1320 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007

An Approximate Analysis of Buffered S-ALOHAin Fading Channels Using Tagged User Analysis

Shahzada B. Rasool, Student Member, IEEE, and Asrar U. H. Sheikh, Fellow, IEEE

Abstract— Tagged user analysis (TUA) is a generic approxi-mate method of analyzing random access protocols for finite-user finite-buffer systems. This technique decouples the channelcontention behavior from the user queuing behavior and allowsthe use of classical queuing theory results to be directly applicableto the analysis of finite-user finite-buffer random access methods.In this paper, we extend TUA to analyze finite buffer S-ALOHAoperating over flat fading radio channels and derive expressionsfor system performance indices like throughput, average packetdelay, blocking probability and queue length. It is shown that fora moderate number of active users, the simulation and analyticalresults fit closely.

Index Terms— Channel symmetry, finite buffer, finite user,queuing analysis, rayleigh channel, S-ALOHA, TUA.

I. INTRODUCTION

RECENT proliferation of packet data has resulted insystems where the user terminal must use a finite size

buffer. Also, in any realistic packet transmission system, onlya finite number of users share channel at any time. Markovchain is an exact analysis method of analyzing finite-userfinite-buffer systems, but it results in a very large number ofinter-dependent states even for a moderately small number ofusers and buffer sizes, making the analysis very cumbersome[1]. There have been many attempts to simplify the solutionby reducing the size of state space, in particular by exploitingredundant information in the system and assuming equilibriumconditions [2], [3]. Channel symmetry, that is identical statis-tical behavior of each user in the system is the most importantassumption in [4]–[6].

In [7], the system is described by two Markov chains,called the system Markov chain and the user Markov chain;the interaction among users is partially taken into accountthrough the inherent coupling between the two chains andthe resulting equations are solved numerically to get thesystem performance. Another approach to solve the problem,called equilibrium point analysis (EPA), is reported in [1].In EPA, the size and dimension of the state space are stillquite large but the amount of computation is relatively smallsince no transition probability matrix is required. However,this technique is difficult to apply efficiently to buffered

Manuscript received March 15, 2005; revised June 14, 2006 and August18, 2006; accepted August 22, 2006. The associate editor coordinating thereview of this paper and approving it for publication was Y. Fang. This workwas in part supported by KACST.

S. B. Rasool was with KFUPM, and is currently at the Department ofElectrical and Computer Engineering, Purdue University, West Lafayette, IN(e-mail: srasool@purdue.edu).

A. Sheikh is currently on a leave of absence from KFUPM andis the Rector of Foundation University, Islamabad, Pakistan (e-mail:sheikh.asrar@gmail.com).

Digital Object Identifier 10.1109/TWC.2007.05185.

systems with a complicated queuing behavior. All of the aboveapproaches employ a simplified system model but still useMarkov analysis.

It is interesting to note that though in a random multipleaccess protocol each user can be viewed as a queuing systemfew attempted to use queueing theory results in [8], [9]to solve the multiple access problems but the complexityof the analysis prohibited their wide use. Further, all theseapproaches did not decouple the channel contention fromthat of user queuing behavior, making them difficult to applyon system with complicated queuing behavior. Tagged useranalysis (TUA) [10], [11] is relatively a new technique toanalyze multiple access systems where each user is consideredas a distributed G/G/1/K system. The assumptions of theexistence of statistical equilibrium conditions for all users,which is a viable assumption for practical systems, and thechannel symmetry allow decoupling of channel contentionand the users’ queuing behavior. In a homogeneous system,since all the users statistically behave identically, we mayselect one representative user, call it the tagged user, andanalyze its behavior. This is extrapolated to get a completesystem solution. The effect of multiple access procedure onservice time distribution is taken into account by analyzingthe busy probability of tagged user. In principle, the channelbusy probability depends on the access protocol, channelstate and the packet service time distribution. The parametricinterdependency due to shared channel results in coupled non-linear equations, which are solved numerically to get thesystem performance.

The TUA is a generic method of analyzing the performanceof random access strategies. In this paper we restrict ourselvesto the analysis of homogeneous systems.It has been applied torandom access schemes for homogeneous and heterogeneoussystems integrating different types of traffic and traffic models[12], [13]. In [11] the analysis was limited to time invariantideal channel, this paper extends TUA to a more realisticchannel model that includes fading. Section II formulates thesystem model followed by the TUA analysis in section III. Insection IV, we consider the effects of fading channels on thesystem performance. Finally, simulation results in section VIshow the validity of the analysis.

II. SYSTEM MODEL

Consider a homogeneous centralized slotted ALOHA sys-tem with a randomly distributed finite population, N , of userseach with a buffer capacity of L packets and same systemparameters e.g. packet arrival rate and channel permissionprobability. All transmissions start and end at slot boundary;

1536-1276/07$25.00 c© 2007 IEEE

RASOOL and SHEIKH: AN APPROXIMATE ANALYSIS OF BUFFERED S-ALOHA IN FADING CHANNELS USING TAGGED USER ANALYSIS 1321

packets of one slot time of duration τ are transmitted to acentral receiver through a common symmetric channel. Thearrival process for each user is modeled as a Bernoulli processwith average arrival rate of λ; the generation of a packet isassumed to occur at the end of a slot. All messages are servedon First Come First Serve (FCFS) basis. Arriving packetsfinding queue full are dropped. A packet that arrives in thelast slot of a service time when the buffer is full, is rejectedfollowing the late arrival with delayed replacement rule [14].Defer First Transmission (DFT) principle with geometricchannel permission (and retransmission) probability, p, isemployed [1]. Hence the re-transmission policy is memoryless.All transmitted packets reach the receiver through independentuncorrelated slowly fading Rayleigh channel. In case of morethan one transmissions during a slot, the packet with powerexceeding the total power of the other contending packets bythe capture threshold z0 is successfully received [15], [16].It is assumed that no other transmission is attempted unlessthe acknowledgment (positive or negative) for the precedingattempt is received. The acknowledgment for a packet isassumed to be received error free D − 1 slots after itstransmission.

III. TAGGED USER ANALYSIS MODEL

Under the assumption of symmetric channel and homoge-neous system, each user has identical behavior at steady stateconditions. This allows us to arbitrarily select a user and an-alyze its behavior in the presence of interference due to otherusers. The tagged user can be modeled as a G/G/1/K queuingsystem consisting of a queue and a virtual server which isoperating at its equilibrium state. In FCFS service discipline,the head-of-line packet is the one in the virtual server. Sincethe arrival process and queue discipline is already known, weonly need virtual packet service time distribution (VPST) andsome standard queuing theory results to obtain performancemeasures of the tagged user (or equivalently the entire systemsystem by extrapolation) [14], [17], [18]. Unfortunately, dueto interactions between all users, VPST is equal to the packetplus a random channel access delay, which is determined bythe number of users and the transmission protocol. Hence,finding the distribution of VPST for the tagged user is the keycontention analysis. Once this is obtained and the the queuingsystem of the tagged user is identified, corresponding resultsfrom the queuing theory are applied to solve for the systemperformance measures(termed as queuing analysis).

The state flow graph technique [19] has been used todetermine the distribution of VPST. It is apparent that theVPST will be a function of the interference from other usersand their busy probabilities, channel conditions and protocolsemantics. Given the packet service time distribution andarrival process, we can use queuing theory to obtain userbusy probabilities. In general, these functional relationshipsare non-linear, but these are sufficient to enable us to determineVPST and user busy probabilities.

A. Channel Contention Analysis

Fig. 1 shows the state flow graph of the tagged user at steadystate conditions, where S0, Stx, Sbk, and Se are packet-ready

DeparturePacket Arrival

S

(1-p

)z

p

1

tx

SesD

(1-p)z

p zs D

Sbk

S0Geo/G/1/K

Fig. 1. State flow diagram of tagged user.

state, transmission-start state, back off state, and departurestate of the tagged user, respectively [11].

At the start of a new slot, when a new packet arrives oris already waiting for transmission in the queue, the statechanges from S0 to Sbk. Assuming a DFT rule, the usertransmits with channel permission probability p and enters Stx

or defers for one more slot with probability 1− p. Followingtransmission of the packet the user waits for D − 1 slots toreceive an acknowledgment. If the transmission is successful,it enters the departure state Se, and removes the packet fromits queue. In case of unsuccessful transmission (Negative ACKor No ACK), the user at the end of D slots enters theback-off state, Sbk. The probability that the transmission issuccessful, ps, depends on the interference from other usersand the channel conditions. The (virtual) packet service time(VPST), or response time, is defined as the time durationfrom the packet entering entering S0 to its exiting Se. Usingsignal flow graph techniques, the expression for the ProbabilityGenerating Function for VPST over steady channels, for DFTrule, is given as [20]

B(z) =ppsz

D

1 − (1 − p)z − p(1 − ps)zD(1)

A busy user after a packet transmission can be in one oftwo modes: CON (contending for channel) or waiting foracknowledgment (WAI), which takes D − 1 slots. Denotingby pc the probability that the user is at CON, we have

ps =N−1∑i=0

(N − 1

i

)(ppc)i(1 − ppc)N−1−ips|i (2)

where, ps|i is the probability that the tagged user transmissionis successful given i other simultaneous transmissions and isdependent on channel conditions and receiver characteristics.Hence B(z) is a function of pc. Since queuing theory wouldgive us another functional relationship between B(z) and userbusy probability pb, we need to develop a relation between pc

and pb. From (1) the mean packet service time is given by[21]

b�= B′(1) =

D − 1ps

+1

pps(3)

From this we observe that on average the user waits foracknowledgment for D−1

psslots, hence

pc =pb

b

(b − D − 1

ps

)(4)

1322 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007

B. Queueing Analysis

From (1) we note that VPST has a general distributionand consequently the tagged user should be modeled as aGeo/G/1/K system. From queuing theory results, we alreadyknow that in any queuing system the user busy probabilityp b is always determined by packet service time distributionB(z) [14], [18]. Using this relationship of p b and B(z) fromqueuing theory of Geo/G/1/K systems and relationships ob-tained in the contention analysis, we can solve these equationsiteratively to find p b and B(z). Once we solve these non-linearequations, we obtain all performance measures of a multipleaccess system.

Results from [14] for Geo/G/1/K case show that for ak,the probability that k packets arrive at the tagged user bufferduring a packet service time with a PGF

A(z) =L−1∑k=0

akzk = B(1 − λ + λz) (5)

since at most L−1 packets arrive in one service time. Denotingπi, i = 0, 1, . . . , L−1, as the probability that a leaving packetsees i packets in queue, a recursive algorithm is developed tosolve the system.

Using total probability theorem, π0 =(∑L−1

k=0 π′k

)−1, and

letting p k, 0 ≤ k ≤ L, the probability that there are k packetspresent in the system at any slot boundary,

p k =πk

π0 + ρ0 ≤ k ≤ L − 1 (6)

pL = 1 − 1π0 + ρ

(7)

where we define

b = B′(1) (8)

ρ = λb (9)

Equations (1)–(4) show us that B(z) is a function of pc

or equivalently p b, while queuing theory results of (5)–(8)affirm the dependence of p b on B(z). These relationships areindependent and hence they can be solved simultaneously.However, due to non-linearity of the equations, numericalmethods have to be used, and for this purpose an algorithmdeveloped in [11] is used. If the system has a unique operating(global equilibrium) point, different initial values for pc willconverge to the same result, while in the case of multipleoperating (local equilibrium) points (e.g., bistable behavior)the results will be affected by the initial value used in thealgorithm.

IV. FADING EFFECTS

In a fading channel environment, the strength of the receivedsignal is a random variable with a certain pdf. Assuminga transmission slot duration of τ seconds, the tagged usertransmission would be unsuccessful only if

Pt < z0 Pn during tw 1 ≤ n ≤ N − 1

where z0 is the receiver signal capture ratio and tw is theminimum time duration needed to lock on to the desiredtransmission. Pt is the received power of the tagged user

transmission and Pn denotes the total interference power.Hence the conditional success probability of tagged usertransmission can be written as

ps|i = Pr[Pt > z0Pi] = 1 − Pr[Pt

Pi< z0

]= 1 − Pr[Zi < z0] i = 0, . . . , N − 1

(10)

Using independence of Pt and Pi, and simple transformationof random variables, we get

ps|i = 1−FZi(z0) =∫ ∞

z0

dz

∫ ∞

0

fPt(zw)fPi(w)wdw (11)

Moreover, if the transmitters are randomly distributed inthe cell, we have a case of Rayleigh fading superimposedover a randomly varying mean signal amplitudes. In this casewe average the instantaneous signal power (due to multipatheffects) over the conditional density of the mean signal power.Therefore, the received power pdfs of the tagged and theinterfering users take the form

fPt(pt) =∫ ∞

0

fPt(pt|pt)fP t(pt)dpt (12)

fPi(pi) =∫ ∞

0

fPi(pi|pi)fP i(pi)dpi (13)

where fP k(pk), k = 0, . . . , N is the local mean received

power pdf of mobile terminal k, which is dependent on thespecific distance between transmitter and receiver and thechannel path loss model.

A. Uniform Rayleigh Fading

Assuming all packets are received through independentRayleigh channel with equal mean power P k = P0, 1 ≤ k ≤N , all received signal amplitudes Rk(t) will be identicallyRayleigh distributed, i.e.,

fRk(t) =2r

P0exp(−r2/P0) (14)

Then the power of kth received signal, Pk(t), is exponentiallydistributed,

fPk(pk) =

1P0

exp[− pk

P0

](15)

For phase-modulated signals, we have two possible scenariosfor the received interference signals - adding coherently ornon-coherently [22], [23].

1) Non-coherent Addition: This is the case when the thereceiver is unable to phase synchronize the signals and theseare non-coherently detected. In this case the pdf of jointinterference power is found by i-fold convolution of (15) tobe the gamma distribution

fPi(pi) =1P0

(pi/P0)i−1

(i − 1)!exp

[− pi

P0

](16)

Substituting (15) and (16) in (11), and carrying out theintegration we obtain

ps|i =1

(z0 + 1)i(17)

RASOOL and SHEIKH: AN APPROXIMATE ANALYSIS OF BUFFERED S-ALOHA IN FADING CHANNELS USING TAGGED USER ANALYSIS 1323

TABLE I

SIMULATION VALUES FOR SLOTTED ALOHA WITH L = 16.

Parameter N = 10, p = 0.125 N = 100, p = 0.0125

(per user) Mean Std. Dev. Mean Std. Dev.

Successful transmissions 6997.30 5.4396 6998.1 9.0579

Packet delay (slots) 35.5505 0.2028 382.02 1.5849

Waiting time (slots) 20.0197 0.1861 219.00 1.4823

Queue length 1.2442 0.0076 1.3372 0.0063

User busy probability 0.5434 0.0008 0.5704 0.0010

Blocking probability 0.0001 0.0000 0.0001 0.0000

2) Coherent Addition: In a slow fading environment, therandom phases of the interference signals hardly vary overthe packet transmission time, and the received power of thedesired signal is sufficiently high, coherent demodulation canbe used and coherent addition of phasors is more appropriate.In this case a phasor addition of signal amplitudes is usedinstead of a power sum. Since the sum of any number ofRayleigh phasors is also a Rayleigh phasor [24], we have thepdf of coherently added interference signal amplitude Ri(t)as,

fRi(t) =2r∑i P0

exp[−r2/

∑i P0

]1 ≤ i ≤ N (18)

Therefore, the pdf of the total interference power is simply

fPi(pi) =1∑i P0

exp[− pi∑

i P0

](19)

Using this result and (15) in (11), we get

ps|n =1

(nz0 + 1)(20)

V. PERFORMANCE MEASURES

In a multiple access network, the performance measures ofinterest such as channel throughput, user queue length, packetresponse time, and channel inter-departure times are easilyfound using TUA,

1) Packet Blocking ProbabilityAn arriving packet is rejected (blocked) if it finds thebuffer full, which occurs with probability pL. Hence,using (7),

pB = pL = 1 − 1π0 + ρ

(21)

2) System ThroughputSince all of the packets accepted into user queue wouldbe sooner or later successfully transmitted, the userthroughput is equal to the average information bitsaccepted by the queue in one bit time. Hence the userthroughput is

θ = λ(1 − pB)T (22)

since λ(1−pB) is the average rate of packet acceptanceinto the tagged user buffer and T is the mean packettransmission time (assumed to be one slot). Employingchannel symmetry, the overall system throughput Θ is

Θ = Nθ (23)

3) Queue LengthIf Iq is the number of packets in the user buffers, thenemploying (6) we get

E(Iq) =L∑

k=0

k.pk (24)

where we have used our assumption that all behaviorof all users are statistically equivalent at the equilibriumpoint.

4) Waiting Time and Response TimeThe packet waiting time is the time elapsed from packetarrival to its being ready for transmission. The packetresponse time is the time that a packet stays in thesystem from its arrival to departure. Similarly, the packetaccess delay is the time elapsed from the packet beingready for transmission to the start of its successfultransmission. These times can only be defined for thepackets accepted into the system. If we let tw and tras the packet waiting time and packet response time inslots, respectively, we have from Geo/G/1/K theory [14]

W (z) =π0(1 − z)

1 − λ − z + λB(z)

(1 −

[ λB(z)z − 1 + λ

]L)

(25)

+[B(z)

]L−1L−1∑j=0

πj

( λ

z − 1 + λ

)L−j

and

E(tw) = W ′(1) =E(Iq)

λ(1 − pB)− b (26)

Furthermore, since R(z) = W (z)B(z), therefore

E(tr) = R′(1) =E(Iq)

λ(1 − pB)(27)

VI. NUMERICAL RESULTS

In this section, we test the accuracy of TUA by consideringtwo cases of slotted ALOHA systems: an immediate acknowl-edgment system with non-coherent addition, and a systemwith acknowledgment delay. Prior to running simulations, wetest the validity of our assumption of steady-state conditions(i.e., statistical equilibrium) for each user. Table I lists themean values and the standard deviation of different systemparameters for two homogeneous systems with L = 16 andan overall arrival rate N.λ = 0.35 for different values of Noperating in an ideal non-fading symmetric channel.

1324 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5λ = 0.0045, N=100, D=1, Capture@4dB

Channel Access Probability

Thr

ough

put

L = 1L = 8L = 32

L = 32

L = 8

L = 1

Lines: TUA AnalysisMarkers: Simulation Values

(a) Throughput

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0410

1

102

103

104

105

λ = 0.0045, N=100, D=1, Capture@4dB

Channel Access Probability

Pac

ket R

espo

nse

Tim

e (T

otal

Del

ay)

L = 1L = 8L = 32

L = 1

L = 32

L = 8

Lines: TUA AnalysisMarkers: Simulation Values

L = 32

L = 8

(b) Total Delay

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9λ = 0.0045, N=100, D=1, Capture@4dB

Channel Access Probability

Pac

ket B

lock

ing

Pro

babi

lity

L = 1L = 8L = 32

L = 32

L = 1

L = 8

Lines: TUA ResultsMarkers: Simulations

(c) Packet Blocking Probability

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0410

1

102

103

104

105

λ = 0.0045, N=100, D=1, Capture@4dB

Channel Permission Probability

Wai

ting

Del

ay

Lines: TUA AnalysisMarkers: Simulation Values

L = 32

L = 32

L = 8

L = 8

L = 32

(d) Average Waiting Delay

Fig. 2. Analytical and Simulation results for an S-ALOHA system with immediate acknowledgment and non-coherent addition. N = 100, λ = 0.0045.

For the system with N = 10, the simulation length waschosen to be 200,000 slots, while for N = 100, simulationlength was 2,000,000 slots. The results were averaged over100 independent runs. The results show that all the users havenearly identical behavior, which permits us to consider onerepresentative user, evaluate its performance and extrapolateit to evaluate the overall system performance. From Table I,it is also observed that as the number of users in the systemgrows, the system tends to converge to a stable equilibriumcondition, and hence TUA estimates are expected to be moreaccurate for a larger number of users. This may be attributedto an improved condition of symmetry.

A. Immediate Acknowledgment System

We consider a S-ALOHA system operating in a slowlyfading Rayleigh channel with an equal mean received powerat the receiver, which is equivalent to distribution of users in aring or to implementation of some form of power control. Weassume that the receiver employs non-coherent demodulation

with a signal capture threshold of z0 = 4dB. Figure 2 showsthe TUA results and simulation values for for a range ofchannel permission probabilities and buffer lengths of a systemwith 100 users with packet arrival rate λ = 0.0045. Channelthroughput, packet response time (or total delay), blockingprobability and average waiting delay results are shown. In thefigures, the lines are the TUA analytical results while markersdenote simulation results. The simulation values depicted byempty symbols were obtained under the initial condition thatthe user buffers at the start of simulation were empty, whilesolid symbols denote the simulations values obtained undera full-buffer initial condition. From the figure, we observethat the system is globally stable and has a single operatingpoint, however it becomes bistable for p > 0.0275 for L = 8and p > 0.0225 for L = 32. Further, when the system isbistable as predicted by analytical results, the system staysin only one of the two locally stable points during eachsimulation run. The reason may be that the ”potential well”associated with these two locally stable points is very large

RASOOL and SHEIKH: AN APPROXIMATE ANALYSIS OF BUFFERED S-ALOHA IN FADING CHANNELS USING TAGGED USER ANALYSIS 1325

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55 N=100, D=13, Capture@4dB

Channel Access Probability

Thr

ough

put

L = 1L = 8

Nλ = 0.65

Nλ = 0.45

(a) Throughput

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

1

102

103

104

N=100, D=13, Capture@4dB

Channel Access Probability

Pac

ket R

espo

nse

Tim

e

L = 1L = 8

Nλ = 0.45

Nλ = 0.65

Nλ = 0.65

(b) Total Delay

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 N=100, D=13, Capture@4dB

Channel Access Probability

Pac

ket B

lock

ing

Pro

babi

lity

L = 1L = 8

N.λ = 0.45

N.λ = 0.65

(c) Packet Blocking Probability

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

0

101

102

103

104

L=8, N=100, D=13, Capture@4dB

Channel Permission Probability

Wai

ting

Del

ay

N.λ = 0.45

N.λ = 0.65

(d) Average Waiting Delay

Fig. 3. Analytical and Simulation results for a system with acknowledgment delay and non-coherent addition. N = 100, D = 13, z0 = 4dB.

compared with the simulation run time. The differences in theresults for full buffer and empty buffer conditions is relatedto packet arrival rate and the packet service time, whichin turn depends on the permission probability and delay inreceiving acknowledgment. When the arrival rate is higherthan the successful packet transmission rate, the system withfull buffer at the start will go into condition of higher droppedpackets, which will lead to unstable condition sooner. Moreimportantly, TUA is able to predict the bistable nature of thesystem, a conclusion reached earlier in [11].

B. System with Acknowledgment Delay

Our second example is that of an S-ALOHA system with ac-knowledgment delay. The channel is a slowly fading Rayleighchannel with non-coherent addition at the receiver. Fig. 3shows the results for a system of 100 users with D = 13and z0 = 4dB.

The results are shown for N.λ = 0.45 and 0.65. Solidlines are plots for L = 1 and dotted lines designate TUA

results for L = 8. From Fig. 3, we observe that as weincrease packet arrival rate, system stability improves, thoughoptimal region of operation reduces as well. Furthermore,TUA and simulation results match very well. It is also be notedthat delay in acknowledgment introduces extra memory intothe system, so other analysis methods already too complexbecome even more involved and intractable if memory isincluded. However, in the case of TUA, the extra delayintroduces a small computational overhead. Unfortunately,TUA does not provide us a way to know the existence of asingle operating point for the system with specific parameters;the equations in TUA are non-linear and the solution is highlydependent on the initial guess. Nevertheless, the most practicaland desirable operating point of a multiple access systemis that where it is globally stable and has a good channelthroughput, in which case TUA provides accurate systemperformances.

1326 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007

VII. CONCLUSIONS

In this paper, we extended TUA application to a homo-geneous system operating in slow fading Rayleigh channel.An approximate analysis of finite user finite buffer slottedALOHA system operating in a Rayleigh fading channel usingTUA was presented. A good match between the analyticaland simulation results verifies the channel symmetry andequilibrium probability approximations made in the model.TUA can be easily extended to include other wireless trans-mission effects like shadowing, inter-cell interference and evenchannel frequency selectivity. The analysis can be extended toheterogeneous systems. The case of heterogeneous systems notonly arises for different system parameters e.g. buffer lengthsor permission probabilities, but also comes up when wehave non-identical fading effects for different users. Anotheroccasion is when the users are spatially distributed in clustersso that the users in each cluster have identical parametersand can be grouped together. In short, TUA is a fairly goodapproximate technique to evaluate random access systemswith finite users having finite buffer size under differenttransmission disciplines.

ACKNOWLEDGMENT

The authors acknowledge the support of KFUPM to carryout this work. The work reported in this paper is supportedby KACST under project Number LG 6-8. Authors alsoacknowledge the guidance of Dr. Zaki Al-Akhdar in designingsimulation experiments.

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Shahzada Basharat Rasool received his B.S. de-gree in electrical engineering from UET Lahoreand Masters degree from King Fahd Universityof Petroleum and Minerals, in telecommunicationengineering in 2005. He is currently working towardhis PhD at Purdue University. His research interestsinclude random access protocols and low complexityscheduling algorithms in wireless networks.

Asrar U. H. Sheikh graduated from the Universityof Engineering and Technology, Lahore, Pakistan,with first class honors and received the M.Sc. andPh.D. degrees from the University of Birmingham,Birmingham, U.K. He held faculty and adminstra-tive positions at universities in several countriesincluding Pakistan, Iran, Libya, Canada, Hong Kongand Saudi Arabia before returning to Pakistan asthe Rector of Foundation University Islamabad.Dr. Sheikh established research laboratories at Car-leton (PCS Research Lab), Hong Kong Polytechnic

(WISR Centre) and King Fahd University of Petroleum & Minerals (TRL).Dr. Sheikh is a recipient of several teaching and research awards including thebest paper award from IERE (UK) in 1984. He chaired the Technical Programof VTC98, ISWSN’03 and ISWSN’05 and was vice-chair TPC for IEEE-GCC2006. He has been on the editorial staff of several journals including IEEECommunications Magazine, IEEE Transactions on Wireless Communications,and is on the editorials boards of WPC (Springer) and WCMC (Wiley). Hehas been a consultant to many private companies and government agencies.His current interests are in signal processing in communications, mitigationof interference in wireless systems, digital cellular systems, spread spectrum,and personal communication systems. He is a Fellow of the IEEE and IEE.He is listed in Marqui’s Who’s Who in the world as well as in Science andEngineering.