GoS Evaluation forOFDMA CellularNetworksgssst.iitkgp.ac.in/uploads/faculty/File25.pdf · GoS Evaluation for OFDMA Cellular Networks 3 users exceeds the maximum supported in the network

Noname manuscript No.(will be inserted by the editor)

GoS Evaluation for OFDMA Cellular Networks

Subhendu Batabyal · Suvra Sekhar Das

Received: date / Accepted: date

Abstract Analytical methods on grade of service (GoS, quantified by theblocking probability which is denoted as Pb) evaluation in orthogonal fre-quency division multiple access cellular networks for real-time traffic are im-portant for cell-dimensioning. Existing works use simplified spectral efficiency(SE) models that lead to conservative outcomes. For computing cell-wide andcell-average Pb the exact but mathematically intractable Kaufman RobertsAlgorithm (KRA) is indispensable. The Erlang approximation (EA) tractablycomputes only cell-average Pb but ignores bandwidth allocation granularityleading to an error of around 25% in typical scenarios.

In this work, by using an accurate SE model we show a significant re-duction in projected base station (BS) density requirement for maintainingsatisfactory cell-average Pb. We show that cell-average Pb does not completelydepict the GoS situation within a cell since there is significant variation of Pbinside it. Further, we enhance the EA in two ways. Firstly, we improve its accu-racy to within 2% of KRA by accounting for bandwidth allocation granularity.Secondly, we extend it to compute cell-wide Pb like the KRA. The resultingmethod helps to explain properties such as the linearly increasing trend ofregion-wise Pb vs. region-index in case of low bit-rate streaming calls. We alsouse the aforementioned linearly increasing trend to arrive at another alterna-tive to KRA. The development of these two alternatives to KRA highlightscertain interesting properties of region-wise and cell-average Pb as well as the

Subhendu BatabyalG. S. Sanyal School of TelecommunicationsIndian Institute of Technology, KharagpurTel.: +91-801-674-6110Fax: +91-322-228-2266E-mail: [email protected]

Suvra Sekhar DasG. S. Sanyal School of TelecommunicationsIndian Institute of Technology, KharagpurE-mail: [email protected]

2 Subhendu Batabyal, Suvra Sekhar Das

EA. The error in estimated BS density requirement with these alternatives incase of 12.2 kbps voice in the interference limited situation is negligibly small.We employ discrete event simulations to validate our models.

1 Introduction

The cellular systems such as GSM [1], cdmaOne etc. are designed to carryvoice. Such traffic is real-time and delay sensitive. The radio resource alloca-tion (RRA) entity in such systems allocates fixed amount of radio resourceunits (RRUs) in terms of time and bandwidth to voice users during their call.The underlying technology is entirely circuit switched. Dedicated resourcesare allocated for the full duration of the real-time call, even when voice in-formation is not being generated. Such dedicated resource allocation is ofteninefficient and wastes network resources. Real-time traffic is characterized withcall arrival rate and call holding time1. The product of the mean arrival rateand mean holding time is defined as the network load in Erlangs. In this con-text the capacity of 2nd and 3rd Generation circuit switched cellular systemsis defined as the maximum offered traffic intensity in Erlangs for which theblocking probability (denoted by Pb), which quantifies the Grade of Service(GoS), is less than a target threshold.

Modern (4G) cellular networks use Orthogonal Frequency Division Multi-ple Access (OFDMA) as their dominant transmission scheme [2,3]. Examplesof such networks are WiMAX and 3GPP LTE. In contrast to 2G/3G systemsthese networks are packet switched and are designed primarily with the goalto carry best effort traffic which is delay tolerant. The capacity of these net-works for carrying data traffic is measured in terms of area spectral efficiency(bits/sec/Hz/m2). Real time traffic such as voice and video are also carriedas IP packets in such networks. Such all IP packet switching is supposed toimprove the overall efficiency of these networks.

Unlike data traffic, real-time traffic does not fit seamlessly into such packetswitched networks since it is delay sensitive. However, in terms of percentagetraffic volume, real-time traffic is expected to be predominant in such networks[4]. Such predominance is mainly due to a large number of subscribers usingreal-time services such as streaming video (e.g. for watching their favoritesports event live on the mobile phone etc.). The question that naturally arisesis how to analyze the performance of real-time traffic in these modern packetswitched all IP networks. The key performance indicators of such traffic inthese networks include capacity (number of supportable users or supportableErlang load), outage rates (packet error rate from link perspective as well ascall dropping and blocking rates from system perspective) and delay or delayvariation.

In 4G networks the VoIP capacity is defined in terms of the maximumnumber of users that can be accommodated in the system while ensuring thatat least 95% of the users are satisfied [2], where a satisfied user is one whosuccessfully receives at least 98% of the transmitted packets. If the number of

GoS Evaluation for OFDMA Cellular Networks 3

users exceeds the maximum supported in the network there is a catastrophicfall in QoS [5]. To prevent such a fall there is a need for restricting the numberof simultaneously ongoing calls. Other than an upper limit on the number ofsimultaneously ongoing calls there is also a limitation on the maximum sup-portable Erlang load. In fact for cell dimensioning and call admission controlErlang capacity is still the key performance indicator for real-time traffic insuch networks. Erlang load analysis can lead to estimating supported numberof users in a cell (eNodeB / base station). The Erlang capacity per cell is foundbased on the maximum Erlang load that can be sustained while maintainingthe cell-average Pb below an upper limit. This capacity value when divided bythe offered Erlang traffic per user [6] yields the number of users supportable percell. This is essential information for call admission control (CAC) to maintaingood QoS of calls. This also determines the number of eNodeBs or base sta-tions (BSs) to be rolled out per unit area as per the projected subscriber base.This is especially important in case of 4G networks as the number of radioresource units allocated to users is dynamic in nature unlike GSM/cdmaOnesystems. Therefore we must use the measure of Erlang to define the capacityof a 4G network for carrying real-time traffic. Hence one of the items of ourfocus is to evaluate the Erlang capacity of OFDMA based cellular networksfor real time services. In fact all our focus areas center around evaluation ofPb for real-time traffic in cellular networks.

1.1 State of the Art

In order to arrive at realistic Erlang capacity estimates the usage of an accuratespectral efficiency (SE) model is of critical importance [7]. All SE models areprimarily based on the well known channel capacity formula by Shannon [8],which places an upper bound on the SE achievable at a given signal to noiseratio (SNR) - the SE of practical systems being always less than this upperbound. Such reduction in SE is partly due to the transmission of pilot symbols,control information, and training sequences etc. Other than this the primaryreason for reduced SE in real-world systems is the performance limitations ofreal world forward error correction (FEC) mechanisms. This limitation resultsin a SNR penalty that reduces the effective SNR and creates a performancegap from Shannon limit. The SNR penalty is usually modeled as a constantterm in the context of Erlang capacity analysis [9]. In reality, however, theSNR penalty varies significantly with average SNR [10]. The reason for suchvariation is the dependence of FEC performance on the coding rate and codeblock length which are selected differently at different SNRs. Ignoring suchvariation leads to significant inaccuracy in Erlang capacity and Pb studies. Inthis context we show that the required BS density for a given traffic load goesdown significantly (by a factor of 3 or more) when an improved SE model isused for mapping SNR to SE.

Existing works on Erlang capacity analysis [9,11–13] show cellular capac-ity from a GoS point of view, and study the cell-average blocking probability.


They treat the entire cell as a single group which experiences the cell-averagePb. This treatment is despite the fact that there are multiple classes of usersfor the same bit-rate traffic requiring distinct number of subcarriers for a call.All users belonging to a class require the same number of subcarriers to makea call by virtue of the range within which the signal to interference plus noiseratio (SINR) experienced by these users belongs. The lower the SINR range,the higher is the corresponding number of required subcarriers. The KaufmanRoberts Algorithm2 (KRA) [9] yields one value of Pb per user class therebyconcretely quantifying the varying Pb within a cell. The Erlang approxima-tion (EA) method is an alternative to KRA which computes the cell-averagePb. This EA does not take into account frequency allocation granularity asa result of which the GoS evaluated by it is rendered inaccurate. In spite ofits inaccuracy the EA is important as it involves an explicit mathematicalexpression which is analytically tractable unlike the recursive KRA. Hencewe consider accounting for frequency allocation granularity for an improvedmethod of cell-average Pb computation.

However, cell-average Pb does not yield a comprehensive view of GoS. Eventhough present Erlang capacity analysis and cell-dimensioning ignore the vari-ation of Pb within a cell, it is important to quantify such variations [7,9]. Ifwe visualize subcarriers as servers, then it is evident that calls requiring morenumber of subcarriers will face a higher blocking probability due to a reducedvalue of the effective number of servers in their case. For example, users atthe cell-edge experience a stronger interference and weak desired signal. As aresult, they require a higher number of subcarriers compared to those at thecell center. This in turn leads to a higher Pb at the cell-edge. The consequentvariation of Pb within the cell is detrimental to user satisfaction. Now-a-dayscellular network users expect a consistent network QoS throughout the cellwhich, however, is violated by the varying Pb. It is therefore important tostudy cell-wide Pb in a comprehensive manner. As discussed before, the KRAestimates cell-wide Pb values, but is intractable. In this context, we show thatit is possible to compute cell-wide Pb with reasonable accuracy by algorithmsother than KRA [14,15]. Such algorithms have more explicit mathematical ex-pressions and are vital for a deeper perspective into intra-cell Pb variations. Wepresent two such algorithms both of which employ the Erlang approximation.The development of these algorithms enables us to explore some interestingproperties of the Erlang approximation as well as cell-wide and cell-averagePb.

Earlier works have not addressed these topics. The works [9,11,16] buildan analytical framework for evaluating the Erlang capacity of OFDMA cel-lular networks but ignore the intra-cellular variation of Pb and the effect ofshadowing. The SE model used in these works is inaccurate considering theSE performance of real-world systems such as LTE [10]. This leads to highlyinaccurate Erlang capacity estimates. The work [12] analyzes the Erlang ca-pacity of OFDMA networks in presence of shadowing but completely ignoresboth fading as well as the performance gap from Shannon limit of real-worldFEC. It also ignores the intra-cell variation of Pb. The work [13] ignores the


variation of the resource requirement for a call from various regions of the celland uses the approximate Erlang ‘B’ formula [17] instead of the exact KRA.It models interference with only a single tier of interfering cells. It also ignoresthe effect of fading as well as the performance gap from Shannon limit. Noneof these works has dealt with the topic of analytically tractable computation[9] of cell-wide as well as cell-average GoS.

Considering the gaps identified in the state of the art presented in thissection, we first focus on the analytical evaluation of GoS in case of downlinkOFDMA with an accurate SE model. The required BS density for a giventraffic load is shown to go down significantly when such a model is applied.We also show that it is possible to improve the accuracy of the Erlang approx-imation substantially by taking into consideration the bandwidth allocationgranularity. With such consideration the error in estimated cell-average Pbgoes down from 25% to 2% in typical scenarios. After this we show that thecell-average Pb does not depict the GoS situation completely since there is sig-nificant variation of Pb within the cell. We then present two mathematicallytractable alternatives to KRA for class-wise and cell-average Pb evaluation.The error in evaluated BS density with these methods in case of voice in theinterference limited situation is negligibly small. In case of 64 kbps video whereonly one of these methods may be applied the estimation error is approx 2.3%in the interference limited situation. We simultaneously study the propertiesof cell-wide Pb such as the linearly increasing property of class-wise Pb as afunction of class-index. Our GoS studies also employ discrete event simulations(DES) for validation of our models.

1.2 Organization of the paper

The rest of the paper is organized as follows. Section 2 describes the systemmodel. Section 3 first sets up the background for improved GoS evaluation anddimensioning in downlink of OFDMA cellular networks and then presents im-proved results considering a more accurate SE model and an enhanced Erlangapproximation. Section 4 presents two mathematically tractable alternativesto KRA for cell-wide and cell-average GoS evaluation. Finally Section 5 con-cludes the paper.

2 System Model

In this section we describe the system model which has been used for the GoSevaluation of streaming traffic. In the OFDMA system under consideration,let there be Nsc subcarriers over a system bandwidth W Hz. The subcar-rier bandwidth is given by ∆fsc = W

Nsc, where the abbreviated subscript ‘sc’

indicates ‘subcarrier’. Let D denote the set of BSs. Let d ∈ D denote thedesired BS whose capacity and coverage are being analyzed with a focus onthe GoS of real-time (streaming) traffic. Let d

′

∈ D \ d denote any BS which


adds cochannel interference to the desired link. Let u → d denote an user inthe network attached to BS d. Each BS d has an omnidirectional antenna.The total transmit power of BS d is PTd while the signal component of thetotal power is represented as PSd . We denote the system noise power overbandwidth W as PN . We consider log-normal shadowing on top of a sim-ple path-loss model to constitute large scale fading. The effect of small scalefading is implicitly accounted for in the SINR to SE mapping described here-after. Let k (the multiplicative component) and np (the path-loss exponent) bethe parameters constituting the path-loss model. Given a zero-mean Gaussiandistributed random variable (RV) ξ, let eξ represent the random log-normalshadowing coefficient. The wideband average SINR of user u at distance ru,dfrom BS d is given by:

γu =PSd e

ξu,d(kru,d)−np

PN +∑

d′ 6=d P

Td′ eξu,d

′

(kru,d′ )−np

. (1)

In OFDMA systems various radio resource allocation (RRA) algorithmsare known[18]. For real time traffic such as VoIP, which is the main focus ofthis work, semi-persistent (SP) scheduling has been found to be most suitable[5]. In SP scheduling, the amount of radio resource allocated depends on thewide-band SINR of the user [2,5]. Other than SP scheduling there is dynamicscheduling which allocates resources based on the instantaneous sub-band spe-cific SINR of the users. The dynamic scheduler takes into account time vary-ing channel conditions based on which it allocates variable amount of radioresources. One of the possible objectives of such allocation is maximization ofnumber of supportable users; another possible objective is to maximize systemthroughput. In the process of such allocation it is possible to select the bestsubchannel for each transmission based on user specific channel conditions.In case of SP scheduling a fixed amount of bandwidth is required to transmitvoice packets for a given bit-rate depending on the wideband SINR situation.Thus, given the wideband SINR, there is a direct mapping into the bandwidthrequired. On the other hand, in case of dynamic scheduling the bandwidthrequirement of a VoIP packet may vary from one scheduling instant to thenext depending on the channel conditions and the requirements of other en-queued packets which simultaneously seek to be scheduled. However, it hasbeen found through exhaustive simulations [5] that given a particular traffictype (e.g. VoIP), area SINR distribution of users and a scheduling algorithmit is possible to find the average bandwidth requirement as a function of thewideband average SINR of the users even for dynamic scheduling. It has beenverified by applying the mapping to estimate the capacity of the cell for VoIPtraffic for the given scheduler. The function mapping wideband SINR to av-erage bandwidth requirement varies depending on the traffic type, the areaSINR distribution and the specific scheduling algorithm chosen. The averagebandwidth requirement thus computed then aids in calculating other relatedquantities such as maximum number of supportable users over a given systembandwidth etc. Thus with appropriate scheduler characterization it is possibleto easily apply this work to any scheduling algorithm.


We denote the bandwidth allocated to user u as wu, and the achievablespectral efficiency in case of such an user as ϕu. For a bit rate requirement ofνbu bits per second (bps), the number of subcarriers required to be allocatedare

N reqscu

=

⌈

νbuϕu

NscW

⌉

, (2)

wherein ϕu = bϕ log2

(

1 +γu

aϕ(γu)

)

, (3)

in which bϕ and aϕ represent the link bandwidth efficiency and the SINRimplementation efficiency factors [10] respectively which together model thegap from Shannon limit of real-world systems such as LTE. Here aϕ, which is afunction of SINR (γu), models the combined effect of the performance limits ofreal-world Forward Error Correction (FEC) and (small-scale) fading. Karray’smodel of these factors works by increasing νbu to 180 kbps instead of 12.2 kbpsand uses the original Shannon formula, which renders the model inaccurate.The variation of aϕ as a function of SINR (γ), which is ignored by Karray, isillustrated in Fig. 2 of [10]. It can be observed from this Fig. that aϕ lies withinthe range 4-10 dB for most SINR values. The model for mapping SINR to SEgiven by (3) along with the SINR to aϕ mapping labeled as ‘SISO TU’ in Fig.2 of [10] constitute the accurate SE model that we consider in this work.

Two important constraints as in [9] are required to be satisfied for theaforementioned allocation of N req

scusubcarriers to be made. The RA policy

must satisfy (a) the information theoretic constraint given by :

νbu ≤ wuϕu ∀u→ d (4)

and (b) the maximum bandwidth constraint given by:∑

u→d

wu ≤W. (5)

The set up for Erlang capacity evaluation assumes that the desired BSd receives streaming calls whose inter-arrival times are i.i.d exponential RVswith rate λ (mean 1

λ). The duration of a call has mean 1

µ, the Erlang load

β being given by λµ. We note here that the queuing theoretic model that has

been considered - including details of the call arrival process (whose rate is λ)and the call service process (whose rate is µ) - is same as in [9,19]. The discreteevent simulator generates call arrival events and processes calls as soon as theyarrive in the first come first served (FCFS) fashion.

Finally we note that the usage of a constant effective bit-rate for eachtype of streaming calls as in [9,11,12] is possible even when the calls are tobe processed as intermittent packet streams as in 4G cellular networks [2,20].This is possible because packet switched streaming calls can be classified by

their equivalent bit-rate νb [21,22]. Henceforth in our analyses the specifiedbit-rate (12.2 kbps, 64 kbps etc.) always indicates the considered value of the

effective bit-rate νb.


3 GoS in downlink OFDMA

Given a single class of traffic (which is not the case for cellular networks butuseful as an approximation and included for completeness), and further givenNsrv servers with a policy that an arriving call which does not find a free serveris instantaneously blocked, we have the Erlang loss model to compute blockingprobability as a function of Erlang load. In queuing theory, the Erlang load oroffered load (symbolized by β) is defined as the product of the mean arrivalrate of calls into the system (symbolized by λ) and the reciprocal of the meanservice rate (symbolized by 1

µ), the latter being the average time required

by a call to be completed in the system. The unit of offered load is Erlangs,a dimensionless measure. To compute the blocking probability as a functionof offered load in such a system we may use the Erlang B formula [17]. LetψNsrv

(β, i) represent the Erlang loss formula for computing the probability ofi occupied servers out of a total of Nsrv available servers given a load β; herewe may write:

ψNsrv(β, i) =

βi

i!∑Nsrv

l=0βl

l!

. (6)

Sometimes we will use a more compact notation ψ(β, i) for the Erlang Lossfunction omitting the parameter Nsrv which will, however, be implied. Wealso note that when Nsrv ≫ 1, ψ(β, i) ≈ e−ββi/i! which is independent of theexact value of Nsrv. The Erlang B formula for computing Pb is simply givenby substituting i = Nsrv in ψ(β, i) (6):

Pb(β,Nsrv) = ψNsrv(β,Nsrv) =

βNsrv

Nsrv !∑Nsrv

l=0βl

l!

. (7)

In a cellular system it is possible to divide the incoming calls into classesbased on subcarrier requirement such that calls from each class require a dis-tinct number of subcarriers. These classes are referred to as SINR classes.They are also referred to as SINR regions based on their spatial source withina cell. Here the proportion of users belonging to each class will be referred toas the class-wise user density and symbolized by ζl for the l th SINR class.Similarly the subcarrier requirement of each class will be referred to as theclass-wise subcarrier requirement and symbolized by N req

scl. Let Nocc

sc denotethe the total number of occupied subcarriers. Let q(n) denote the probabilityPr(Nocc

sc = n), i.e. the probability of occupancy of n subcarriers. To analyzeblocking probability in such a situation where multiple classes of traffic exist(even for the same bit-rate requirement) (7) cannot be applied (as is) sincethe Erlang Loss model assumes a single class of arriving calls. In such a casethe Kaufman Roberts recursion [14] to evaluate q(n) efficiently is given by

nq(n) =M∑

l=1

βζlNreqscl

q(n−N reqscl

), (8)


whereM is the number of SINR classes and ζl, Nreqscl

the class-wise user densityand subcarrier requirement respectively. Also βl = βζl gives a more compactnotation by denoting class-wise offered load. After computing q(n) through(8), the region-wise Pb are given by:

Pb(l) = 1−

Nsc−Nreqscl

∑

n=0

q(n)

=

Nsc∑

n=Nsc−Nreqscl

+1

q(n) ∀l ∈ 1, 2, · · · ,M, (9)

and the cell-average Pb by:

Pb =M∑

l=1

ζlPb(l). (10)

In this section, we discuss two improvements to the results from analysis ofQoS for real-time traffic in downlink OFDMA cellular networks. The firstresult is related to the Erlang approximation for which a method to increaseits accuracy is discussed; the second is related to the improved SE modelconsidered in this paper and its impact on predicted Erlang capacity.

The Erlang Approximation method as in [9] first computes the cell-average

bandwidth requirement wu of wu =νbu

ϕuand then computes the number of

available logical channels (servers) as:

Nsrv =

⌊

W

wu

⌋

, (11)

W being the system bandwidth. Finally, the Erlang loss formula for a queuewith Nsrv logical channels is used to compute (approximate) Pb at given loadβ. Thus Pb is given by:

Pb ≈ ψ(β,Nsrv), (12)

where the function ψ(β, i) is defined as in (6). This method ignores the band-width allocation granularity which must be accounted for in order to let theapproximation work with sufficient accuracy. Our suggested modification tothis method works by first computing the average subcarrier requirement fora call as N req

sc =∑

lNreqscl

ζl and then the effective number of servers based onthis by:

N′

srv =

⌊

Nsc

N reqsc

⌋

. (13)

We note that in computing the number of servers we have used the ‘floor’operation instead of ‘round’ or ‘ceil’. This is significant as the Erlang Ap-proximation in general underestimates Pb. After this modified initial step theErlang B formula (12),(6) is used as before for computing the cell-average ofPb (Pb) from the Erlang load β and the number of available servers N

′

srv. This


modified approach to the Erlang approximation improves its accuracy as willbe highlighted by some of the results in Section 3.1. We refer to this modifiedapproach as the improved Erlang approximation (IEA).

The second improvement considered in this section is the result of usingthe more accurate spectral efficiency model described earlier (3). Karray [9]uses the approach of increasing the effective bit rate from νb1 = 12.2 kbps toνb2 = 180 kbps in order to model a flat SINR gap from Shannon (aϕ) at allSINRs. In terms of the Shannon formula, the impact is given by

νb1 =W log2(1 + γu1) (14)

and

νb2 =W log2(1 + γu2) (15)

where γu1is the effective SINR and γu2

the required (raw) SINR. From thisit can be shown that

aϕ = γu2/γu1

=2

νb2

W − 1

2νb1

W − 1. (16)

Given W = 5 MHz, this implies that a flat SINR scaling factor aϕ of ≈ 11.7dB is chosen by Karray for all SINRs, which is very high (see Fig. 2 in [10])and likely to result in pessimistic estimates of Erlang capacity, as we shallobserve in the following subsection.

3.1 Performance Analysis of IEA and Dimensioning with Improved SEModels

In Table 1 we list some important parameter values [9,23] used for performanceevaluation. In the table, σξ indicates standard deviation of the normal RVξ capturing the effect of log-normal shadowing which appears in (1). Theparameter Ξaa denotes inter-site shadowing correlation. Pbmax denotes themaximum allowed value of Pb.

An important parameter that is needed for simulations as well as for theanalysis of Pb and its variations (which we perform subsequently) is the region-wise user density array ζ. The parameter ζ may be evaluated simply by com-puting the number of subcarriers required for a streaming call on the basis of(2) at various randomly chosen points in the cell and then classifying theminto regions with specific subcarrier requirement. One observation that mayaid such classification is that in general for streaming traffic with low bit-ratein the interference limited situation

N reqscl

= l. (17)

Thus ζl in such cases is simply the proportion of users that require l subcarriersto make a call. We further note that in the interference limited situation for


Table 1 Simulation Parameters

Parameter(s) Chosen Value

νb,W 12.2 kbps, 5 MHzbϕ, aϕ(γ) 0.83, TU SISO (Fig. 2 in [10])

Pbmax,Pamin 0.02(2%), 0.98(98%)Nsc 128 as in [9], or 512 as in LTE

PTd 52 dBm (including 9 dB antenna gain)PN −103 dBm

Frequency Reuse Factor 1Cell Layout Regular Hexagonal Layout (19 cell layout)

Cell Radius (R) 200 m (default)σξ, Ξaa 4 dB, 0.5

Correlation distance 10 mk, np 8.667 m−1, 3.38 [9]

12.2 kbps streaming rate, under the TU-SISO channel, ζ without shadowing,which we denote as ζ, has M = 18 elements and is given by

ζ(1 : 6) = [0.1470 0.2452 0.1105 0.0831 0.0704 0.0608],

ζ(7 : 12) = [0.0557 0.0462 0.0435 0.0405 0.0376 0.0244],

ζ(13 : 18) = [0.0149 0.0087 0.0059 0.0038 0.0016 0.0003].

(18)

With shadowing ζ may have up to M′

= 24 elements. The average of ζ withshadowing, which we denote as ζ is given in the same situation (12.2 kbps ratein the interference limited situation under TU-SISO channel) by

ζ(1 : 6) = [0.1385 0.2392 0.1051 0.0826 0.0673 0.0583],

ζ(7 : 12) = [0.0521 0.0472 0.0425 0.0382 0.0326 0.0264],

ζ(13 : 18) = [0.0209 0.0156 0.0114 0.0080 0.0056 0.0036],

ζ(19 : 24) = [0.0022 0.0013 0.0007 0.0004 0.0002 0.0001].

(19)

Fig. 1 captures ζ and ζ in plot form.Now we present results related to performance analysis of the IEA as well as

those related to dimensioning with the improved SE model (3). First few resultsare generated as per evaluation condition in [9] for the sake of benchmarkingour simulator; Nsc = 128 is assumed. These evaluation conditions are referredto by the term ‘Karray model’ and include considering 180 kbps as the bit-rate requirement for voice calls (instead of the actual 12.2 kbps) and usage ofthe original Shannon formula in SE evaluation. As stated earlier, this impliesconsidering a flat gap of 11.7 dB from the Shannon limit at all SINRs (16).Another attribute of the ‘Karray model’ is that it ignores shadowing. Fig. 2accordingly shows the variation of Pb vs. offered load (Erlang) for differentcases. Curve ‘A’ is generated using KRA and shows a capacity of 24 Erlangs


0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

Region index (l)

ζ l

Without ShadowingWith Shadowing

Fig. 1 Region-wise user density with and without shadowing in the interference limitedsituation (R = 200 m).

(denoted as βmax) at 2%Pb. Curve ‘B’ represents the results using EA methodas described by (12) in [9]. It is found to be away from ‘A’ by about 25%.On the other hand curve ‘C’, which is obtained using IEA (12),(6),(13) isfound to be within 2% of ‘A’. Hence it clearly shows that IEA is a far superiorapproximation compared to EA. The curve labeled ‘D’ is obtained using KRAbut SE model is as described in this work (3). It shows that results in [9] arepessimistic by as much as 64%.

20 30 40 50 60 70 800

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

β (Erlangs)

E[P

b]

KRA + Karray model (A)

EA + Karray model (B)

IEA + Karray model (C)

KRA +Improved SE model (D)

A

B

C

D

Fig. 2 Variation of Pb with Erlang load in absence of shadowing.


Now we include shadowing and improved SE model (3) in the analysisdescribed earlier. Fig. 3 shows the variation of Pb with Erlang load in the newset up in which we assume Nsc = 512 as per LTE specifications for 5 MHzbandwidth. Again IEA (‘F’) is found to closely match the result obtainedusing KRA (‘E’), while the one obtained using EA (‘G’) is off by 25%. Wecan further note (by comparing with Fig. 2) that by using proper models,the capacity estimates are much higher (≈ 85 Erlangs) than the 24 Erlangsobtained in [9].

65 70 75 80 85 90 95 100 1050

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

β (Erlangs)

E[P

b]

IEA (F)

KRA (E)

EA (G)

E

F

G

Fig. 3 Variation of Pb with Erlang load considering improved SE model in presence ofshadowing.

Fig. 4 shows the variation of Erlang capacity with cell radiusR. Of the threecurves the one using ‘Karray model’ shows the lowest Erlang capacity (less

than 13

rdof the other two). The other two curves both employ the improved

SE model (3), one of them considering shadowing and the other ignoring it.Both these curves show a 200% or higher increase in Erlang capacity comparedto the first one. It is further observed that all of these three curves have aconsistent nature. Each may be split into three regions - the first ‘flat’ region,the second or middle sloping region and the third or last sloping region. Theslope of the third or last region is the maximum out of the three while thefirst region (R < 1.5 - 2 km) is almost flat with zero slope. The flat portionrepresents the interference limited situation where a change in cell radius doesnot change the SINR distribution or the Erlang capacity. Comparing the twocurves employing the improved SE model it is observed that shadowing with4 dB standard deviation impacts Erlang capacity by ≈ 10%.

Fig. 5 shows the effect of the usage of proper models on cell dimensioning. Itcan be seen that a three-fold reduction in BS density requirement is predictedby the improved SE model. This result can heavily influence cell dimensioning


100

10

20

30

40

50

60

70

80

90

100

Cell Radius (R) in Km

Erla

ng

Ca

pa

city

(β m

ax)

No shadowing and improved SE

With shadowing and improved SE

No shadowing and Karray model

Second ’sloping’ region

First ’flat’ region

Third ’sloping’region

Fig. 4 Variation of Erlang capacity with cell radius given Nsc = 512 subcarriers.

100

102

10−2

10−1

100

Traffic (Erlang/km 2)

No

. o

f B

Ss

(pe

r km

2)

No shadowing and Karray model

With shadowing and improved SE

No shadowing and improved SE

Fig. 5 Dimensioning for 12.2 kbps voice.

by significantly reducing the associated capital expenditure and opening up thepossibility of higher operational profit. This also influences the cost-per-bit ofthe rolled out network and hence billing which in turn affects user satisfactionsince users enjoy good QoS at a lesser price. Overall this result has hugeimplication on investment planning or service provisioning for network serviceproviders.


4 Alternatives to KRA

For computing the region-wise Pb or user group-wise Pb the KRA algorithm isvital. However, the KRA uses a recursive procedure and lacks a more explicitmathematical expression. In this sense the KRA is like a black-box that acceptsthe region-wise user density and subcarrier requirement as well as the overallcell load as inputs (8) and generates region-wise Pb (9) and Pb (10) as outputs.This is a disadvantage since the network designer may need a closed form andexplicit mathematical representation aiding pen-and-paper analysis to tweakparameters to influence cell-wide GoS rather than only be able to compute theregion-wise and cell-average GoS through a computer program for a given setof inputs (network parameters). In this context we introduce two fast, non-recursive and analytically tractable methods for computing region-wise as wellas cell-average Pb. The development of these methods enables us to exploresome interesting properties of SINR class-wise and cell-average GoS as well asof the Erlang approximation. Such properties are not evident from the analysisby KRA alone. The results are validated using DES method.

4.1 Analysis of SINR class-wise variation of Pb

The KRA (8) requires as its inputs the SINR class wise user density (i.e. thetraffic density distribution) and subcarrier requirement as well as the Erlangload. With these inputs it is able to compute the class wise Pb (9) which maythen be used to compute the cell-average Pb (10). Instead of looking only atthe final output of KRA (which is the cell-average Pb), we also investigate theintermediate region-wise output. In absence of shadowing, each SINR classoccupies an annular ring-like region on the surface of the cell over which Pb isconstant and the ring-wise Pb increases with distance of the ring from the BS.

Fig. 6 shows the Pb as a function of distance for 12.2 kbps for multipleErlang loads in the vicinity of peak load (Pb = 0.02, βmax ≈ 93.5 Erlangs).The β values considered are 92, 94 and 96 Erlangs; Nsc is chosen as 512. Theregion-wise Pb values are obtained from KRA and the region-specific radialdistance range from a classification of users on the basis of their subcarrierrequirement. For every class of users the maximum and minimum radial dis-tance are selected. It is observed from such a plot that the Pb increases (in astaircase manner) with distance, and that the Pb at the cell-edge is the highestwithin the cell. While moving away from the cell center, as long as the subcar-rier requirement remains constant, so does the Pb. As soon as the subcarrierrequirement increases, the effective number of servers (13) reduces and hencethe Pb increases to a new level in a stepwise manner.

Fig. 7 shows Pb as a function of distance for different streaming rates (12.2,64 and 256 kbps) considering Nsc = 128 total subcarriers. In case of higherrate streaming traffic, specifically 256 kbps, the cell-edge Pb is as high as 25%(more than 12 times the commonly accepted upper limit of 2%) which mayresult in very low level of user satisfaction at these regions for such rates.


40 50 60 70 80 90 100 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Distance between BS and MS

Blo

ckin

g P

rob

ab

ility

(P

b)

β = 94

β = 92

β = 96

Fig. 6 Variation of Pb with distance between MS and BS in case of 12.2kbps voice for Inter-site Distance of 200m (cell radius R ≈ 115.5m) evaluated by KRA. The Pb correspondingto β = 94 is ≈ 0.02.

100 200 300 400 5000

5

10

15

20

25

Distance of MS from BS (m)

Pb

(%

)

r = 12.2 kbps ( βmax = 66.6)

r = 64 kbps ( βmax = 11.7)

r = 256 kbps ( βmax = .9)

Fig. 7 Variation of Pb with distance between MS and BS in case of 12.2, 64 and 256kbps streaming traffic for cell radius of 500m evaluated by KRA. For all of the three plotsPb = 2%.

It is also meaningful to consider the class-wise Pb as a function of class-index as shown in Fig. 8 for 12.2 kbps voice without shadowing at variousErlang loads (β = 70 to 100 Erlangs). A near-linear variation of Pb is ob-served across classes. For the case with shadowing the class-wise Pb variesfrom realization to realization. Hence the quantity to be studied is the averageclass-wise Pb where the average is computed over several independent realiza-tions of shadowing. Fig. 9 considers this average class-wise Pb with shadowingfor 12.2 kbps voice at various Erlang loads (β = 70 to 100 Erlangs) and showsthat this quantity too, varies in a near-linear fashion vs. class-index. The lin-


ear variation of Pb vs. class index both with and without shadowing will beexplained later on in this section.

5 10 15

0.02

0.04

0.06

0.08

0.1

0.12

0.14

SINR Class (l)

Cla

ss−

wis

e P

b: P

b(l)

β = 70

β = 80

β = 90

β = 100

Fig. 8 Region/class-wise Pb variation in absence of shadowing for cell-radius of 200m incase of 12.2 kbps voice-calls.

5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SINR class (l)

Ave

rage

Pb c

lass

−wis

e : (

E[P

b(l)

])

β = 70

β = 80

β = 90

β = 100

Fig. 9 Region/class-wise Pb variation in presence of shadowing for cell-radius of 200min case of 12.2 kbps voice-calls. The plot shows the averages of class-wise Pb over severalinstances of shadowing.


4.2 Analytically tractable GoS evaluation

In the previous section we have established the need for a re-look at theSE models used in analytical GoS evaluation and dimensioning for modernOFDMA networks. Furthermore we have shown that the EA can be improvedto match its results closely with that of KRA. This is very useful as it gives usa tool that enables easy calculation of first cut Erlang capacity with accuracyand analytical tractability. With such analytically tractable GoS model we areable to make predictions such as a single system with 2Nsc subcarriers is bet-ter than two separate identical systems with Nsc subcarriers each by virtue ofthe fact that:

ψ(2β, 2Nsrv) < ψ(β,Nsrv) (20)

even though in the first case the single system needs to handle twice the loadas each of the individual systems in the second. This is important since itprovides a motivation for spectrum sharing among operators. Instead of us-ing independent (non-overlapping) spectrum and competing with each other(two independent systems with the same capacity) operators may cooperate bysharing their spectrum (a single merged system with greater combined capac-ity) resulting in greater user satisfaction (lower Pb) and potentially increasedrevenues. The question that naturally comes to mind is that since we havesuccessfully computed cell-average Pb with reasonably good accuracy (with-out KRA) can we go one step further and compute the cell-wide Pb as well insome similar fashion? We discuss this issue in this section.

4.2.1 GoS evaluation with the extended Erlang loss model

The disadvantage of the KRA is its recursive form which is not analyticallytractable. The network designer may seek to impose bounds on or restrictparameters (such as cell-wide or region-wise load) to influence the GoS andthe rate of its variation within the cell through pen-and-paper analysis. Thedesigner may still use the KRA, but then for every step of the way he orshe would have to depend on results generated by a computer program, whichmay be cumbersome for him or her. For this the network designer needs a non-recursive algorithm involving a more explicit mathematical expression [9]. Anexplicit mathematical representation of this sort has also been shown to enablethe network performance analysis at a high level, e.g. with respect to spec-trum sharing (20). We shall further present an example of how the networkdesigner, with the availability of a closed form expression for GoS parameters,can choose to restrict the rate of variation of GoS within the cell by load con-trol later on in this work. An analytically tractable algorithm for cell-wide andcell-average GoS evaluation keeping such analysis and design goals in mind canbe arrived at starting from the Erlang approximation (6),(11),(12). There aretwo shortcomings in the original Erlang Approximation. Firstly, (11) ignoresfrequency allocation granularity, thereby leading to inaccurate Pb estimates.Secondly, ψ(β, i) (6),(12) is evaluated only for i = Nsrv, ignoring the other


values i ∈ 0, 1, · · · , Nsrv−1 which may carry significant information regard-ing bandwidth/subcarrier occupancy. We have already described a method tobypass the first shortcoming in Section 3. To overcome the second shortcom-ing, we look for correspondence between q(n) (8) evaluated by KRA and thevalue ψ(β, i) for i ∈ 0, 1, · · · , Nsrv. Firstly from Fig. 10 which shows q(n)as a function of n evaluated by KRA, we observe that the peak of q(n) occurs

always at n ≈ βN reqsc (N req

sc being ≈ 4.7 and Nsrv ≈ 108). Now the peak of

ψ(β, i) (6) occurs at i ≈ β and i servers are equivalent to iN reqsc subcarriers

(each server encapsulates N reqsc subcarriers). Thus there appears to be a re-

lation between ψ(β, i) and Pr(Noccsc = iN req

sc ). This appears to be intuitively

satisfying since i active servers implies iN reqsc occupied subcarriers.

100 200 300 400 5000

2

4

6

8

10

12

x 10−3

X: 280Y: 0.008683

X: 424Y: 0.0076

n

q(n

) =

Pro

b(N

occ

sc =

n)

β = 60

β = 90

4.7x60 = 282 4.7x90 = 423

PeakPoints

Fig. 10 Probability distribution of the number of occupied subcarriers (Noccsc ) considering

12.2 kbps voice traffic for various β in a cell with radius 200 m and M = 18 SINR classes inabsence of shadowing evaluated by KRA.

However, we note thatN reqsc is not usually integer valued. Typically ⌊N req

sc ⌋ <

N reqsc < ⌈N req

sc ⌉ = ⌊N reqsc ⌋ + 1. Given that Nocc

sc subcarriers are occupied,let an equivalent number of active servers or logical channels provided bythese be denoted by Neq

srv. Since there is a mapping between subcarriers and

servers (given that Nsc subcarriers act as⌊

Nsc

Nreqsc

⌋

servers (13)) we can say

that a certain number of occupied subcarriers (Noccsc ) act as a certain equiva-

lent number of active servers (Neqsrv). Let us assume that since on an average

N reqsc subcarriers are required to make a call, a set of ⌈N req

sc ⌉ subcarriers will

constitute a single server or logical channel. Thus if Noccsc < ⌈N req

sc ⌉, thereare no servers available (Neq

srv = 0), i.e. there are too few occupied subcar-

riers to constitute a single server. Hence if Noccsc ∈ 0, 1, 2, · · · , ⌈N req

sc ⌉ − 1

then Neqsrv = 0. Similarly if Nocc

sc ∈ ⌈N reqsc ⌉, ⌈N req

sc ⌉ + 1, · · · , ⌈2N reqsc ⌉ − 1

then there are sufficient subcarriers to constitute a single server only and


hence Neqsrv = 1. Proceeding in this fashion it is possible to show that if

Noccsc ∈ ⌈iN req

sc ⌉, ⌈iN reqsc ⌉ + 1, · · · , ⌈(i + 1)N req

sc ⌉ − 1 then Neqsrv = i. Let

Ωi denote the set of values of Noccsc that correspond to Neq

srv = i. Thus

Ω0 ≡ 0, 1, 2, · · · , ⌈N reqsc ⌉ − 1, Ω1 ≡ ⌈N req

sc ⌉, ⌈N reqsc ⌉ + 1, · · · , ⌈2N req

sc ⌉ − 1,

and Ωi ≡ ⌈iN reqsc ⌉, ⌈iN req

sc ⌉+ 1, · · · , ⌈(i + 1)N reqsc ⌉ − 1. Now the probability

of i active servers, denoted as Pr(Neqsrv = i), may be written as:

Pr(Neqsrv = i) = Pr(Nocc

sc ∈ Ωi) (21)

= q(⌈iN reqsc ⌉) + q(⌈iN req

sc ⌉+ 1) + · · ·

+q(⌈(i+ 1)N reqsc ⌉ − 1)

=∑

n∈Ωi

q(n), (22)

where q(n) = Pr(Noccsc = n). Alternatively Pr(Neq

srv = i) may be inferred fromthe Erlang loss formula as:

Pr(Neqsrv = i) ≈ ψ(β, i). (23)

Therefore combining (22) and (23) we may write:

∑

n∈Ωi

q(n) ≈ ψ(β, i). (24)

Now from Fig. 10 we observe that q(n) varies slowly with n, in other wordsthe neighboring values of q(n) are approximately equal. By assuming that

q(⌈iN reqsc ⌉) ≈ q(⌈iN req

sc ⌉+ 1) ≈ q(⌈iN reqsc ⌉+ 2) · · · ≈ q(⌈(i + 1)N req

sc ⌉ − 1) it ispossible to approximate q(n) as:

q(n) ≈ψ(β, i)

|Ωi|, ∀n ∈ Ωi. (25)

In this approximation |Ωi| denotes the cardinality of the set3 Ωi. As shownin Fig. 11 it turns out that this is a reasonably good approximation, espe-cially for q(n) between4 q(Nsc− (M − 1)) and q(Nsc), M being the number ofregions. After computing q(n), region-wise Pb(l) may be computed using (9)and cell-average Pb using (10). We refer to this algorithm, which uses the Er-lang loss model to compute approximate subcarrier occupancy probabilities,as Erlang2Qn. It offers an alternative to KRA with a more explicit mathemat-ical expression (6),(13),(25), enabling us to compute both region-wise (9) andcell-average Pb (10).

4.2.2 GoS evaluation with the linear region-wise GoS and other simplified

models

Fig. 12 illustrates the variation of Pb with offered load (β) for 12.2 kbps voicewhen the cell-radius R = 200 m, evaluated by various techniques : KRA, EA,IEA, DES and Erlang2Qn. Out of these techniques only KRA yields the exact


200 300 400 500

2

4

6

8

10

12

x 10−3

n

q(n

) =

Pro

b(N

occ

sc =

n)

β = 60, KRA

β = 90, KRA

β = 60, Erlang2Qn

β = 90, Erlang2Qn

q(Nsc

)

38%maxerror

q(Nsc

− (M−1))

Fig. 11 Probability distribution of the number of occupied subcarriers (Noccsc ) considering

12.2 kbps voice traffic for various β in a cell with radius 200 m and M = 18 SINR classes inabsence of shadowing evaluated by KRA and Erlang2Qn.

85 90 95 100 1050

0.01

0.02

0.03

0.04

0.05

β (Erlangs)

Ave

rag

e B

lock

ing

Pro

ba

bili

ty (

E[P

b])

IEAKRAP

bmax

= .02

EADESErlang2Qn

Fig. 12 Pb as a function of offered load in Erlangs for 12.2 kbps voice and a cell radius of200 m in absence of shadowing.

value of Pb at a given load β. It is therefore meaningful to compare the othermethods in terms of their accuracy in computing Pb. This accuracy may bequantified by means of the root-mean-square (RMS) error in the Pb evaluatedby these methods vs. exact. For evaluating RMS estimation error we choosethe range of β given by 88 ≤ β ≤ 100 as the region of interest since the Pb

for β in this range varies between Pbmax

2 = 0.01 and 2Pbmax = 0.04 (observe

Fig. 12) which is the region of typical Pb values close to Pbmax (0.02). TheRMS error may be normalized by Pbmax and expressed as a % value. A first


scrutiny of Table 2 reveals that IEA has nearly 5.5 times higher accuracy (or,correspondingly, lower estimation error) compared to EA (whose normalizedRMS error is close to 100%), which gives a strong motivation to prefer theformer method over the latter. DES accuracy is 3 times and Erlang2Qn ac-curacy ≈ 1.4 times that of IEA. Among EA, IEA, DES and Erlang2Qn, DEShas the best accuracy, which makes this method important for GoS evalua-tion in cellular networks even though it tends to be slow compared to theother methods. Now, going ahead, since Pb may be evaluated exactly by KRAand approximately by IEA, we shall use the superscripts ‘KRA’ to denote the

first (PKRAb ) and ‘IEA’ to denote the second (P IEAb ). The βmax evaluatedby Erlang2Qn almost exactly matches that evaluated by KRA. Thus from adimensioning point of view Erlang2Qn has almost perfect accuracy (negligiblysmall error in evaluated BS density requirement). From the same viewpointIEA has an error of approx. 1.2% in evaluated BS density.

From Fig. 12 we observe that P IEAb starts off at a lower level compared

to PKRAb but slowly approaches PKRAb and crosses it between β = 95 andβ = 100 Erlangs. Thus there appears to be a regular pattern in the behavior

(rate of increase etc.) of P IEAb relative to that of PKRAb . Establishing a math-

ematical relationship between P IEAb and PKRAb may enable us to evaluate the

exact PKRAb from the approximate P IEAb itself. This motivates us to study

the relation of P IEAb to PKRAb as a function of β. Fig. 13 shows the ratio ofPb evaluated by IEA to the exact value from KRA for varying β. This ratio isapproximately a linear function of β, with a slope that varies with the effective

bit-rate νb.

Table 2 Normalized Root Mean Square Estimation Error (%) in Pb estimation by variousmethods compared against the exact result obtained by KRA for 88 ≤ β ≤ 100 considering

voice traffic with νb = 12.2 kbps and cell radius 200 m

Method RMS Estimation Error

Pbmax

× 100%

EA 99IEA 18

IEA-scaled 0.785DES 6

Erlang2Qn 13

The linear behavior ofP IEA

b

PKRAb

vs. β (as observed from Fig. 13) is a very

useful property since by using appropriate scaling Pb evaluated by IEA willyield nearly exact Pb estimates. Thus, using the straight line approximationwe may write

P IEAb (β)

PKRAb (β)≈ cmβ + c0, (26)


80 85 90 95 1000

0.2

0.4

0.6

0.8

1

β

E[P

IEA

b]/

E[P

b]

E[P IEAb

]/E[Pb]

Straight−line fit

Fig. 13 Linearity of the ratio of Pb evaluated by IEA to the exact value obtained fromKRA for varying β.

where Pb(β) designates Pb from either the KRA or IEA (depending on thesuperscript) as a function of β, cm is a constant term signifying the slope, andc0 another constant signifying the y-intercept of the straight line variation ofP IEA

b

PKRAb

vs. β. Then we can obtain a fairly accurate estimate of exact value of

Pb by the fact that from (26)

PKRAb (β) ≈P IEAb (β)

cmβ + c0(27)

which is a scaled version of the Pb computed through IEA (12) and hence maybe expressed as:

PKRAb (β) ≈ψ(β,Nsrv)

cmβ + c0. (28)

The accuracy of this scaled version of IEA is demonstrated by the curve titledIEAscaled in Fig. 14 and by its normalized RMS estimation error as capturedin Table 2. The RMS estimation error of IEA-scaled is the lowest among allthe methods, and more than an order of magnitude better compared to IEA(two orders of magnitude better compared to EA).

An explanation of the linear behavior ofP IEA

b

PKRAb

vs. β for 12.2 kbps voice

in the interference limited situation is provided in Observation 1 below. Weshall give several such explanations of observed properties of region-wise andcell-average GoS as well as the Erlang approximation with the help of theErlang2Qn model (25) in this section. In these Observations we shall mostlyfocus on the interference limited situation (R < 1.5 km) as this is most com-monly encountered in practical urban cellular scenarios. We note that withcommonly used σξ = 4 dB, there is a certain level of agreement between theresults with and without shadowing (βmax evaluated in the two cases differ


85 90 95 100 1050

0.01

0.02

0.03

0.04

0.05

β (Erlangs)

Ave

rag

e B

lock

ing

Pro

ba

bili

ty (

E[P

b])

IEAKRAP

bmax

= .02

IEA scaled

Fig. 14 Pb as a function of offered load in Erlangs for 12.2 kbps voice and a cell radius of200 m in absence of shadowing measured by IEA-scaled derived from IEA.

by ≈ 10%). Thus both of these have significance, and we shall consider bothcases in many of the Observations that follow. In these Observations, unlessotherwise specified, we shall consider 12.2 kbps voice for which Nsrv = 98and βmax = 84 with shadowing while Nsrv = 108 and βmax = 93.5 withoutshadowing. We note here that Nsrv may easily be evaluated using (13) wherein

N reqsc =

∑

lNreqscl

ζl =∑

l lζl (17) while incorporating the appropriate value of ζ

- ζ (18) for the case without shadowing and ζ (19) for the case with shadowing.

Observation 1: The ratio of Pb evaluated by IEA to the exact Pb evaluated

by KRA, i.e.P IEA

b

Pbis linear in β for low streaming rates, specifically 12.2 kbps

voice in the interference limited situation under the TU-SISO channel.

Explanation: Let PKRAb denote the exact value of Pb obtained from

KRA(9),(10). Now from (9), (17) Pb(l) =∑l−1m=0 q(Nsc −m). Therefore, using

(10)

PKRAb =

M∑

l=1

ζl

l−1∑

m=0

q(Nsc −m). (29)

Given two integers i and l such that l ≥ i, let [i, l] denote the set of integersi, i + 1, i + 2, · · · , l. We first consider 12.2 kbps voice in the interferencelimited situation without shadowing, withNsc = 512,Nsrv = 108 andM = 18.Changing the order of summation in (29), we may write:

PKRAb =M−1∑

m=0

q(Nsc −m)(1−m∑

l=1

ζl) (30)


From (25) we have for q(n) with M = 18 and Nsc − (M − 1) ≤ n ≤ Nsc (seeFig. 11):

q(Nsc) ≈ψ(β,Nsrv)

5

q(Nsc −m) ≈ψ(β,Nsrv − 1)

4∀m ∈ [1, 4]


5∀m ∈ [5, 9]


5∀m ∈ [10, 14]


4∀m ∈ [15, 17] (31)

From (30) and (31) we have

PKRAb ≈ψ(β,Nsrv)

5+ψ(β,Nsrv − 1)

4

∑

m∈[1,4]

(1−

m∑

l=1

ζl)

+ψ(β,Nsrv − 2)

5

∑

m∈[5,9]

(1−

m∑

l=1

ζl)

+ψ(β,Nsrv − 3)

5

∑

m∈[10,14]

(1 −

m∑

l=1

ζl)

+ψ(β,Nsrv − 4)

4

∑

m∈[15,17]

(1 −m∑

l=1

ζl) (32)

Since (1−∑ml=1 ζl) decreases with increasing m, the first three terms involving

m = 0, m ∈ [1, 4] and m ∈ [5, 9] dominate5 (32). Using (18) in (32) we maywrite:

PKRAb ≈ψ(β,Nsrv)

5+ 2.37

ψ(β,Nsrv − 1)

4

+1.17ψ(β,Nsrv − 2)

5(33)

= ψ(β,Nsrv − 1)(0.2β

Nsrv+ 0.59 + 0.23

Nsrv − 1

β). (34)

Now since Nsrv ≫ 1 we may approximate PKRAb (34) as:

PKRAb ≈ ψ(β,Nsrv − 1)(0.2β

Nsrv+ 0.59 + 0.23

Nsrvβ

). (35)

= ψ(β,Nsrv − 1)g0(β) (36)

where g0(β) is given by

g0(β) = (0.2β

Nsrv+ 0.59 + 0.23

Nsrvβ

). (37)


Now differentiating g0(β) with respect to β we obtain

g′

0(β) = (0.2

Nsrv− 0.23

Nsrvβ2

), (38)

where the superscript′

indicates the first derivative. Now Nsrv = 108 ≫ 0.2.Also when β is in the neighborhood of βmax (= 93.5) β2 ≫ 0.23Nsrv (since93.52 ≫ 0.23× 108). Therefore

g′

0(β) ≈ 0, (39)

which implies that g0(β) is approximately constant against β in the chosensetting. Substituting β = Nsrv in (37) we obtain: g0(Nsrv) = 0.2+0.59+0.23 =1.02 ≈ 1. Alternatively, substituting β = βmax = 93.5 in (37) we obtaing0(βmax) = 0.2(93.5108 ) + 0.59 + 0.23( 108

93.5 ) = 1.029 ≈ 1. Thus the approximateconstancy of g0(β) is also verified. Since g0(β) effectively equates to 1 for β in

the neighborhood of βmax, we may approximate PKRAb as (36):

PKRAb ≈ ψ(β,Nsrv − 1) =Nsrvβ

ψ(β,Nsrv) (40)

Since P IEAb = ψ(β,Nsrv), Observation 1 follows from (40). Fig. 13 confirmsthe result. For the case with shadowing, we may consider a single instanceof shadowing with M ′ = 24 regions. In this case, the equations for q(n) withNsc − (M ′ − 1) ≤ n ≤ Nsc include those in (31) as well as:


5∀m ∈ [19, 23] (41)

The explanation for this case is very similar as the case without shadowingand hence omitted for the sake of brevity. This concludes the explanation ofObservation 1.

Aside from KRA and Erlang2Qn another method for computing region-wise Pb that will be elaborated shortly is to utilize the linearity property ofregion-wise Pb vs. region-index (Pb(l) vs. l) which is shown in Fig. 8 (withoutshadowing) and Fig. 9 (with shadowing). To explain this linear variation weuse (17) and then proceed as follows. From (9) and (17) we get

Pb(l + 1)− Pb(l) = q(Nsc − l) and Pb(l)− Pb(l − 1) = q(Nsc − l + 1). (42)

Linearity of Pb(l) is finally explained by observing that q(n) varies veryslowly with n (as seen from Fig. 15, which zooms in on the useful ‘tail’ portionof q(n) values in Fig. 10) which implies that q(Nsc − l) ≈ q(Nsc − l + 1)∀l ∈2, 3, · · · ,M − 1. Based on this we may also say that q(Nsc − l) is approxi-mately constant for l ∈ 0, 1, · · · ,M − 1, i.e.

q(Nsc) ≈ q(Nsc − 1) ≈ q(Nsc − 2) · · · ≈ q(Nsc − (M − 1)). (43)

From (43) we may infer a necessary condition for linearity of Pb(l) vs. l as:

q(Nsc) ≈ q(Nsc − (M − 1)), (44)


which holds for 12.2 kbps voice (Fig. 15) as we shall show here, but does nothold for higher streaming rates such as 64 kbps, as we shall show later inObservation 3.

495 500 505 510

0

0.5

1

1.5

2

2.5

3

3.5

x 10−3

n

q(n

) =

Pro

b(N

occ

sc =

n)

β = 60, KRA

β = 90, KRA

Fig. 15 Probability distribution (q(n)) of the number of occupied subcarriers (Noccsc ) con-

sidering 12.2 kbps voice traffic for various β in a cell with radius 200 m and M = 18 SINRclasses in absence of shadowing evaluated by KRA. In this plot we focus on the values ofq(n) that are actually used in region-wise Pb (Pb(l)) evaluation for voice, i.e. for the range(Nsc − (M − 1) ≤ n ≤ Nsc).

It is easy to explain (44) approximately as follows. We consider 12.2 kbpsvoice in the interference limited situation with Nsc = 512 and 5 MHz sys-tem bandwidth without shadowing. For this case, Nsrv = 108, M = 18 andβmax ≈ 93.5. From (17), (18) we have N req

sc ≈ 4.78. Now q(Nsc − 1) ≈ψ(β,Nsrv−1)|ΩNsrv−1|

and q(Nsc − (M − 1)) ≈ ψ(β,Nsrv−4)|ΩNsrv−4|

where M − 1 = 17 unoc-

cupied subcarriers correspond to ⌈M−1

Nreqsc

⌉ = 4 unoccupied servers. Since |Ωi| =

either ⌊ Nsc

Nsrv⌋ or ⌊ Nsc

Nsrv⌋+1 ∀i ∈ 1, 2, · · · , Nsrv, |ΩNsrv−1| ≈ |ΩNsrv−4|. Thus

q(Nsc−(M−1))q(Nsc−1) ≈ ψ(β,Nsrv−4)

ψ(β,Nsrv−1) =1β3 (Nsrv − 1)(Nsrv − 2)(Nsrv − 3). Considering

β in the neighborhood of βmax, this ratio ≈ 1.5. This explains (44) approx-imately in absence of shadowing. The explanation considering shadowing issimilar.

Linearity of Pb(l) vs. l is true for low bit-rates. Let ρβ denote the slope ofPb(l) vs. l curve. Being a function of β,

ρβ ≈Pb(i, β)− Pb(1, β)

i− 1∀i ∈ 2, 3, · · ·M. (45)


Thus (observing Fig. 8 and Fig. 9 and noting that the lines have constantslope)

ρβ ≈ Pb(2)− Pb(1) ≈Pb(3)− Pb(1)

2· · · ≈

Pb(M)− Pb(1)

M − 1. (46)

Here ρβ may be evaluated by the KRA which yields the values of Pb(l) ∀l ∈1, 2, · · · ,M. We refer to ρβ evaluated from KRA as ρKRAβ . To find an alter-native to KRA for evaluating ρβ and subsequently Pb(l) vs. l at given β, weneed to find a function mapping β to ρβ . Since from (46)

ρβ ≈ Pb(2)− Pb(1) = q(Nsc − 1) ≈ q(Nsc), (47)

we may approximate ρβ by q(Nsc). Since q(Nsc) is the probability of all subcar-riers (implying all servers) being occupied, it is possible to generate an estimateof it (hence of ρβ) based on the Erlang loss formula (6). Thus we may say thatthe rate of increase of Pb(l) vs. l, i.e. the rate of increase of Pb from one region tothe next, which is a sort of measure of network congestion, may be derived fromthe Erlang loss formula which captures the server occupancy level. In other

words since from (31) in an approximate sense q(Nsc) ∝ ψ(β,Nsrv) = P IEAb

it should be possible to evaluate ρβ approximately from P IEAb .For this we first plot ρKRAβ vs. β in Fig. 16. As expected it is an increasing

function of β. The question that now arises is whether there is a relationship

between ρKRAβ (Fig. 16) and P IEAb (Figs. 12 and 14) that will allow easy andaccurate evaluation of the former from the latter.

In order to find an answer we plot the ratio of P IEAb to ρKRAβ as a functionof β in Fig. 17. This ratio is seen to vary approximately linearly with β, havinga slope that varies with the bit-rate. An explanation of this property for 12.2kbps voice in the interference limited situation is provided in Observation 2below.

Observation 2: The ratio of Pb evaluated by IEA to the ρβ evaluated by

KRA, i.e.P IEA

b

ρKRAβ

, is linear in β for low streaming rates, specifically 12.2 kbps

voice in the interference limited situation under the TU-SISO channel.Explanation: We have from (46) ρβ ≈ Pb(M)−Pb(1)

M−1 . From (9), (17) wehave

ρβ ≈

∑M−1l=1 q(Nsc − l)

M − 1, (48)

since Pb(M) = q(Nsc) +∑M−1

l=1 q(Nsc − l) = Pb(1) +∑M−1

l=1 q(Nsc − l). Con-sidering 12.2 kbps voice in the interference limited situation in absence ofshadowing, we have M = 18. Now using (31) in (48) we have

ρβ ≈

∑3l=1 ψ(β,Nsrv − l) + 3

4ψ(β,Nsrv − 4)

M − 1(49)

≈ψ(β,Nsrv − 1)(1 + Nsrv

β+

N2

srv

β2 + 34N3

srv

β3 )

M − 1(50)


85 90 95

1

2

3

4

5

6

7

x 10−3

β

ρ β

ρKRAβ

Fig. 16 Plot of ρKRAβ (≈

Pb(M)−Pb(1)M−1

, the slope of Pb(l) vs. l) obtained from KRA.

80 85 90 95 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

β

E[P

b]/

ρ β

E[Pb] from IEA/ ρβ from KRA

Linear fit

Fig. 17 Straight line fitting forP IEAbρβ

.

Now with steps similar to those used for arriving at (40) from (36) we canshow that for β in the neighborhood of βmax:

ρβ ≈4.65ψ(β,Nsrv − 1)

M − 1. (51)

Using P IEAb = ψ(β,Nsrv) in (51), we obtain:

P IEAb

ρβ≈

(M − 1)β

4.65Nsrv. (52)


This explains the Observation. Fig. 17 confirms the result.The explanation considering a single instance of shadowing is very similar.

In this case, from (46), (31) and (41) we have

ρβ ≈Pb(M)− Pb(1)

M − 1≈

∑5l=1 ψ(β,Nsrv − l)

M − 1, (53)

where we have used the fact that there are M = 24 SINR classes with shad-owing because of which Pb(M) = q(Nsc) +

∑M−1l=1 q(Nsc − l) = Pb(1) +

∑23l=1 q(Nsc − l). Using the method employed to arrive at (51) from (49), it is

possible to show that:

ρβ ≈7

(M − 1)ψ(β,Nsrv − 1) =

7Nsrv(M − 1)β

ψ(β,Nsrv) (54)

from which the result follows. This concludes the explanation of Observation2.

Here we observe that the analysis of GoS by Erlang2Qn has put a tool inthe network designer’s hand whereby he can choose to control the variation ofGoS within the cell and not just the cell-average GoS. As a simple example,using (54), the network designer may find the maximum load the network cansustain if it is to maintain the slope of Pb(l) vs l curve (i.e. ρβ) within a certainlimit. If ρmaxβ represents the chosen value of upper limit on ρβ , then βmax is

given simply by the solution to the equation ψ(β,Nsrv − 1) ≈ (M−1)7 ρmaxβ .

The linear behavior ofP IEA

b

ρKRAβ

vs. β as explained above is also a very useful

property since by using appropriate scaling Pb evaluated by IEA will yield goodestimates of ρβ (observe the curve labeled ρIEAβ in Fig. 18). Thus, similar to

the method of accurate estimation of Pb(β) from the linear model ofP IEA

b

PKRAb

,

we may also approach the accurate estimation of ρβ from the linear model ofP IEA

b

ρKRAβ

. Following such an approach, we get

ρKRAβ ≈ψ(β,Nsrv)

c′mβ + c′

0

, (55)

where c′

m represents the slope and c′

0 the y-intercept of the linear model ofP IEA

b

ρKRAβ

.

From the linear model of Pb(l) vs. l, considering M SINR classes/regionswhose indices comprise the set ΩR := 1, 2, · · · ,M, we have

Pb(l, β) ≈ Pb(1, β) + ρβ(l − 1) ∀l ∈ ΩR, (56)

where Pb(l, β) denotes the Pb of the l th region at an Erlang load of β. Also,

Pb(β) ≈ Pb(1, β)− ρβ + ρβl, (57)


85 90 95−1

0

1

2

3

4

5

6

7

x 10−3

β

ρ β

ρIEAβ

ρKRAβ

Fig. 18 Plot of ρβ (≈Pb(M)−Pb(1)

M−1, the slope of Pb(l) vs. l) obtained from KRA as well as

(through scaling) from IEA.

where

l =

M∑

l=1

ζll (58)

is independent of β and is identical to N reqsc for low bit rate streaming traffic

(17).Finally, for evaluating Pb(l, β) ∀l ∈ ΩR, a non-recursive algorithm which

we refer to as ‘Alternative to KRA’ and abbreviate as Alt2KRA is suggestedbased on the above observations as follows

1. Evaluate Pb(β) from IEA. Use the linear model ofP IEA

b

PKRAb

vs. β to make

this accurate (27),(28). Since the slope (cm) and y-intercept (c0) of thislinear variation depends on the bit-rate, it is assumed that these have beenpre-computed for the chosen bit-rate from KRA using a plot such as Fig.13. The resulting method till this stage is also referred to as IEA-scaled.The Pb evaluated by Alt2KRA is obtained at this step.

2. Evaluate ρβ from P IEAb using the linear model ofP IEA

b

ρKRAβ

vs. β (similar to

the method used for evaluating accurate Pb in the previous step) by (55) .

Here also the slope and y-intercept ofP IEA

b

ρKRAβ

vs. β need to be precomputed

for the chosen bit-rate from KRA using a plot such as Fig. 17.3. Evaluate l(58). Then evaluate Pb(1, β) from (57) using Pb(β), ρβ and l thus

obtained.4. Evaluate Pb(l, β) ∀l ∈ ΩR \ 1 using (56).

Fig. 19 shows the SINR region/class-wise Pb variation for β in the neighbor-hood of βmax = 93.5 in case of voice evaluated by KRA, Alt2KRA, Erlang2Qnand DES for cell-radius of 200m. Here the results from DESmatch closely those


from KRA, thereby confirming the validity of our models. Alt2KRA and Er-lang2Qn are seen to provide a mathematically tractable, accurate alternativeto KRA for region-wise Pb evaluation. Erlang2Qn is, however, seen to be lessaccurate at β = 92 Erlangs (with percentage gap in evaluated Pb in case ofthe outermost regions coming close to 14.7%).

0 5 10 150

0.02

0.04

0.06

0.08

0.1

0.12

l (Region/Class index)

Pb(l)

Alt2KRA

KRA

Erlang2Qn

DES

β = 92

β = 96

Fig. 19 SINR region/class-wise Pb (Pb(l) vs. l) around peak load (βmax ≈ 93.5 Erlangs) for12.2 kbps and a cell radius of 200 m evaluated by KRA, DES, and the proposed algorithmsAlt2KRA and Erlang2Qn in absence of shadowing.

KRA

IEA

Erlang2Qn

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

l (Region/Class index)

Pb(l)

KRAErlang2Qn

Fig. 20 SINR region/class-wise Pb (Pb(l) vs. l) at peak load (βmax = 13.5 Erlangs) for 64kbps and a cell radius of 200 m in absence of shadowing evaluated by KRA and Erlang2Qn.

Fig. 20 shows the region-wise Pb curve for 64 kbps at peak load (βmax =13.5 Erlangs) and a cell-radius of 200 m evaluated by KRA and Erlang2Qn. For


this situation there areM = 91 SINR classes. As seen from the plot Pb(l) vs. lis non-linear and hence Alt2KRA method is not applicable. An explanation ofthe non-linear nature of region-wise Pb variation for 64 kbps and higher ratesis provided in Observation 3 below. The Erlang2Qn method is found to befairly accurate. The maximum error in estimated Pb(l) is found to be ≈ 10%.The usage of this method is not limited to low bit rates. Bit rates higher than64 kbps may also be considered when using Erlang2Qn. However, since wehave shown that at bit rates equal to or higher than 64 kbps (Observation 3)Alt2KRA cannot be employed, we have chosen this rate here. Fig. 21 shows Pbas a function of β evaluated by KRA, IEA and Erlang2Qn. Erlang2Qn is foundto be more accurate than IEA in Pb evaluation for 64 kbps. Correspondingly,the βmax evaluated by Erlang2Qn is closer to the exact value obtained fromKRA than that evaluated by IEA. Thus from a dimensioning viewpoint theerror in evaluated BS density is only 2.3% with Erlang2Qn and 9.5% withIEA.

10 11 12 13 14 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

β (Erlangs)

E[P

b]

KRA

IEA

Erlang2Qn

Fig. 21 Pb for 64 kbps and a cell radius of 200 m as a function of offered load in Erlangsin absence of shadowing evaluated by KRA, IEA and Erlang2Qn.

Observation 3: Pb(l) vs. l is not linear for streaming bit-rates νb ≥ 64kbps.

Explanation: Consider 64 kbps streaming traffic in the interference lim-ited situation. For this case, M = 91 SINR classes exist and have Pb varyingbetween Pb(1) = q(Nsc) and Pb(M) =

∑Ml=1 q(Nsc − (l − 1)). In this case

N reqsc ≈ 22.75 and Nsrv = 22 (13). The maximum Erlang load is given by

βmax ≈ 13.5. For linearity of Pb(l) vs. l it was shown in (44) that a necessarycondition is q(Nsc − 1) ≈ q(Nsc − (M − 1)). We can show that in this case,q(Nsc − 1) ≪ q(Nsc − (M − 1)), and therefore Pb(l) vs. l cannot be linear.

In this case, with Nsc = 512, we have q(Nsc−1)q(Nsc−(M−1)) = q(511)

q(422) ≈ ψ(β,22)ψ(β,18) =


β4

∏4

l=1(22−(l−1))

(25). At β ≈ βmax, this evaluates toq(Nsc−1)

q(Nsc−(M−1)) ≈ 0.2 which

implies that q(Nsc − 1) ≪ q(Nsc − (M − 1)). Hence Pb(l) vs. l is not linearfor 64 kbps. By extrapolation this must be true for all higher rates, for whichM > 91 and βmax < 13.5. Fig. 20 confirms the result. This concludes theexplanation of Observation 3.

Note that for the Alt2KRA method to work, the linear variations ofP IEA

b

PKRAb

andP IEA

b

ρKRAβ

need to be studied in advance using KRA. Thereafter the method

does not depend on KRA anymore and provides a simple closed form mathe-

matical expression of the form (28) ψ(β,Nsrv)cmβ+c0

for evaluation of Pb(β) as well as

ρβ (leading to Pb(l)) which can be employed for the chosen bit-rate as often asrequired. The slopes and y-intercepts need to be computed for each relevantbit-rate only once in advance in the interference limited situation. In the noiselimited situation (which is relevant only for large cells with R > 1.5 km), theslope and y-intercept need to be computed for each cell radius. Here we notethat unlike Alt2KRA, Erlang2Qn does not require such pre-computation andis completely independent of the KRA. The accuracy of Alt2KRA, however,is better in capturing the actual variation of Pb(l) vs. l as we have just seen.

Figs. 19 and 20 show that at peak load (βmax), irrespective of the streamingrate involved,

Pb(M) > 4Pbmax while Pb(1) <Pbmax

4(59)

which implies that users at the cell periphery encounter significantly higherPb than the cell-average in contrast to those at the cell center. This suggests apotential issue with user satisfaction at the cell periphery as well as with GoSfairness.

5 Conclusion

In this paper it has been shown that the existing analytical framework for GoSevaluation and cell dimensioning gives very conservative estimates of Erlangcapacity due to usage of inaccurate SE models. A reduction of BS density bya factor close to 3 may be achieved if appropriate models are used. It has alsobeen shown that the accuracy of the existing Erlang approximation is poor(off by > 25%). The accuracy of the proposed improved version of the Erlangapproximation, referred to as IEA, is significantly higher (off by only < 2%).

Thereafter we have shown that the region-wise Pb values obtained fromKRA indicate significantly higher Pb at the cell-edge and a continuously in-creasing class-wise Pb. The class-wise Pb increases in a linear fashion withclass-index in case of low bit-rate streaming calls. A mathematical study ofuser satisfaction and GoS fairness requires evaluation of SINR region/class-wise Pb. The KRA evaluates region-wise Pb through a recursive equation. Thisis mathematically intractable and inadequate for a more meaningful study of


these topics. We meet the ensuing requirement for a more explicit mathemati-cal expression for region-wise and cell-average Pb by proposing two algorithmsas alternatives.

The first of these proposed alternative algorithms titled Erlang2Qn extendsthe use of the Erlang loss model in a novel way to model subcarrier occupancyprobabilities. Once the approximate subcarrier occupancy probabilities havebeen computed, region-wise and cell-average Pb are easy to evaluate. The sec-ond of these algorithms titled Alt2KRA is applicable for low bit rate streamingtraffic and uses the approximately linear variation of Pb(l) vs. region-index lalong with the Erlang approximation to compute both region-wise and cell-average Pb. Both these algorithms employ the Erlang approximation in itsmodified form accounting for the frequency allocation granularity for greateraccuracy. Through the use of the models involved in these algorithms, sev-eral interesting properties such as the linearity of Pb(l) vs. l for low streamingrates can be explained. This is an added advantage. Of the two proposedalgorithms, Erlang2Qn is usable for higher bit-rates such as 64 kbps and iscompletely independent of KRA, although less accurate in evaluated Pb(l) forvoice. In the context of cell-dimensioning for voice, however, Erlang2Qn showsa good accuracy (almost exactly matching KRA) while IEA shows an error ofapprox 1.2%. With Erlang2Qn the error in evaluated BS density for video too,is sufficiently small, being approx 2.3%, while IEA shows an error of approx.9.5%. On the other hand, Alt2KRA is usable only for lower bit-rates such as12.2 kbps voice and needs the KRA for pre-computing parameters such as the

slope of theP IEA

b

PKRAb

straight line curve. Still its accuracy is higher compared

to Erlang2Qn in terms of the evaluated Pb(l). Its accuracy from dimension-ing perspective is almost perfect since it evaluates cell-average Pb through anappropriate scaling of the IEA result.

Finally, by employing DES in our GoS studies we have validated our mod-els.

Notes

1In case of packet switched real-time traffic the packet arrival rate and the packet sizealso characterize the call.

2This algorithm will be described in Section 3.

3If Nreqsc is integer valued then |Ωi| = N

reqsc ∀ i ∈ 0, 1, · · · , Nsrv.

4Using (9) and (17) we have Pb(1) = q(Nsc) and Pb(M) =∑Nsc

n=Nsc−M+1 q(n) which

implies that only the values of q(n) for n varying between Nsc − (M − 1) and Nsc (the lastM values of q(n)) are relevant for Pb computation. We call this the useful ‘tail’ portion ofq(n) since it involves the last few values of q(n) that are the only ones actually used.

5The fourth term is given by 0.045ψ(β, Nsrv −3) and the fifth term by 0.002ψ(β, Nsrv −4), both terms being evidently small compared to the first three which are 0.2ψ(β,Nsrv),0.593ψ(β, Nsrv − 1) and 0.234ψ(β, Nsrv − 2) respectively.


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