IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999 153
On the Teletraffic Capacity ofCDMA Cellular Networks
Jamie S. Evans and David Everitt
Abstract—The aim of this paper is to contribute to the un-derstanding of the teletraffic behavior of code-division multiple-access (CDMA) cellular networks. In particular, we examinea technique to assess the reverse link traffic capacity and itssensitivity to various propagation and system parameters.
We begin by discussing methods of characterizing interferencefrom other users in the network. These methods are extremelyimportant in the development of the traffic models of latersections. We begin with a review of several existing approaches tothe problem of handling other-cell interference before presentinga novel characterization of the interference in the form of ananalytic expression for the interference distribution function inthe deterministic propagation environment.
We then look at extending the capacity analyses that assumea fixed and equal number of users in every cell to handle therandom nature of call arrivals and departures. The simplestway to do this is by modeling each cell of the network asan independentM=G=1 queue. This allows us to replace thedeterministic number of users in each cell by an independentPoisson random variable for each cell. The resulting compoundPoisson sums have some very nice properties that allow us tocalculate an outage probability by analyzing a single random sum.This leads to a very efficient technique for assessing the reverselink traffic capacity of CDMA cellular networks.
I. INTRODUCTION
CODE-DIVISION multiple access (CDMA) is an alterna-tive multiple-access strategy to frequency-division and
time-division multiple access. Provided the synchronizationand power control problems can be overcome, CDMA isa very attractive technique for wireless communications. Itsadvantages over other multiple-access schemes include higherspectral reuse efficiency, greater immunity to multipath fading,gradual overload capability, simple exploitation of sectoriza-tion and voice inactivity, and more robust handoff procedures[1], [2].
As early as 1978, a CDMA system had been proposedfor mobile communications [3], however, interest was limiteduntil Qualcomm demonstrated the feasibility of implementingsuch a system in the late 1980’s [4]–[6]. Since then, there has
Manuscript received February 27, 1996; revised October 15, 1996. Thiswork was supported in part by the Australian Telecommunications andElectronics Research Board, the Australian Research Council, and a TelstraResearch Laboratories Postgraduate Fellowship. The material in this paper waspresented in part at GLOBECOM, Singapore, 1995, and the IEEE VehicularTechnology Conference, Atlanta, GA, 1996.
J. S. Evans was with the Department of Electrical and Electronic Engineer-ing, University of Melbourne, Parkville 3052, Victoria, Australia. He is nowwith the Department of Electrical Engineering and Computer Science, Univer-sity of California, Berkeley, CA 94720 USA (e-mail: [emailprotected]).
D. Everitt is with the Department of Electrical and Electronic Engineering,University of Melbourne, Parkville 3052 Victoria, Australia.
Publisher Item Identifier S 0018-9545(99)00712-4.
been an explosion in CDMA research mainly concentrating onthe design and performance analysis of receivers, coding andmodulation techniques and power control algorithms. Higherlayer issues such as call admission control, analysis of softhandoff, and the effects of gradual overload and imperfectpower control on capacity have also begun to receive attention(see [7] and [8]). Yet to be properly examined, however, is theteletraffic behavior of cellular networks employing CDMA.
The traffic modeling of orthogonally channelized reuse-based cellular systems, such as those employing frequency-division or time-division multiple access, is well developed[9]. The behavior of networks employing fixed channel as-signment and dynamic channel assignment has been studiedand several approaches to analyzing handoff have been putforward. Much of the success in this area results from theseparation of traffic analysis from transmission issues whichallows the mobile network to be treated as a conventionalcircuit switched or open queueing network. Unfortunately,in CDMA the separation between traffic and transmissionissues is not so clear with capacity being determined by theinterference caused by all transmitters in the network.
The goal of this paper is to contribute to the developmentof a deeper understanding of the traffic behavior of CDMAcellular networks through the determination of analytic toolsfor performance analysis and design of these networks. Suchan understanding is vital to sensible network operation underthe stochastically varying loads that characterize teletraffic.
The paper is organized as follows. In Section II, the systemstructure and propagation models used throughout the paperare introduced. Section III examines methods of quantifyingthe interference produced by mobiles in other cells of thenetwork. The main result is an analytic expression for thedistribution function of the interference from a mobile whoseposition is a random variable in another cell of the network.
Most of the literature on traffic modeling of CDMA cellularnetworks is based on modeling each cell as an independent
queue. This literature is reviewed in Section IVbefore a new model based on this assumption is presentedin Section V. From the network operator’s point of view, themodel corresponds to a system where no calls are blockedand no calls are terminated prematurely. From a mathematicalpoint of view, the number of users in each cell becomes aPoisson random variable and the total interference can be mod-eled as a compound Poisson sum. Methods for approximatingtail probabilities associated with these sums are discussed inSection VI. These methods lead to a very efficient techniquefor assessing the reverse link traffic capacity of CDMA cellular
0018–9545/99$10.00 1999 IEEE
154 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
Fig. 1. Standard cellular layout.
networks and for investigating its sensitivity to propagationand system parameters. Numerical examples are presented inSection VII before the paper is concluded in Section VIII.
Before proceeding, it is important to highlight that the ideaof modeling cells in a CDMA network by queues isnot new and has indeed been proposed by several authors asdiscussed in Section IV. The main contribution of this paper isthe concept of working with distribution functions of other-cellinterference rather than just means and variances. This allowsthe Chernoff bound to be employed in the cellular context andprovides an alternative to the Gaussian approximation. We alsostate and prove limit theorems for the case of a random numberof mobiles in the system which demonstrate the asymptoticaccuracy of the Chernoff bound and Gaussian approximation.
II. SYSTEM AND PROPAGATION ISSUES
Throughout this work, we consider the standard uniformhexagonal layout as shown in Fig. 1 with a base station (BS)at the center of every cell. The forward and reverse links usedisjoint frequency bands and can thus be analyzed indepen-dently. We only consider the reverse link as it is generallyaccepted to be the limiting factor in capacity calculations. Inthe sequel, all mention of path loss, signal to interference ratio(SIR), and capacity refers to the reverse link. We also note thatwe are solely concerned with direct-sequence CDMA systems.
Unless otherwise stated, a mobile connects to the BS thatoffers the least path loss at any given time. The chosenBS employs power control to maintain the received signalpower at a constant level. We also assume the system isinterference limited and that background noise is negligible.In real systems, the background noise provides the referencefrom which absolute signal powers are set.
Without loss of generality, we will work with normalizedvalues of distance, power, and interference. In particular, allpower and interference values are normalized to the fixed valueof the target received signal power. Furthermore, all distancesare normalized by the distance between closest BS’s in thenetwork of Fig. 1. Thus, the target received signal power is one(normalized) unit of power and adjacent BS’s are separated byone (normalized) unit of distance.
The simplest model for the mobile radio channel is apropagation loss inversely proportional to the distance between
the transmitter and the receiver raised to an exponent [10],[11]. If the transmitter and receiver are separated byunits,then the received power is given by
(1)
where is the transmit power and and are independentof distance. is a function of carrier frequency, antennaheights, and antenna gains, and we assume it is constant for allpaths between a mobile and a BS.is the path-loss exponent(PLE) which varies with antenna heights and is typically inthe range two–six.
The simple model of (1) is accurate for distances from 1to 20 km with BS antenna heights greater than 30 m andin areas with little terrain profile variation. Thus, the modelis reasonable for conventional cellular systems in flat serviceareas but is not accurate in city microcells which employ smallcells and low antennas.
Empirical results have illustrated that the deviation from (1)is normally distributed on a log–log plot [12, pp. 105–107].The errors are due primarily to variations in terrain contour andto shadowing from buildings. Incorporation of this deviation,commonly calledlognormal shadowing, leads to
(2)
where and are as before and is a zero-mean Gaussianrandom variable with standard deviation typically in therange six to twelve. is now a random variable withlognormal density
where and .The spatial correlation between shadowing random variables
is significant over a distance of several meters [13] givingrise to a local mean over small areas. Another importantpropagation effect is a fast fading about this local mean. Thefast fading is due to the arrival of several replicas of the signalwith varying time delays and is characterized by a Rayleighdistribution for the received signal amplitude. The fading isbasically independent over distances greater than half a carrierwavelength.
In this paper, we do not model multipath fading. It isgenerally assumed that the use of techniques such as inter-leaving, diversity reception and equalization, as well as theemployment of a RAKE receiver, greatly mitigate fast fading.At any rate, it is reasonable to assume that the effects ofthe fast fading are encapsulated in the requirementsof the system. This means that the propagation models usedcenter on distance-driven path loss like (1) and the inclusionof lognormal shadowing as in (2).
III. I NTERFERENCECHARACTERIZATION
A. Introduction and Review
In this introduction, we review several approaches to thecharacterization of interference in cellular CDMA networks.While simulation studies allow a great deal of complexity
EVANS AND EVERITT: ON THE TELETRAFFIC CAPACITY OF CDMA CELLULAR NETWORKS 155
to be included we are solely concerned with analytical andnumerical treatments of the problem.
The first paper to give an analysis of other-cell interferencein spread spectrum mobile systems was [3]. Although theauthors study a frequency hopped system their interferenceanalysis in terms of SIR is applicable to direct-sequence sys-tems. The propagation model of (1) is assumed, which coupledwith perfect power control leads to a simple expression for theinterference at the desired BS from a mobile station (MS) withknown position in the network. The total interference from acell other than the desired one is calculated by integratingthe above interference expression mixed with a continuousand uniform user density over a circular region approximatingan hexagonal cell. An analytic result is only possible forrestricted values of the PLE and so the authors use numericalintegration to calculate the interference levels. The overallother-cell interference results after summing the contributionsfrom all interfering cells apart from the desired one. Thispaper does not deal with the randomness of the user locationsand is equivalent to calculating expected values when eachuser is independently and uniformly distributed over the cellof concern.
In [14], a very similar analysis to the above is presentedwith the exception that the fixing of the PLE at four leads toanalytic expressions for the interference from the circular cells.This is extended to an analytic result for the variance in [15].
An extension of [3] which includes the effects of shadowingand voice activity monitoring is found in [16]. A standardhexagonal cellular layout is assumed with the propagationmodel of (2) that includes lognormal shadowing taken tobe independent on distinct paths. The total interference ata target BS is examined assuming that there are an equalnumber of users per cell () spread evenly and continuouslyover each cell. MS’s are initially assumed to connect tothe BS offering the least path loss. If this BS is the targetthen the interference is the fixed constant power specified bythe power control, otherwise, the interference is a lognormalrandom variable with mean dependent on the position of theMS. To simplify the analysis, an MS decides between theclosest BS (not including the target) and the target BS only.An expression for the interference dependent upon the MSposition is then multiplied by the user density (not a probabilitydensity function) and integrated over the network to give thetotal other-cell interference. This total interference is a randomvariable due to the lognormal shadowing, and in the paperits mean and variance are calculated numerically as functionsof the user density. Approximating the probability densityfunction as a Gaussian, the other-cell interference is fullycharacterized.
An extension of the reverse link analysis of [16] is discussedin [17] and [18]. First, the propagation model is extended totake into consideration the dependence of the shadowing froman MS to different BS’s. Second, rather than choosing betweenthe target BS and the closest BS, an MS can connect to anyof the nearest BS’s. This involves a fairly straightforwardextension of the analysis in [16] although the computationalcomplexity increases considerably to the extent that only meanvalues for the interference are calculated. The results show a
dramatic drop in the mean other-cell interference fromto for typical values of the shadowing variance, whilethe improvement is small for .
In [19], there is no modeling of shadowing, but more de-tailed and accurate versions of (1) are employed. The analysisassumes a circular target cell plus wedge-shaped adjacent cells,this geometry allowing a fairly simple investigation into thesensitivity of other-cell interference to user density profilevariation. Power control errors are not examined and the resultsare all numerical.
In all of the above treatments, the randomness in userlocation within a cell has not been dealt with. Rather, some(usually uniform) continuous user density has been assumedand its product with an interference function integrated overthe network. An alternative approach is to look at the inter-ference as a function of a random position vector as in [20],where the MS location is assumed uniformly distributed overeach cell. In this paper, however, only the mean and varianceof the interference are required since a Gaussian approximationis used. This means the treatment is identical to [16], and itis only because of the slightly different angle taken that it ismentioned here.
Section III-B discusses a novel method of characterizingother-cell interference in CDMA cellular networks. As with theabove work, Rayleigh fading is not studied and perfect powercontrol is assumed. To begin, we work with the propagationmodel of (1). Given uniformly distributed users and circularapproximation of the hexagonal cells, the distribution functionof the interference from a MS in another cell is calculatedanalytically as a function of the PLE and the location of thecell. The calculation of this distribution function compares to[14] and [15], which assume a similar geometry and propa-gation model yet only derive an expression for the mean andvariance respectively for a fixed PLE of four. The results canbe extended to include lognormal shadowing similarly to [16],however, unlike it, an expression for the distribution functionof the interference is constructed. This distribution functionmust be calculated numerically. Our numerical examples dealexclusively with the deterministic path-loss model, however,for completeness the details for the model including lognormalshadowing are given in Appendix II.
B. Deterministic Path Loss
Consider the situation shown in Fig. 2. Note that all coor-dinates and distances are normalized to the distance betweenadjacent BS’s as discussed in Section II. An MS is located at
within a hexagonal cell of the standard two-dimensional(2-D) layout of Fig. 1. The MS is connected to the BSwith coordinates and causes interference to the BS at
. Based on the power control assumptions and (1), the(normalized) interference is
(3)
and we would like to be able to calculate the distributionfunction of the random variable given the jointdistribution function of the random variables and .
156 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
Fig. 2. Interference in 2-D network.
Fig. 3. Interference in 2-D network—approximation.
This is in general a complicated problem, so to simplifymatters let us assume the joint density of and isuniform over the hexagon. Due to the large number of possibleorientations of the hexagonal cell and to the dependence of
and , the analysis remains exceedingly tedious. Theseproblems can be eradicated by approximating the hexagonalcells by circles of (normalized) radiusas shown in Fig. 3.The orientation difficulty clearly vanishes and by havingand
as parameters, a great deal of flexibility results.The derivation is carried out in the Appendix I and leads to
the distribution function of the interference received atfrom an MS that has a uniformly distributed location withinthe circle of radius and center the origin. The distributionfunction is given below
(4)where
and
Fig. 4. Interference: approximation versus simulation.
In all future numerical calculations, is chosen such thatthe areas of the hexagonal cell and approximating circle areequal which gives . Using this value for , wecompare the approximate analytic distribution function withthat obtained via Monte Carlo simulation for the hexagonalcell in Fig. 4. The points shown on the graph are obtainedfrom simulation while the solid lines are the correspondingdistribution functions from (4). As the values ofand PLE arevaried, the analytic approximation remains in good agreementwith simulation.
IV. THE APPROXIMATION
A. Introduction and Review
In this section, we present a technique that allows the tele-traffic capacity of CDMA cellular networks to be estimated.The simplicity of the technique arises from modeling eachcell of the network as an independent queue andconsequently the theory of this section provides no input to theunderstanding of how calls should be admitted to the system.A more advanced network model which does impact on calladmission control schemes is presented in [21] and [22].
We begin with a review of several papers [20], [23]–[25]that employ the approximation and compare them tothe model presented in this paper and in [26]. Note that in thesepapers the generally distributed holding times are replaced byholding times with a negative exponential distribution giving
queues. However, all their results apply in thegeneral case since it is only the stationary distribution of thenumber of mobiles in the system that is used.
The first of the above papers to appear was [23]. The paperlooks only at the limiting reverse link and has as its aimsthe development of a model that deals with variability in thenumber of users per cell, voice activity, and variablerequirements. Concentrating on a single cell (or sector), theauthors assume that no new call requests are denied and assuch model the cell as an queue. The number ofusers in the cell is thus modeled as a random variable with
EVANS AND EVERITT: ON THE TELETRAFFIC CAPACITY OF CDMA CELLULAR NETWORKS 157
a Poisson distribution having mean equal to the cell offeredtraffic. Voice activity is included simply by assuming eachmobile is gated on with probability and off with probability
. Voice activity is thus modeled by a Bernoulli randomvariable .
Based on fitting of empirical data on the receivedvalues required to maintain frame error rates below 1%,the authors model the required by each mobile as alognormal random variable. Since the data involves actualreceived values required for acceptable performance it mustinclude the effects of imperfect power control and varyingpropagation conditions.
With the above aims achieved, the authors essentially defineblocking to occur when the instantaneous requirementsof all users cannot be met. The blocking probability is boundedfor a range of cell offered traffics using a modified Chernoffbound and compared to results from a Gaussian approximationand simulation. Based on this comparison and the relative easewith which it is calculated, the Gaussian approximation is usedin the extension to multiple cells.
The first assumption of the extension is that the number ofusers in each cell remains equal. The second assumption is thatthe users in every outer cell produce a combined interferenceequivalent to users in the inner cell whereis an expectedouter-cell interference fraction obtained from [17]. Acceptingthe assumptions the multiple cell case reduces to the singlecell problem with an equivalent number of active mobiles of
where as before is a Poisson random variable. Theanalysis in the cellular situation is thus a combination of adynamic single cell capacity analysis with the static, multiplecell capacity results of [17].
In [24], a computationally intensive procedure is presentedfor the evaluation of the teletraffic capacity of both forwardand reverse links in a CDMA cellular system. Each cell ismodeled as an independent queue and the qualityof service (QoS) criterion evaluated is the outage probabilityor the probability that the SIR of a link is below a certainthreshold. A uniform hexagonal layout, a uniform density forthe mobile location within each cell, and a propagation modelincluding lognormal shadowing are other features of the modelpresented. The main disadvantage of the approach presented inthis paper is the extreme computational effort required and itis debatable whether the approach is any more valuable thena straight out simulation.
In [25], a teletraffic model of the reverse link is con-sidered. The assumptions include uniform hexagonal layout,equal traffic offered to every cell, uniform density for mobilelocations within cell, two layers of interfering cells considered,deterministic propagation loss only, and perfect power controlof received signal strength.
Despite the initial discussion of a model including a finitenumber of modems, trunk reservation for handovers, andmobility, the subsequent analysis does not allow for mo-bility and assumes traffic levels which reduce the new callblocking probability to a negligible figure. Thus, the systemis actually modeled as a network of independentqueues. Once more the QoS measure concerns the probabilityof the SIR being below a given threshold which with the
assumption that each mobile is received at a fixed powerlevel involves calculating the probability that the interferencegets too large. The total interference is calculated as thesum of inner-cell interference and outer-cell interference. Inline with the assumption, the contribution fromwithin the desired cell is taken as a Poisson random variablewith mean equal to the cell offered traffic. The outer-cellinterference is approximated as a Gaussian random variablewith mean and variance obtained via simulation. A numericalconvolution of the Gaussian and Poisson densities then leadsto the density function for the total interference. The trafficcapacity corresponding to two QoS values is presented as afunction of the PLE of the propagation model.
Many of the assumptions of [20] are as in [25]. Only thereverse link is considered, the QoS is based on a minimum SIRrequirement, perfect power control of received signal strengthis assumed, and each cell acts as an independentqueue. The internal and external interference are again treatedseparately—the internal a Poisson random variable and theexternal a Gaussian random variable. The mean and varianceof the external interference are calculated by analytical andnumerical methods based on the treatment in [16], whichincludes lognormal shadowing in the propagation model. Theblocking or outage probability is then given in the form of aconvolution as in [25].
The analysis of reverse link traffic capacity for CDMAcellular networks developed in this paper and in [26] sharesmany of the features of the above papers. In particular, weemploy the independent queue model for each cell,the service requirement is in terms of SIR, and each mobile ispower controlled to a fixed and equal power.
In the most general development [26], arbitrary networklayouts, user distributions, and traffic profiles are allowed, andlognormal shadowing is included in the propagation model. If,however, a symmetric structure is imposed, the calculation ofthe service measure reduces to evaluating the probability of acompound Poisson sum exceeding a certain threshold. If thepropagation model does not include shadowing, an analyticexpression is available for the distribution of the randomsummands.
The service measure is approximated using a standardGaussian approximation and bounded with the Chernoff boundand results are presented for various propagation environmentsand system bandwidths.
The approach in [20] is closest in spirit to this work,but differs in several aspects. First, [20] treats internal andexternal interference separately thereby requiring a numericalconvolution at the last step. This is avoided in our approachwhere there is no distinction made. Second, they give noanalysis of, or justification for, the Gaussian approximationwhile we prove a central limit result for compound Poissonsums. Third, our analysis is strengthened with the use ofthe Chernoff bound and an illustration of its asymptoticaccuracy. Finally, we present several results that explore howthe service quality varies with the offered traffic per cell,system bandwidth, and PLE. Such results are not given inany of the above papers.
158 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
B. Model Description
In this section, we discuss the assumptions leading to, andjustification for, modeling the CDMA cellular network as acollection of independent queues. The assumptionsof the traffic model we will use are as follows.
• The call initiation processes in each cell are modeled asindependent Poisson streams.
• All arriving calls are accepted into the network andremain in the network for the full call duration (noblocked or dropped calls).
• Call durations are generally distributed and independentof the arrival processes and other holding times.
• Mobility is not modeled, and, thus, the mobile is asso-ciated with the cell of its call initiation for the durationof the call.
The first and third points are standard assumptions fromteletraffic engineering that have been employed for severaldecades to model the stochastic nature of call arrivals totelephone exchanges and their circuit holding times. Thesecond assumption is reasonable for systems operating withCDMA since there is no theoretical hard limit on the numberof quasi-orthogonal codes available to assign to users. From amore practical point of view, it is reasonable to assume thatthere are enough codes available so that the new call blockingprobability is negligible for moderate offered traffic. The finalassumption is a good approximation when the cell size is largecompared to the distance a typical mobile will travel duringa call.
The assumptions imply that each cell of our network be-haves like an independent queue [27]. This is oneof the most basic queueing models and has a particularlysimple form for the steady-state distribution of the numberof active calls. If the mean time between call arrivals is
s and the mean call holding time is s, then thetraffic to the system is Erlangs. Let be therandom variable representing the number of active calls inthe system at steady state. Then,has the Poisson distribu-tion
The independence of each cell in the network implies thatthe joint steady-state distribution for the number of activecalls in each cell is simply a product of Poisson distribu-tions.
C. Inclusion of Voice Activity Effects
Now let us suppose that once a mobile call is connectedto the network the mobile user is ON with probabilityandOFF with probability . This model results when voiceactivity monitoring is included and the subsequent suppressionof transmission by a mobile after voice inactivity is detected.We are now interested not in the number of mobiles connectedto a BS, but in the number of mobiles in a cell that are ON.
Let this number be . We have [20]
which is again Poisson distributed but with reduced trafficload . Thus, the gains from voice activity detection enterthe formulation in a simple multiplicative manner.
Before proceeding to the next section, we make one finalpoint. In the rest of this paper, it is assumed that the trafficoffered to every cell of the network is equal. We emphasizethat this equality applies to the parameters of a stochasticmodel and is distinctly different to assuming an equal staticload in every cell. This along with the infinite, symmetrical,cellular layouts, and uniform user distributions that we haveassumed allows all calculations to be performed for one cell ofthe network only. The extension of this work to asymmetricallayouts, offered traffic, and user distributions is straightforwardfrom a theoretical point of view [26] and is not included here.
V. OUTAGE PROBABILITY : DEFINITION
In this section, we develop a simple expression for a QoSindicator which we call the outage probability. Calculationof the outage probability reduces to the evaluation of theprobability that a compound Poisson random variable exceedsa given threshold. The analysis of such an expression is leftto the following section.
A. Definition
The outage probability is defined as the probability that amobile achieves an insufficient SIR. We recall that a similarperformance measure is called blocking probability in [23],however, we prefer to use the term outage so as not to confusethis performance measure with that related to blocking of newcall requests.
To calculate the outage probability, we must determinethe stationary probability that an arbitrary mobile anywherein the cellular network receives a reverse link SIR that isinsufficient for acceptable QoS. If certain symmetries exist,then will be the same for mobiles at any point in the networkand we may just as well consider calls that are connected toa particular BS. Associate with this target BS and its cell theindex 1.
Because of the standard power control assumptions, mobilesare received at BSwith one unit signal power. We can thuseasily translate the SIR requirement into a constraint on thetotal interference at BS. That is,
(5)
where is a random variable representing the total powerreceived at an arbitrary BS in the network.is a measure ofthecapacityof the CDMA system and is related to the system
EVANS AND EVERITT: ON THE TELETRAFFIC CAPACITY OF CDMA CELLULAR NETWORKS 159
bandwidth ( Hz), the data rate ( bps), and the required bitenergy to interference density ratio by [16]
B. Interference
Suppose that there are cells apart from cell thatgenerate significant interference at the target BS with labels
. Assume that the interference random variables formobiles in cell are independent and identically distributed(iid) with distribution function and the interference randomvariables from different cells are independent. Remember thatfor our model the possible sources of randomness in theinterference include location, shadowing and voice activity.In the example of Section V-C and in the numerical re-sults of Section VII we focus on randomness due to positiononly with voice activity readily incorporated as discussed inSection IV-C.
Given calls in cell , the total power received at BSisgiven by
where is the interference from theth mobile in cell andthe are independent Poisson random variables with mean
.It is readily shown using characteristic functions that
(6)
where is a Poisson random variable with parameterand is a random variable with distribution function
being a finite mixture of the original distribu-tion functions. The symbol indicates equality in distribution.In (6), the interfering cells are combined and the total trafficinto the conglomeration considered.
Combining (6) and (5), we arrive at a simple expression forthe outage probability in the network
(7)
We now present an example to illustrate and clarify the ideasof the last sections.
C. Example
Consider the standard 2-D layout of Fig. 1 withErlangs oftraffic offered to each cell. Assume the mobile locations withineach cell are iid random variables uniformly distributed overeach cell and that the propagation environment is governed by(1). The interference resulting at some target BS from a mobilerandomly located in a cell of the network is characterized bythe approximate distribution function of (4). If only the targetcell and the first two surrounding rings of cells are taken tocontribute significantly to the total interference at the target
BS we have , where is a Poisson randomvariable with mean and the are a sequence of iidrandom variables with distribution function
(8)
In the above, is the unit step function and aregiven in (4).
To calculate the outage probability as a function of theoffered traffic per cell we are faced with evaluating (7). Twomethods of approximating this probability are described in thenext section.
VI. OUTAGE PROBABILITY : APPROXIMATIONS
In this section, we consider techniques for approximating
where are iid random variables and is aPoisson random variable with meanthat is independent ofthe .
Let us define and denote the mean andvariance of by and ,respectively. Then, if and
A. Normal Approximation
1) The Approximation:The normal approximation is
where is a zero-mean unit variance normal randomvariable. We thus have
where
2) Asymptotic Behavior: Integral A:To examine theasymptotic behavior of this approximation as , wefirst assume that takes nonnegative integral values only.Since the sum of Poisson random variables is also Poisson,we have
where are iid random variables and
160 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
The sum with a random number of summands has beentransformed into a standard deterministic sum of iid randomvariables for which we can apply the central limit theorem(CLT) in its simplest form. The CLT [28] states that provided
is finite and is positive and finite
as
The symbol refers to convergence in distribution.3) Asymptotic Behavior: Real A:We now consider the
case when is a positive real. As no appropriate resultcould be found in the literature we prove the required CLThere using characteristic functions. For convenience, we usethe following notation:
so that and .Theorem 1: If is finite and is positive and finite,
then
as
Proof: Let , with
where . This follows by conditioning anduse of elementary properties of characteristic functions.
Because
where as . Thus
Fix so that
by continuity of the exponential at zero.Finally, by the Continuity Theorem for characteristic func-
tions as required.
B. Large Deviations Bound
We now give an upper bound on the outage probability usingthe Chernoff bound. The asymptotic behavior of this bound isdiscussed in the context of elementary large deviations theory.
Consider first the case when takes on positive integervalues and the Poisson sum can be rewritten as the determin-istic sum . The large deviation rate functionis defined by
where is real and
is the log moment generating function (LMGF) of thewhichis related as shown to the LMGF of the .
Provided for all and that is not a boundedrandom variable in the sense that for allfinite and , then from Cramer’s Theorem [29]
(9)
for . Moreover, for all positive integral
The above bound is commonly called the Chernoff bound andis directly applicable for any positive real value of. Toextend the limit result of (9) to the case whenis real ismore involved, but is readily accomplished either by modifieduse of Cramer’s Theorem or by direct application of the morepowerful Gartner–Ellis Theorem [30], [31].
Applying the above to our problem, we have
and from Cramer’s Theorem the bound becomes tight aswith held constant.
VII. N UMERICAL EXAMPLES
In this section, we use our previous results to examinethe traffic performance of our CDMA cellular system. Afterdescribing the network models used, we compare calculatedoutage probabilities by simulation with both the Chernoffbound and Gaussian approximation for some representativecases. The variation in performance with both PLE and systemsize is then investigated.
A. Network Model
The network model used in calculations is as inSection V-C. In particular, the distribution function ofthe interferers is given by (8) with . It shouldbe remembered that this distribution function is implicitlydependent on the PLE.
In what follows, we consider outage probabilities in therange 0.01% to 10% with the offered traffic limits altered toproduce this range for each scenario considered. The two mainparameters we have to vary are the PLE and. The PLE usedlies in the set with four being a typical value forexisting macrocellular systems.takes values inwhich might correspond to systems with dB,
kbps, and and MHz, respectively.
EVANS AND EVERITT: ON THE TELETRAFFIC CAPACITY OF CDMA CELLULAR NETWORKS 161
Fig. 5. Outage probability versus normalized traffic per cell(� = 100;
PLE = 4).
In all plots, the ordinate represents the base 10 logarithmof outage probability while the abscissa corresponds to theoffered traffic per cell divided by . The traffic axis isthus normalized by the size of the system making capacitycomparisons for different straightforward. Simulation pointsare accurate to within plus or minus 20% with 95% confidence.
For a given set of system parameters and an offered trafficvalue, the simulation point is generated directly using aMonte Carlo technique. This involves repeatedly generatinga random (Poisson) number of users for each interfering celland a random location (uniform over each cell) for eachmobile. In each trial, the total interference at the target BS isdetermined from which the outage condition can be checked.The simulated outage probability is then obtained by takingthe ratio of the number of outage events to the total numberof trials.
B. Comparison of Bound and Approximation with Simulation
In Figs. 5 and 6, we compare the Chernoff bound andGaussian approximation to simulation for the 2-D networkwith PLE and , respectively.
The following points are evident.
• The bound overestimates outage probability by about anorder of magnitude in both cases. This translates to underestimating traffic capacity by about 10% in Fig. 5 and15% in Fig. 6.
• The accuracy of the approximation decreases as theoffered traffic, and, thus, the outage probability decreases.The effect is less severe for the largersince in this casewe are effectively summing a larger number of randomvariables and thus getting a better approximation to thetail of the sum.
The above points give some heuristic tips on when theGaussian approximation is reasonable. Clearly, for large valuesof and high-outage probabilities, the approximation is ex-cellent, however, for low values of 20 and or low-outageprobabilities 0.1% the accuracy of the approximation may
Fig. 6. Outage probability versus normalized traffic per cell(� = 20;
PLE = 4).
Fig. 7. Variation of outage probability with PLE (� = 100, Chernoff bound).
deteriorate rapidly. In the latter case, the bound is a muchsafer and more robust technique.
C. Variation of Bound with System Parameters
Figs. 7 and 8 show how the traffic capacity varies withPLE and respectively for the 2-D network. In these plots,the Chernoff bound was used to obtain values for the outageprobability. We make the following points.
• The capacity (for a fixed outage probability) is signifi-cantly reduced as the PLE decreases.
• The economy of scale for systems with largeresults insignificant increases in normalized traffic capacity. Thisis important in comparing narrow-band CDMA (low)to wide-band CDMA (high ).
VIII. C ONCLUSION
In this paper, we have presented an analysis for the reverselink traffic capacity of CDMA cellular networks.
162 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
Fig. 8. Variation of outage probability with� (PLE= 4, Chernoff bound).
Initially, we provided a characterization of other-cell in-terference in CDMA cellular networks that was crucial tothe development of the subsequent traffic analysis. The endproducts were expressions for the distribution functions of theinterference when a mobile’s location is a random variablewithin a cell. These expressions are analytic for the determinis-tic propagation environment, but involve numerical integrationwhen shadowing is introduced.
The remaining analysis and results contributed toward theunderstanding the traffic capabilities of CDMA cellular net-works. The key assumption was that each cell can meaning-fully be modeled as an independent queue. After dis-cussing the justification for and consequences of themodel, an expression for outage probability was developed interms of a compound Poisson random variable. Two techniqueswere then applied to approximate the outage probability alongwith corresponding asymptotic results. The numerical resultsgave an initial estimate of the traffic capacity of CDMAnetworks and demonstrated the sensitivity to propagationparameters and system processing gain.
The primary shortcoming of the preceding analysis is that itprovides no information on how a network operator shouldcontrol call admissions to the network so as to provide amore robust quality of service. This issue is addressed in [21]and [22].
APPENDIX IDERIVATION OF INTERFERENCEDISTRIBUTION FUNCTION
Before proceeding, note that by symmetry it is only neces-sary to consider the upper semicircle in Fig. 3. Furthermore,only the ratio of and is relevant and in this Appendix weset and without loss of generality. The finaldistribution function is readily transformed back in terms ofand by setting . It is also expedient to work in polarcoordinates since the random variablesand defined by
and are then independent.The situation is then as shown in Fig. 3. If the MS is at location
in polar coordinates and is connected to the BS at the
origin, the interference caused at the BS with locationis given by
and the problem now is to calculate the distribution function ofthe random variable where and are independentrandom variables with readily calculated distributions. Theproblem is formalized below.
A. Problem Formulation
Define the nonnegative, real valued functionby
(10)
where and . We will always assume that.
Let and be independent random variables with marginaldistribution functions
(11)
and
(12)
Aim: Find the distribution function of the random variable.
Solution: Fix and define .Lemma 2: is strictly increasing on for all
.Proof:
for
is strictly increasing from to , and stan-dard transformation techniques can thus be applied to calculatethe distribution function of , where is distributed asin (11). In particular, we have
where the inverse function is well defined onbecause of the monotonicity of . It is calculated
by solving
for taking into consideration the allowed values of thevariables involved.
We are thus led to the following lemma.
EVANS AND EVERITT: ON THE TELETRAFFIC CAPACITY OF CDMA CELLULAR NETWORKS 163
Lemma 3:
where
with the provision that when.
We can consider as a distribution function condi-tioned on the value of . That is, . Asimple unconditioning then allows us to write
where
If , then
The integrals involve terms that can be integrated usingelementary techniques. We leave the details to the interestedreader. We are thus led to the following result:
Theorem 4: The distribution function of , whereis defined in (10) and where and are independent
random variables with marginal distributions given in (11) and(12), is given by
(13)
where
and
APPENDIX IIINCLUSION OF LOGNORMAL SHADOWING
In the following, we take as the position vector andas the Euclidean norm. Equation (3) becomes
which represents the interference in the non shadowing envi-ronment to a BS at BS from an MS at connected toa BS at BS .
We are interested in extending the interference results ofSection III-B and Appendix I, which were based on thepropagation model of (1), to include shadowing effects asgiven in (2). With reference to the latter equation, we assume
and are constant over all paths and that the shadowingrandom variables are independent for different paths.
Initially assume that an MS at connects to the closestBS, BS, and suppose we are interested in the subsequentinterference , produced at a target BS, BS. Ifthen the interference is clearly one unit since the MS willconnect to the target BS and be power controlled to one unitsignal power. If , then
(14)
where as the difference of two independent zero-meanGaussian random variables is a zero-mean Gaussian randomvariable with variance .
Suppose, however, that rather than connecting to the closestBS, an MS is linked to the BS offering the least path loss. Itis practically infeasible to allow connection to any BS in alarge network and it is sensible to consider choosing betweenonly the closest. In any case can be chosen so that thereis an arbitrarily high probability of thebestBS belonging tothe closest. This is the approach taken in [17] and [18]although as already mentioned, a fairly complicated analyticaland numerical procedure, results only in mean values for theother cell interference. We prefer the simpler approach of [16]and [20], where the choice is made between the closest BSand the target BS. As most of the other cell interference tothe target BS comes from MS’s near its cell boundaries thisseems a reasonable approximation.
Once more we assume that if the target BS BSis theclosest BS, then the MS connects to it and causes one unitof interference. Failing that and with BSthe closest BS, theinterference produced by an MS atis given by
(15)
164 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999
which is simply a truncated version of (14). One significantadvantage of (15), apart from being a more accurate model ofthe system operation, is that the moment generating function ofthe interference exists only in the truncated case. The momentgenerating function is used for obtaining bounds on sums ofinterferers in later sections.
Given (15) our aim, just as in the last sections, is to allowthe position vector to be a random variable taking valuesin the cell corresponding to the BS at. Denote this randomposition vector and assume it has distribution functiondefined on cell. Our problem then is to find the distributionfunction of , which means calculating the distributionfunction of .
Given the position vector , the distribution function of therandom variable , which we treat as a distributionfunction conditioned on , is
(16)
where
Alternatively, we can view (16) as a distribution functionconditioned on
(17)
In the above, and is the standarddeviation of in (15).
Equation 16 can be unconditioned ongiven
cell(18)
while (17) can be unconditioned on given its distri-bution function
(19)
In the above, comes from (4) (remembering that thisis an approximation and is only valid when is uniformlydistributed) with replaced by .
The final form for the distribution function of isgiven by
(20)
and results because of the operator in (15).In summary, (20) gives the distribution function of the
interference produced at BSfrom one MS with location incell distributed as . It is equivalent to (3) when lognormalshadowing is modeled as above.
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Jamie S. Evans, for a photograph and biography, see this issue, p. 46.
David Everitt , for a photograph and biography, see this issue, p. 46.
