MILCOM 95, SAN IEGO, CA, NOV. 995 Pefomance analysis of ARQ Go-Back-N potocol in fading mobile adio channels Michele Zozi y, Ramesh R. Rao z, Lauence B. Milstein z y ipatimento di Elettonica e Infomazione, Politecnico di Milano, Italy z epatment of Electical and Compute Engineeing, Univesity of Califonia San iego, USA Abstact In this pape, an ARQ Go-Back-N potocol with uneliable feedback and time-out mechanism is consideed. Both the fowad and the evese channels ae subject to fading impaiments. A theshold model fo successes and failues of both packets and feedback infomation is assumed, and the esulting success/failue pocesses ae modeled as Makov. Thoughput and delay ae analytically evaluated. Fo the fading channel model and this potocol, only simulation esults fo the thoughput ae available in the ecent liteatue, wheeas delay has not been studied. The match between analytical and simulation esults shows that the Makov appoximation is indeed vey good. The analysis is useful in that it povides a fast and accuate method fo thoughput and delay evaluations, and can be used to choose design paametes, such as the time-out peiod. In paticula, it is shown that thee exists an optimum value fo the time-out peiod that is diffeent fom the one commonly assumed in the liteatue. I. INTROUCTION In ecent times, consideable eseach is being done on the design and implementation of mobile communication systems. The pimay goal is usually the tansmission of voice signal, which is by fa the dominant pat of the taffic. Howeve, data sevices have begun to appea in these systems, and ae expected to gow in the nea futue. In fact, many applications aleady available in fixed netwoks ae expected to have a mass maket in mobile systems as well, such as facsimile, file tansfe, electonic mail, image tansmission, to mention but a few. Also, mobile units ae evolving fom thei oiginal fom as cellula phones to some moe sophisticated teminals, e.g., laptop computes. The access to ISN and even B-ISN will theefoe be a pimay equiement in ode to povide a satisfactoy sevice to uses of this sot. In this view, eliable data exchange will assume a geate impotance, and techniques to achieve the desied tansmission quality will be equied. Among them, fowad eo coection (FEC) achieves good quality though edundancy and eo coection, wheeas automatic epeat equest (ARQ) potocols guaantee eliability though eo detection and etansmission of the eoneous blocks. Some hybid techniques ae also possible. In this pape, we will focus on ARQ techniques, and in paticula on the study of one of them, namely Go-Back-N with time contol []. In most of the existing liteatue on ARQ, two majo simplifications ae made: eos in distinct blocks ae assumed independent and feedback is assumed eo-fee. Of couse, these conditions ae not satisfied in a mobile adio envionment, whee the fading pocess is usually highly coelated, and a simila channel is used to tansmit feedback infomation. Recently, some papes, that discuss the pefomance of ARQ potocols unde moe geneal conditions, have appeaed. Kanal and Sasty [2] eview the dependent channel, but focus moe on FEC techniques. Towsley This wok has been patially suppoted by the NSF unde gant NCR-9234, and by CNR and MURST, Italy. [3] consides a Makov fowad channel model, and focuses on queueing pefomance. Leung et al. [4] also give esults fo the ideal feedback case. A genealized Makov model of highe ode is consideed in [5], in which the effect of uneliable feedback is not included but two othe classic ARQ techniques, namely Stopand-Wait (SW) and Selective-Repeat (SR) ae studied. Kim and Un appea to have been the fist to study the effect of a Makov model fo the feedback channel as well [6], combining the effects of dependent tansmissions and uneliable feedback. Simila esults ae found fo some moe complex potocols, which include contol mechanisms to esolve uncetainties that may aise when feedback infomation is lost [7]. In this last pape, fo analytical convenience, some ules of the actual potocols ae disegaded in the analysis. Also, a simulation study of GBN and SR in Rayleigh fading channels is pesented in [8, 9], along with a thoughput estimation technique. Finally, in [, ], a technique based on a Makov chain analysis is pesented, which allows one to accuately study the pefomance of the GBN potocol with time contol in Makov channels with uneliable feedback, and can be potentially extended to the study of othe ARQ potocols involving memoy. Many eal-wold channels can be modeled by means of a twostate Makov model (fist poposed by Gilbet [2]), and this fact makes esults such as those pesented in the cited papes valuable. Recently, the Makov chaacte of the Rayleigh fading pocess has been assessed [3]. Also, simila esults ae discussed in [4], with efeence to the binay pocess of successes/failues on the channel. In this pape, following the analysis in [, ], we will study in detail the thoughput and delay pefomance of the GBN with time contol, using the Makov model fo both fowad and feedback channels, and showing that this appoximation is vey good and matches simulation esults vey well. The pape is oganized as follows. In Section 2, the channel model is discussed, with efeence to both the fading pocess and the block successes/failues binay sequence. A Makov appoximation fo the latte is consideed (Gilbet s model [2]) and the paametes fo it ae computed. Sections 3 and 4 descibe the potocol ules and the Makov chain appoach, espectively. Finally, in Section 5, some numeical esults ae pesented and discussed. A. Fading Model II. CHANNEL MOEL In the liteatue, the flat fading channel is modeled as a multiplicative complex andom pocess, (t). A popula model is that of a Gaussian andom pocess with a given mean and covaiance function [5]. On the time scale of the fading vaiations, the pocess can be consideed as being stationay. Theefoe, with no loss in geneality, we will nomalize its powe to. Given the stationaity, the eal and imaginay axes can be chosen so that the mean, = E[(t)], is eal. Also, we define the covaiance function, K( ) as K( ) = E[((t + )? ) ((t)? )]: ()
MILCOM 95, SAN IEGO, CA, NOV. 995 2 Note that if =, the envelope of (t) is Rayleigh distibuted fo any t, and the envelope squaed has an exponential distibution. On the othe hand, when >, we ae in the pesence of the socalled Rician fading; this latte model accounts fo the pesence of a line-of-sight component, and is often moe accuate in mico- and picocells. When the LOS component is absent, o has negligible powe, the Rician model degeneates into the Rayleigh one. In a widely accepted model, the Gaussian pocess is assumed to have a bandlimited non-ational spectum, given by [5] S(f) = S() "? f f 2 #?=2 ; fo jfj < f ; (2) and zeo othewise. f is efeed to as the opple fequency, and measues the coelation of the pocess. In paticula, its invese, called the coheence time, gives an estimate of the time duing which the pocess can be consideed as constant [6]. The spectum (2) coesponds to the covaiance function K( ) = J (2f jj); (3) whose physical meaning has been investigated in [5, 7]. J () is the modified Bessel function of the fist kind and of zeoth ode. Note that (3) coesponds to a pocess with unit powe; this implies that S() = 2=. Note also that the mean and the covaiance, along with the assumption that eal and imaginay pat ae independent, give a complete statistical desciption of the pocess, since it is Gaussian. The covaiance function, K( ), depends on the poduct f jj. When f jj is small (< :), two samples sepaated by may be quite coelated ( slow fading); on the othe hand, fo lage values of f jj (> :2), similaly spaced samples of the channel will be almost independent ( fast fading). In the pesent context, we ae inteested in slow fading, which causes the pefomance to be significantly diffeent fom the iid case; fast fading, on the othe hand, can be appoximated faily well by a model with independent eos. B. Quantization of the fading pocess In the pevious section, we analyzed some featues of the complex Gaussian andom pocess which models the fading phenomenon. On the othe hand, fom a communications point of view, the elevant quantity is not actually the channel complex value, (t), but athe some function of it such as the eo pobability of a block of bits. In this case, we can define a new andom pocess, (t), which depends on (t), as (t) = '((t)); (4) in which, fo simplicity, we assume an instantaneous (i.e., memoyless) elationship between the two pocesses. If the fading pocess can be assumed oughly constant duing the j-th block of bits (e.g., a codewod), we have (t) ' j. In this case, Eq. (4) becomes a elationship between the values duing a block, i.e., between the sampled vesions of the pocess. Hee, following [8, 9], we focus on the single-use pefomance, i.e., multiple access intefeence is eithe absent o not a significant facto. This coesponds, fo example, to a cellula stuctue designed in such a way that intefeence is almost negligible, and can be modeled as noise impaiing the intended signal. Also, we ae inteested in the fading pocess, which vaies much faste than othe impaiments such as shadowing and distance-dependent attenuations. These last effects will be neglected, since they can be compensated fo by powe contol. As an example, conside the following commonly adopted appoximation, [8], fo the opeation of the eo detecting mechanism. In this scheme, the success of a block is detemined by compaing the signal powe to a theshold; if the eceived powe is above a cetain theshold, the block is successfully decoded with pobability ; othewise, it is lost with pobability. In this case, ' is a step function, and the binay pocess which descibes block successes and failues on the channel, j, is obtained by quantization of the magnitude of the complex Gaussian desciption, i.e., if jj j 2 > b; j = if j j j 2 (5) b; whee b is the powe theshold, whose invese, =b, is sometimes called fading magin [8], and stands fo a packet failue. Note that, even if the sequence j wee a Makov pocess, this would not be necessaily tue, in pinciple, fo the new sequence, j. In fact, the latte is a dastically simplified vesion of the fome, in which much infomation has been lost. By doing so, the infomation caied by the most ecent sample of j is much less than that of the last sample of j : thus, j will in geneal not be Makov. Howeve, the possibility of pedicting j based only on a limited (and possibly small) numbe of pevious samples deseves some moe investigation: it can be conjectued, in fact, that fo highly coelated fading the quantized vesion will also be highly coelated, allowing fo some simplified model. This possibility is also appealing because thee exist in the liteatue a numbe of esults which apply to Makov channels, and which could be used in the pesent context. C. Gilbet s Model Without going into details, we appoximate the sequence of the j s with a binay Makov pocess, whose paametes can be detemined accoding to the fading model and the chaacteistics of the communications scheme. A deepe discussion about the goodness of this appoximation, not addessed hee, can be found in [4]. Hee, we limit ouselves to assuming it to begin with, and to check it though simulation. Also, it will be assumed that both the diect and the evese channels ae subjected to fading, so that a Makov eo patten is consideed fo both packets and feedback infomation. The channel is theefoe modeled as in [7], i.e., as a Gilbet channel in each diection [2]. Thus the pattens of packet and feedback eos follow two independent fist-ode Makov models, which ae descibed by the tansition matices M F (x) = [M F ()] x and M B (x) = [M B ()] x, with M F (x) = p(x) (x) M F () = p q(x) s(x) q s ; M B (x) = ; M B () = a(x) a c(x) c b d b(x) d(x) ; (6) ; (7) whee p(x) =? q(x) ((x) =? s(x)) is the pobability that the fowad slot i is successful given that the fowad slot i?x was successful (unsuccessful), and similaly fo the enties of M B (x), with efeence to the backwad channel. Note that = and =c epesent the aveage lengths of the busts of eos, which ae descibed by geometic.v. s. Given the matices in (7), the channel popeties ae completely chaacteized. Fom these matices it is possible to deduce that the maginal pobability that a packet is in eo, ", is given by " =?? p + ; (8) and analogously fo the maginal pobability of an eo on feedback. In the pesent context, we ae inteested in vey ough tansmission conditions, whee the maginal pobability of having an
MILCOM 95, SAN IEGO, CA, NOV. 995 3 unsuccessful slot is faily high (say, moe than 5- %) and the length of an eo bust may span a consideable numbe of packets. As discussed late, this aspect, which may seem unealistic in wied netwoks, is quite common in mobile adio channels, due to fading [8, 9]. We assume, in the following, that the packet length is a constant, equal to one time unit (slot). The ound-tip delay is m? slots fom the end of a tansmission to the eception and decoding of the coesponding feedback infomation, so that if a packet tansmitted in slot i is negatively acknowledged, it will be etansmitted in slot i+m. Positive (ACK) and negative (NAK) acknowledgements can neve be confused with each othe, i.e., the effect of backwad eos is to map the ACK and NAK symbols to an Easue symbol. This can be accomplished though suitable encoding of the ACK/NAK messages. Also, each ACK/NAK caies the identity of the last coectly eceived packet. This implies that a packet whose ACK is lost may be subsequently acknowledged by feedback eceived in the futue.. Computation of the paametes In ode to cay out the pefomance analysis, the paametes of the Makov model pesented in the pevious section ae to be found. This can be done in two ways, i.e., though analysis o simulation. In this pape, as discussed in the following, we will adopt the latte appoach. In the notation, we will efe to the fowad channel: of couse, eveything is the same with efeence to the backwad link. Thee ae essentially two independent paametes to compute, e.g., p and, o " and. Following most of the liteatue, we will study " and, wheeas p can be found fom (8). This choice is motivated by the fact that these two paametes have an immediate physical significance: ", as aleady mentioned, is the aveage block eo ate, and measues how often a block is in eo; = is the aveage length of an eo bust, and gives an idea egading how clusteed the eos tend to be. The Makov paametes wee obtained by simulation. We note hee that they could have been found analytically fom the Gaussian model pesented above. Howeve, we chose to simulate them fo the following easons. Fist, the analytical appoach involves numeical computations of its own. Simulating the fading pocess is simple, less sensitive to modelling discepancies, and the compute time involved is compaable to that equied by analysis. Secondly, the simulation appoach can be easily extended to othe fading models, wheeas the analytical appoach citically depends on the Gaussian assumption which, although theoetically nice, may not always coespond to eality. E. Antenna divesity It has been obseved that Rayleigh fading gives ise to deep fades which seveely impai the tansmissions. An effective way to combat this effect is povided by antenna divesity; due to the fading popeties, two (o moe) antennas will expeience independent fading, if they ae adequately spaced [6]. The two signals ae then appopiately combined, to obtain a signal which is demodulated and decoded. Hee we will choose a vey simple stategy, i.e., maximum signal selection, in which fo each block the antenna with the stonge signal powe is selected [8, 9]. Since the noise powe is assumed the same at both antennas, this coesponds to selecting the antenna with the lage useful signal. Theefoe, the fading vaiable is now the maximum between two independent Rayleigh fading values. Again, the Makov paametes fo this case can be computed following an analytical appoach [4] o via simulation. III. PROTOCOL ESCRIPTION In the following, we conside the Go-Back-N (GBN) ARQ potocol with time contol, as descibed in [7]. The eceive follows the standad ules of ARQ GBN [], i.e., it sends an ACK fo evey coectly eceived packet. When it eceives an incoect packet, it sends a NAK instead, and discads evey successive packet, until a coect copy of the negatively acknowledged packet is eceived. The tansmitte acts accoding to the following ules: It sends packets in ode, as long as it eceives ACKs on the backwad channel. Upon eception of NAK i, it goes back to packet i, etansmitting in ode all packets fom packet i. If a gabled feedback (easue symbol) is eceived it is ignoed. It is possible that a futue ACK/NAK will povide infomation that may acknowledge packet i. In fact, the ACK/NAK of packet k contains implicit acknowledgement of all packets i < k as well, by vitue of the ules of the potocol. Howeve, it is possible that the lost feedback was a NAK (in which case no moe feedback will be sent, and the uncetainty could last fo eve), o that the feedback channel is undegoing a vey long bust of eos, so that a lage numbe of ACKs get lost, and the esulting delay is unacceptable. To exclude such cases, a time-out mechanism is used. This is povided by the use of a counte, which causes the tansmitte to etansmit packet i in slot i + t (and all packets following i aftewads), if by that time it is still unknown whethe o not it was coectly eceived. This allows the tansmitte to avoid deadlock situations o buffe oveflow (note, in paticula, that the tansmitte buffe needs only contain t packets). Of couse, it must be that t m (if t = m we have the classic GBN scheme, in which if an ACK is not eceived at the pope time, etansmission is immediately pefomed). IV. MARKOV CHAIN APPROACH The poblem of evaluating the pefomance of the potocol descibed in the pevious section has been completely solved in [, ], fo a system chaacteized by Makovian eos on both packets and feedback. In the fome of these papes [], some ecent esults on the thoughput pefomance have been eviewed, and a new technique has been pesented, which allows one to exactly study the potocol pefomance (pevious esults have been shown to be actually bounds). The analysis is caied out via a Makov appoach, in which a Makov chain is used in ode to take into account both the state of the two channels (diect and feedback) and the memoy of the potocol. With modeate complexity, it is possible to compute the aveage thoughput fo any choice of the matices (7) and fo easonable values of the ound-tip delay and time-out peiod, m? and t, espectively. The latte pape [] deals with the delay pefomance of the potocol, which has neve been addessed befoe. Again, a Makov appoach is taken, even though the state space is now lage, since additional infomation needs to be tacked. Fom the analysis epoted in [], it is possible to compute the aveage packet delay fo any value of the Makov paametes, m and t. The complexity gows as (t? m) 2, and is not pohibitively lage as long as t? m is of the ode of -2. In the delay analysis in [], only the efficiency of the potocol is studied, without taking into account possible queueing outside the ARQ system. The aival pocess is theefoe neglected, i.e., an infinite supply of packets is assumed. The delay hee is defined as the time elapsed fom when a packet entes the ARQ system (i.e., it is tansmitted fo the fist time), to when it depats (i.e., it is coectly tansmitted and acknowledged). Without giving any moe details about the analysis, we will use the esults in [, ] to assess the pefomance of the ARQ GBN potocol in the pesent context, whee the Makov paametes, as aleady mentioned, ae computed accoding to the fading model.
MILCOM 95, SAN IEGO, CA, NOV. 995 4.5 ε.8.6..4.5 -p.2. 2.5 5 7.5 2.5 5 7.5 2 Fig.. Makov paametes of the Rayleigh fading channel, ", and? p, vs. the fading magin, F, with (dotted) and without (solid) antenna divesity; ft = :244. 5 4 2.5 5 7.5 2.5 5 7.5 2 V. NUMERICAL RESULTS Based on the model descibed above, in this section some numeical esults will be pesented. All esults efe to a slow fading case, with f T = :244 [8, 9], whee T is the time inteval between two consecutive blocks (i.e., the sampling peiod). In this pape, we will conside Rayleigh fading only: some esults fo Rician fading ae given in [4]. Fig. shows the Makov paametes, ", and? p, vs. the fading magin, denoted by F. Note that? p has an analogous meaning to ; = is the aveage length of a bust of eos, wheeas =(? p) is the aveage length of an eo-fee un of slots. As expected,? p, in geneal, except fo vey small F. Also, " deceases and inceases when F gets lage, meaning that a lage value of the fading magin causes the eos to be ae and less clusteed. The use of divesity induces a simila elationship between and In Fig. 2, thoughputand delay ae plotted vs. the fading magin, F, with and without divesity. These analytical esults, obtained as in [, ], have been compaed with simulation esults (obtained by simulating the fading pocess and the potocol and not shown hee), and the Makov appoximation was seen to be indeed vey good. Fom the figue, it can be seen the significant impovement yielded by the use of divesity, especially in tems of smalle delay. In Fig. 3, thoughput and delay ae plotted vs. the time-out peiod, t. It can be seen that, fo given m and F, the choice of the time-out is a tade-off, and thee exists an optimum value, t opt. This is to be expected, since t should be shot in ode not to waste too many slots in the event of an uncetainty to be esolved by timeout expiation, but, on the othe hand, a lage t enhances the time divesity featue. In the pesence of divesity, the pefomance is geneally bette, but exhibits the same qualitative behavio. Note that the choice t = 7, often assumed in the liteatue [8, 9], is not the best. The analysis of the pefomance vs. the value of the timeout has not been addessed completely in pevious contibutions dealing with this type of channel [7, 8, 9]. Howeve, such a study is impotant, since t is the one design paamete which can be easily alteed. In fact, fo a given potocol, thee is some (but not vey much) flexibility in choosing the data ate, packet and fame size (which affect the paametes " and ). Also, " and can be deceased and inceased, espectively, by inceasing the fading magin if the system is limited by noise; on the othe hand, in intefeence-limited systems, F is imposed by design choices, such as the modulation fomat and the cellula fequency euse 3 2 2.5 5 7.5 2.5 5 7.5 2 Fig. 2. Aveage thoughput,, and aveage packet delay,, vs. the fading magin, F, with (dotted) and without (solid) antenna divesity; ft = :244, m = 5, t = 7. patten. Theefoe, it seems that the paamete t can be moe easily chosen, based on the above analysis. On the othe hand, it must be obseved that t diectly affects the delay pefomance, and theefoe cannot be abitaily chosen. In fact, since is geneally inceasing up to t = 2? 25 (see Fig. 3), it seems that the choice of t is essentially detemined by delay consideations. Howeve, fo values of the fading magin F > 5 db, the delay incease is vey limited fo t not too lage. The choice of t seems theefoe to be not vey citical. An inteesting combination of the esults of Fig. 3 is povided in Fig. 4, whee thoughput is plotted vs. delay. Each cuve is dawn fo a value of F, and each point of the cuve coesponds to a value of t. It can be seen that, fo small fading magins (as in Fig. 4a), an incease in thoughput can be achieved, at the pice of an inceased delay. On the othe hand, fo lage values of the fading magin (as in Fig. 4b), inceasing t can simultaneously impove the two pefomance measues. Also, even when thoughput and delay constitute a tade-off, the delay pefomance is not vey sensitive, at least fo F not too small. VI. CONCLUSIONS In this pape, the ARQ Go-Back-N potocol with time-out contol has been consideed, with efeence to a Rayleigh fading envionment, which is typical in mobile adio channels. The fading coelation fom slot to slot is taken into account, along with the uneliability of the feedback infomation, which is tansmitted on a faded channel as well. A Makov model is developed fo the sequence of successes/failues of data blocks in the two diections.
MILCOM 95, SAN IEGO, CA, NOV. 995 5.8 (a) 8.6 6.4 4 2 5 2 25 3 35 4.4.6.8 t. 7 5. (b) 6.75 6.5 2. 6.25. 5 6 5.75 5.5 2 5.25 6 8 2 4 6 8 2 Fig. 3. Aveage thoughput,, and aveage packet delay,, vs. the time-out peiod, t, fo vaious values of F ; ft = :244, m = 5, no divesity; F =, 2, 4, 6, 8,, 8 db. t 5.86.88.9.92.94.96.98 Fig. 4. Aveage thoughput,, vs. aveage delay,, fo some values of F and t; ft = :244, m = 5, t = 6 to 2, no divesity; F =, 2, 4, 6, 8,, 2 db (a); F = 2, 4, 6, 8 db (b). Once the model paametes ae computed, the thoughput and delay pefomance of the scheme can be analytically computed. Some optimization issues ae also addessed, showing that the time-out duation can be optimized, even though it has a modeate effect on the global pefomance. Futhe eseach will addess the study of othe ARQ potocols. Also, based on the thoughput/delay pefomance hee given, some intenetwoking issues can be studied. REFERENCES [] A.S. Tanenbaum, Compute Netwoks, Englewood Cliffs, NJ: Pentice-Hall, 989. [2] L.N. Kanal and A.R.K. Sasty, Models fo channels with memoy and thei applications to eo contol, Poc. of the IEEE, vol. 66, pp. 724-744, Jul. 978. [3]. Towsley, A statistical analysis of ARQ potocols opeating in a nonindependent eo envionment, IEEE Tans. Commun., vol. COM-29, pp. 97-98, Jul. 98. [4] C.H.C. Leung, Y. Kikumoto, S.A. Soensen, The thoughputefficiency of the Go-Back-N ARQ scheme unde Makov and elated eo stuctues, IEEE Tans. Commun., vol. COM-36, pp. 23-234, Feb. 988. [5].L. Lu and J.F. Chang, Pefomance of ARQ potocols in nonindependent channel eos, IEEE Tans. Commun., vol. COM-4, pp. 72-73, May 993. [6] S.R. Kim and C.K. Un, Thoughput analysis fo two ARQ schemes using combined tansition matix, IEEE Tans. Commun., vol. COM-4, pp. 679-683, Nov. 992. [7] Y.J. Cho and C.K. Un, Pefomance analysis of ARQ eo contols unde Makovian block eo patten, IEEE Tans. Commun., vol. COM-42, pp. 25-26, Feb.-Ap. 994. [8] L.F. Chang, Thoughput estimation of ARQ potools fo a Rayleigh fading channel using fade- and intefade-duation statistics, IEEE Tans. Veh. Tech., vol. VT-4, pp. 223-229, Feb. 99. [9] J.C.-J. Chuang, Compaison of two ARQ potocols in a Rayleigh fading channel, IEEE Tans. Veh. Tech., vol. VT-39, pp. 367-373, Nov. 99. [] M. Zozi, R.R. Rao, Pefomance of ARQ Go-Back-N potocol in Makov channels with uneliable feedback: Thoughput analysis, submitted to IEEE Tans. Netwoking [] M. Zozi, R.R. Rao, Pefomance of ARQ Go-Back-N potocol in Makov channelswith uneliable feedback: elay analysis, submitted to IEEE Tans. Netwoking [2] E.N. Gilbet, Capacity of a bust-noise channel, Bell System Tech. Jounal, vol. 39, pp. 253-266, Sept. 96. [3] H.S. Wang, On veifying the fist-ode Makovian assumption fo a Rayleigh fading channelmodel, in Poc. ICUPC 94, pp. 6-64, San iego, CA, Sep. 994. [4] M. Zozi, R.R. Rao, L.B. Milstein, ARQ eo contol on fading mobile adio channels, submitted to IEEE Tans. Veh. Tech. [5] W.C. Jakes, J., Micowave mobile communications, New Yok: John Wiley & Sons, 974. [6] J.G. Poakis, igital communications, New Yok: McGaw-Hill, 989. [7] R.H. Clake, A statistical theoy of mobile adio eception, Bell System Tech. Jounal, vol. 47, pp. 957-, Jul. 968.