request(MRC-GBn-ARQ)protocol with limited retransmission.AnN+2-state Markov transition model is constructed for analyzing arbitrary packet transmission process with limited retransmission.Then the system’s steadystate distribution is derived by using the Kolmogorov equations. Further,the mean packet transmission delay and startup delay are obtained respectively;the channel utilization is derived based on discrete-time Geom/G/1 queue model with startup.Also,we get the energy efficiencof the MRC-GBn-ARQ based on energy analytical model.Especially,the presented models are contrary to the classical ARQ protocols of always allowing infinitretransmission,and our analysis of the MRC-GBn-ARQ protocol with limited retransmission provides theoretical results for reducing the system’s delay and energy consumption.Finally,analysis is validated through some numerical simulations.

steady-state distribution,mean packet transmission delay,channel utilization,energy efficienc.

1.Introduction

The cooperative automatic repeat request(ARQ)is a promisingtechnologyof combinationcooperativecommunication and ARQ,which can effectively increase the diversity gains while guaranteeinghigh reliability data transmission.The cooperative ARQ protocol has already been applied in currently emerged networks e.g.,Ad hoc networks[1].However,few references deal with cooperative ARQ to improvechannelutilization while maintaininglow energy consumption.

In[2],a single relay cooperative ARQ retransmission strategy is proposed for cellular access networks,and it is shown that the retransmission rate of source and the implementation of acknowledgement signals from relays are greatly impacted on the system throughput,but the system delay and channel utilization are not given.Reference[3] suggested an incorporate ARQ with sub-optimal multiple input multiple output(MIMO)detection to maintain theoptimal/near-optimaldetectionperformancewhileguaranteeing low complexity.Reference[4]proposed a twodimensional discrete time Markov chain,which is used to obtain the probability generating function,the average latency and saturated throughput.However,the analytic process is too complicated.Reference[5]proposed a delay model for single source and single relay cooperative selective repeat ARQ in slotted radio networks with non-instantaneous feedback,which is used to estimate the queuing and transmission delay based on Poisson arriving frames,but the number of retransmission is not limited in the paper.Reference[6]gave a Markov analysis of ARQ systems with variable rtt.methodologyaside,a notable quantitative conclusion is that the queueing delay statistics is onlymarginallyaffectedby the rtt.distribution. Reference[7]proposed an enhanced RARQ mechanism, called the local E2E(LE)RARQ.Its basic idea behind the scheme is to facilitate RSs with relay error control by appendingRACKs or RNACKs to acknowledgementpackets sothat ARQ states oneachRS canbeknownbytheBS and the MS.The proposed RARQ in[7]appends error control information only when error packets are presented so that theRARQ workloadcanbeconsiderablyreduced.Theperformanceofhybridautomaticrepeatrequest(HARQ)technique in an Ad hoc network is analyzed in[8],in which both time-correlated interference and feedback errors are taken into account in the analysis.Based on the theory of point processes,outage probability after thenth retransmission,delay limited throughput and mean transmission time are derived in closed forms in[8].

Recently,although we have researched many different types of ARQ protocol fully in[9–11],and obtained their performances such as system mean delay,system steadystate distribution,it is necessary continuously to research the performance of cooperative ARQ protocol with multi-ple relays under the conditions of limited retransmission. Especially analysis of the system channel utilization and energy efficien y is very meaningful to design collaboration scheme in the cooperative communication.In this paper,we consider a multiple relays cooperative go backnARQ(MRC-GBn-ARQ)protocol with limited retransmission in wireless sensor networks.By analyzing the transmission mechanism of MRC-GBn-ARQ,a discrete-time Geom/G/1 queue model with startup is established,anN+2-state Markov transition model is constructed to analyze the packet transmission process,and we obtainedthe steady-state distributionof the system.After the numberof packets entering the system is assumed to be independent and identically distributed random variables,we derived the mean packet transmission delay and the mean number of transmission that a packet has gone through when it is successfully received by the receiver.When the system reaches a steady state,the channel utilization is also derived based on discrete-time Geom/G/1 queue model with startup.Besides,the expression is derived for the energy efficien y of MRC-GBn-ARQ.

This paper is organized as follows.In Section 2,the system description and queue model of the MRC-GBn-ARQ protocol are introduced.In Section 3,anN+2-state Markov transition model is presented,and the system steady-state distribution is derived too.In Section 4,the mean packet transmission delay,startup delay and channel utilization of the system are obtainedrespectively.Section 5 gives the energy efficien y of MRC-GBn-ARQ protocol.In Section 6,some numerical simulations are presented to validate our theoretical analysis.

2.Description of the MRC-GBn-ARQ protocol

Now we consider an MRC-GBn-ARQ protocol with the limited retransmission in wireless sensor networks,which is consisted of one source node(S),Nrelay nodes and one destination node(D)(see Fig.1).In this MRC-GBn-ARQ protocol,the source node broadcasts packets continuously to relay nodes and destination node.If the transmitted packets are successfully received by the destination node, the destination node feedbacks ACK to the source node, and the source node transmits new packets in the next time slot.Otherwise,if the destination node feedbacks NACK and one relay node feedbacks ACK to the source node(we assume that the other relay nodes feedback NACK),the relay node and the source node will retransmit the packets respectively in the next time slot.However,if the packets sent by the relay nodes are erroneous,the source nodemovesnsteps back alongthe sendingpacket window and sends all the following packets again.We assume that the time axis is divided into equal size slots,and the acknowledgement(ACK/NACK)messages are received reliably.

Fig.1MRC-GBn-ARQ model

Here we firs defin the notations used in the MRCGBn-ARQ system as follows:the number of relay nodes is denoted byN;theith relay node is denoted byRi(i= 1,2,...,N);the retransmission probabilities of packet in source nodeSand relay nodeRiare denoted bypsand

pRirespectively;the packet error rate(PER)of the channel fromrelaynodeRito destinationnodeDis denotedby

pRiD;the PER of the channel from source nodeSto relay nodeRiis denoted bypSRi;the PER of the channel from source nodeSto destination nodeDis denoted bypSD.

In the protocol,we usually assume that the packets are transmitted on firs come firs service(FCFS)basis,then the packet service delay is generally distributed and not dependent on the arrival process;packet arrival at sending node randomly occurs according to a Bernoulli process with rateλ(0＜λ＜1);μdenotes the packet service rate,ρ=λ/μ,which represents the traffiintensity.Once the system without packets,the service facilities are shut down until a new packet arrival.However, after a period of startup delayVt,the system begins service for the data packets.The mean packet transmission delaySt,whichis theintervalfromthebeginningofpacket transmission to the time of the packet received successfully by the receiver,follows general probability distribution.Therefore,the data transmission system with the proposed MRC-GBn-ARQ protocol can be described with a discrete-time Geom/G/1 queue model with startup,and the transition probabilities are determined by the PER of each channel.

3.Markov model of MRC-GBn-ARQ and its steady-state distribution

For the MRC-GBn-ARQ protocol,according to the different cases of ACK feed backed by all nodes,we can construct anN+2-state Markov chain with state space

Ω={0,1,...,N+1}to describe the feedback mechanism of MRC-GBn-ARQ system(see Fig.2).That is,state0 represents the destination node and all relay nodes feedback NACK without ACK feed backed to the source node; statei(1≤i≤N)represents the relay nodeRifeedbacks ACK and the destination node feedbacks NACK to thesourcenode;stateN+1representsthedestinationnode feedbacks ACK and all relay nodes feedback NACK to the source node.

Under the MRC-GBn-ARQ,on which we focus in this paper,the source node broadcasts packets non-stop,the relay nodes are assumed not to discard packets.If the system is in the state 0,the source node interrupts transmission,goesnsteps back and sends all the following packets again;when the system turns into statei,the relay node instead of the source node retransmission the packets in the next time slot,until the packets are successfully received by the destination node;once the system enters the stateN+1,no matter the relay nodes feedbackACK or NACK, the source node transmission new packets in the next time slot.

Fig.2 N+2-state transition model of the MRC-GBn-ARQ

The one-step transition probabilities are given by

We express the one-step transition probability matrix as

Theorem 1When the MRC-GBn-ARQ is in the statel(l=1,2,...,N),the system steady-state distributionπ={πi0,...,πil,...,πi(N+1)}can be given by the following functions

ProofBecause of the Kolmogorov equations for the steady-state distributions of the system[πi0,...,πil,..., πi(N+1)]=[πi0,...,πil,...,πi(N+1)]Pand the normalizing conditionthe proof of this theorem is immediately obtained.

4.Delay and channel utilization

For the MRC-GBn-ARQ,the system PER can be define as

Theorem 2The mean packet transmission delay and startup delay of the MRC-GBn-ARQ protocol are respectively given by

wheretTis the one packet transmission time,tpis the one packet propagation time,α=tT/tpis the relative transmission delay,jis the number of the arrival packet in startup period.

ProofLetKdenote the number of retransmission the system permitted,kdenote the number of transmissions. So the mean number of transmissions that a packet has gone through when it is successfully received by the receiver is

Lemma 1[12] Ifρ=λE(S),the system reaches a steady state,which can be described as a discrete-time Geom/G/1 queue model with startup.Therefore,the expression for mathematical expectation of busy periodBis given as

Theorem 3Whenρ＜1,the channel utilization of the MRC-GBn-ARQ is given by

ProofThe firs moment ofStandVtare respectively derived as

Substituting Lemma 1,(4)and(5)into(6)completes the proof.

5.Energy efficienc

Energy efficien techniques typically focus on minimizing the transmission energy[13].In order to analyze the energy efficien y of the MRC-GBn-ARQ protocol,we assume that the nodes use the same transmission powerPt, and the nodes use the ideal Nyquist pulse and M-QAM modulation mode.The modulation level isb=log2M, the bit transmission rate isRb=Rsb,andRsdenotes the symbol rate.Letβbe the power amplifie loss factor,PSobe the circuit power overhead of source node,PDebe the circuit power overhead of destination node.

Lemma 2[14]Considerthe meanenergyconsumptionEfor the packet transmission and the PER of MRC-GBn-ARQ protocol,the energy efficien yηhas the expression

Theorem 4LetηEbe the energy efficien y of MRCGBn-ARQ,then we have

ProofLetLdenote one packet length.For the MRCGBn-ARQ protocol,the mean energy consumption for the packet transmission is given by

Substituting(1)and(7)into Lemma 2 completes the proof.

6.Numerical results and analysis

In this section,we present some numerical simulations to validate the above analysis.Here we analyze the results about the mean packet transmission delay,startup delay,channel utilization and energy efficien y of the MRCGBn-ARQ protocol.In all plots,we choosepSD=0.8,pSRi=0.5,pRiD=0.5,Rb=10,Pt=10?1,PSo=10?2,PDe=5×10?2,λ=0.15,tT=1.2Lin Matlab simulation.The simulation results are given as Fig.3–Fig.6.

Fig.3 Mean packet transmission delay vsαandK

Fig.4 Startup delay vsαandj

Fig.5 Channel utilization vsαandn

Fig.6 Energy efficienc vsNandβ

Fig.3 shows the mean packet transmission delay of the MRC-GBn-ARQ protocol versus relative transmission delayαandthenumberofretransmissiontimesKthesystem permitted.It can be seen that the mean packet transmission delay decrease with the increasing values ofα.Besides,the mean packet transmission delay is enhanced asKincreases.Fig.4 gives the startup delay versusαand the numberofthearrivalpacketjinstartupperiod.Whenthejincreasing,the startup delay increases rapidly,while theαbrings less effect on the startup delay of MRC-GBn-ARQ.

Fig.5 depicts the channel utilization versusαand sliding windowlengthn.It is observedthat channelutilization increase with decreasing the lengthn.While theαbrings less effectonthechannelutilization.Fig.6plotstheenergy efficien y of MRC-GBn-ARQ protocol versus the number ofrelaysNandpoweramplifie loss factorβ.Itis seenthat the system energy efficien y increases with the increasing values ofN(N＜6),but the energy efficien y decreases with the increasing ofN(N＞6).Besides,the energy efficien y is reduced asβincreases.

7.Conclusion

An MRC-GBn-ARQ protocol in wireless sensor networks is proposed in this paper.TheN+2-states Markov chain model presented in this paper is general and useful for arbitrary packet transmission process with limited retransmission.Based on the model,the one-step transition probability matrix and steady-state distribution can be calculated well.And the mean packet transmission delay and startup delay are obtained respectively.The discrete-time Geom/G/1/∞queue model is established sententiously to analyze the channel utilization.In addition,the energy efficien y of MRC-GBn-ARQ protocol is derived based on energy analytical model.Interestingly,the presented models are contrary to the classical ARQ protocols of always allowinginfinit retransmission,and we analyze the MRCGBn-ARQ protocol with limited retransmission for reducing the system’s delayand energyconsumption.Our future work will focus on the analysis of the best relay selection of the cooperative ARQ.

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Biographies

Suoping Liwas born in 1965.He is a professor of School of Science at Lanzhou University of Technology(LUT),China.He received his Ph.D. degree in signal and information processing from Beijing Jiaotong University,China,in 2004.He got his M.S.degree in stochastic model with applications from Lanzhou University in 1996 and B.S.degree in Mathematics from Northwest Normal University in 1986.He was a visiting scholar at Swiss Federal Institute of Technology(ETH)Zurich from 2007 to 2008.He was also a visiting professor at East Texas Baptist University(ETBU)from August to December 2011.His primary research interests include stochastic control theory and applied stochastic process,error control theory and data communication,hybrid dynamic systems modeling,control and simulation.

E-mail:lsuop@163.com

Yongqiang Zhouwas born in 1986.He received his M.S.degree in operations research and control theory and Ph.D.degree in control theory and control engineering from Lanzhou University of Technology(LUT),China,in 2011 and 2014 respectively. His research interests are in the areas of modeling analysis of wireless communication systems and system modeling,control and simulation.

E-mail:zhoupaper@163.com

Xijuan Yangwas born in 1981.She received her M.S.degree in computer technology and application from LanZhou JiaoTong University,China,in 2007. Sheis currently a Ph.D.student in control theory and control engineering at Lanzhou University of Technology(LUT),China.Her research interests are in the areas of optimization system and combinatorial optimization.

E-mail:yangxj331@163.com

Zufang Douwas born in 1984.She received her B.S.degree in applied mathematics from Central China Normal University,China,in 2005.Now she is a Ph.D.student in control theory and control engineering at Lanzhou University of Technology.Her research interests are in the areas of stochastic modeling analysis of wireless communication systems and hybrid dynamic systems modeling,control and simulation.

E-mail:douzufang@163.com

10.1109/JSEE.2015.00132

Manuscript received August 04,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61167005).