Performance Analysis of SR-ARQ Based on Geom/G/1/ Queue over Wireless Link

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1 Al. Math. Inf. Sci. 7, No. 5, (013) 1969 Alied Mathematics & Information Sciences An International Journal htt://dx.doi.org/ /amis/ Performance Analysis of SR-ARQ Based on Geom/G/1/ Queue over Wireless Link Suoing Li 1,3, and Yongqiang Zhou,3 1 School of Science, Lanzhou University of Technology, Lanzhou, China School of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, China 3 Key Laboratory of Gansu Advanced Control for Industrial Processes, Lanzhou, China Received: 9 Jan. 013, Revised: 30 May. 013, Acceted: 31 May. 013 Published online: 1 Se. 013 Abstract: Selective-Reeat ARQ (Automatic Reeat request) rotocol is the most efficient error control technology in wireless because it allows the transmitter to retransmit only negatively acknowledges (NACK) ackets. In this aer, by analyzing the transmission mechanism of SR-ARQ, the discrete-time Geom/G/1/ queue model is established sententiously based on equivalent delay under the condition that the numbers of ackets entering the transmitter are assumed to be indeendent and identically distributed random variables. With the method of embedded Markov chain, the exression formulations of the acket mean waiting delay, system mean delay and channel utilization are exlicitly obtained. And then, the sliding window length control model is built adatively according to the change of transmission conditions. Furthermore, the influences of acket length, the successful transmission robability and acket arrival rate on system mean delay are analyzed by numerical simulation. The numerical simulation results show that the exression formulations are valid for SR-ARQ rotocol. Keywords: SR-ARQ, Geom/G/1/ Queue Model, System Mean Delay, Channel Utilization, Window Length Control Model 1 Introduction In wireless data communication systems, error control techniques have been used extensively to enhance the reliability of data transmissions. There are two tyes of error control techniques: Forward Error Control (FEC) and Automatic Reeat request (ARQ) rotocols [1]. In FEC, redundancy is introduced in order to correctly decode a corruted acket. In ARQ rotocols, erroneous data are retransmitted. If an error is detected by the error-detecting code, a negative acknowledgement (NACK) will be sent back to the transmitter via the feedback channel. The transmitter, uon the receit of the NACK, retransmits the acket until a ositive acknowledgment is received. The ARQ rotocols have been widely used because of their high reliability. Nowadays, as the demand for a wireless transmission is getting larger, they are even of greater imortance than ever. Generally, there are three basic ARQ schemes: sto-and-wait (SW), go-back-n (GBN) and selective-reeat (SR). In order to imrove the erformance of ARQ rotocols, many imortant erformance measures are analyzed by building its stochastic models. Literature [] resents some new SW-ARQ rotocols that significantly imrove the throughut, while retaining the simle imlementation of the classical SW-ARQ. In [3], by using the exression for the robability generating function (PGF) of the buffer content working under a SW-ARQ retrans-mission rotocol, the queue length characteristics and the mean acket delay are derived. But SW-ARQ is inherently inefficient due to the idle time sent on waiting for an acknowledgment of each transmitted acket. In GBN-ARQ, the transmitter does not wait for an acknowledgment but sends ackets to the receiver continuously and receives acknowledgements as well. Literature [4] gives an exact analysis of the transmission delay of a GBN-ARQ rotocol which simultaneously transmits multile frames using multile arallel channels. Literature [5] researches the erformance of an adative GBN-ARQ rotocol in time-varying channel environments with random or correlated feedback errors. The throughut of the three-mode GBN (TM-GBN) (that includes standard GBN (SGBN), n-coy GBN (ngbn) Corresonding author lsuo@163.com c 013 NSP

2 1970 S. Li, Y. Zhou: Performance Analysis of SR-ARQ Based on... and continuous GBN (CGBN)) are analyzed in [6] and [7]. As oosed to the other two rotocols, in the SR-ARQ rotocol the transmitter sends ackets continuously and retransmits the NACK acket without re-sending the transmitted ackets following it, so the SR-ARQ rotocol is the most efficient one. Certainly, SR-ARQ is considered esecially interesting. In [8], the erformance of the basic SR-ARQ rotocol has been analyzed, and the two different methods (by using an exact Markov state model and a substantially simler stochastic model) for the evaluation of the queue length and the block delay are resented. Literature [9] studies the waiting time and queue length distributions for GBN-ARQ and SR-ARQ rotocols, rovides a erformance analysis of ARQ rotocols in connection-oriented transmission, and obtains the PGF of acket waiting time and the mean erformance measures. The exressions to calculate the throughut for both ACK-NACK-based SR-ARQ and NACK-based SR-ARQ with multi-coy retransmission are derived in [10]. Another delay erformance of the unit of delivery to a higher layer is analyzed in [11], which derives an exact closed-form SDU delay exression and its aroximation in a much simler form. The queuing delay at the transmitter and the resequencing delay at the receiver are resented in [1], which analyzes the transmitter queuing statistics, and obtains the joint distribution of the transmitter and the receiver buffer occuancies. The average acket transmission delay of SR-ARQ using the signal-flow grah is analyzed in [14]. However, the acket arrival rocess is not considered. Shumji Fujii and Yukuo Hayashida analyze the erformance of SR-ARQ rotocol with limited retransmissions in [15], and the distribution of the acket transmission delay and the required receiver buffer caacity are deduced. The erformance of SR-ARQ is analyzed by using a Markov two-state channel with Bernoulli block arrivals [16]. Literature [17] describes the tail asymtotic of the robability mass function of the resequencing buffer content for SR-ARQ. In this work, based on the mentioned works, we focus on the analysis of the link utilization, the system mean delay, the window length control model and the channel utilization of SR-ARQ rotocol with startu time and stochastic vacation. For this urose, the acket service delay is the time interval from the beginning of transmission to the receiving acket successfully at the receiver, which can be considered an equivalent service delay. The startu time can be regarded as a eriod of holiday time triggered by the arriving ackets in idle eriod, which can be considered an equivalent holiday delay. We devise the discrete-time Geom/G/1/ queue model sententiously based on equivalent service delay and equivalent holiday delay. The stochastic decomosition of the steady queuing length and steady waiting time are got based on the results of [13], then the PGFs of the additional queuing length and the additional latency are obtained resectively. The first moment and second moment of the equivalent service delay and equivalent holiday delay are also solved resectively. Then the recise formulation of the system mean delay, channel utilization and the exact analysis of delay for the SR-ARQ rotocol are obtained too. Furthermore, the sliding window length control model is built adatively according to the change of the system erformance index. The system mean delay and channel utilization are examined by using numerical simulation, and the results show that the delay erformance can be imroved by changing the acket length, the successful transmission robability and the acket arrival rate. The outline of the aer is as follows. In section, by analyzing the scheme of the SR-ARQ, we calculate the maximum mean link utilization. In section 3, we establish the Geom/G/1/ queue model based on equivalent service delay and equivalent holiday delay. The system mean delay of SR-ARQ will be derived based on Geom/G/1/ model. In section 4, the best sliding window length control model will be built for SR-ARQ, and the channel utilization will be researched too. In section 5, the influences of the successful transmission robability, acket length and acket arrival rate on system mean delay are comarative analyzed by numerical simulation. Moreover, the influences of acket length and the successful transmission robability on channel utilization and three different channel utilization of ARQs are exlained by numerical simulation too. Finally, in section 6, conclusions and future work are given. SR-ARQ scheme In ARQ rotocols, the data are sent in terms of ackets, each of which is encoded for error detection by the receiver. The ackets that arrive at the transmitter are assigned subsequent numbers that identify them uniquely and are referred to as identifies. Under the SR-ARQ rotocol, on which we focus in this aer, the transmitter continuously sends new ackets and the receiver accets every acket that arrives error-free. Uon receit of a NACK by the transmitter, only the corresonding acket is retransmitted. Assuming a fixed acket length, let T D be the transmission delay, T A be the resonse information transmission delay, and T P be the roagation delay, then d T D + T P + T A denotes the time interval from the beginning of transmission to the receiving the feedback from the receiver. Obviously, for the SR-ARQ transmission rotocol, the required time of transmitting n ackets continuously is nt D. When nt D > d, the resonse information can be fed back to the sender within nt D. At this moment, if the acket is error free, the transmitter can send the acket continuously, and the maximum link utilization of SR-ARQ u to 1. If a NACK arrives at the transmitter, in which case the transmitter retransmits the negatively acknowledge acket without re-sending the transmitted c 013 NSP

3 Al. Math. Inf. Sci. 7, No. 5, (013) / ackets following it. When nt D < d, the resonse information can not be fed back to the transmitter within nt D, and the maximum link utilization of transmission is nt D /d. We assume a acket which is retransmitted for k times, and let be the successful transmission robability of a acket in channel. Because the transmitter only retransmits the NACK ackets, the required mean number of ackets that successfully transmitted a acket can be considered as N f k0 (k+ 1)(1 ) k 1. (1) Thus, the maximum mean link utilization is 1/N f, nt D d, n 1+α µ nt D N f d, nt D < d n 1+α, nt D < 1+α () Where α (T P + T A )/T D. Aarently, the maximum mean link utilization relates to the relative transmission delay α, the successful transmission robability and the length of the window n. Numerical analysis shows that the best window length is aroximately equal to 1 + α in the case of given [1]. Where x denotes the smallest ositive integers greater than or equal to x. 3 Queuing model and erformance analysis 3.1 Mathematical model We consider a transmitter and a receiver, which communicate data ackets of fixed lengths through a slotted channel, and where acket transmission times equal one slot. The transmitter and receivers are synchronized with the channel and acket transmissions start at slot beginnings. To investigate the delay erformance of SR-ARQ rotocol farther by modeling, our analysis is based on the following assumtions. Packets are transmitted on FCFS (First Come First Service) basis, so that transmission of a message starts after all ackets in the revious message is successfully transmitted; the acket service delay is generally distributed and not deendent on the arrival rocess; ackets arriving at the transmitter are stored in a buffer of an infinite caacity; the feedback channel for acknowledge is assumed to be error free; acket arrival randomly occurs according to a Bernoulli rocess with rate λ, 0 < λ < 1; µ denotes the acket service rate, ρ λ/µ. The number of arrival ackets in the M eriod to obey binomial distribution. for i0,1,...,m. P{A(M)i} ( M i ) λ i (1 λ) M i (3) In this aer, the discrete-time Geom/G/1/ queuing model with startu is resented. Once the system without ackets, the service facilities are shut down until a new acket arrival. However, after a eriod of startu time V, the system begins service for the data ackets. Called Geom/G/1(ES, SU). P{V k} V k,k 1. (4) Startu time can be regarded as a eriod of holiday time triggered by the arriving ackets in idle eriod, which can be considered an equivalent holiday delay. While the acket service delay S which is the interval from the beginning of transmission to the time of the acket is successfully received at the receiver, which can be considered an equivalent service delay (that includes the first transmission delay T 1 and the retransmission delay T r ), which follows general distribution and can be described as a Geom/G/1/ queuing model. b r P{Sk},k 1. (5) When ρ < 1, L n is the remaining number of ackets after the nth acket leaves, and {L n,n 1} is the embedded Markov chain of the queue length rocess. { Ln 1+C, L n 1 L n (6) C, L n 0 Where C is the number of ackets entering the system in a service interval, we get the robability distribution as k j P{C j} r j P{Sr} P{ j ackets entering the system during slot r} ) r b r( λ j (1 λ) r j, j 0 r j j The PGF of C as C(z) j0 z j k j j0z j ) r b r( λ j (1 λ) r j r j j ( ) r (zλ) j (1 λ) r j j r r1b j0 r1 b r (1 λ + λz) r. When ρ < 1, Geom/G/1(ES, SU) have stochastic decomosition [13] : the steady queuing length and steady waiting time are resectively as (7) (8) L + L L d, W W 0 +W d. (9) c 013 NSP

4 197 S. Li, Y. Zhou: Performance Analysis of SR-ARQ Based on... Where L 0 + is the steady queuing length of classical Geom/G/1, L d is additional queuing length; W 0 denotes the waiting time of classical Geom/G/1, W d denotes additional latency. The PGF of L d and W d are resectively as 1 zv[γ(z)] L d (z) [1+λE[V]](1 z), (10) W d (s) λ (λ (1 s))v(s) [1+λE(V)](1 s). (11) Eq. (10) and (11) can be rewritten as 1 L d (z) 1+λE(V) + zλe(v) 1 V[γ(z)] 1+λE(V) λe(v)(1 z), (1) W d (s) V(s) 1+λE(V) + λe(v) 1 V(s) 1+λE(V) E(V)(1 s). (13) It shows that W d consists of comlete startu time with robability (1+λE(V)) 1 and surlus startu time with robability 1. L d consists of one acket within the comlete startu time with and the number of ackets within surlus startu time with robability1. The mean value exressions are easily obtained by the stochastic decomosition as follows E(L d ) λe(v)+λ E(V(V 1)), [1+λE(V)] (14) E(V)+λE(V(V 1)) E(W d ), [1+λE(V)] (15) E(L + )ρ+ λ E(S(S 1)) (1 ρ) + λe(v)+λ E(V(V 1)), (16) [1+λE(V)] E(W) λe(s(s 1)) E(V)+λE(V(V 1)) +. (17) (1 ρ) [1+λE(V)] Therefore, the mean delay of the system is as follows E(T)E(W)+E(S) λe(s(s 1)) (1 ρ) E(V)+λE(V(V 1)) + + E(S). [1+λE(V)] 3. Performance analysis (18) For SR-ARQ rotocol, the robability that a retransmission acket is located in the ith sliding window is 1/N. So the mean service delay in the first transmission at the receiver can be calculated as T 1 1 N l(n+ 1) l[n+(n 1)+ ++1]. (19) Where N denotes the length of sliding windows, and l denotes the acket length. Under receit of a NACK by the transmitter of the SR-ARQ, only the corresonding acket is retransmitted. Therefore, when a acket is retransmitted for k times, the equivalent retransmission delay T r is T r lk+(k+ 1)(T D + T P + T A ). (0) Hence, the equivalent service delay that a acket is successfully transmitted is S k T 1 + T r with robability (1 ) k (one times successful transmission, k times retransmission). That is, the robability distribution of stochastic variable S k is l(n+ 1) P{S k + lk+(k+ 1)(T D + T P + T A )} (1 ) k. (1) Assume that the retransmission times of the arrival acket in idle eriod is j 1, then the equivalent holiday delay is V k l j with robability of (1 ) j 1. That is, the robability distribution of stochastic variable V k is as P{V k l j} (1 ) j 1. () The first moment of S k and V k are resectively derived [ l(n+ 1) E(S) + lk k0 ] +(k+ 1)(T D + T P + T A ) (1 ) j 1 E(V) l(n+ 1) + j1 Also, we obtain l(1 ) + T D+ T P + T A, (3) l j(1 ) j 1 l. (4) E(S(S 1)) {[ l(n+ 1) ] + lk+(k+ 1)(T D + T P + T A ) k0 [ l(n+ 1) ]} + lk+(k+ 1)(T D + T P + T A ) 1 (1 ) k {[ l(n+ 1) ] +(l+ T D + T P + T A )k+ T D + T P + T A k0 [ l(n+ 1) ]} + lk+(k+ 1)(T D + T P + T A ) 1 (1 ) k [ l(n+ 1) ] + T D + T P + T A + (l+ T D+ T P + T A ) (1 )( ) l (N+ 1)+l(1 )+(T D+ T P + T A ) c 013 NSP

5 Al. Math. Inf. Sci. 7, No. 5, (013) / [l(n+ 1)+(T D+ T P + T A )] (l+ T D+ T P + T A )(1 ), (5) E(V(V 1)) j1 l j (1 ) j 1 j1 l j(1 ) j 1 l ( ) 1. (6) Let ρ λe(s), when ρ < 1, the system reaches a steady state. Then, substituting formula (3)-(3.) into (18), the mean delay of the system is given by E(T) λ[k 1 + K (1 )+K 3 (1 )( )] λ [l (N+ 1)+l(1 )+d] l (N+ 1)+l(1 )+d + + l +λ[l ( ) l ] (. (7) + λl) Where d T D + T P + T A, [ l(n+ 1) ], K 1 + T D + T P + T A K [l(n+ 1)+(T D + T P + T A )](l+ T D + T P + T A ), K 3 (l+ T D + T P + T A ). The normalized maximum throughut is γ λ max (T 1 + T D + T P + T A ) l(n+ 1)+(T D+ T P + T A ) E(T) (8) Where E(T) is a function of N, l, λ,, d and ρ. λ max 1/E[T] denotes the maximum throughut under steady state. So we have E(T) f(n,l,λ,,d,ρ) (9) 4 The best sliding window length and channel utilization The best sliding window length is very imortant to design SR-ARQ system. By (9), the sliding window length N should be the function as follows N g(l,λ,,d,ρ). (30) When ρ < 1, we can deduce that the best sliding window length is Where b± b N best min 4ac a. (31) aλl, { b λl l( 3) d(+ )+4(λ 1) + + λ max λ l +λ[l ( ) l ] }, +λl c4λ(l+ d)(1 )[l d (l+ d)( )] + λ (l 4d ) 8l (1 )+4d(λd+ λl ) { λ l( 3) d(+ ) 4(1 λ) + + λ max λ l +λ[l ( ) l ] } +λl { + 4[ λ(1 ) λd] λ max l +λ[l ( ) l ] +λl Then, the best sliding window length control model is built adatively according to the change of E(T), l, λ,, d and ρ. We can design the best sliding window for SR-ARQ system in terms of (31). At the discrete-time Geom/G/1/ queuing model with startu, the busy eriod of model can be obtained as follows by [13] }. B v (z)q b [B(z)]. (3) Where Q b denotes the number of ackets in the system at the busy eriod begins, and B(z) denotes the PGF of the busy eriod in the classical system. Hence, the mean value of busy eriod is given by E[B v ] [1+λE(V)]E(S). (33) 1 ρ Thereby, the channel utilization is given by E(B v ) η E(V)+E(B v ) [1+λE(V)]E(S) E(V)+E(S) {( 1+ λl )[ l(n+ 1) l(1 ) + [ l(n+ 1) l( ) + (+λl)[l+l(1 )+d] [L+l( )+d] ] + T D+ T P + T A Where d T D + T P + T A, Ll (N+ 1). + T ]} D+ T P + T A (34) c 013 NSP

6 1974 S. Li, Y. Zhou: Performance Analysis of SR-ARQ Based on... 5 Numerical results and analysis This section validate our theoretical analysis through numerical simulation. Here we analyze the results about the system mean delay and channel utilization of SR-ARQ rotocol based on Eq.(7)-(34) resectively. Note that there are five arameters which affect the system mean delay and channel utilization. Viz., the successful transmis-sion robability, the acket length l, the arrival rate λ, the length of sliding windows N, and the time interval from the beginning of transmission to the receiving the feedback from the receiver d. In all lots in this section, we choose N 3 and d 1.l. As first results, in Fig.1 and Fig., we reort the system mean delay statistics relying on the successful transmission robability. In both grahs, the statistics obtained by simulation are comared against the two erformance arameters, i.e., the acket length l (l 1, l, l 3) and the arrival rate λ (λ 0.001, λ 0.005, λ 0.01). Fig.1 shows when the system is in stable state and the same arrival rate (λ 0.01), the system mean delay decreases as the acket length decreases. Similarly, the system mean delay decreases as the arrival rate decreases at the same acket length (l ) is shown in Fig.. But when the successful transmission robability is larger (>0.6), the arrival rate bring on less effect on the system mean delay. Fig. : System mean delay versus successful transmission robability and arrival rate. the system mean delay decreases as the arrival rate decreases at the same successful transmission robability (0.5) is shown in Fig.4. But when the acket length is smaller (l < 3), the arrival rate bring on less effect on the system mean delay. Fig. 1: System mean delay versus successful transmission robability and acket length. Fig. 3: System mean delay versus acket length and successful transmission robability. In the following Fig.3 and Fig.4, we deict the system mean delay statistics relying on the acket length l. In both grahs, the statistics obtained by simulation are comared against the successful transmission robability ( 0.5, 0.6, 0.7) and the arrival rate λ (λ 0.001, λ 0.005, λ 0.01). Fig.3 shows when the system is in stable state and the same arrival rate (λ 0.01), the system mean delay decreases as the successful transmission robability increases. Similarly, Last, in Fig.5 and Fig.6, we describe the channel utilization by numerical simulation. Fig.5 shows when the system is in the same arrival rate λ 0.01, the channel utilization increases as the successful transmission robability and acket length increases. Fig.6 shows when the system is in the same acket length l and the same arrival rate λ 0.005, the channel utilization of ARQs versus successful transmission robability. When < 0.5, the GBN-ARQ rotocol has the biggest channel c 013 NSP

7 Al. Math. Inf. Sci. 7, No. 5, (013) / Fig. 4: System mean delay versus acket length and arrival rate. Fig. 6: Channel utilization of ARQs versus successful transmission robability. utilization. While, the SR-ARQ rotocol has the biggest channel utilization when > 0.5. obtained. Last, the influences of acket length, the successful transmission robability and the acket arrival rate on system mean delay, and three different channel utilization of ARQs rotocols are comarative analyzed by numerical simulation. The resented method of calculation could be alied on delay analysis of all ARQ rotocol. The system mean delay is influenced by the times of retransmission, which is the subject that remains for future work. Also, the distribution of the acket delay and more general system models will be investigated in the following studies. Acknowledgements Fig. 5: Channel utilization versus successful transmission robability and acket length. 6 Conclusion In this aer, we analyze the erformance of SR-ARQ rotocol. The required mean number of ackets that successfully transmitted a acket and the maximum average link utilization of SR-ARQ rotocol are derived. Furthermore, the discrete-time Geom/G/1/ queue model is established sententiously based on equivalent service delay and equivalent holiday delay, from which the exression formulations of the acket mean waiting delay, system mean delay, the sliding window length control model and channel utilization are exlicitly This research was suorted by grant from the Natural Science Foundation of China, grant Z from the Chun-hui Program from the Ministry of Education of China, grant 0809RJZA019 from the Natural Science Foundation of Gansu Province and grant from the Science Foundation for Graduate Teachers of Gansu Province. The authors thank the reviewers for their comments. References [1] Pingyi Fan and Chongxi Feng, Fundamentals of advanced communications (II) networking technology. Tsinghua University Press, (007). [] Giuliano Benelli and Andrea Garzelli, New modified sto-and-wait ARQ rotocols for mobile communications. Wireless Personal Communications, 1, (1995). [3] De Vuyst S, Tworus K and Wittvrongel S, et.al, Analysis of sto-and-wait ARQ for a wireless channel. A Quarterly Journal of Oerations Research, 7, (009). c 013 NSP

8 1976 S. Li, Y. Zhou: Performance Analysis of SR-ARQ Based on... [4] Shumji Fujii, Yukuo Hayashida and Masaharu Komatsu, Exact analysis of delay erformance of go-back-n ARQ scheme over multile arallel channels. Electronics and Communications in Jaan, 84, 7-41 (001). [5] S. S. Chakrabory, M. Liinaharja and P. Lindroos, Analysis of adative GBN schemes in a Gilbert-Elliott channel and otimisation of system arameters. Comuter Networks, 48, (005). [6] Ranko Vojinović and Zoran Petrocić, A novel three-state ARQ scheme for variable error-tate channels. International Journal of Electronics and Communications (AEU), 56, (00). [7] Ranko Vojinović, Grozdan Petrović and Zoran Petrocić, The analysis of the adative three-mode ARQ GBN scheme using retransmission cycles mechanism. AEU International Journal of Electronics and Communications, 60, (006). [8] Miltiades E. Anagnostou and Emmanuel N. Protonotarios, Performance analysis of the selective reeat ARQ rotocol. IEEE Transactions on Communications, 34, (1986). [9] Masakazu Yoshimoto, Tetsuya Takine, et.al, Waiting time and queue length distributions for go-back-n and selective-reeat ARQ rotocols. IEEE Transactions on Communications, 41, (1993). [10] Moon Young Choi, Yudong Yao and Harry Heffes, Throughut analysis of a class of selective reeat ARQ with multi-coy retransmissions. IEEE VTS 53rd, 3, (001). [11] Wei Luo, Krishna Balachandran, Sanjiv Nanda and Kirk K. Chang, Delay analysis of selective-reeat ARQ with alications to link adatation in wireless acket data systems. IEEE Transactions on Wireless Communications, 4, (005). [1] Zvi Rosberg and Moshe Sidi, Selective-Reeat ARQ: The joint distribution of the transmitter and the receiver resequencing buffer occuancies. IEEE Transactions on Communications, 38, (1990). [13] Naishuo Tian, Xiuli Xu and Zhanyou Ma, Discrete time queuing theory. Science Press, (008). [14] Lu D. L. and Chang J. F., Analysis of ARQ rotocols via signal flow grahs. IEEE Transactions on Communications, 37, (1989). [15] Shumji Fujii and Yukuo Hayashida, Exact analysis of erformance of selective-reeat ARQ scheme with limited retransmissions. Electronics and Communications in Jaan, 87, 1-11 (004). [16] R. Fantacci, Queuing analysis of the selective reeat automatic reeat request rotocol wireless channel. IEEE Transactions on Vehicular Technology, 45, (1996). [17] Yun Han Bae, Bara Kim, et.al, Tail asymtotic behaviour of resequencing buffer content for selective reeat ARQ. Oerations Research Letters, 39, 7-77 (011). Suoing Li is a rofessor of School of science at Lanzhou University of Technology (LUT), China. He received the Ph.D. degree in Signal and Information Processing from Beijing Jiaotong University, China, in 004. He also holds a M.Sc. degree in Stochastic Model with Alications from Lanzhou University in 1996, and B.Sc. degree in Mathematics from Northwest Normal University in He was a visiting scholar at Swiss Federal Institute of Technology (ETH) Zurich from May 007 to May 008. He was also a visiting rofessor at East Texas Batist University (ETBU) from Aug to Dec 011. His rimary research interests include stochastic control theory and alied stochastic rocess; error control theory and data communication; hybrid dynamic systems modeling, control and simulation. Yongqiang Zhou received the MS degree in oerations research and control theory from Lanzhou University of Technology. He is currently a Ph.D. student in School of Electrical and Information Engineering, control theory and control engineering. His research interests are in the areas of stochastic modeling analysis of wireless communication systems and hybrid dynamic systems modeling, control and simulation. c 013 NSP