Markov-based channel characterization for tractable performance analysis in wireless packet networks

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004 821

Markov-Based Channel Characterization forTractable Performance Analysis in Wireless

Packet NetworksMohamed Hassan, Marwan M. Krunz, Senior Member, IEEE, and Ibrahim Matta

Abstract—Finite-state Markov chain (FSMC) models haveoften been used to characterize the wireless channel. The fittingis typically performed by partitioning the range of the receivedsignal-to-noise ratio (SNR) into a set of intervals (states). Differentpartitioning criteria have been proposed in the literature, butnone of them was targeted to facilitating the analysis of thepacket delay and loss performance over the wireless link. Inthis paper, we propose a new partitioning approach that resultsin an FSMC model with tractable queueing performance. Ourapproach utilizes Jake’s level-crossing analysis, the distribution ofthe received SNR, and the elegant analytical structure of Mitra’sproducer–consumer fluid queueing model. An algorithm is pro-vided for computing the various parameters of the model, whichare then used in deriving closed-form expressions for the effectivebandwidth (EB) subject to packet loss and delay constraints.Resource allocation based on the EB is key to improving theperceived capacity of the wireless medium. Numerical investiga-tions are carried out to study the interactions among various keyparameters, verify the adequacy of the analysis, and study theimpact of error control parameters on the allocated bandwidthfor guaranteed packet loss and delay performance.

Index Terms—Markov modeling, performance analysis,Rayleigh fading, wireless channels.

I. INTRODUCTION

WIRELESS networks are characterized by time-varyingchannels whose bit-error rates (BERs) vary dramatically

according to the received signal-to-noise ratio (SNR). Dueto their mathematical tractability, finite-state Markov chain(FSMC) models have often been used to characterize the BERbehavior [1]–[3]. Typically, an FSMC model is constructed bypartitioning the range of the received SNR into a set of nonover-lapping intervals .

Manuscript received December 12, 2001; revised December 20, 2002; ac-cepted March 19, 2003. The editor coordinating the review of this paper andapproving it for publication is K. K. Leung. The work of M. Hassan and M. M.Krunz was supported in part by the National Science Foundation under GrantsANI-9733143, CCR-9979310, and ANI-0095626, and in part by the Centerfor Low Power Electronics (CLPE) at the University of Arizona (NSF GrantEEC-9523338). The work of I. Matta was supported in part by the NSF grantsANI-0095988, EIA-0202067, and ITR ANI-0205294, and in part by grants fromSprint Laboratories and Motorola Laboratories. An abridged version of thispaper was presented at the MMT 2002 Workshop, Rennes, France, June 2002.

M. Hassan and M. M. Krunz are with the Department of Electrical and Com-puter Engineering, The University of Arizona, Tucson, AZ 85721-0104 USA(e-mail: [email protected]; [email protected]).

I. Matta is with the Department of Computer Science, College of Arts and Sci-ences, Boston University, Boston, MA 02215 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2004.827729

Each interval is represented by a nominal BER, which in turnrepresents a certain channel quality. In our study, the selectionof the thresholds , , and their correspondingnominal BERs is done in a manner that leads to tractableanalysis of the packet loss and delay performance over thewireless link.

Several studies have addressed the modeling of the wirelesschannel through appropriate partitioning of the received SNR[6]–[10]. In [6], a Markov model was presented based onexperimental measurements of real channels. To maximize thethroughput while minimizing the probability of packet error,Rice and Wicker [7] represented the time-varying channel by anFSMC model in which the thresholds , , wereobtained by counting the number of retransmission requestsduring the so-called observation or frame intervals. In [8],an FSMC model was constructed by optimizing the systemperformance in the sense of maximizing the channel capacity.By requiring all states to have the same mean durations, Zhangand Kassam [9] employed an FSMC model for the Rayleighfading channel. Using the Nakagami-m distributions as thebasis for partitioning, the authors in [10] characterized thedynamics of amplitude variations of time-varying multipathfading. They also built an FSMC model whose states representthe different intervals of fading amplitude. None of the abovemodels was designed to enable tractable analysis of packet-levelperformance degradations. More specifically, the use of packetbuffering at the transmitter side of a wireless link introducesvariable queueing delays and occasional packet loss (due tobuffer overflow). Prior assessment of such degradations is akey to providing quality of service (QoS) guarantees and tothe design of online admission control policies. Hence, a goodmodel should not only reflect the physical characteristics of thechannel, but it should also facilitate analytical investigation ofits performance.

A key problem in the design of wireless networks is howto efficiently allocate their scarce resources to meet applica-tions’ packet loss and delay requirements. Such efficient allo-cation can be achieved using the concept of effective bandwidth(EB). In general, the EB refers to the minimum amount of net-work resources (in bits per second) that if allocated to a giventraffic flow would guarantee a certain level of QoS (typically,in terms of the packet loss rate). The EB has been extensivelystudied for wireline packet networks and has been widely ac-cepted as a basis for connection admission control (CAC) andresource allocation (e.g., [14], [18], [22], [23], [25], and [26]).In [11] and [15], the authors presented an analytical framework

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822 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Fig. 1. Wireless-link model.

for studying the packet loss and delay performance over a wire-less link. Based on this framework, they derived closed-form ex-pressions for the EB subject to packet loss and delay constraints.The analysis, however, was conducted under the assumption ofa two-state FSMC channel model (a slightly more general ver-sion of the Gilbert–Elliot model). Arguably, a two-state modelprovides a coarse approximation of the channel behavior andmay not always be acceptable. In fact, as shown in this paper,a two-state model leads to a highly conservative (and unneces-sary) estimate of the EB. Therefore, the motivation behind ourstudy is to construct a multistate FSMC model that lends itself toanalytical investigation of the EB in the wireless environment.

The main contributions of this paper are twofold. First, wedevise a new partitioning approach of the received SNR rangethat leads to an analytically tractable multistate Markov channelmodel. Second, we use this model to derive closed-form expres-sions for the EB subject to either packet loss or packet delayconstraints. Such expressions can be instrumented in the CACmodule at a base station of a cellular network to determine on-line the admissibility of an incoming call request. Our numericalresults indicate that such use of the EB results in a significantbandwidth gain.

The rest of the paper is organized as follows. Section IIpresents our analytical framework. In Section III, we presentthe adopted SNR partitioning approach. Section IV provides thequeueing model and the corresponding wireless EB analysis.Section V reports the numerical results. Finally, Section VIsummarizes the results of this study and outlines our futurework.

II. WIRELESS-LINK MODEL

A. Preliminaries

Consider Fig. 1, which represents a simplified representationof a wireless link. Arriving packets at the transmitter are storedtemporarily in a first-in first-out (FIFO) buffer, which is drainedat a rate that depends on the state of the channel at the receiver.Throughout this work, we refer to the draining (or service) ratewhen the received SNR is by . After departing the buffer,a packet undergoes a strong CRC encoding followed by partialforward error correction (FEC) that allows for correcting only afraction of packet errors. This hybrid automatic-repeat-requestapproach is often used to enhance the efficiency of the wireless

link by minimizing the number of retransmissions. In practice,the transmission path also includes packet and bit interleavers,possibly in conjunction with multiple FEC encoders (e.g., outerand inner encoders). For simplicity, we do not directly accountfor the impact of interleaving, but assume that such impact hasalready been incorporated in the FEC “box” in Fig. 1. In otherwords, this box could consist of multiple stages of encoding andinterleaving.

In packet networks, a traffic source is often viewed as an al-ternating sequence of active and idle periods. During an activeperiod, one or more network-layer packets (e.g., Internet Pro-tocol datagrams) arrive back-to-back, forming a burst (an ac-tive period). This so-called ON–OFF model has the advantageof being able to capture the bursty nature of various types ofnetwork traffic, including computer data, voice, and variablebit rate video [12]. Its appropriateness in analyzing the per-formance over wireless links was demonstrated in several pre-vious studies (e.g., [11] and [13]). When a network-layer packetarrives at a transmitting node, it is typically fragmented intofixed-size link-layer (LL) packets that are then transmitted overthe wireless interface. The sheer difference between the burstand LL-packet time scales makes it reasonable to separate thetwo by adopting a fluid approximation of the arrival traffic. Sucha decomposition approach, which has been successfully appliedin wireline networks [14], [16], [18], [20], allows us to empha-size the time scale of most relevance to network-layer perfor-mance (e.g., packet loss rate and queueing delay).

Accordingly, we consider incoming traffic sources, eachof which is modeled as a fluid source with exponentially dis-tributed ON and OFF periods. The means of the ON and OFF pe-riods are and , respectively. When the source is active,it transmits at a peak rate . The channel is modeled by the

-state FSMC model shown in Fig. 2. This particularstructure of the Markov chain is chosen because it lends itself tothe queueing analysis of Mitra’s producer–consumer model [16](to be described later), which allows us to evaluate the packetloss and delay performance over the wireless channel and com-pute closed-form expressions for the EB.

Let be the steady-state probability that the channel is instate , . The FSMC stays in state for an expo-nentially distributed time with mean . It is assumed that biterrors within any given state are mutually independent. For anFEC code with a correction capability of bits per code block

HASSAN et al.: MARKOV-BASED CHANNEL CHARACTERIZATION FOR TRACTABLE PERFORMANCE ANALYSIS 823

Fig. 2. Markov chain in Mitra’s producer–consumer model.

(packet), the probability of an uncorrectable error in a receivedpacket when the channel is in state is given by

(1)

where is the number of bits in a code block, including the FECbits, is the BER when the instantaneous SNR is (theform of depends on the underlying modulation scheme),and is the “nominal” SNR value in state (its calculationwill be discussed later). The packet transmission–retransmis-sion process can be approximated by a Bernoulli process [1].We assume that the transmitter always gets the feedback mes-sage from the receiver before the next transmission slot, and apacket is retransmitted persistently until it is successfully re-ceived. The nominal service rate in state , denoted by , canbe approximated by the inverse of the mean of the geometricallydistributed retransmission process

(2)

where is the error-free service rate, is the FEC over-head, and is the number of information bits in a code block.We take (the nominal service rate during the worst state).

Wireless transmission of continuous waveforms in obstacleenvironments is prone to multipath fading, which results in ran-domly varying envelope for the received signal. This random-ness has been shown to follow a Rayleigh distribution. In thepresence of additive Gaussian noise, the instantaneous receivedSNR is proportional to the square of the signal envelope [5].Accordingly, the SNR is exponentially distributed with proba-bility density function

(3)

where . An important parameter that reflects the be-havior of the SNR process at the receiver is the level-crossingrate (LCR), defined as the average rate at which the signal en-velope crosses a given SNR level . For the Rayleigh fadingchannel, the LCR at an instantaneous SNR is given by [5]

(4)

where is the Doppler frequency, is the speedof the electromagnetic wave, is the speed of the mobile user,and is the carrier frequency. We assume slow fading with re-spect to symbol transmission time. Furthermore, we assume thattransitions between channel states take place only at the end ofa packet transmission. As mentioned before, the FSMC modelthat represents the time-varying behavior of the Rayleigh fadingchannel will be obtained by partitioning the received SNR into

intervals (states). Letbe the thresholds that define

the partitioning. The steady-state probability that the FSMC isin state is given by

(5)

B. Mitra’s Producers–Consumers Fluid Model

In this section, we briefly describe Mitra’s producers–con-sumers fluid queueing model [16]. This model facilitates the an-alytical investigation of communication systems possessing ran-domly varying statistical properties. According to this model,the fluid produced by producers is supplied to a FIFO bufferthat is drained by consumers. Each producer and consumeralternates between independent and exponentially distributedactive and idle periods. Let and denote the mean of theidle and active periods of a consumer, respectively. When active,a consumer drains fluid from the buffer at a constant rate, whichis the same for all consumers. It is easy to see that the numberof active consumers fluctuates in time according to the Markovchain in Fig. 2. In [16], Mitra analyzed the steady-state behaviorof this queueing system. We use his analysis as the basis to de-rive the wireless EB. In our wireless model, correspondsto the number of incoming ON–OFF sources at the transmittingnode, while corresponds to the ratio between the nominalservice rate when the channel is in the best state (state ) andthe corresponding service rate when the channel is in the secondworst state (state 1). (Recall that the nominal service rate in state0 is zero.) In other words, one can think of the nominal servicerate in state 1 as the unit of bandwidth, and of as the numberof units of bandwidth that can be offered when the channel is inthe best state.

To analyze the packet loss and delay performance, we mustfirst partition the wireless channel in a manner that produces thesame Markovian structure of Fig. 2. In other words, we matchthe service rate at the transmitter buffer to the total instantaneousconsumption rate in Mitra’s model. This requires that we choosethe partitioning thresholds such that each state corresponds to agiven number of active consumers. Letbe the ratio between the service rate at a received SNR and itsvalue when the channel is error-free (note that dueto the FEC overhead). We form an FSMC model based on therequirement that in each SNR interval , there exists apoint , , that satisfies the following relation-ship:

(6)

824 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Note that in the producer–consumer model, the consumptionrate in state , , is times the consumption ratein state 1. The BER that corresponds to is called the nominalBER associated with state . We set and

(i.e., ).When , the model defaults to the standard two-state

Gilbert–Elliot model with and , and nopartitioning is needed. Hereafter, we concentrate on the case

.

III. SNR PARTITIONING

It is easy to see that in the producer–consumer model, thesteady-state probability distribution is binomial, i.e.

(7)

For our wireless channel to fit the Markov chain in [16], thepartitioning must be done so that no more than one state falls inthe “good” (BER close to zero) and “bad” (BER close to one)regions of the SNR space, since either scenario will lead to anunrealistic number of states. This can be justified as follows.For , must satisfy (6). Selecting two states inthe “bad” region implies that

which results in a very large . Likewise, selecting two statesin the “good” region will lead to

where . Hence

which also leads to a large .To completely specify the underlying Markov chain, we need

to determine , , and the thresholds . Ourprocedure for computing these quantities is outlined as follows.First, by equating (5) and (7), we get an expression for interms of and . Using the level-crossing analysis and thestructure of the embedded Markov chain, we then obtain an ex-pression for in terms of and . Combining the ob-tained expressions, we can express in terms of andonly. Then, by selecting an appropriate value for , the valueof the threshold can be obtained. After obtaining , theother thresholds can be obtained recursively, as follows: Thesteady-state probability that the channel is in state is given by(5), from which, and after obtaining the th threshold , the

th threshold can be obtained using the following expres-sion:

(8)

The details of the above procedure are now presented. Letbe the (irreducible) Markov process that repre-

sents the state of the channel. The time spent in any state is

exponentially distributed with mean . The parameterrepresents the total rate out of state . Consider states 0 andin Fig. 2. The total rate out of state 0 can be approximated by

the LCR at , i.e.

(9)

Similarly, the total rate out of state can be approximated bythe LCR at

(10)

Dividing (9) by (10), we get the following expression for :

(11)

The above equation relates , , and . We still need torelate to these parameters. This can be done as follows. From(7), the steady-state probability that the channel is in state 0 isgiven by

(12)

From (5), is also given by

(13)

(recall that ). From (12) and (13), we have

(14)

Similarly, the steady-state probability that the channel is in stateis given by

(15)

From (5), and noting that , can also be writtenas

(16)

From (15) and (16), an expression for the ratio in terms of, , and can be obtained

(17)

From (15), (16), and (11), we can write the following equationfor :

HASSAN et al.: MARKOV-BASED CHANNEL CHARACTERIZATION FOR TRACTABLE PERFORMANCE ANALYSIS 825

Fig. 3. Behavior of r , r , and �. (a) f(r ; r ; N) versus r for different values of r (� = 4). (b) r versus � for different values of r . (c) r versus rfor different values of �.

Taking the logarithm of both sides, we end up with

(18)

To explicitly express in terms of and , we replacein (14) by its expression in (11). Equation (18) can now

be solved to obtain provided that a suitable is chosen.Recall that by a suitable , we mean a value that will not leadto the situation of two states both being in the good or in thebad regions. By arranging the terms in (18), we arrive at thefollowing nonlinear equations that can be solved numericallyfor and (assuming that and are already known):

(19)

(20)

In our analysis, we choose such that the service rate at stateis almost equal to the error-free service rate (this depends on

the relationship between the SNR and the BER, which in turn isdependent on the modulation scheme). This is done by solvingfor in

(21)

where is a predefined control parameter. In (21),the expression for depends on the deployed modulationscheme. Using the obtained , we numerically solve (19) for

. Then from and , we get using (20). Fig. 3(a) de-picts as a function of for different values of

(with ). It is easy to show that as , ap-proaches . Since is continuous in , a sufficient condi-tion for the existence of a solution for (19) in the rangeis that . This limit is given by

(22)

Accordingly, a solution for (19) is guaranteed to exist if. This gives a lower bound on the value of that is needed

to ensure the existence of a solution. Fig. 3(b) depicts re-sulting from the partitioning versus for various values of .For a given , increasing results in a larger , and subse-quently smaller (the function saturates faster). As shownin Fig. 3(c), for a fixed , an increase in results in a decreasein . This, in effect, leads to a larger separation between and

on the SNR axis, and consequently to a higher value.The algorithm in Fig. 4 summarizes the steps that are needed

to obtain the parameters of the FSMC model. Note that the pro-cedure Parameterize-FSMC takes as inputs the parameters ofthe coding scheme ( , , ), the modulation-dependent BERfunction , , and . It returns the number of states , thepartitioning thresholds , and the nominal SNR values

(recall that and ).Note that in Step 9 of the algorithm, if for some

state , then there is nofor which , and the partitioning does not fulfillthe requirements of the producer–consumer model. If that hap-pens, we decrement the value of and repeat the computa-tions (intuitively, decrementing increases the ranges of thevarious states, which improves the likelihood of finding appro-priate nominal SNR values). The recursion in Step 8 is obtainedfrom . Note that the algorithm isguaranteed to return a solution, since for , the two nom-inal SNR values, and , are given (the partitioning reducesto the two-state Gilbert–Elliot model).

IV. PERFORMANCE ANALYSIS

A. Queueing Model

Once the parameters of the FSMC are obtained, we proceedto compute the queueing performance. The underlying queueingmodel can be described by a -state Markov

826 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Fig. 4. Algorithm to calculate the parameters of the FSMC model.

chain. The state space of this chain is given by, where a state indicates

that there are active sources and the channel is in state . Recallthat the transition rates for the FSMC are chosen such that theanalysis in [16], [17] stays applicable.

The evolution of the buffer content follows the following first-order differential equation [17]:

(23)

where is the generator matrix of the underlying Markovchain, is the diagonal drift matrix, and is the probabilitydistribution vector for the buffer content, defined as

(24)

where is the queue length at steady state.Throughout the paper, boldfaced notation is used to indicate

matrices and vectors. The solution of (23) is generally given bythe following spectral decomposition:

(25)

where ’s are constant coefficients, and the pairs ,, are the eigenvalues and left eigenvectors of the ma-

trix (see [16] and [17] for details).

B. Wireless EB

Determining the EB amounts to determining the minimumvalue of (the error-free service rate before accounting for the

FEC overhead) that guarantees a desired QoS requirement. Inthis section, we determine the wireless EB subject to packetloss and delay constraints. The computation of this EB has beenperformed in [11] and [15] for the case of a two-state channelmodel. As we show later on, this results in an unnecessarilyconservative allocation of network bandwidth.

For the packet loss case, the QoS requirement is given by the( , ), where . The corresponding EB is definedby

(26)

Similarly, in the case of the delay requirement, the EB is definedas

(27)

where the pair represents the delay constraint. Hence-forth, we use to indicate either or , depending onthe context. We obtain both quantities in terms of the source, thechannel, and the error control parameters.

A common approximation for the EB, which becomes exactas the buffer size goes to infinity, is based on the dominanteigenvalue in (25). Essentially, is the eigenvalue with thesmallest positive real part. In [14], it is shown that is theunique positive solution for the following equation:

(28)

where and are the Gärtner-Ellis limits for the ac-cumulative arrival process and the accumulativeservice process , respectively, defined as

and similarly for . Given , as the buffer size goes to infinitythe probability distribution functions for buffer overflow andpacket delay are approximately given by [14], [15], [18]

(29)

(30)

For an aggregate of two-state sources, is given by[14]

(31)Recall that and are the means of the ON and OFF pe-riods of the incoming traffic flow, respectively, and is the peaksource rate.

As for the multistate service process, it can be viewed as a“negative-flow” arrival process, where each consumer repre-sents a fictitious flow with arrival rate that alternates betweenzero and . Since the negative-flow sources are indepen-dent, for the total consumption process is given by [14,Lemma 9.2.1]

(32)

HASSAN et al.: MARKOV-BASED CHANNEL CHARACTERIZATION FOR TRACTABLE PERFORMANCE ANALYSIS 827

where is the Gärtner-Ellis limit for one consumer, andis given by

(33)Using (28), (31), and (33), one can solve for . As shown in

the Appendix, is found to be one of the roots of a quadraticpolynomial in (hence, it is given in closed form).

Next, we compute . Let .From (29), can also be written in terms of the packet lossrequirement ( , ) as . Substituting(31) and (33) in (28) and setting , we arrive at the followingequation:

(34)

Let be defined as follows:

Note that is not a function of , hence, can be treated as aconstant with respect to . Equation (34) can now be written as

Rearranging and squaring both sides of the above equation,we get

(35)

Solving for , we obtain a closed-form expression for thewireless EB subject to a packet loss constraint

(36)

Next, we consider the delay case. From (30), we have. Equating to the value of

given in (32) and (33), we obtain

(37)After some manipulations, the above equation can be ex-

pressed in terms of

(38)

which gives as a function of the channel parameters and thedelay constraint. Likewise, can be expressed in terms of thesource parameters and the delay constraint: Substituting for

in (28), with replaced by its value in (31), weobtain

(39)

which can be expressed in terms of

(40)

Equating (38) and (40), and solving for , we get aclosed-form expression for the wireless EB subject to a prede-fined delay constraint

(41)

V. NUMERICAL RESULTS

In this section, we provide numerical results obtained basedon the previously presented analysis. A modulation schemeneeds to be specified in order to obtain the BER curve and thecorresponding service ratios. In our examples, we mostly focuson the binary phase-shift keying (BPSK) and the differentialphase-shift keying (DPSK) modulation schemes. For BPSKmodulation with coherent demodulation, the BER is given by

(42)

where is the instantaneous received SNR and is the com-plementary error function. For DPSK modulation, the BER isgiven by

(43)

In all of our examples, we let and , and weuse Reed–Solomon (RS) code for error correction. We also take

and Hz. Unless specified otherwise, wetake and (hence, for the RS code). Thefirst two examples are intended to illustrate the partitioning ap-proach. The first example uses BPSK modulation. Following thealgorithm in Fig. 4, we first obtain . From this value,we obtain and , which we truncate tothree, i.e., the channel is represented by a four-state FSMC. Wethen recompute the values of and . The resulting partitionis given by

with for . This partition is found ade-quate (i.e., it satisfies the service-ratio criterion).

For the second example, we consider DPSK modulation. Inthe first pass, we obtain , , and

. Setting , we find that the resulting partition doesnot satisfy the required service ratios. Hence, we decrementand recompute the calculations. For , the partition isgiven by

with for . This partition is found to beappropriate.

Fig. 5(a) depicts versus for three different modula-tion schemes (BPSK, DPSK, and frequency-shift keying). Theircontrasting behavior indicates that the channel partitioning isdependent on the modulation scheme. Fig. 5(b) depicts

828 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Fig. 5. Service ratio �(r) versus r. (a) Different modulation schemes (� = 5). (b) Different correction capabilities (BPSK).

Fig. 6. N versus key channel parameters �, r , and r . (a) N versus r for different values of � (r = 1:0). (b) N versus r for different values of � (r =9:7). (c) N versus average received SNR � (r = 9:7).

versus for different values of with BPSK modulation. It canbe noted that the larger the value of (stronger FEC code), thefaster approaches its asymptotic value, leading to a smallervalue of . As shown later, a larger implies higher bandwidthallocation efficiency (less EB for a given QoS constraint). Soone should always try to use the maximum possible subjectto the ability to enforce the requirements of the producer–con-sumer model. The impact of the choice of on is shownin Fig. 6(a) for different values of ( fixed at 1.0). It can beseen that there exists an “optimal” value for at which ismaximized (and as shown later, the EB is minimized). Fig. 6(b)shows the variation of as a function of for different valuesof with fixed at 9.70. From Fig. 6(a) and (b), it is clearthat depends on the separation between the thresholds and

(recall that for a given , increasing results in a smaller). Fig. 6(c) shows the effect of on . It is observed that as

increases, so does , suggesting the possibility of using as ameans of controlling (since can be controlled by adjustingthe signal power at the transmitter).

Next, we study the impact of our channel partitioning ap-proach on EB-based allocation subject to either a packet loss

or a packet delay constraint. For brevity, we only show the re-sults for BPSK modulation. The source parameters are set to

packets/second (about 1.1 Mb/s when using424-bit packets), s, and s. Un-less noted otherwise, the results are obtained based on the pre-viously described BPSK example with default parameter valuesfor , , and . We vary most of the remaining parameters andexamine their impact on the EB.

Fig. 7 depicts the impact of on the EB. In theory, thechannel partition, and subsequently the values of and , de-pend on . However, in order to isolate the effect of andgauge its impact on the EB separately from other factors, we fix

and at their values in the BPSK example (with ). Forboth the packet loss and delay cases, it is clear that for a givenQoS constraint the EB decreases dramatically with an increasein . This corroborates our intuition of the conservative natureof the popular two-state Markov channel model. For example,by using four states instead of two, subject to apacket loss rate (PLR) of is reduced by almost 40% (andby 46% from the source peak rate). The reduction in the EB canbe explained by the fact that characterizing the channel with a

HASSAN et al.: MARKOV-BASED CHANNEL CHARACTERIZATION FOR TRACTABLE PERFORMANCE ANALYSIS 829

Fig. 7. EB versus N . (a) c (bu�er size = 50). (b) c (t = 2 ms).

Fig. 8. c versus the PLR p (N = 3).

larger reflects its error characteristics more accurately, andhence, leads to more efficient allocation.

Fig. 8 depicts the EB as a function of the PLR constraintusing different buffer sizes with . The figure showsthat even with a small buffer, typical PLR requirements (e.g.,

– ) can be guaranteed using an amount of bandwidththat is less than the source peak rate. The significance of our EBanalysis is that it allows the network operator to decide before-hand the amount of resources (buffer and bandwidth) needed toprovide certain QoS guarantees. A reduction in the per-connec-tion allocated bandwidth translates into an increase in the net-work capacity (measured in the number of concurrently activemobile users). Fig. 9 depicts a similar behavior for the delaycase. In fact, the reduction in the required bandwidth is evenmore pronounced in this case.

Fig. 10 depicts the wireless EB versus the number of cor-rectable bits for the delay and PLR cases. As increases,the EB decreases up to a certain point , which one maycall the “optimal” FEC. Beyond the trend is reversed (i.e.,the overhead of FEC starts to outweigh its benefits).

Fig. 9. c versus � (N = 3).

Fig. 11 demonstrates the impact of mobility on the EB for thedelay case ( s, , and GHz). A change inthe mobile speed affects the Doppler frequency , whichin turn affects the level crossing rate . While directlyimpacts the values of and , it does not impact the ratio .Hence, the channel partitioning is not impacted by . Interest-ingly, the EB is sensitive to for small values of . As the mo-bile speed exceeds a certain threshold, barely impacts the EB.In this case, the channel becomes fast varying, stabilizing theamount of resources needed to ensure a specific level of QoS.

VI. CONCLUSION

In this paper, we have presented a new approach for parti-tioning the received SNR range that enables tractable analysisof the packet loss and delay performance over a time-varyingwireless channel. This approach is based on adapting amultistate embedded Markov channel model to Mitra’s pro-ducer–consumer fluid model, which has known queueingperformance. Our analysis exploited several properties of a

830 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Fig. 10. EB versus the number of correctable bits (�). (a) c (t = 1). (b) c (bu�er size = 50).

Fig. 11. c versus mobile speed (v).

slowly varying wireless channel, including its LCR and theRayleigh distribution of the signal envelope. We provided analgorithm for iterative computation of the various partitioningthresholds and the “nominal” BERs for the various states. Wethen investigated the wireless EB subject to packet loss anddelay constraints. Closed-form expressions were derived forthe EB in each case. Besides capturing the channel fluctuations,our analysis also accommodates the inherent burstiness in thetraffic through the use of appropriate fluid source models.Numerical examples showed that the allowable number ofstates in the Markovian model depends on the underlyingmodulation scheme, the average SNR, and the separationbetween the thresholds and . The larger the value of

, the higher is the channel efficiency (in terms of the EB).This observation establishes the true impact of the paper, asit demonstrates the conservative nature of the widely populartwo-state model. The provided closed-form expressions forthe EB can be used as part of admission control and serviceprovisioning in cellular wireless packet networks.

APPENDIX

DOMINANT EIGENVALUE

According to [14], is the unique positive solution of (28).Replacing and in (28) by their expressions in(31) and (32), respectively, we obtain

(44)

After some straightforward algebraic manipulations, the aboveequation can be reduced into a cubic polynomial in

(45)

where

(46)

(47)

(48)

Clearly, is one of the solutions to (45). The othertwo solutions, denoted by and , are given by the roots

HASSAN et al.: MARKOV-BASED CHANNEL CHARACTERIZATION FOR TRACTABLE PERFORMANCE ANALYSIS 831

of the quadratic polynomial in (45). Note that for a stablesystem, at least one of these two roots must be positive. Hence,if , . Otherwise, if both roots are positive,

.

ACKNOWLEDGMENT

The authors would like to express their thanks to the anony-mous reviewers for their constructive comments.

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Mohamed Hassan received the B.Sc. degree inelectronics and communications and the M.S. degreein computer engineering from Cairo University,Cairo, Egypt, in 1990 and 1996, respectively. In2000, he received the M.S. degree in electricalengineering from the University of Pennsylvania,Philadelphia. He is working toward the Ph.D.degree in electrical and computer engineering at theUniversity of Arizona, Tucson.

Since August 2000, he is associated with theBroadband Networking Laboratory as a Research

Assistant at the University of Arizona, Tucson. His current research interestsinclude developing and evaluating theoretical foundations for providing QoSguarantees over wireless packet networks.

Marwan M. Krunz (S’93–M’95–SM’04) receivedthe Ph.D. degree in electrical engineering fromMichigan State University, Michigan, in 1995.

From 1995 to 1997, he was a Postdoctoral Re-search Associate with the Department of ComputerScience and the Institute for Advanced ComputerStudies (UMIACS), University of Maryland, CollegePark. In January 1997, he joined the Universityof Arizona, where he is currently an AssociateProfessor of Electrical and Computer Engineering.He served as a reviewer and a panelist for NSF pro-

posals, and is a consultant for several corporations in the telecommunicationsindustry.

Dr. Krunz is a recipient of the National Science Foundation CAREER Award(1998–2002). He currently serves on the editorial board for the IEEE/ACMTRANSACTIONS ON NETWORKING, the Computer Communications Journal, andthe IEEE Communications Interactive Magazine. He was a Guest Coeditor of aspecial issue on Hot Interconnects in IEEE Micro and of a Feature Topic on QoSRouting in IEEE Communications, June 2002. He was the Technical ProgramCochair for the 9th Hot Interconnects Symposium, Stanford University, August2001. He has served and continues to serve on the executive and technical pro-gram committees of many international conferences.

Ibrahim Matta received the Ph.D. degree in com-puter science from the University of Maryland, Col-lege Park, in 1995.

He is currently an Assistant Professor in theComputer Science Department of Boston University,Boston, MA. His research involves the design andanalysis of QoS and wireless architectures andprotocols, and Internet topology and traffic analysis.

Dr. Matta received the National Science Foun-dation CAREER Award in 1997. He was GuestCoeditor of two special issues on Transport Protocols

for Mobile Computing in the Journal of Wireless Communications and MobileComputing, February 2002, and on Quality of Service Routing in the IEEECommunications Magazine, June 2002. He was Technical Program Cochairof the Workshop on Wired/Wireless Internet Communications (WWIC 2002),Publication Chair of IEEE INFOCOM 2003, and Tutorial and Panel Chairof Hot Interconnects 2001. He was the representative of the IEEE TCCC forGLOBECOM 1999.