Effective capacity: A wireless link model for support of ...in admission control and resource...

630 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 4, JULY 2003

Effective Capacity: A Wireless Link Model forSupport of Quality of Service

Dapeng Wu, Student Member, IEEE,and Rohit Negi, Member, IEEE

Abstract—To facilitate the efficient support of quality of service(QoS) in next-generation wireless networks, it is essential to modela wireless channel in terms of connection-level QoS metrics such asdata rate, delay, and delay-violation probability. However, the ex-isting wireless channel models, i.e., physical-layer channel models,do not explicitly characterize a wireless channel in terms of theseQoS metrics. In this paper, we propose and develop a link-layerchannel model termedeffective capacity(EC). In this approach, wefirst model a wireless link by two EC functions, namely, the prob-ability of nonempty buffer, and the QoS exponent of a connection.Then, we propose a simple and efficient algorithm to estimate theseEC functions. The physical-layer analogs of these two link-layerEC functions are the marginal distribution (e.g., Rayleigh–Riceandistribution) and the Doppler spectrum, respectively. The key ad-vantages of the EC link-layer modeling and estimation are: 1) easeof translation into QoS guarantees, such as delay bounds; 2) sim-plicity of implementation; and 3) accuracy, and hence, efficiencyin admission control and resource reservation. We illustrate theadvantage of our approach with a set of simulation experiments,which show that the actual QoS metric is closely approximated bythe QoS metric predicted by the EC link-layer model, under a widerange of conditions.

Index Terms—Delay, Doppler spectrum, fading, quality of ser-vice (QoS), queueing theory, wireless channel model.

I. INTRODUCTION

T HE NEXT-GENERATION wireless networks such as thethird-generation (3G) and the fourth-generation wireless

systems are targeted at supporting diverse quality of service(QoS) requirements and traffic characteristics [9]. The successin the deployment of such networks will critically depend uponhow efficiently the wireless networks can support traffic flowswith QoS guarantees [10]. To achieve this goal, mechanisms forguaranteeing QoS (e.g., admission control and resource reser-vation) need to be efficient and practical [6].

Efficient and practical mechanisms for QoS support requireaccurate and simple channel models [10]. Toward this end, it isessential to model a wireless channel in terms of QoS metricssuch as data rate, delay, and delay-violation probability. How-ever, the existing channel models (e.g., Rayleigh-fading modelwith a specified Doppler spectrum) do not explicitly charac-terize a wireless channel in terms of these QoS metrics. To usethe existing channel models for QoS support, we first need to

Manuscript received March 30, 2001; revised January 2, 2002 and June 3,2002; accepted June 4, 2002.The editor coordinating the review of this paperand approving it for publication is Y. Fang. This work was supported in part bythe National Science Foundation under Grant ANI-0111818.

The authors are with the Department of Electrical and Computer Engi-neering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TWC.2003.814353

estimate the parameters for the channel model, and then extractQoS metrics from the model. This two-step approach is obvi-ously complex, and may lead to inaccuracies due to possibleapproximations in extracting QoS metrics from the models.

To address this issue, we propose and develop a link-layerchannel model termed theeffective capacity(EC) model. Inthis approach, we first model a wireless link by two EC func-tions, namely, the probability of nonempty buffer, and the QoSexponent of the connection. Then, we propose a simple andefficient algorithm to estimate these EC functions. The phys-ical-layer analogs of these two link-layer EC functions are themarginal distribution (e.g., Rayleigh–Ricean distribution) andthe Doppler spectrum, respectively. The key advantages of EClink-layer modeling and estimation are: 1) ease of translationinto QoS guarantees, such as delay bounds; 2) simplicity of im-plementation; and 3) accuracy, and hence, efficiency in admis-sion control and resource reservation. Simulation results showthat the actual QoS metric is closely approximated by the esti-mated QoS metric obtained from our channel estimation algo-rithm, under a wide range of conditions. This demonstrates theeffectiveness of the EC link-layer model in guaranteeing QoS.

Conventional channel models directly characterize thefluctuations in the amplitude of a radio signal. We call thesemodels physical-layer channelmodels, to distinguish themfrom thelink-layer channelmodel we propose. In this paper, weconsider a small-scale fading model [12] for the physical-layerchannel. Small-scale fading models describe the characteristicsof generic radio paths in a statistical fashion. Small-scale fadingrefers to the dramatic changes in signal amplitude and phasethat can be experienced as a result of small changes (as small asa half-wavelength) in the spatial separation between a receiverand a transmitter. Small-scale fading can be slow or fast,depending on the Doppler spread. The statistical time-varyingnature of the envelope of a flat-fading signal is characterizedby distributions such as Rayleigh, Ricean, Nakagami, etc. [12].

Physical-layer channel models provide a quick estimate ofthe physical-layer performance of wireless communicationssystems [e.g., symbol error rate versus signal-to-noise ratio(SNR)]. However, physical-layer channel models cannot beeasily translated into complex link-layer QoS guarantees for aconnection, such as bounds on delay. The reason is that thesecomplex QoS requirements need an analysis of the queueingbehavior of the connection, which is hard to extract fromphysical-layer models. Thus, it is hard to use physical-layermodels in QoS support mechanisms, such as admission controland resource reservation.

Recognizing that the limitation of physical-layer channelmodels in QoS support is the difficulty in analyzing queues

1536-1276/03$17.00 © 2003 IEEE

WU AND NEGI: EFFECTIVE CAPACITY: A WIRELESS LINK MODEL FOR SUPPORT OF QoS 631

Fig. 1. Packet-based wireless communication system.

using them, we propose moving the channel model up theprotocol stack from the physical layer to the link layer. Wecall the resulting model an EC link model because it capturesa generalized link-level capacity notion of the fading channel.Fig. 1 illustrates the difference between the conventionalphysical-layer and our proposed link-layer model.1 For sim-plicity, we interchange “physical-layer channel” with “physicalchannel” and interchange “link-layer channel” with “link” inthe rest of the paper.

To summarize, the EC link model that we propose aims tocharacterize wireless channels in terms of functions that can beeasily mapped to link-level QoS metrics, such as delay-boundviolation probability. Furthermore, we propose a novel channelestimation algorithm that allows practical and accurate measure-ments of the EC model functions.

The remainder of this paper is organized as follows. In Sec-tion II, we elaborate on the QoS guarantees that motivate us tosearch for a link-layer model. We describe usage parameter con-trol (UPC) traffic characterization, and its dual, the service curve(SC) network service characterization. We show that these con-cepts, borrowed from networking literature, lead us to considerthe EC model of wireless channels. In Section III, we formallydefine the EC link model in terms of two functions: probabilityof nonempty buffer and QoS exponent. We then describe an es-timation algorithm, which accurately estimates these functions,with very low complexity. Section IV shows simulation resultsthat demonstrate the advantage of using the EC link model to ac-curately predict QoS, under a variety of conditions. This leads toefficient admission control and resource reservation. Section Vconcludes this paper and points out future research directions.Table I lists the notations used in this paper.

1In Fig. 1, we use Shannon’s channel capacity to represent the instantaneouschannel capacity. In practical situations, the instantaneous channel capacity islog(1+SNR=� ), where� is determined by the modulation scheme andthe channel code used.

II. M OTIVATION FOR USING LINK-LAYER CHANNEL MODELS

Physical-layer channel models have been extremely suc-cessful in wireless transmitter–receiver design, since they canbe used to predict physical-layer performance characteristicssuch as bit/frame error rates as a function of SNR. These arevery useful for circuit switched applications, such as cellulartelephony. However, future wireless systems will need to handleincreasingly diverse multimedia traffic, which are expected tobe primarily packet switched. For example, the new widebandcode-division multiple-access (WCDMA) specifications makeexplicit provisions for 3G networks to evolve over time, fromcircuit switching to packet switching. The key difference be-tween circuit switching and packet switching, from a link-layerdesign viewpoint, is that packet switching requiresqueueinganalysis of the link. Thus, it becomes important to characterizethe effect of the data traffic pattern, as well as the channelbehavior, on the performance of the communication system.

QoS guarantees have been heavily researched in thewirednetworks (e.g., asynchronous transfer mode (ATM) and Internetprotocol (IP) networks). These guarantees rely on the queueingmodel shown in Fig. 2. This figure shows that the source trafficand the network service are matched using a first-in first-outbuffer (queue). Thus, the queue prevents loss of packets thatcould occur when the source rate is more than the service rate,at the expense of increasing the delay. Queueing analysis, whichis needed to design appropriate admission control and resourcereservation algorithms [1], [13], requires sourcetraffic charac-terizationandservice characterization. The most widely usedapproach for traffic characterization is to require that the amountof data (i.e., bits as a function of time) produced by a sourceconform to an upper bound, called thetraffic envelope . Theservice characterization for guaranteed service is a guaranteeof a minimum service (i.e., bits communicated as a function oftime) level, specified by anSC [7]. Functions and

are specified in terms of certain traffic and service param-eters, respectively. Examples include the UPC parameters usedin ATM [1] for traffic characterization, and the traffic specifi-cation (T-SPEC) and the service specification (R-SPEC) fieldsused with the resource reservation protocol [2], [7] in IP net-works.

To elaborate on this point, a traffic envelope charac-terizes the source behavior in the following manner: over anywindow of size , the amount of actual source traffic doesnot exceed (see Fig. 3). For example, the UPC parametersspecify by

(1)

where is the peak data rate, the sustainable rate, andthe leaky-bucket size [7]. As shown in Fig. 3, the curve

consists of two segments; the first segment has a slope equal tothe peak source data rate , while the second segment, has aslope equal to the sustainable rate , with . isthe axis intercept of the second segment. has the propertythat for any time .

Just as upper bounds the source traffic, a network SClower bounds the actual service that a source will


TABLE INOTATIONS

Fig. 2. Queueing system model.

receive. has the property that for any time .Both and are negotiated during the admission controland resource reservation phase. An example of a network SC isthe R-SPEC curve used for guaranteed service in IP networks

(2)

where , is the constant service rate, andthe delay error term (due to propagation delay, link sharing,

and so on). This curve is illustrated in Fig. 3. consists oftwo segments; the horizontal segment indicates that no packet isbeing serviced due to propagation delay, etc., for a time intervalequal to the delay error term , while the second segment hasa slope equal to the service rate . In the figure, we also ob-serve that: 1) the horizontal difference between and ,denoted by , is the delay experienced by a packet arrivingat time ; and 2) the vertical difference between the two curves,

Fig. 3. Traffic and service characterization.

denoted by , is the queue length built up at time, due topackets that have not been served yet.

In contrast to packet-switched wireline networks, providingQoS guarantees in packet-switched wireless networks is a chal-lenging problem. This is because wireless channels have low re-liability, and time-varying capacities, which may cause severeQoS violations. Unlike wireline links, which typically have aconstant capacity, the capacity of a wireless channel depends


upon such random factors as multipath fading, cochannel inter-ference, and noise disturbances. Consequently, providing QoSguarantees over wireless channels requires accurate models oftheir time-varying capacity, and effective utilization of thesemodels for QoS support.

The simplicity of the SCs discussed earlier motivates us todefine the time-varying capacity of a wireless channel as in(2). Specifically, we hope to lower bound the channel serviceusing two parameters: the channel sustainable rate, and themaximum fade duration .2 However, physical-layer wire-less channel models do not explicitly characterize the channelin terms of such link-layer QoS metrics as data rate, delay, anddelay-violation probability. For this reason, we are forced tolook for alternative channel models.

A tricky issue that surfaces is that a wireless channel has acapacity that variesrandomlywith time. Thus, an attempt toprovide a strict lower bound [i.e., the deterministic SC ,used in IP networks] will most likely result in extremely conser-vative guarantees. For example, in a Rayleigh or Ricean fadingchannel, the only lower bound that can bedeterministicallyguaranteed is a capacity3 of zero. This conservative guaranteeis clearly useless. Therefore, we propose to extend the conceptof deterministic SC to a statistical version, specified asthe pair . The statistical SC specifies thatthe service provided by the channel, denoted as , willalways satisfy the property that . Inother words, is the probability that the wireless channel willnot be able to support the pledged SC . For most practicalvalues of , anonzeroSC can be guaranteed.

To summarize, we propose to extend the QoS mechanismsused in wired networks to wireless links, by using the trafficand service characterizations popularly used in wired networks,namely, the traffic envelope and the SC , respectively.However, recognizing that the time-varying wireless channelcannot deterministically guarantee a useful SC, we propose touse a statistical SC .

As mentioned earlier, it is hard to extract a statistical SCusing the existing physical-layer channel models. In fact, in Sec-tion III-D, we show how physical-layer channel models can beused to derive in an integral form. There, the readerwill see that: 1) it is not always possible to extractfrom the physical-layer model (such as, when only the Dopplerspectrum, but not the higher-order statistics are known); and 2)even if it is possible, the computation involved may make the ex-traction extremely hard to implement. This motivates us to con-sider link-layer modeling, which we describe in Section III. Thephilosophy here is that we want to model the wireless channelat the layer in which we intend to use the model.

2� and� are meant to be in a statistical sense. The maximum fade dura-tion� is a parameter that relates the delay constraint to the channel service; itdetermines the probabilitysup PrfS(t) < (t)g. We will see later that�is specified by the source with� = D , whereD is the delay boundrequired by the source.

3The capacity here is meant to be delay-limited capacity, which is the max-imum rate achievable with a prescribed delay bound (see [8] for details).

III. EFFECTIVECAPACITY MODEL OFWIRELESSCHANNELS

Section II argued that QoS guarantees can be achieved if astatistical SC can be calculated for the given wireless link. Thus,we need to calculate an SC such that, for a given ,the following probability bound on the channel service issatisfied:

(3)

Further, is restricted to being specified by the parame-ters , as below [(2), which we reproduce, for con-venience]

(4)

Therefore, the statistical SC specification requires that werelate its parameters to the fading wirelesschannel. Note that a (nonfading) additive white Gaussian noise(AWGN) channel of capacity can be specified by thetriplet ; i.e., an AWGN channel can guaranteeconstant data rate.

At first sight, relating to the fading wirelesschannel behavior seems to be a hard problem. However, at thispoint, we use the idea that the SC is a dual of the trafficenvelope . A rich body of literature exists on the so-calledtheory of effective bandwidth[3], which models the statisticalbehavior oftraffic. In particular, the theory shows that the rela-tion

(5)

is satisfied for large , by choosing two parameters (which arefunctions of the channel rate) that depend on the actual datatraffic, namely, the probability of nonempty buffer, and the ef-fective bandwidth of the source.Thus, a source model definedby these two functions fully characterizes the source from a QoSviewpoint. The duality between (3) and (5) indicates that it maybe possible to adapt the theory of effective bandwidth to SCcharacterization. This adaptation will point to a new channelmodel, which we call theEC link model. Thus, the EC linkmodel can be thought of as the dual of the effective bandwidthsource model, which is commonly used in networking.

The rest of this section is organized as follows. In Sec-tion III-A, we present the theory of effective bandwidthusing the framework of Chang and Thomas [3]. An accurateand efficient source traffic estimation algorithm exists [11],which can be used to estimate the functions of theeffectivebandwidth source model. Therefore, we use a dual estimationalgorithm to estimate the functions of the proposedEC linkmodel in Section III-B. In Section III-C, we provide physicalinterpretation of our link model. Section III-D shows that inthe special case ofRayleigh-fading channel at low SNRs, it ispossible to extract the SC from a physical-layer channel model.For Rayleigh-fading channels at high SNRs, the extraction iscomplicated, whereas the extraction may not even be possiblefor other types of fading. Therefore, our link-layer EC modelhas substantial advantage over physical-layer models, inspecifying SCs, and hence QoS.


(a) (b)

Fig. 4. (a) Effective bandwidth function�(u) and (b) EC function� (u).

A. Theory of Effective Bandwidth

The stochastic behavior of a source traffic process can bemodeled asymptotically by its effective bandwidth. Consider anarrival process , , where represents the amountof source data (in bits) over the time interval . Assume thatthe asymptotic log-moment generating function of , definedas

(6)

exists for all . Then, theeffective bandwidth functionofis defined as

(7)

See [3] for details.Consider a queue of infinite buffer size served by a channel of

constantservice rate (see Fig. 2), such as an AWGN channel.Due to the possible mismatch between and , the queuelength (see Fig. 3) could be nonzero. Using the theory oflarge deviations, it can be shown that the probability ofexceeding a threshold satisfies [3]

as (8)

where means that . How-ever, it is found that for smaller values of, the following ap-proximation is more accurate [4]:

(9)

where both and are functions of channel capacity.According to the theory, is theproba-bility that the buffer is nonemptyfor randomly chosen time,while theQoS exponent is the solution of . Thus,the pair of functions model the source. Note that

is simply the inverse function corresponding to the effec-tive bandwidth function .

If the quantity of interest is the delay experienced by asource packet arriving at time(see Fig. 3), then the probabilityof exceeding a delay bound satisfies4

(10)

Thus, the key point is that for a source modeled by the pair, which has a communication delay bound of

, and can tolerate a delay-bound violation probabilityof at most , the effective bandwidth concept shows that theconstant channel capacity should be at least, where is thesolution to . In terms of the traffic envelope

(Fig. 3), the slope and .Fig. 4(a) shows a typical effective bandwidth function. It can

be easily proved that is equal to the average data rate of thesource, while is equal to the peak data rate. From (10),note that a source that has a more stringent QoS requirement(i.e., smaller or smaller ), will need a larger QoS expo-nent .

Reference [11] shows a simple and efficient algorithm to es-timate the source model functions and . In the fol-lowing section, we use the duality between traffic modeling( ), and channel modeling to propose an EC linkmodel, specified by a pair of functions . Itis clear that we intend to be the channelduals of the source functions . Just as the constantchannel rate is used in source traffic modeling, we use the con-stantsource traffic rate in modeling the channel. Furthermore,we adapt the source estimation algorithm in [11] to estimate thelink model functions .

B. EC Link Model

Let be the instantaneous channel capacity at time. De-fine , which is the service provided by thechannel. Note that the channel service is different from theactual service received by the source; only dependson the instantaneous channel capacity and, thus, is independent

4�(r) in (10) is different from� (r) in (9). The relationship between themis �(r) = � (r) � r [15, pp. 57].


Fig. 5. Relation between SC and delay-bound violation.

of the arrival . Paralleling the development in Section III-A,we assume that

(11)

exists for all . This assumption is valid, for example, fora stationary Markov-fading process . Then, theEC functionof is defined as

(12)

Consider a queue of infinite buffer size supplied by a datasource ofconstantdata rate (see Fig. 2). The theory of effec-tive bandwidth presented in Section III-A can be easily adaptedto this case. The difference is that whereas in Section III-A thesource rate was variable while the channel capacity was con-stant, in this section, the source rate is constant while the channelcapacity is variable. Similar to (10), it can be shown that theprobability of exceeding a delay bound satisfies

(13)

where are functions of source rate. Theapproximation in (13) is accurate for large , but we willshow later, in the simulations, that this approximation is alsoaccurate even for smaller values of .

For a given source rate, is againtheprobability that the buffer is nonemptyat a randomly chosentime , while theQoS exponent is defined as

, where is the inverse function of . Thus,the pair of functions model the link.

So, when using a link that is modeled by the pair, a source that requires a communication

delay bound of , and can tolerate a delay-bound violationprobability of at most , needs to limit its data rate to a max-imum of , where is the solution to .In terms of the SC (Fig. 3), the channel sustainable rate

and . It is clear that when the bufferis empty, the extra service will be lost, i.e.,not used for data transmission (see Fig. 5). So, we have

for any time . Furthermore, we know that theevent and the event are the

same. This can be illustrated by Fig. 5: whenever the curveis below , the horizontal line will cross the line ;i.e., we have an event , since the horizontaldistance between and is . Then, we have

(14)

where follows from the fact thatand are the same event, and fromthe fact that for any . From (14), it canbe seen that implies

.Fig. 4(b) shows that the EC decreases with increasing

QoS exponent ; that is, as the QoS requirement becomes morestringent, the source rate that a wireless channel can supportwith this QoS guarantee, decreases. The channel sustainable rate

is upper bounded by the AWGN capacity and lowerbounded by the minimum rate 0. Fig. 4(a) and (b) together il-lustrate the duality of the effective bandwidth source model andthe EC link model.

The function pair defines our proposedEC link model. The definition of these functions shows that theEC model is a link-layer model, because it directly characterizesthe queueing behavior at the link layer. From (13), it is clear thatthe QoS metric can be easily extracted from the EC link model.

Now that we have shown that QoS metric calculation istrivial, once the EC link model is known, we need to specifya simple (and hopefully, accurate) channel estimation al-gorithm. Such an algorithm should estimate the functions

from channel measurements, such as themeasured SNR or channel capacity .

Let us take a moment to think about how we coulduse the existing physical-layer channel models to estimate

. An obvious fact that emerges is that if achannel model specifies only themarginal probability densityfunction (pdf) at any time (such as Ricean pdf), along withthe Doppler spectrum (such as Gans Doppler spectrum),which issecond-order statistics, then the model does not haveenough information to calculate the EC function (12). Indeed,such a calculation would need higher order joint statistics,which cannot be obtained merely from the Doppler spectrum.Thus, only approximations can be made in this case. For aRayleigh-fading distribution, the joint pdf of channel gainswill be complex Gaussian, and hence, the Doppler spectrumis enough to calculate the EC. This result is presented inSection III-D. However, it will be shown there that even inthe Rayleigh-fading case, the calculation is complicated, and,therefore, not likely to be practical.

Assume that the channel-fading process is stationaryand ergodic. Then, a simple algorithm to estimate the functions

is the following (adapted from [5], [11]):

(15)

(16)


and

(17)

where is the average remaining service time of a packetbeing served. Note that is zero for a fluid model (assuminginfinitesimal packet size). The intuition in (15) is that, since thedistribution of is approximately exponential for large[see (13)], then is given by (15). Now, the delayis the sum of the delay incurred due to the packet already inservice, and the delay in waiting for the queue to clear. Thisresults in (16), using Little’s theorem. Substitutingin (13) results in (17).

Solving (16) for , we obtain

(18)

Equations (17) and (18) show that the functionsand canbe estimated by estimating , , and .The latter can be estimated by taking a number of samples,say , over an interval of length , and recording the fol-lowing quantities at theth sampling epoch: the indicator ofwhether a packets is in service ( ), the numberof bits in the queue (excluding the packet in service), andtheremaining service time of the packet in service (if there is onein service). The following sample means are computed:

(19)

(20)

and

(21)

Then, from (18), we have

(22)

Equations (19)–(22) constitute our channel estimationalgorithm, to estimate the EC link model functions

. They can be used to predict the QoSby approximating (13) with

(23)

Furthermore, if the ultimate objective of EC link modeling isto compute an appropriate SC , then as mentioned ear-lier, given the delay bound and the target delay-boundviolation probability of a connection, we can find

by: 1) setting ; and 2) solving (23)for and setting . A fast binary search procedure thatestimates for a given and , is shown in the Appendix.

This section introduced the EC link model, which is parame-terized by the pair of functions . It was shownthat these functions can be easily used to derive QoS guarantees(13), such as a bound that uses . Furthermore, thissection specified a simple and efficient algorithm [(19)–(22)]

Fig. 6. Marginal CDF and (�) versus source rate�.

to estimate , which can then be used in (13).This completes the specification of our link-layer model.

The EC link model and its application are summarized below.

EC link model:is the EC link model, which exists

if the log-moment generating function in (11)exists [e.g., for a stationary Markov-fading process

In addition to its stationarity, if is also ergodic,then can be estimated by (19) through(22).

Given the EC link model, the QoS canbe computed by (23), where

The resulting QoS corresponds directlyto the SC specification with

and

C. Physical Interpretation of Our Model

We stress that the model presented in the previous section,, is not just a result of mathematics (i.e., large

deviation theory). But rather, the model has direct physical inter-pretation; i.e., corresponds to marginal cu-mulative distribution function (CDF) and Doppler spectrum ofthe underlying physical-layer channel. This correspondence canbe illustrated as follows:

• The probability of nonempty buffer is similar tothe concept of marginal CDF (e.g., Rayleigh–Ricean dis-tribution), or equivalently outage probability (the proba-bility that the received SNR falls below a certain speci-fied threshold). As shown later in Fig. 6, different mar-ginal CDF of the underlying physical-layer channel cor-responds to different . However, the two functions,marginal CDF (i.e., outage probability) and , arenot equal. The reason is that the probability of nonemptybuffer takes into account the effect of packet accumula-tion in the buffer, while the outage probability does not


Fig. 7. � versus Doppler ratef .

(i.e., an arrival packet will be immediately discarded if theSNR falls below a threshold). Therefore, the probability ofnonempty buffer is larger than the outage probability, be-cause buffering causes longer busy periods compared withthe nonbuffered case.

From Fig. 6, we observe that and marginal CDFhave similar behavior: 1) both increases with the sourcerate ; and 2) a large outage probability at the physicallayer results in a large at the link layer. Thus,

does reflect the marginal CDF of the underlyingwireless channel.

• , defined as the decay rate of the probability, corresponds to the Doppler

spectrum. This can be seen from Fig. 7. As shown inthe figure, different Doppler rates give different .In addition, the figure shows that increases withthe Doppler rate. The reason for this is the following.As the Doppler rate increases, the time diversity of thechannel also increases. This implies lower delay-violationprobability , which leads to alarger decay rate . Therefore, does reflectthe Doppler spectrum of the underlying physical channel.

Note that a stationary Markov-fading process (as is com-monly assumed, for a physical wireless channel), willalways have a log-moment generating function .Therefore, the EC link model is applicable to such a case. Onthe other hand, in Section IV-B, the simulation results showthat the delay-violation probabilitydoes decrease exponentially with the delay bound (seeFigs. 8–10). Therefore, our link model is reasonable, not onlyfrom a theoretical viewpoint (i.e., Markovian property of fadingchannels) but also from an experimental viewpoint (i.e., theactual delay-violation probability decays exponentially withthe delay bound).

D. QoS Guarantees Using Physical-layer Channel Models

In this section, we show that in contrast to our approach,which uses a link-layer model, the existing physical-layer

channel models cannot be easily used to extract QoS guar-antees. In [14], the authors attempt to model the wirelessphysical-layer channel using a discrete state representation.For example, [14] approximates a Rayleigh flat-fading channelas a multistate Markov chain, whose states are characterizedby different bit-error rates. With the multistate Markov chainmodel, the performance of the link layer can be analyzed, butonly at expense of enormous complexity. In this section, weoutline a similar method for the Rayleigh flat-fading channelat low SNRs.

As mentioned earlier, cannot be calcu-lated using (12), in general, if only the marginal pdf at any time

and the Doppler spectrum are known. However, such an ana-lytical calculation is possible for a Rayleigh flat-fading channelin AWGN, albeit at very high complexity.

Suppose that the wireless channel is a Rayleigh flat-fadingchannel in AWGN with Doppler spectrum . Assume thatwe have perfect causal knowledge of the channel gains. For ex-ample, the Doppler spectrum from the Gans model [12] isthe following:

(24)

where is the maximum Doppler frequency, is the carrierfrequency, and .

We show how to calculate the EC for this channel. Denotea sequence of measurements of the channel gain, spacedat a time-interval apart, by , where

are the complex-valued channel gains( are, therefore, Rayleigh distributed). Without loss ofgenerality, we have absorbed the constant noise variance intothe definition of . The measurement is a realization ofa random variable sequence denoted by, which can bewritten as the vector . The pdf of arandom vector for the Rayleigh-fading channel is

(25)

where is the covariance matrix of the random vector,the determinant of matrix , and the conjugate of

. Now, to calculate the EC, we first need to calculate

(26)

where approximates the integral by a sum from thestandard result on Gaussian channel capacity (i.e.,


(a) (b)

Fig. 8. Actual delay-violation probability versusD for various Doppler rates. (a) Rayleigh fading and (b) Ricean fading.

(a) (b)

Fig. 9. Prediction of delay-violation probability when the average SNR is (a) 15 and (b) 0 dB.

, where is the modulus of ), and from(25). This gives the EC (12) as

(27)

In general, the integral in (27) is of high dimension (i.e.,dimensions) and it does not reduce to a simple form, except forthe case of low SNR, where approximation can be made. Next,we show a simple form of (27), for the case of low SNR. Wefirst simplify (26) as follows:

(28)

where using the approximation for(26) (if is small, that is, low SNR), by the definition ofthe norm of the vector , and by the relation(where is identity matrix). We consider three cases of interestfor (28).

• Case 1 (Special Case)Suppose , whereis the average channel capacity. This case happens


Fig. 10. Prediction of delay-violation probability, whenf = 5 Hz.

when a mobile moves very fast with respect to the sampleperiod. From (28), we have

(29)

where follows from the fact that the sample periodis .

As the number of samples , we have

(30)

Thus, in the limiting case, the Rayleigh-fading channel re-duces to an AWGN channel. Note that this result would notapply at high SNRs because of the concavity of thefunction. Since Case 1 has the highest degree of diversity,it is the best case for guaranteeing QoS; i.e., it providesthe largest EC among all the Rayleigh-fading processeswith the same marginal pdf. It is also the best case for highSNR.

• Case 2 (Special Case)Suppose , wheredenotes matrix transpose and . Thus,all the samples are fully correlated, which could occur ifthe wireless terminal is immobile. From (28), we have

(31)

Since Case 2 has the lowest degree of diversity, it is theworst case. Specifically, Case 2 provides zero EC becausea wireless terminal could be in a deep fade forever, makingit impossible to guarantee any nonzero capacity. It is alsothe worst case for high SNR.

• Case 3 (General Case)Denote the eigenvalues ofmatrix by . Since is sym-metric, we have , where is a unitarymatrix, is its Hermitian, and the diagonal matrix

. From (28), we have

(32)

Case 3 is the general case for a Rayleigh flat-fadingchannel at low SNRs.

We now use the calculated to derive the log-mo-ment generating function as

(33)

where follows from (32), follows from the fact that thefrequency interval and the bandwidth , and

from the fact that the power spectral densityand that the limit of a sum becomes an integral. This gives theEC (12) as

(34)

Thus, the Doppler spectrum allows us to calculate .The EC function (34) can be used to guarantee QoS using (13).

Remark 1: We argue that even if we haveperfectknowledgeabout the channel gains, it is hard to extract QoS metrics fromthe physical-layer channel model, in the general case. The ECfunction (34) is valid only for a Rayleigh flat-fading channel,atlow SNR. At high SNR, the EC for a Rayleigh-fading channelis specified by the complicated integral in (27). To the best ofour knowledge, a closed-form solution to (27) does not exist. Itis clear that a numerical calculation of EC is also very difficult,because the integral has a high dimension. Thus, it is difficult toextract QoS metrics from a physical-layer channel model, evenfor a Rayleigh flat-fading channel. The extraction may not evenbe possible for more general fading channels. In contrast, theEC link model that we have proposed can be easily translatedinto QoS metrics for a connection, and we have shown a simpleestimation algorithm to estimate the EC model functions.


Fig. 11. Queueing model used for simulations.

IV. SIMULATION RESULTS

In this section, we simulate a queueing system and demon-strate the performance of our algorithm for estimating the func-tions of the EC link model. Section IV-A describes the simula-tion setting, while Section IV-B illustrates the performance ofour estimation algorithm.

A. Simulation Setting

We simulate the discrete-time system depicted in Fig. 11. Inthis system, the data source generates packets at aconstantrate

. Generated packets are first sent to the (infinite) buffer at thetransmitter, whose queue length is , where refers to the

th sample interval. The head-of-line packet in the queue istransmitted over the fading channel at data rate. The fadingchannel has a random channel gain(the noise variance is ab-sorbed into ). We use a fluid model that is the size of a packetis infinitesimal.

We assume that the transmitter has perfect knowledge ofthe channel gain (the SNR, really) at each sample interval.Therefore, it can use rate-adaptive transmissions and strongchannel coding to transmit packets without errors. Thus, thetransmission rate is equal to the instantaneous (time-varying)capacity of the fading channel, as below

(35)

where is the channel bandwidth.The average SNR is fixed in each simulation run. We define

as the capacity of an equivalent AWGN channel, whichhas the same average SNR, i.e.

SNR (36)

where SNR is the average SNR, i.e., . Then, we caneliminate using (35) and (36) as

SNR(37)

In our simulations, the sample interval is set to 1 ms. Thisis not too far from reality, since 3G WCDMA systems alreadyincorporate rate adaptation on the order of 10 ms [9].

Each simulation run is 1000 s long in all scenarios. Since thechannel sample rate is 1000 samples/s, 1 000 000 samples ofRayleigh–Ricean flat-fading were generated for each 1000-srun, using a first-order autoregressive (AR) model. Specifically,

is generated by the following AR(1) model:

(38)

TABLE IISIMULATION PARAMETERS

where the noise is zero-mean complex Gaussian with unitvariance per dimension and is statistically independent of .The coefficient can be determined by the following procedure:1) compute the coherence time by [12, pp. 165]

(39)

where the coherence time is defined as the time, over which thetime autocorrelation function of the fading process is above 0.5;2) compute the coefficient by5

(40)

where is the sampling interval.Table II lists the parameters used in our simulations.

B. Performance of the Estimation Algorithm

We organize this section as follows. In Section IV-B1, we es-timate the functions of the EC link modelfrom the measured . Section IV-B2 provides simulation re-sults that demonstrate the relation between the physical channeland our link model. In Section IV-B3, we show that the esti-mated EC functions accurately predict the QoS metric, under avariety of conditions.

1) EC Model Estimation: In the simulations, aRayleigh flat-fading channel is assumed. We simulate fourcases: 1) SNR dB and the maximum Doppler rate

Hz; 2) SNR dB and Hz; 3)SNR dB and Hz; and 4) SNR dBand Hz. Figs. 12 and 13 show the estimated ECfunctions and . As the source rate increasesfrom 30 to 85 kb/s, increases, indicating a higher bufferoccupancy, while decreases, indicating a slower decayof the delay-violation probability. Thus, the delay-violationprobability is expected to increase with increasing source rate

. From Fig. 12, we also observe that SNR has a substantialimpact on . This is because higher SNR results in larger

5The autocorrelation function of the AR(1) process is� , wheren is thenumber of sample intervals. Solving� = 0:5 for �, we obtain (40).


Fig. 12. Estimated function̂ (�) versus source rate�.

Fig. 13. Estimated function̂�(�) versus source rate�.

channel capacity, which leads to smaller probability that apacket will be buffered, i.e., smaller . In contrast, Fig. 12shows that has little effect on . The reason is thatreflects the marginal CDF of the underlying fading process,rather than the Doppler spectrum.

2) Physical Interpretation of Link Model : To illus-trate that different a physical channel induces different parame-ters , we simulate two kinds of channels, i.e., a Rayleighflat-fading channel and a Ricean flat-fading channel. For theRayleigh channel, we set the average SNR to 15 dB. For theRicean channel, we set the factor6 to 3 dB. We simulate twoscenarios: A) changing the source rate while fixing the Dopplerrate at 30 Hz; and B) changing the Doppler rate while fixing thesource rate, i.e., kb/s.

The result for Scenario A is shown in Fig. 6. For comparison,we also plot the marginal CDF (i.e., Rayleigh–Ricean CDF) ofthe physical channel in the same figure. The marginal CDF for

6TheK factor is defined as the ratio between the deterministic signal powerA and the variance of the multipath2� ; i.e.,K = A =(2� ).

Rayleigh channel, i.e., the probability that the SNR falls belowa threshold SNR , is

SNR SNR SNR SNR (41)

Similar to (37), we have the source rate

SNRSNR

(42)

Solving (42) for SNR , we obtain

SNR SNR (43)

Using (41) and (43), we plot the marginal CDF of the Rayleighchannel, as a function of source rate. Similarly, we plot themarginal CDF of the Ricean channel as a function of source rate

.As shown in Fig. 6, different marginal CDF at the physical

layer yields different at the link layer. We observe thatand marginal CDF have similar behavior: 1) both increases withthe source rate; and 2) if one channel has a larger outage prob-ability than another channel, it also has a larger than theother channel. For example, in Fig. 6, the Rayleigh channel hasa larger outage probability and a larger than the Riceanchannel. Thus, the probability of nonempty buffer is sim-ilar to marginal CDF, i.e., outage probability.

Figs. 7 and 8 show the result for Scenario B. From Fig. 7,it can be seen that different Doppler rate at the physical layerleads to different at the link layer. In addition, the figureshows that increases with the Doppler rate. This is rea-sonable since the increase of the Doppler rate leads to the in-crease of time diversity, resulting in a larger decay rate ofthe delay-violation probability. Therefore, corresponds tothe Doppler spectrum of the physical channel.

Fig. 8 shows the actual delay-violation probabilityversus the delay bound ,

for various Doppler rates. It can be seen that the actualdelay-violation probability decreases exponentially with thedelay bound , for all the cases. This justifies the use of anexponential bound (23) in predicting QoS, thereby justifyingour link model .

3) Accuracy of the QoS Metric Predicted byand : Inthe previous section, the simulation results have justified theuse of in predicting QoS. In this section, we evaluatethe accuracy of such a prediction. To test the accuracy, weuse and to calculate the delay-bound violation probability

[using (23)], and then comparethe estimated probability with the actual (i.e., measured)

.To show the accuracy, we simulate three scenarios. In the first

scenario, the source ratesare 75/80/85 kb/s, which loads thesystem as light/moderate/heavy, respectively. For all three cases,we simulate a Rayleigh flat-fading channel with SNR

dB, kb/s, and Hz. Fig. 9(a) plotsthe actual and the estimated delay-bound violation probability

as a function of . As predicted by(23), the delay-violation probability follows an exponential de-crease with . Furthermore, the estimated

is close to the actual .In the second scenario, we also set kb/s and

Hz, but change the average SNR to 0 dB. Fig. 9(b)


shows that the conclusions drawn from the first scenario stillhold. Thus, our estimation algorithm gives consistent perfor-mance over different SNRs also.

In the third scenario, we set SNR dB andkb/s, but we change the Doppler rate to 5 Hz. Fig. 10

shows that the conclusions drawn from the first scenario stillhold. Thus, our estimation algorithm consistently predicts theQoS metric under different Doppler rate .

In summary, the simulations illustrate that our EC link model,together with the estimation algorithm, predict the actual QoSaccurately.

V. CONCLUDING REMARKS

Efficient bandwidth allocation and QoS provisioning overwireless links demand a simple and effective wireless channelmodel. In this paper, we have modeled a wireless channel fromthe perspective of the communicationlink layer. This is in con-trast to existing channel models, which characterize the wirelesschannel at thephysical layer. Specifically, we modeled the wire-less link in terms of two “EC” functions, namely, the probabilityof nonempty buffer and the QoS exponent . TheQoS exponent is the inverse of a function which we call EC.The EC link model is the dual of the effective bandwidth sourcetraffic model used in wired networks. Furthermore, we devel-oped a simple and efficient algorithm to estimate the EC func-tions . Simulation results show that the actualQoS metric is closely approximated by the QoS metric predictedby the EC link model and its estimation algorithm, under var-ious scenarios.

We have provided key insights about the relations betweenthe EC link model and the physical-layer channel, i.e.,corresponds to the marginal CDF (e.g., Rayleigh–Ricean distri-bution) while is related to the Doppler spectrum. The EClink model has been justified not only from a theoretical view-point (i.e., Markov property of fading channels) but also froman experimental viewpoint (i.e., the delay-violation probabilitydoes decay exponentially with the delay).

The QoS metric considered can be easily translated intotraffic envelope and SC characterizations, which are popularin wired networks, such as ATM and IP, to provide guaranteedservice. Therefore, we believe that the EC link model, whichwas specifically constructed keeping in mind this QoS metric,will find wide applicability in future wireless networks thatneed QoS provisioning.

In summary, our EC link model has the following features:simplicity of implementation, efficiency in admission control,and flexibility in allocating bandwidth and delay for connec-tions. In addition, our link model provides a general framework,under which physical-layer fading channels such as AWGN,Rayleigh-fading, and Ricean-fading channels can be studied.

Armed with the new link model, we are now investigating itsuse in designing admission control, resource reservation, andscheduling algorithms, for efficient support of a variety of trafficflows that require guaranteed QoS. We are also exploring theimplementation of our link model in 3G wireless systems, andits implications in QoS support for such networks.

APPENDIX

We show that the functions that specifyour EC link model can be easily used to obtain the SC specifi-cation . The parameter is simply equalto the source delay requirement . Thus, only the channelsustainable rate needs to be estimated. is the sourcerate at which the required QoS (delay-violation probability)is achieved.

The following binary search procedure estimates fora given (unknown) fading channel and source specification

. In the algorithm, is the error between the targetand the estimated , the precisiontolerance, the source rate, a lower bound on the sourcerate, and an upper bound on the source rate.

Algorithm 1 (Estimation of the channelsustainable rate )

/* Initialization */Initialize , , and ;/* E.g., ; ; ; */

; /* An obvious lower bound on therate */

; /* An obvious upper bound onthe rate is the AWGN capacity */

;/* Binary search to find a that isconservative and within */While (( ) or ( )) {

The data source transmits at theoutput rate ;Estimate and using (19) to (22);Use (23) to obtain ;

;if ( ) {/* Conservative */

if ( ) {/* Too conservative */; /* Increase rate */

;}

}else {/* Optimistic */

; /* Reduce rate */;

}}

.

Algorithm 1 uses a binary search to find the channel sustain-able rate . An alternative approach is to use a parallel search,such as the one described in [11]. A parallel search would re-quire more computations, but would converge faster.

REFERENCES

[1] “Traffic management specification (version 4.0),” ATM Forum Tech.Committee, 1996.

[2] L. Zhang, S. Berson, S. Herzog, and S. Jamin, “Resource ReSerVationProtocol (RSVP) version 1, functional specification,” inRFC 2205, R.Braden, Ed: Internet Eng. Task Force, Sept. 1997.


[3] C.-S. Chang and J. A. Thomas, “Effective bandwidth in high-speed dig-ital networks,”IEEE J. Select. Areas Commun., vol. 13, pp. 1091–1100,Aug. 1995.

[4] G. L. Choudhury, D. M. Lucantoni, and W. Whitt, “Squeezing the mostout of ATM,” IEEE Trans. Commun., vol. 44, pp. 203–217, Feb. 1996.

[5] A. E. Eckberg, “Approximations for bursty and smoothed arrival delaysbased on generalized peakedness,” inProc. 11th Int. Teletraffic Con-gress, Kyoto, Japan, Sept. 1985.

[6] P. Ferguson and G. Huston,Quality of Service: Delivering QoS on theInternet and in Corporate Networks. New York: Wiley, 1998.

[7] R. Guerin and V. Peris, “Quality-of-service in packet networks: Basicmechanisms and directions,”Computer Netw. ISDN, vol. 31, no. 3, pp.169–179, Feb. 1999.

[8] S. Hanly and D. Tse, “Multiaccess fading channels: Part II: Delay-lim-ited capacities,”IEEE Trans. Inform. Theory, vol. 44, pp. 2816–2831,Nov. 1998.

[9] H. Holma and A. Toskala,WCDMA for UMTS: Radio Access for ThirdGeneration Mobile Communications. New York: Wiley, 2000.

[10] B. Jabbari, “Teletraffic aspects of evolving and next-generation wirelesscommunication networks,”IEEE Pers. Commun., vol. 3, pp. 4–9, Dec.1996.

[11] B. L. Mark and G. Ramamurthy, “Real-time estimation and dynamicrenegotiation of UPC parameters for arbitrary traffic sources in ATMnetworks,” IEEE/ACM Trans. Networking, vol. 6, pp. 811–827, Dec.1998.

[12] T. S. Rappaport,Wireless Communications: Principles & Prac-tice. Englewood Cliffs, NJ: Prentice-Hall, 1996.

[13] S. Shenker, C. Partridge, and R. Guerin, “Specification of guaranteedquality of service,” inRFC 2212: Internet Eng. Task Force, Sept. 1997.

[14] Q. Zhang and S. A. Kassam, “Finite-state Markov model for Rayleighfading channels,”IEEE Trans. Commun., vol. 47, pp. 1688–1692, Nov.1999.

[15] Z.-L. Zhang, “End-to-end support for statistical quality-of-service guar-antees in multimedia networks,” Ph.D. dissertation, Dept. Comput. Sci.,Univ. Massachusetts, 1997.

Dapeng Wu (S’98) received the B.E. degree inelectrical engineering from Huazhong Universityof Science and Technology, Wuhan, China, and theM.E. degree in electrical engineering from BeijingUniversity of Posts and Telecommunications,Beijing, China, in 1990 and 1997, respectively. FromJuly 1997 to December 1999, he conducted graduateresearch at Polytechnic University, Brooklyn, NewYork. Since January 2000, he has been workingtoward the Ph.D. degree in electrical and computerengineering at Carnegie Mellon University, Pitts-

burgh, PA.During summers 1998, 1999, and 2000, he conducted research at Fujitsu Lab-

oratories of America, Sunnyvale, CA, on architectures and traffic managementalgorithms in the Internet and wireless networks for multimedia applications.His current interests are in the areas of networking, information theory, commu-nication theory, and multimedia communications over the Internet and wirelessnetworks.

Mr. Wu received the IEEE Circuits and Systems for Video Technology(CSVT) Transactions Best Paper Award for 2001.

Rohit Negi (S’98–M’00) received the B.Tech. degreein electrical engineering from the Indian Institute ofTechnology, Bombay, India, in 1995. He received theM.S. and Ph.D. degrees in electrical engineering fromStanford University, Stanford, CA, in 1996 and 2000,respectively.

Since 2000, he has been with the Electrical andComputer Engineering Department at CarnegieMellon University, Pittsburgh, PA, where he is anAssistant Professor. His research interests includesignal processing, coding for communications

systems, information theory, networking, and cross-layer optimization.Dr. Negi received the President of India Gold Medal in 1995.

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