probabilidad de erlang

ESCUELA SUPERIOR POLITÉCNICA DE CHIMBORAZO

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““DDiissttrriibbuucciióónn ddee pprroobbaabbiilliiddaadd ddee EErrllaanngg yy ssuuss

aapplliiccaacciioonneess””

NOMBRE:

Sebastián Cárdenas (264)

Luis Tumalli (190)

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EESSCCUUEELLAA DDEE IINNGGEENNIIEERRIIAA EELLEECCTTRROONNIICCAA EENN

TTEELLEECCOOMMUUNNIICCAACCIIOONNEESS YY RREEDDEESS

Ingenieria Electronica, Telecomunicaciones y redes

T E L E F O N I A D I G I T A L

1

TEMA: Distribución de probabilidad de Erlang y sus aplicaciones

OBJETIVOS

OBJETIVO GENERAL

Realizar una investigación acerca de la distribución de probabilidad de Erlang

además de sus principales aplicaciones para entender su importancia en las

redes telefónicas.

OBJETIVOS ESPECÍFICOS

Definir la probabilidad de Erlang además de las principales aplicaciones

que esta pueda tener.

Determinar el uso de las probabilidad de Erlang B y C dentro de las redes

de telefonía

MARCO TEORICO

La distribución de probabilidad de Erlang puede definirse como una distribución

de probabilidad continua, cuya función principal se centra en determinar el

número de llamadas telefónicas que se podrán realizar en un tiempo por parte

de los operadores. [1]

La distribución de Erlang se aplicara principalmente en redes telefónicas que

tengan un tráfico masivo, es asi que para esta se tiene la ecuación general que

se plantea a continuación [1]:

Mediante esta probabilidad se mide el tiempo transcurrido entre la recepción de

llamadas, sobre todo en caso en los que existen elevados tiempos de espera

para la ocurrencia de cada uno de los eventos, es asi que me mediante este

indicador podremos determinar la probabilidad de retardo y bloqueo, para

conseguir esto se plantean dos tipos de distribuciones de Erlang, que serán las

que más aportan en la telefonía asi estas son:



2

1. Erlang B: Esta nos permitirá calcular la probabilidad de bloqueo, si se tiene

una pérdida del sistema, es decir no se atiende a una solicitud, por fallas

del sistema o colapso de los servidores, se considerara que le número de

usuarios es infinito: [2]

B: probabilidad de bloqueo.

N: número de circuitos.

A: intensidad de tráfico

2. Erlang C: Esta nos permite calcular la probabilidad de que una llamada

entretrante no sea atendida inmediatamente, de igual manera se

considerara que el número de usuarios es infinito y que se tiene un numero

N de fuentes, para calcular esta probabilidad tenemos la siguiente

ecuación [3]:

Pc: probabilidad de retardo.

N: número de circuitos.

A: intensidad de tráfico

Tanto la probabilidad de Erlang B como la C serán de gran utilidad para lograr

determinar el rendimiento de la red, sobre todo considerando que el tráfico es de

gran volumen.

CONCLUSIONES:

La probabilidad de Erlang se utilizara en redes telefónicas que poseen

tráfico de gran volumen o que tienen un elevado nivel de crecimiento.

Erlang B nos permitirá conocer el nivel o probabilidad de bloqueo en una

red que se considera de infinitos usuarios, mientras que Erlang C nos da

una idea de la probabilidad de que una llamada sufra retardo cuando

hay fallas del sistema.

RECOMENDACIONES:

Buscar ejemplos en los que se aplique la probabilidad de Erlang para de

esta forma entender el funcionamiento de estas formulas

Utilizar estas ecuaciones en redes que tengan un alto nivel de tráfico.



3

REFERENCIAS:

[1] S. Rappaport, Calculation of Some functions Arising in Problems of

Queueing and Communications Traffic, 2013. [LINK]:http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1094249&queryText%3

DCalculation+of+Some+functions+Arising+in+Problems+of+Queueing+and+Communication

s+Traffic

[2] Insoo Koo, Erlang Capacity Analysis of CDMA Systems Supporting Voice

and Delay-Tolerant Data Services Under the Delay Constraint [LINK]:http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=4277079&queryText%

3DErlang+Capacity+Analysis+of+CDMA+Systems+Supporting+Voice+and+Delay-

Tolerant+Data+Services+Under+the+Delay+Constraint

[3] E, Chromy. Erlang C formula and its use in the call centers [LINK]:https://dspace.vsb.cz/bitstream/handle/10084/84489/AEEE-2011-9-1-

7chromy.pdf?sequence=1

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 1 , JANUARY 1979 249

Calculation of Some Functions Arising in Problems of Queueing and Communications Traffic

STEPHEN S. RAPPAPORT

Abstract-Useful formulas are developed for the iterative calculation of the inverse of blocking probability for Erlang loss and Erlang delay systems, and for the determination of offered traffic and blocking probability for Erlang loss systems when the carried traffic and number of servers are given. These formulas which arise in many problems of queueing and communications traffic can be easily programmed. In tests they gave rapid convergence to high accuracy over a broad range of parameter values much greater than the range of usual display.

1. INTRODUCTION

In various problems of queueing or communications traffic it is often required or convenient to calculate certain inverse functions for which it is not possible to obtain an explicit expression. Examples of wide interest include, 1) determination of offered load and blocking probability for an Erlang loss system when carried load and number of (servers) trunks are given, 2) determination of carried load and offered load for an Erlang loss system when blocking probability and number of servers are given, and 3 ) determination of carried load for an Erlang delay system when blocking probability and number of servers are given. It is frequently necessary to resort to reading the required quantities from published graphs or tables [ 1, 21, sometimes entering these (in reverse) with the quantity usually taken as the dependent variable. Easily programmable numerical solutions are advantageous in many instances. This is especially true in view of the advent and proliferation of programmable calculators and microcomputers. Such solutions are presented here.

2 . CALCULATIONS FOR ERLANG LOSS SYSTEMS

The Erlang B formula gives the probability that a call or “customer” is blocked when Poisson traffic is offered to a group of .?I servers and blocked calls are cleared from the system (i.e., M / G / h / b queue). The blocking probability, 0, is given by B(n, a) where

2 u k / k ! k = O

is the number of servers, and “a” is the traffic offered to the server group. Values of B(b, a) can be conveniently calculated

Paper approved by the Editor for Communication Theory of the IEEE Communications Society for publication without oral presentation. Manuscript received September 9, 1977; revised June 14, 1978. This work was supported in part by the National Science Foundation under Grant ENG-76-09001.

The author is with the Department of Electrical Engineering, State University of New York, Stony Brook, NY 11794.

using the recursion formula [ 1 ]

with B(0, U ) 2 1.

number of busy servers and can be shown to be The traffic carried by the system is defined as the average

It is often desired to find the offered traffic ‘‘a” and blocking probability when the carried traffic and number of servers are given. Mathematically this is equivalent to solving the implicit equation (3) for “a”. If one divides both sides of (3) by [ l - B(h, a)] the equation can be written in the form a = f(u), and one can attempt a solution using successive substitutions [ 3 ] . However, closer examination shows that iteration in this form works well only for small values of a,.

Instead, using the recursion formula ( 2 ) one finds that (3) can be written in the form u = f(u), where

Differentiation of (4) gives

and we define

The desired solution to (3) (i.e., u,), when U, and 6 are given, can then be found using the iteration formulas

beginning with uo = a,/[ 1 - (u,/b)l. Rapid convergence is usually achieved provided that f’(a,) # 1, and accuracy to six significant figures is generally attained by stopping the iteration when

I (ai+ 1 - Ui)/ui+ 1 I < 1 0-6. (10)

A recursion formula for B’(& a) which can be easily derived from (2) is

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250 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 1 , JANUARY 1979

and the required values of B‘(6, ai) and B(6, ai) in the iteration formulas (7)-(9) can be generated recursively using (2) and (1 1) jointly, beginning with B(0, a) = 1 and B‘(0, a) = 0. The blocking probability can then be determined using a,, the solution to (3), obtained by this iteration procedure. Specifi- cally, p = B(6, a,.). The iteration scheme described was tested on a UNIVAC 11 10 computer at approximately 800 points in the region, lop5 < ( c L , / ~ ) < ,999999, 1 < 6 < 500. Successive iterates in the scheme were monitored. At several points (where ~ ‘ ( c L , ) 1) for which these iterates were not converg- ing, the magnitude of 0 was decremented from its previous iterated value and fixed. The scheme was resumed with the fixed 0 and this decrementing process was repeated if necessary. In most cases tried the correct solution within an accuracy of six significant figures was obtained in less than fifteen iterations. Often less than five or six iterations were required.

The second calculation described in section 1 is the determination of a, when 0 and 6 are given. Mathematically this is equivalent to taking = a C / ( l - 0) in (3) and solving the resulting equation for a,. For A = 1 the solution is a, = 0, and for 6 = 2 i t is a, = 0 + doz 20( 1 - p), but for arbitrary 6 an explicit solution is unobtainable and a numerical technique can be used.

The equation to be solved for a, (given 6 and 0) is

F ( u , ) ~ B 6,- -/3= 0. ( 1 3 (12)

In this calculation the solution a, must lie in the interval 0 < U, < 6. Also for 6 > 1, F(u,) in (1 2) has an inflection point somewhere in this interval. To obtain a scheme applicable over a wide range of 0 and 6 the method of false position was used [ 3 1 . To reduce the number of iterations required a simple one dimensional search (requiring at most 20 evalua- tions of F ) was first used to isolate the solution to within an interval of b/lOO. Then beginning with c ~ ~ i - 1 and U,i as the leftmost .and rightmost points of the interval, respectively, a new value CL,~+~ was generated using

The process was continued until the absolute value of the normalized difference between the two most recent iterates was smaller than This results in a nominal accuracy of six significant figures. The traffic offered to the system can then be obtained using U = a,/( 1 - 0).

The scheme was programmed on a UNIVAC 11 10 computer and tested over a grid of approximately 800 points spanning the range lop5 < 0 < .999999, 1 < b < 500. The explicit expressions were used for 6 = 1 and 2. For larger 6, six significant figures were obtained at all points tried in no more than ten iterations after the initial search.

3. CALCULATIONS FOR ERLANG DELAY SYSTEMS

In dealing with Erlang delay systems a similar calculation is encountered. The Erlang C formula gives the blocking probability for Poisson arrivals to the server group when blocked arrivals are held until service is complete (i.e. M/M/6 queue). The blocking probability is given by [ 1 ]

.I ”

As before the number of servers is 6, but here all the traffic “a” offered to the group is carried. Therefore 0 < d < 6. For arbitrary 6 and 0 it is in general not possible to find an explicit expression for the inverse function u = Cl(6 , 0) although it can be easily found that CI(1, 0) = 0, and Cr(2, 0) = (0 + 4-0)/2. For > 2 a numerical scheme was used. C(6, a) can be easily calculated using the identity [ 11

together with the recursive formula (2). The equation to be solved for “u” can be written in the form

F(u) P C(6, u) - p = 0. (18)

Of aCipl and c(,i that value was saved which produced an F For 6 = 1 and 2 the explicit expressions for C I ( 6 , 0) were differing in sign from F(c%,i+l). The smaller of the two values used. For larger values of 6, the method of false position was saved was identified as u,.~-~ and the larger as aCi. Equation used preceded by an initial search as in the second calculation (1 3) was then used again. This process was repeated until prox- described previously. The only differences were that (1 8) imity to the solution was detected by the denominator of (1 3) defines the function used in the initial search, (1 3) should be becoming smaller than .OOlfi. After this occurred (say on the replaced by

with the constant p given by and (1 5) should be replaced by

(15)

P = u k - a k - 1

c(6, a h ) - c(& a h - 1 ’

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 1 , JANUARY 1979 25 1

Proximity to the solution was detected by the denominator (1 5) becoming smaller than .OlP. As before the scheme was tested over a grid of approximately 800 points spanning the range loA5 < fl< ,999999, 1 < < 500. In all cases six significant figures were obtained in less than ten iterations after the initial search.

The formulas presented here have been found useful in many instances (viz. [4] ) and the general approach can be readily applied to other cases.

REFERENCES Cooper, R. B., Introduction to Queueing Theory, Macmillan Co.: New York, 1972. Reference Data for Radio Engineers, 6th ed. Howard W. Sams & Co.: Indianapolis, Ind. 1975. Hildebrand, F. B., Introduction to Numerical Analysis, McGraw- Hill: New York, 1956. Rappaport, S . S., “Traffic Capacity of DAMA Systems Using Col- lision Type Request Channels”, h o c . National Telecommunica- tions Conference, vol. 3, December 1977.

Concerning the Recovery of a Bandlimited Signal or Its Spectrum from a Finite Segment

RICHARD G. WILEY

Abstruct-The relationship between the iterative techniques of Sandberg and Papoulis is clarified. The conditions of Sandberg’s theorem are not satisfied in general when the problem is that of extrapolating a bandlimited signa outside of a known segment (a problem that is ‘‘ill- posed”). For an interesting special case, the iteration may be applied and the signal recovered exactly.

In [ 11, we showed that the iteration proposed by Papoulis [2] for recovery of a bandlimited signal from a finite segment was of the same form as the contraction mapping of Sandberg [ 3 ] , and that the constant in this iteration could be taken as unity. The key inequality which must be satisfied t o apply Sandberg’s theorem when the constant in the iteration is unity is1

m m

(Qx - Qv)(x - Y > d t 2 k ( x - y I 2 d t

m

2 (PQx - P Q Y ) ~ d t (1)

Paper approved by the Editor for Communication Theory of the IEEE Communications Society for publication without oral presentation. Manuscript received April 27, 1978; revised August 25, 1978.

The author is with the Syracuse Research Corporation, Syracuse, NY 13210.

In [ 1 1 , the center term of this inequality was not shown.

where:

Q is a distortion operator which transforms a bandlimited

x , y are any pair of (square-integrable) bandlimited signals P is the projection operator which transforms a square

integrable signal into a bandlimited one k is a positive constant which depends on Q and the

bandwidth of the subspace of bandlimited signals, but not on x and y .

signal into a signal that is square integrable

When Q corresponds t o multiplication in the time domain by the function

it is not possible to determine a constant, k , such that the left inequality of (1) holds for all x, y . We demonstrate this by means of an example. Suppose x ( t ) is a sinc pulse centered at +to and y ( t ) is a sinc pulse centered at - t o . Then, since

m T I-- L ( Q l x - Q l y ) ( x - Y ) d t = (x - Y I 2 d t , ( 3 )

it is clear that the left side of (1) can be made arbitrarily small by choosing t o 9 T . At the same time,

is essentially equal to the combined energy of the two sinc pulses. This shows that while

m m Lm ( Q l x - Q l v > ( x -u) d t 2 ( P Q l x - P Q I Y ) ~ d t , m

( 5 )

nevertheless, Sandberg’s theorem does not apply when Q corresponds to multiplic’ation by el, i.e., when trying t o recover a bandlimited signal from a finite segment.

Indeed, the failure of the left side of (1) is consistent with the fact that this problem is “ill-posed’’ [4] in the sense of Hadamard [ 5 1 . This means, roughly, that there are numerous bandlimited functions which are arbitrarily “close” to the desired function over the given segment, but which are. quite different from the desireh function outside of the given segment. If our observation of the finite segment is slightly noisy or cor- rupted slightly by tlie measurement or recording or sampling process, the corresponding differences between the extrapolated signal and the original signal outside the interval of observation can be very large.

Note that this is due to the nature of the problem of trying to extrapolate (or estimate the spectrum of) a bandlimited signal from a finite segment and is not due to the iterative approach. Any other “solution” (e.g., the maximum entropy method [ 21 ) faces the same difficulty.

In practice, what one obtains by applying the iteration is the (square-integrable) bandlimited signal which matches the known segment and which has the minimum amouni of ad- ditional energy (as allowed by the specified bandwidth) outside of the known segment. If the original bandlimited function

.0090-6778/79/0100-025 1$00.75 0 1979 IEEE

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2375

Erlang Capacity Analysis of CDMA Systems SupportingVoice and Delay-Tolerant Data Services

Under the Delay Constraint

Insoo Koo, Jeong Rok Yang, and Kiseon Kim

Abstract—In this paper, we analyze the Erlang capacity of a code-division multiple-access system supporting voice and delay-tolerant dataservices. The novelty here is to consider the characteristic of delay-toleranttraffic in terms of a delay confidence. The delay confidence is defined as theprobability that a new data call is accepted within the maximum tolerabledelay without being blocked. In this case, the Erlang capacity is confinednot only by the required blocking probability of voice call but also bythe required delay confidence of data call. For the performance analysis,we develop a 2-D Markov model, which is based on the First-Come–First-Serve service discipline, and further present a numerical procedureto analyze the Erlang capacity. According to the procedure, we can makea balance between the Erlang capacities with respect to the blockingprobability of voice call and with respect to the delay confidence of datacall, in order to accommodate extra Erlang capacity. We demonstrate thebalancing by properly selecting the size of the designated queue for datatraffic.

Index Terms—CDMA, delay constraint, Erlang capacity, voice and data.

I. INTRODUCTION

Next-generation mobile communication systems are primarily de-signed to provide users with multimedia services such as voice, inter-active data, file transfer, Internet access, and image in the affordableway as today’s wired communication systems do. Multimedia trafficwill have different and multiple quality of service (QoS) requirements.In terms of system operation, it is a very important task to analyzethe capacity of system that can be supportable while multiple ser-vice requirements of multimedia traffic are being satisfied since thecapacity can be used as a measure of system resource. For thesereasons, much effort has been taken to analyze the capacity of a code-division multiple-access (CDMA) system. Typically, the capacity ofa CDMA system has been defined as the maximum number of usersor the Erlang capacity [1]–[4]. The former and latter definitions ofthe capacity are used for estimating a supportable size of the systemat a time and for measuring the economic usefulness of the system,respectively [1]. In [1] and [2], the outage probability was presumed tobe the call blocking probability, and the call blocking probabilities ofdifferent traffics in the system were represented identically. By using amultidimensional Markov loss model, which is based on the maximumnumber of supportable current users, the call-blocking probabilities ofdifferent traffics were considered separately, and the Erlang capacitywas analyzed with respect to the required blocking probabilities ofdifferent traffics [8], [9].

Multimedia traffic can be roughly classified into “delay-intolerant”and “delay-tolerant” traffic. To achieve higher capacity using thedelay-tolerant characteristic, delay-tolerant traffic can be queued untilthe required resources are available in the system. The blocking

Manuscript received June 19, 2004; revised July 26, 2005 and January 15,2006. This work was supported in part by the Ministry of Commerce, Industry,and Energy and in part by Ulsan Metropolitan City through the Network-basedAutomation Research Center at the University of Ulsan. The review of thispaper was coordinated by Prof. C. Lin.

I. Koo is with the University of Ulsan, Ulsan 680-749, Korea.J. R. Yang is with the Korean Intellectual Property Office, Daejeon 302-701,

Korea.K. Kim is with the Kwang-Ju Institute of Science and Technology, Gwangju

500-712, Korea.Digital Object Identifier 10.1109/TVT.2007.897655

probability and the average delay have been typically considered to beperformance measures for the delay-tolerant traffic [8], [10]. However,a more meaningful measurement for delay-tolerant traffic is the delayconfidence rather than the average delay, where the delay confidence isdefined as the probability that a new data call gets a service within themaximum tolerable delay requirement without being blocked. Notingthat the previous works [1], [8]–[10] have not considered the delayconfidence when evaluating the Erlang capacity, in this paper, we adoptthe delay confidence as a performance measure of delay-tolerant traf-fic. Further, we analyze Erlang capacity of a CDMA system supportingvoice and data services since voice and data calls are typical delay-intolerant and delay-tolerant traffic, respectively. Here, the Erlangcapacity is defined as a set of average offered traffic loads of voiceand data calls that can be supported in the system while the requiredblocking probability of voice call and the required delay confidenceof data call are being satisfied simultaneously. To analyze the Erlangcapacity, we develop a 2-D Markov model, which is based on the First-Come–First-Serve (FCFS) service discipline, where the queue withfinite size is exclusively allocated for delay-tolerant data calls. Basedon the Markov model, we present a numerical procedure to analyze thecall blocking probabilities of voice and data traffic and the delay distri-bution and delay confidence of data traffic, which are performance ma-trices necessary to analyze the Erlang capacity. In addition, a procedureselecting the proper size of queue length for data traffic is suggested,in order to accommodate extra Erlang capacity in the system.

The remainder of this paper is organized as follows. In Section II,we describe system model and a call admission control (CAC) scheme.In Section III, we develop a 2-D Markov model and analyze theblocking probabilities of voice and data calls. In Section IV, we presentan analytical approach to derive the cumulative distribution function(CDF) of delay and delay confidence of data calls in order to analyzethe Erlang capacity. In Section V, we consider a numerical example.Finally, we draw conclusions in Section VI.

II. SYSTEM MODEL

In this paper, we are concerned with the traffic analysis of theCDMA systems. In order to focus on the traffic analysis of the CDMAsystems, we consider the number-based admission control with pre-determined CAC threshold, and as a reference for the predeterminedCAC threshold, we utilize the link capacity that details the numbers ofmobiles of each class in each cell that the system operator should allowin order to maintain an acceptable QoS. The link capacity reflects theCDMA aspects and gives performance guarantees that overcome thevariability in the interference levels that are characteristic of CDMAnetworks such that the CDMA mobile network, operating within theadmissible region described previously, has a very similar form toa circuit-switched network. The similarity allows the existing trafficmodeling techniques and network management strategies for generalloss networks to be applied to CDMA systems.

Let us first consider the link capacity. In CDMA systems, althoughthere is no hard limit on the number of concurrent users in the linklevel, there is a practical limit on the number of supportable concurrentusers in order to control the interference among users having the samepilot signal; otherwise, the system may fall into the outage state, whereQoS requirements of users cannot be guaranteed. In order to considerthe link capacity of a CDMA system, let Qv and Qd be the linkqualities such as frame error rate that individual voice and data usersexperience in the CDMA, respectively, and let Qv,m and Qd,m be aset of minimum link quality level of each service. Then, for a certainset of system parameters, such as service quality requirements and

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2376 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

link propagation model, the admissible region of CDMA system withrespect to the simultaneous number of users satisfying service qualityrequirements in the sense of statistic S can be defined as

S = (i, j)|Pr(Qv ≤ Qv,m and Qd ≤ Qd,m) ≥ β%

= (i, j)|0 ≤ fn(i, j) ≤ 1 and i, j ∈ Z+ (1)

where i and j are admissible number of voice and data users in theCDMA, respectively, β% is the system reliability defined as minimumrequirement on the probability that the link quality of the current usersin the system is larger than the minimum link quality level, whichis usually given between 95% and 99%, and fn(i, j) is the normal-ized capacity equation. In the single cell case, the fn(i, j) can begiven as [4]

fn(i, j) = γvi+ γdj, i and j ≥ 0 (2)

where γv = ((W/Rvqv) + 1)−1 and γd = ((W/Rdqd) + 1)−1,which are the amount of system resources that are used by onevoice and one data user, respectively. W is the allocated frequencybandwidth. qv and qd are the required bit energy to interferencepower spectral density ratio for voice and data calls, respectively, toachieve the target bit error rate at the base station (BS). Rv and Rd

are the required information data rates of the voice and data calls,respectively. For the link capacity of a multiple cell CDMA system,the effect of other cell interference can be included in (1), and thenormalized capacity equation fn(i, j) is given as [4]

fn(i, j) = γv · i+ γd · j +E(z) + 2.33√

(var(z)) (3)

where the term E(z) + 2.33√

(var(z)) includes the effect of intercellinterference on the link capacity. The link capacity of the multiple cellcase is confined by a lower hyperplane than that of the single cell casedue to the effect of the other cell interference. Such linear link capacitybounds on the total number of users of each class, which can be sup-ported simultaneously while maintaining adequate QoS requirements,are commonly found in other literature on CDMA systems supportingmulticlass services [5], [6]. Further, provided the network state lineswithin the admissible region, then the QoS requirement of each userwill be satisfied with β% reliability. Here, we are concerned with thetraffic analysis of the CDMA systems operating with the specifiedCAC algorithm such that we consider the single cell case in order tofocus on the traffic analysis of the CDMA systems.

Each user is classified by QoS requirements such as the requiredinformation data rate and the required bit energy to interferencespectral density ratio, and all users in same service group have the sameQoS requirements. Equations (2) and (3) indicate that the calls withdifferent services take different amount of system resources accordingto their QoS requirements. We also assume that the system employsa circuit switching method to handle the transmission of voice anddata calls, which means that once a call request is accepted in thesystem, the call occupies the required amount of system resources andtransmits the information without any delay throughout the duration ofthe call.

In the aspects of network operation, it is of vital importance toset up a suitable policy for the acceptance of an incoming call inorder to guarantee a certain QoS. In general, CAC policies can bedivided into two categories: 1) number-based CAC (NCAC) and2) interference-based CAC (ICAC) [7]. NCAC implies that a call willbe accepted or not, depending on the condition whether the number ofconcurrent users is greater than a threshold. In the case of ICAC, a BS

determines whether a new call is acceptable or not by monitoring theinterference level on a call-by-call basis, whereas the NCAC utilizesa predetermined CAC threshold. In this paper, we adopt an NCAC-type CAC due to its simplicity, although the NCAC generally suffersfrom a slight performance degradation over the ICAC [7]. That is, theCDMA mobile network, which is operated according to the NCACwith the threshold stipulated by link capacity, has a very similar formto a circuit-switched network. The similarity allows the existing trafficmodeling techniques and network management strategies for generalloss networks to be applied to CDMA systems. Here, we set thecapacity bound, which is stipulated by (2), as a predetermined CACthreshold. However, it is noteworthy that the link capacity reflects thefeature that CDMA is interference limited. Further, we consider thequeue with the finite length of K (K ≥ 1) for delay-tolerant datatraffic to exploit its delay-tolerant characteristic, where K denotesthe queue length, and assume that the service discipline is FCFS. Inthe service model, a voice call experiences call blocking when thereis not enough resource to accommodate the voice call and receiveszero queuing delay. On the other hand, a data call experiences callblocking when there is not enough resource and buffer space toaccommodate the data call and receives some queuing delay. Based onthese assumptions, the CAC, for the case γd > γv , can be summarizedas follows.

• If γvi+ γdj ≤ 1 − γd, then both new voice and new data callsare accepted.

• If 1 − γd < γvi+ γdj ≤ 1 − γv , then new voice calls are ac-cepted, and new data calls are queued.

• If 1 − γv < γvi+ γdj ≤ 1 + (K − 1)γd, then new voice callsare blocked, and new data calls are queued.

• If γvi+ γdj > 1 + (K − 1)γd, then both new voice and newdata calls are blocked.

Here, we set a voice channel to occupy one basic channel. In thiscase, the number of total basic channels in the system is 1/γv , andthe number of basic channels required by one data call is γd/γv ,respectively. It is also noteworthy that the number of total basicchannels in the system and the number of basic channels required byone data call are integer numbers in time-division multiple access orfrequency-division multiple access, whereas they are real numbers inCDMA systems [9].

In order to analyze the performance of the system under the CACpolicy, the arrivals of voice and data calls are assumed to be distributedaccording to independent Poisson processes with the average arrivalrate λv and λd, respectively. The service times of voice and data callsare assumed to be exponentially distributed with the average servicetime 1/µv and 1/µd, respectively. Then, the offered traffic loads ofvoice and data calls are expressed as ρv = λv/µv and ρd = λd/µd,respectively.

III. MARKOV CHAIN MODEL AND BLOCKING PROBABILITY

In this section, we develop an analytical model to determine theblocking probabilities of voice and data calls. The model will also beutilized to analyze the delay distribution of data call in Section IV.

According to the CAC rule based on the number of concurrent userswhen the queue with the finite length of queue size K ≥ 1 for delay-tolerant data traffic is employed, the set of possible admissible states isgiven as follows at the medium access control (MAC) layer where calladmission process is performed:

ΩS =(i, j)|0 ≤ i ≤ γ−1

v , j ≥ 0, γvi+ γdj ≤ 1 + γdK. (4)


Fig. 1. State transition diagram for the case that γd > γv .

Here, it is noteworthy that (2) corresponds to the link capacity of aCDMA system supporting voice and data services in the link layer,whereas (4) indicates the set of possible admissible states at the MAClayer. For the case that the buffer is not considered in the BS (K = 0),the link capacity corresponds to the threshold for the admission controlprocess at MAC layer directly. Here, we consider the case K ≥ 1.

Further, the possible admissible states can be divided into five dis-tinct regions, which are represented by the following sets, respectively:

ΩA = (i, j)|0 ≤ γv · i+ γd · j ≤ 1 − γd

ΩB = (i, j)|1 − γd < γv · i+ γd · j ≤ 1 − γv

ΩC = (i, j)|1 − γv < γv · i+ γd · j ≤ 1

ΩD = (i, j)|1 < γv · i+ γd · j ≤ 1 + γd · (K − 1)

ΩE = (i, j)|1 + γd · (K − 1) < γv · i+ γd · j ≤ 1 + γd ·K .

(5)

Fig. 1 shows these five distinct regions and a typical call-level statetransition example for each region. Noting that total rate of flowing

into a state (i, j) is equal to that of flowing out, we can get the steady-state balance equation for each state as follows:

Rate-In = Rate-Out

Rate-In =a · Pi+1,j +b · Pi,j+1 + c · Pi−1,j + d · Pi,j−1

Rate-Out =(i+j + k +l) · Pi,j , for all states (6)

where the state transition rates a, b, c, d,i, j, k, and l involved in (6)can be given by

a ≡ transition rate from state (i+ 1, j) to state (i, j)

=

(i+ 1)µv, (i, j) ∈ ΩS

0, otherwise(7)

b ≡ transition rate from state (i, j + 1) to state (i, j)

=

(j + 1)µd, (i, j) ∈ ΩA

j · µd, (i, j) ∈ ΩB ,ΩC(1 − γv · i)γ−1

d · µd, (i, j) ∈ ΩD

0, otherwise

(8)


Fig. 2. Steady-state balance equations corresponding to the voice/data CDMAsystem for the case that γd > γv .

c ≡ transition rate from state (i− 1, j) to state (i, j)

=λv, (i, j) ∈ ΩA,ΩB ,ΩC0, otherwise

(9)

d ≡ transition rate from state (i, j − 1) to state (i, j)

=λd, (i, j) ∈ ΩS

0, otherwise(10)

i ≡ transition rate from state (i, j) to state (i+ 1, j)

=λv, (i, j) ∈ ΩA,ΩB0, otherwise

(11)

j ≡ transition rate from state (i, j) to state (i, j + 1)

=λd, (i, j) ∈ ΩA,ΩB ,ΩC ,ΩD0, otherwise

(12)

k ≡ transition rate from state (i, j) to state (i− 1, j)

=iµv, (i, j) ∈ ΩS

0, otherwise(13)

l ≡ transition rate from state (i, j) to state (i, j − 1)

=

jµd, (i, j) ∈ ΩA,ΩB ,ΩC(1 − γv · i)γ−1

d , otherwise.(14)

Fig. 2 summarizes the steady-state balance equations for the statetransition diagram according to the region that the current state belongsto. If the total number of all possible states is ns, the balance equationsbecome (ns − 1) linearly independent equations. With these (ns − 1)equations and the normalized equation

∑(i,j)∈ΩS

Pi,j = 1, a set

of ns linearly independent equations for the state diagram can beformed as

Qπ = P (15)

where Q is the coefficient matrix of the ns linear equations, π is thevector of state probabilities, and P = [0, · · · ,0,1]T. The dimensionsof Q, π, and P are ns × ns, ns × 1, and ns × 1, respectively. Bysolving π = Q−1P, we can obtain the steady-state probabilities of allstates [8].

Based on the CAC rule, a new voice call will be blocked if the chan-nel resources are not enough to accept the call, and the correspondingblocking probability for voice calls is given by

Pbv =∑

(i,j)∈Ω(nv,blo)

Pi,j (16)

where

Ω(nv,blo) = (i, j)|γvi+ γdj > 1 − γv . (17)

Ω(nv,blo) is composed of the regions C, D, and E in Fig. 1. Similarly,a new data call will be blocked if the queue is full, and the blockingprobability for data calls is given by

Pbd=

∑(i,j)∈Ω(nd,blo)

Pi,j (18)

where

Ω(nd,blo) = (i, j)|γvi+ γdj > 1 + γd(K − 1) . (19)

Ω(nd,blo) corresponds to the region E in Fig. 1.

IV. ERLANG CAPACITY ANALYSIS

In this section, we analyze the Erlang capacity of CDMA systemsupporting voice and delay-tolerant services under the delay con-straint. For this purpose, we will derive the CDF of delay and the delayconfidence of data traffic. First, let us derive the CDF of delay τ , whichis based on the Markov chain model depicted in Fig. 1. The delay isdefined as the time that a data call waits in a queue until being acceptedin the system. For the convenience of analysis, we separate the CDF ofdelay into two parts, corresponding to discrete and continuous parts ofthe random variable τ , i.e.,

Fd(t) = Prτ ≤ t = Fd(0) +G(t) (20)

where Fd(0) = Prτ ≤ 0, and G(t) represents the continuous partof the delay. At first, the discrete part Fd(0) represents the case whenthe delay is zero, and it can be calculated as follows:

Fd(0) = Prτ ≤ 0 = Prτ = 0

=∑

(i,j)∈Ω(nd,acc)

P ′i,j (21)


Fig. 3. Set of states representing the admissible numbers of voice and data calls for the case that W = 1.25 MHz, qv = 7 dB, qd = 7 dB, Rv = 9.6 kb/s,Rd = 19.2 kb/s, and K = 3.

where Ω(nd,acc) is the acceptance region of new data calls, which isgiven as

Ω(nd,acc) = (i, j)|γvi+ γdj ≤ 1 − γd (22)

and

P ′i,j =

Pi,j

1 − Pbd

. (23)

P ′i,j represents the probability that there are i voice and j data calls

in the system just before a new data call is admitted. If the state (i, j)belongs to the blocking region of new data calls Ω(nd,blo), the call willbe blocked.

To investigate the continuous part of delay G(t), let (i′, j′) denotethe number of calls excluding the number of service-completed callswithin time τ from (i, j). Consider the case that (i, j) belongs tothe queueing region of new data calls just before a new data call isadmitted, where the queueing region of new data calls is given as

Ω(nd,que) = (i, j)|1 − γd < γvi+ γdj ≤ 1 + (K − 1)γd. (24)

In order for a new data call to be accepted within the time t accordingto the FCFS service discipline, (i′, j′) should fall into the acceptanceregion of new data calls within the time t. G(t) is the sum of the

probabilities of all cases that a state (i, j) in Ω(nd,que) changes into(i′, j′) in Ω(nd,acc) within the time t, which can be expressed as

G(t) =∑

(i,j)∈Ω(nd,que)

Pr

(i′, j′) ∈ Ω(nd,acc) within time t|

the system state is (i, j)

just before a new data call

is admitted· P ′

i,j

=∑

(i,j)∈Ω(nd,que)

t∫0

w(i,j)(τ)dτ · P ′i,j (25)

where w(i,j)(τ) is the delay distribution for the state (i, j), and itrepresents the probability of a new data call being accepted within timeτ , given that the system state is (i, j) just before the call is admitted.For example, Fig. 3 shows the set of states representing the admissiblenumbers of voice and data calls for the case that W = 1.25 MHz,qv = 7 dB, qd = 7 dB, Rv = 9.6 kb/s, Rd = 19.2 kb/s, and K = 3.Consider the case that there are 17 voice calls and seven data calls inthe system just before a new data call is admitted. In this case, thestate (17, 7) can change into (i′, j′) in Ω(nd,acc) through many pathsin order for a new data call to be accepted. For example, if (i′, j′) is(17, 4), no voice call and three data calls are service completed duringthe time τ , and if (i′, j′) is (16, 4), one voice call and three data callsare service completed. For the more general case where k voice calls


Fig. 4. State transition paths for analyzing the delay distribution. (a) The case that no voice call and (j − (1 − γvi)/γd + 1) data calls are service completedwithin time τ , given that the system state is (i, j) just before a new data call is attempted. (b) The case that one voice call and (j − (1 − γv(i − 1))/γd + 1)data calls are service completed within time τ . (c) The case that two voice calls and (j − (1 − γv(i − 2))/γd + 1) data calls are service completed withintime τ . (d) The case that k voice calls and (j − (1 − γv(i − k))/γd + 1) data calls are service completed within time τ .

get service completed during the time τ , the delay distribution for thestate (i, j) can be expressed as

w(i,j)(τ) =

I∑k=0

w(i,j)k(τ) (26)

where

I = min

(i, i−

⌊1 − γd(1 + j)

γv

⌋). (27)

w(i,j)k(τ) represents the delay distribution multiplied by the prob-

ability that k voice calls get service completed during the time τ ,given that the system state is (i, j) just before a new data call isadmitted. I is the maximum number of service-completed voice callsduring the change, which happens when voice calls are only servicecompleted.

The paths where the state (i, j) in Ω(nd,que) changes into (i′, j′)in Ω(nd,acc) can be generalized, as in Fig. 4. Since the service timedistribution is memoryless and the delay distribution is independentof the current arrival, w(i,j)k

(τ) is the convolution of k independent


exponential random variables, where k corresponds to the numberof service-completed voice calls [11]. Since the Laplace transformsof w(i,j)k

(τ) is equal to the product of the Laplace transforms ofexponential distributions, the Laplace transform of w(i,j)0(τ), for thecase in Fig. 4(a), can be expressed as

W(i,j)0(s) =

⌊1−γvi

γd

⌋µd

iµv +⌊

1−γviγd

⌋µd

j−⌊

1−γviγd

⌋+1

×

⌊1−γvi

γd

⌋µd

s+⌊

1−γviγd

⌋µd

j−⌊

1−γviγd

⌋+1

. (28)

The first term of W(i,j)0(s) in (28) represents the probability fork = 0, which corresponds to the probability that the state (i, j) inΩ(nd,que) is changed into (i′, j′) in Ω(nd,acc), as in Fig. 4(a). In (28),the exponent (j − (1 − γvi)/γd + 1) corresponds to the requirednumber of service-completed data calls for the new data call to beaccepted. The second term of W(i,j)0(s) in (28) comes from theproduct of the Laplace transforms of exponential distributions ofservice time of the service-competed data calls.

For the case that k = 1 which corresponds to Fig. 4(b), there areJ1 different paths, and w(i,j)1(τ) is expressed as the sum of delaydistributions multiplied by the probability that the path is selected out

of all paths. The Laplace transform of w(i,j)1(τ) can be expressed in(29), shown at the bottom of the page, where

J1 =

j −

⌊1−γv(i−1)

γd

⌋, if

⌊1−γv(i−1)

γd

⌋=

⌊1−γvi

γd

⌋

j −⌊

1−γv(i−1)γd

⌋+ 1, otherwise.

(30)

In the case of one service-completed voice call, the number of service-completed data calls should be (j − (1 − γv(i− 1))/γd + 1) for anew data call to be accepted. J1 is selected to avoid the path for thecase of k = 0, and, for example, it takes the path (17, 7) → (17, 4) →(16, 4) in Fig. 3.

By expanding the previous results to the general case of k service-completed voice calls, W(i,j)k

(s) can be obtained in (31), shown atthe bottom of the page, where Jk can be obtained in (32) and (33),shown at the bottom of the page, and

Da(s) =

⌊1−γv(i−a)

γd

⌋µd

(i− a)µv +⌊

1−γv(i−a)γd

⌋µd

×

⌊1−γv(i−a)

γd

⌋µd

s+⌊

1−γv(i−a)γd

⌋µd

. (34)

W(i,j)1(s) =

iµv

iµv +⌊

1−γviγd

⌋µd

(

iµv

s+ iµv

) J1∑j1=0

⌊1−γvi

γd

⌋µd

iµv +⌊

1−γviγd

⌋µd

j1

⌊1−γvi

γd

⌋µd

s+⌊

1−γviγd

⌋µd

j1

·

⌊1−γv(i−1)

γd

⌋µd

(i− 1)µv +⌊

1−γv(i−1)γd

⌋µd

j−⌊

1−γv(i−1)γd

⌋+1−j1

⌊

1−γv(i−1)γd

⌋µd

s+⌊

1−γv(i−1)γd

⌋µd

j−⌊

1−γv(i−1)γd

⌋+1−j1

(29)

W(i,j)k(s) =

(D0(S))j−

⌊1−γvi

γd

⌋+1

, k = 0k−1∏a=0

Va(S)Jk∑

j1=0

Jk∑j2=j1

· · ·Jk∑

jk=jk−1

(D0(S))j1 (D1(S))j2−j1 (D2(S))j3−j2 . . .

(Dk−1(S))jk−jk−1 (Dk(S))j−

⌊1−γv(i−k)

γd

⌋+1−jk

, otherwise

(31)

Jk =

j −

⌊1−γv(i−k)

γd

⌋, if

⌊1−γv(i−k)

γd

⌋=

⌊1−γv(i−(k−1))

γd

⌋j −

⌊1−γv(i−k)

γd

⌋+ 1, otherwise

(32)

Va(s) =

(i− a)µv

(i− a)µv +⌊

1−γv(i−a)γd

⌋µd

(

(i− a)µv

s+ (i− a)µv

)(33)


The left term of Va(s) in (33) is the probability that one voice callis service completed among (i− a) voice and (1 − γv(i− a))/γddata calls in the service state, and the right term is Laplace transformof the time distribution for the voice call to be service completed.On the other hand, the left term of Da(s) in (34) is the probabil-ity that a data call is service completed among (i− a) voice and(1 − γv(i− a))/γd data calls in the service state, and the right termis Laplace transform of the service time distribution for the data callto be service completed. It is noteworthy that the probability of voiceor data call being service completed, and the time distribution for acall to be service completed can be represented by the number ofvoice calls for given average service times of voice and data calls. Itcomes from the fact that the number of data calls in the service stateis determined by the number of voice calls. w(i,j)k

(τ) is the sum ofdelay distribution of all possible paths for k service-completed voicecalls multiplied by the probability that each path is selected. Jk is aparameter to prevent w(i,j)k

(τ) from including the path for (k − 1)service-completed voice calls. Finally, we can get w(i,j)k

(τ) from theinverse Laplace transform of W(i,j)k

(s).Substituting w(i,j)k

(τ) into w(i,j)(τ) and then successively substi-tuting w(i,j)(τ) into G(t), the CDF of delay can be calculated as

Fd(t) =∑

(i,j)∈Ω(nd,acc)

P ′i,j

+∑

(i,j)∈Ω(nd,que)

t∫0

I∑k=0

L−1W(i,j)k

(s)· P ′

i,jdτ (35)

where L−1 denotes the inverse Laplace transform.For a delay-tolerant traffic, an important performance measure is

related with the delay requirement. Typically, the delay requirementof data calls is that the system should provide the required services tousers within the maximum tolerable delay. Considering that the servicebehavior is randomly characterized, we need to introduce the delayconfidence, which is defined as the probability that new data calls areaccepted within the maximum tolerable delay without being blocked,and further, we formulate the delay confidence as follows:

Pc ≡ (1 − Pbd) · Fd(τmax) (36)

where τmax is the maximum tolerable delay requirement. Here, notethat the delay confidence is related to not only the CDF but also theblocking probability of data calls.

As a system level performance measure, we utilize Erlang capacity,which is defined as a set of supportable offered traffic loads of voiceand data that can be supported while service requirements are satisfied,where we consider the required call blocking probability for voicecalls and the required delay confidence for data calls, as the servicerequirements. Then, the Erlang capacity of CDMA system supportingvoice and data services can be formulated as follows:

CErlang ≡(ρv, ρd)|Pbv ≤ Pbv,req , Pc ≥ Pcreq

(37)

where Pbv,req is the required blocking probability for voice calls,and Pcreq is the required delay confidence for data traffic. We alsodefine a set of supportable offered traffic loads of voice and datathat are confined by the required call blocking probability of voicePbv,req as the “voice-limited Erlang capacity” and that by the requireddelay confidence of data Pcreq as the “data-limited Erlang capacity,”respectively. Then, the Erlang capacity of the system is determined asthe overlapped region of the voice-limited Erlang and the data-limitedErlang.

TABLE ISYSTEM PARAMETERS FOR THE NUMERICAL EXAMPLE: A CDMA

SYSTEM SUPPORTING VOICE AND DELAY-TOLERABLE DATA SERVICES

Fig. 5. Voice-limited Erlang capacity and data-limited Erlang capacity with-out allowing the delay in queue. (i) and (ii) represent the voice-limited Erlangand the data-limited Erlang, and the Erlang capacity corresponds to the over-lapped region limited by (i) and (ii), where Pbv,req = 1%, Pcreq = 99%, andK = 0.

V. NUMERICAL EXAMPLE

For a numerical example, we consider a CDMA system supportingvoice and delay-tolerant data services. The considered parameters aresummarized in Table I and the normalized delay τn means that thedelay τ is normalized by average service time 1/µd. Fig. 5 showsthe voice-limited Erlang capacity and the data-limited Erlang capacitywhen K = 0. Lines (i) and (ii) represent the voice-limited Erlangcapacity and the data-limited Erlang capacity, respectively, wherePbv,req = 1% and Pcreq = 99%. The Erlang capacity is determinedas the overlapped region limited by lines (i) and (ii) to satisfy bothservice requirements for voice and data calls at the same time. Forthe case that there is no queue (K = 0), the CDF of delay at the


Fig. 6. Voice-limited Erlang capacity and data-limited Erlang capacity fordifferent values of the required delay confidence Pcreq , where Pbv,req = 1%,τnmax = 0.1, and K = 1.

maximum tolerable delay Fd(τmax) is given 1 such that the delayconfidence Pc becomes (1 − Pbd

), and the required delay confidenceof 99% corresponds to the required blocking probability of 1% ofdata. The Erlang capacity in Fig. 5 corresponds to that analyzed in [9]for the blocking probabilities for voice and data traffic. Fig. 5 showsthat the Erlang capacity is mainly determined by the data-limitedErlang capacity. The gap between the voice-limited Erlang capacityand the data-limited Erlang capacity comes from the difference in theservice requirements for voice and data calls. In this case, the data-limited Erlang capacity is lower than the voice-limited Erlang capacityfor the same blocking probability because a data call requires moresystem resources than a voice call. In reality, the data calls have adistinct characteristic that may allow some delay; however, it waspresented that the data calls behave like the voice calls, which areeventually blocked when there is no instantly available resource.

In order to increase the Erlang capacity, a proper tradeoff is requiredbetween the voice-limited Erlang capacity and the data-limited Erlangcapacity. One of the methods to get the tradeoff is to use queueingfor delay-tolerant data calls. Fig. 6 shows the voice-limited Erlangcapacity and the data-limited Erlang capacity for different values of therequired delay confidence Pcreq , where Pbv,req = 1%, τnmax = 0.1,and K = 1. In this case, a new data call can be queued into thefinite buffer of the size K until the required resources are available.From Fig. 6, we know that the Erlang capacity is mainly determinedby the data-limited Erlang capacity, where Pcreq = 99%. However,the data-limited Erlang capacity gradually increases as the requireddelay confidence Pcreq decreases such that the Erlang capacity islimited by the voice-limited Erlang capacity when Pcreq is given lessthan 98%.

Fig. 7 shows the voice-limited Erlang capacity and the data-limitedErlang capacity for different values of the maximum tolerable delayτnmax , where Pbv,req = 1%, Pcreq = 99%, and K = 1. The Erlangcapacity is mainly determined by the data-limited Erlang capacitywhen the maximum tolerable delay τnmax is less than 0.1. As τnmax

increases, i.e., when the delay allowance for data calls increases, thedata-limited Erlang capacity also increases. Fig. 7 shows that theErlang capacity is limited by the voice-limited Erlang capacity whenτnmax is more than 0.4.

Figs. 8–11 show the effect of the queue size on the Erlang capacity,where Pbv,req = 1%, Pcreq = 99%, and τnmax = 0.1. In the figures,

Fig. 7. Voice-limited Erlang capacity and data-limited Erlang capacity fordifferent values of the maximum tolerable delay τnmax , where Pbv,req = 1%,Pcreq = 99%, and K = 1.

Fig. 8. Effect of the queue size on the Erlang capacity, where Pbv,req =

1%, Pcreq = 99%, τnmax = 0.1, and K = 0, where the solid and dottedlines represent the voice-limited Erlang capacity and the data-limited Erlangcapacity, respectively.

the solid and dotted lines represent the voice-limited Erlang capacityand data-limited Erlang capacity, respectively. We know that the voice-limited Erlang capacity decreases as the queue size increases, whichcomes from the fact that the call blocking probability of voice increasesfor a larger queue size. On the other hand, the data-limited Erlangcapacity increases until the queue size becomes 2, and after that, itdecreases for a larger queue size. It comes from the fact that thecall blocking probability of data and the CDF of delay, respectively,decreases as the queue size increases. Noting that the delay confidencedepends on not only the blocking probability of data call decreases butalso the CDF of delay, we know that the variation of the data-limitedErlang capacity according to the queue size mainly comes from themutual effects between the improvement in the blocking probability ofdata calls and the decrease of the CDF of delay. Figs. 8–11 also showthat the Erlang capacity when K = 3 is less than that when K = 0,which means that the queue size should be properly selected to make






a balance between the voice-limited Erlang capacity and the data-limited Erlang capacity and, further, to accommodate more Erlangcapacity. We observe that the optimum queue size is 1 in the case ofthe numerical example, with respect to the Erlang capacity.

It will be useful to quantify the optimum size of the buffer fora general case while taking all the necessary model parameters intoaccount. However, it is very complicated to find out such generalsolution for a general case, particularly with an analytical fashion. Inthis paper, we only subsequently present an example study to show away to select the proper queue size for a specific case. The procedure,however, can be applied to any case.

VI. CONCLUSION

In this paper, we have analyzed the Erlang capacity of a CDMAsystem supporting voice and delay-tolerant data services by particu-



larly considering a new performance metric of delay-tolerant traffic,which is called delay confidence. For the performance analysis, wedeveloped a 2-D Markov model, which is based on the FCFS servicediscipline, and presented a numerical procedure to analyze the Erlangcapacity. As a result, for the case that there is no queue for data calls,it was observed that the Erlang capacity is mainly determined by thedata-limited Erlang capacity since one data call requires more systemresources than one voice call. For the case that we consider finite sizebuffer for data calls, the data-limited Erlang capacity increases as themaximum tolerable delay increases or the required delay confidencedecreases. Further, the Erlang capacity is mainly limited by the voice-limited Erlang capacity if the required delay confidence and themaximum tolerable delay requirements go beyond certain values. Byobserving the Erlang capacity according to the queue size, we showedthat the queue size should be properly selected to make a balancebetween the voice-limited Erlang capacity and the data-limited Erlangcapacity. For the numerical example case, we demonstrated that aproper queue size was selectable with respect to the Erlang capacityunder a given delay constraint.

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Pulse Distortion Caused by Cylinder Diffraction andIts Impact on UWB Communications

Chenming Zhou, Student Member, IEEE, andRobert C. Qiu, Senior Member, IEEE

Abstract—One of the characteristics of ultrawideband (UWB) signals ispulse distortion, which is inherently determined by their huge bandwidth.Using a cylinder model as an example, pulse distortion and its impactson UWB system performance have been investigated, based on the closed-form impulse response first derived in this paper. Although a lot of papershave addressed the pulse-distortion issue, quantifying the impacts of pulsedistortion on system performance appears to be novel. The simulationresults show that the SNR loss caused by template mismatch could reachas high as 4 dB. It is also found that the range error caused by pulsedistortion is much larger than the Cramer–Rao lower bound; thus, it isanother fundamental source of errors limiting the accuracy of times ofarrival of a received signal. These results have direct applications in timingsynchronization and positioning.

Index Terms—Cylinder diffraction, pulse distortion, ray tracing, timingand positioning, ultrawideband.

I. INTRODUCTION

Emerging applications of ultrawideband (UWB) are foreseenfor sensor networks that are critical to mobile computing [1],[2]. Such networks, combining low medium-rate communications(50 kb/s–1 Mb/s) over distances of 100 m with positioning capabilities,allow a new range of applications [1], including military applications,medical applications (monitoring of patients), family communications/supervision of children, search-and-rescue (communications with firefighters, or avalanche/earthquake victims), control-of-home applica-tions, logistics (package tracking), and security applications (localiz-ing authorized persons in high-security areas).

When a sensor is placed in different environments, a nonline-of-sight (NLOS) propagation is encountered very often, sometimesin military communications [3]. Sometimes, the propagation path is

Manuscript received August 16, 2005; revised March 20, 2006, August 30,2006, and September 11, 2006. This work was supported by the Office ofNaval Research under Grant N00014-07-1-0529, the National Science Foun-dation under Grant ECS-0622125, the Army Research Laboratory, and theArmy Research Office under STIR Grant W911NF-06-1-0349, and DURIPGrant W911NF-05-1-0111. The review of this paper was coordinated byProf. R. Janaswamy.

The authors are with the Center for Manufacturing Research, Departmentof Electrical and Computer Engineering, Tennessee Technological University,Cookeville, TN 38505 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.897640

blocked by objects that can be modeled by a cylinder [4], [10]. Forexample, when a hill is smooth and not covered by trees or houses, thediffraction process is described more accurately in terms of creepingrays [4]. The purpose of this paper is to model such an environment,analyzing the possible pulse distortion and its impacts on systemperformance.

To be mathematically tractable, a simple channel consisting of aperfectly electrically conducting (PEC) cylinder is considered. Thetransceivers are placed such that only diffracted rays are present at thereceiver. Noting that the research of UWB sensors is still in its earlystage, such a mathematically tractable physics-based channel model,although simple, may still bring us a lot of insight.

The work in this paper is different from previous works [5],[7]–[9] in several aspects. First, it is the first time for us to usethe cylinder model, which is mathematically difficult to deal with.Second, the study of pulse distortion from the point of rangingis also new. Third, the results are different from other canonicalstructures. Particularly, we found that the SNR loss and timing errorcaused by pulse distortion in the correlation-based receivers could besignificant and thus deserved special attention. It is thus shown thatphysical mechanisms can be naturally connected with the system-levelparameters, such as SNR, timing errors, etc. This unique featuremotivates the research behind this paper.

The rest of this paper is organized as follows. Section II willanalyze the impacts of pulse distortion on the system performance. Aclosed-form impulse response for a cylinder channel will be derivedin Section III, based on the well-known frequency-domain results.Some numerical results on pulse distortion and its impact on systemperformance will be shown in Section IV. Finally, Section V willconclude this paper.

II. PERFORMANCE DEGRADATION DUE TO PULSE DISTORTION

A. Impact of Pulse Distortion on System Performance

We follow the general system model and its performance expressionin the studies in [5] and [14]. When zero and one are sent withequal probability, the average error probability in the receiver can beexpressed as

Pe = Q(√

SNR) (1)

where Q(x) is defined as Q(x) =∫ ∞

x(1)/(

√2π) exp(−y2/2)dy,

and SNR is the signal-to-noise power ratio at the input to the thresholddevice. The square root of SNR is given by

√SNR =

(s0(t) ∗ q(t)) |t=T0 − (s1(t) ∗ q(t))|t=T0√2N0‖q‖

(2)

where s0(t) and s1(t) are the received signals, which can be singularbut with limited energy, and ∗ denotes convolution operation. While(2) is valid for any binary modulation, the antipodal modulation willbe assumed in this paper.

Let p0(t) and p1(t) denote the transmitted signals, and h(t) denotethe channel impulse response, then it follows that s0(t) = p0(t) ∗ h(t)and that s1(t) = p1(t) ∗ h(t). In (2), q(t) is the local template usedin the correlation-based receiver, and ‖q‖ = [

∫ ∞−∞ q2(t)dt]1/2 is the

norm of q(t). It is known that, for the optimum receiver, q(t) is selectedto be matched to the received signal si(t). However, sometimes, q(t)is matched to the transmitted waveform: p0(t) and p1(t). This impliesthat the signal waveforms will not change as they pass through the

0018-9545/$25.00 © 2007 IEEE

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1, MARCH 2011 7

© 2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119

ERLANG C FORMULA AND ITS USE IN THE CALL CENTERS

Erik CHROMY.1, Tibor MISUTH.1, Matej KAVACKY.1

1 Department of Telecommunications, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19 Bratislava, Slovak Republic

[email protected], [email protected], [email protected]

Abstract. This paper deals with calculation of important parameters of the Call Center using the Erlang C formula and then results are verified through simulations. Erlang C formula is defined as a function of two variables: the number of agents N and the load A. On the base of their values it is possible to determine the probability PC, that the incoming call will not be served immediately, but it will have to wait in the waiting queue. Simulations satisfy the assumption of Markov models.

Keywords Call Center, Erlang C Formula, Markov Models, Quality of Service.

1. Introduction Call Center is dynamical, technical system (package

of technical equipments – hardware, software and human sources) designed for effective connecting people with the requirements for service with operator or with systems able to satisfy their requirements. The core of the Call Center is Automatic Call Distribution (ACD).

Each of the components of the ACD can be described with some precision by means of mathematical tools and causalities. Since the ACD systems process a large number of incoming requests, the majority of models is based on the principles of mathematical statistics. The right choice of a statistical model is able to ensure the sufficient accuracy of the results. It is essential to describe the dependency of input variables and parameters that can greatly affect the accuracy of the results. The modeling of Call Center parameters is possible through Markov models, but also through Erlang formulas.

This paper deals with calculation of important parameters of the Call Center (which affect proper operation of such queuing system) using the Erlang C formula and then results are verified through simulations. These simulations satisfy the assumption of Markov models.

1.1 Erlang C formula and M/M/m/∞ model

Immediate rejection of call by reason of occupation of all agents (as expected in Erlang B formula) is in terms of provided services by Call Center inappropriate solution. This shortness is eliminated in second Erlang formula – Erlang C. In the case that the call cannot be served immediately, the call is placed into the waiting queue with unlimited length. If the release of one of the agents happens, it is automatically assigned to the following call from the queue. If the waiting queue is empty, the agent is free and he waits for next call.

Erlang C formula [5] is in the original form defined as a function of two variables: the number of agents N and the load A. On the base of their values it is possible to determine the probability PC (1), that the incoming call will not be served immediately, but it will have to wait in the waiting queue.

1

0 !!

!,

N

i

Ni

N

C

ANN

NA

i

A

ANN

NA

ANP , (1)

where

A . (2)

Now, we use the relationship between load A (2), the average number of calls per time λ and the average number of requests processed per time μ . Next, we define the variable η, which represents the load of 1 agent as [2, 3, 7]:

N . (3)

By substituting (2) and (3) into equation (1) we have:

8 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1, MARCH 2011


,

1!!

1!

!!

!

,1

01

0

N

i

Ni

N

N

i

Ni

N

C

N

N

i

N

N

N

NN

N

i

NN

N

NP

(4)

what corresponding with the relation for the probability that the request in queuing system M/M/m/∞ will be placed into the queue, i.e. in the system is more than m requests [2, 3, 7]. Now it is analytically derived, that the Erlang C model (1) and the Markov model M/M/m/∞ (4) are identical.

It is possible by using of the basic form for the Erlang C formula (1) to calculate the value of parameter A (maximum load) at a known number of agents N and the probability of waiting PC. Due to complexity of the analytical expression of these unknown parameters, numerical method for solving is used.

By adding the waiting queue into the system, many other parameter variables that can be monitored and also affected will appear. An important factor in terms of caller is queue waiting time. This value is random variable described by distribution function [7]:

0,1 ANCW ePF . (5)

Then it is possible to calculate the average queue waiting time W (average call waiting time in the queue before assigning a call to agent):

AN

PW C

. (6)

and by applying of Little theorem [7] and formula (2) we get the average number of requests in waiting queue as follows:

CC P

AN

AP

ANWQ

. (7)

By using of general definition of distribution function of any statistical distribution [4] and by applying its properties on the distribution function (5), we can derive the formula for calculation of the parameter GoS (Grade of Service) (percentage of calls, that are answered, or assigned to the agent before the defined threshold AWT – Acceptable Waiting Time) by known value of AWT [2, 6]:

AWTANC ePGoS 1 . (8)

The average number of requests in the system K (and also the average number of occupied lines) is [7]:

QAP

AN

AAPNK CC

1, (9)

where we get from Little theorem a value of the average time T, that the requirement spend in the system:

AN

PW

QAKT C

11

. (10)

Other important parameter is the average utilization of agents η [2]:

1

0

1

0

!!

!1

N

i

Ni

N

k

k

ANN

NA

i

A

Akn

kN

. (11)

Erlang C formula is the basis for the analysis of parameters and simulation of the call center. Its shortness is the assumption of the unlimited waiting queue. This is not a problem in terms of available storage capacity, and therefore the waiting queue could be potentially unlimited, but no real caller will wait too long. Therefore, the limitation of waiting period represents other parameter that is under consideration in the special Markov models.

1.2 M/M/m/M model with limited length of waiting queue

A limited number of requirements placed in the queue of queuing system can be described by Markov model M/M/m/M, where the maximum number of requests in the system M is greater or equal to the number of servers m=N [2].

The probability p0 [2], that in the system occurs exactly 0 requirements (i.e. empty) is defined as:

11

1

00

1!

1

!

N

AN

N

AA

k

Ap

NMN

N

k

k (12)

and then we can define the probability of the call rejection PB [2] as:

0

!p

NN

AP

NM

M

B (13)

and the probability, that the call will be assigned into the waiting queue PC [2] as:

0!

pN

AP

N

C . (14)

Furthermore, for the values Q [2] and K [2] can be calculated as

N

A

N

ANM

N

A

ANN

ApQ

NM

NMN

11

1)()!1(

1

2

10

, (15)

0!

1 pNN

AAQK

NM

M

. (16)



The time characteristics can be calculated on the basis of Little theorem [7].

These relations are relatively complicated. It is therefore possible a consideration, if the approximation using standard Erlang C formula is not sufficient, or what are the conditions, that such approximation is sufficiently exact. If the average number of requests in the system K will be far less than the limit M, it is possible to apply Erlang C formula without thinking of the maximum capacity. The more closely will be the value K to the limit, the less accurate results Erlang C formula will provide.

2. The principle of realized simulations The basic of the simulated algorithm consists of three

blocks:

set up of inputs,

traffic simulations,

processing of measured values and their presentation.

2.1 Modeling of inputs

The basic parameters are:

average number of incoming calls into the call center per time λ,

average time of call processing by agent 1/μ,

number of agents N.

The group of time parameters consists of:

the total length of simulated traffic in seconds TSIM,

time step of simulation TSTEP (by default 1 second),

time to steady state TNAB.

Vector with arrival times of each call based on the average number of requests per time unit and length of simulation is created by random generator. There is used a property about exponential distribution of length of the intervals between arrivals. Generated random variables are then tested by the statistical chi-quadrate [4]. There is test of vector consistency with exponential distribution on significance level α = 0,05. In the case of unsuccessful test, the entire vector is randomly generated once again. Number of generated calls is by 20 % higher than the average number of calls that should enter into the call center through the simulated period.

Furthermore, vector with service times of individual calls that will be allocated during the simulation, is generated for each agent based on his average service time.

All these operations are performed in advance by reason of high rate and efficiency of MATLAB by working with vectors and matrices. The generation of set of values

is then faster than a gradual generation of individual values during the simulation.

2.2 Traffic simulation

The core of simulation is realized as a cycle, in which the each iteration represents one time step in simulation. In the each iteration is the vector of incoming calls compared with actual time and if it is necessary, the call enters into the simulated call center in proper timer. Concurrently the time of incoming call is recorded for later calculations. Call is placed into the waiting queue or directly assigned to a free agent. The time index is stored for later calculations (average waiting time) at the moment of call assignment.

Furthermore, in the each iteration the status and the occupancy of all agents are checked. If any of the agents is free, it will be assigned the first call from the waiting queue. If more agents are free, the call is assigned to agent that didn’t work for the longest time. If there is not a call in the waiting queue, the agent remains as free and expects the arrival of another request into the system.

During the one iteration there is possibility of entry into the system and also assignment to the agent of more than one call simultaneously. The number of calls in the system is stored in each step so then it is possible to determine the number of calls in the waiting queue.

The mathematical model of queuing system is defined for the steady state. It means, that in the moment of parameters monitoring, the simulation runs infinitely long time.

The run of the simulation is terminated after predefined simulation time. Only the calls that are terminated (and are thus served by agent) before expiration of the simulated time are included into the statistics.

2.3 Processing of the simulation results

This phase of the algorithm ensures the processing of all measured data during the simulation (time of events generation, number of calls in the system, agent occupancy, …). Based on these the particular parameters monitored in the call center are calculated. Thus obtained simulation results can be then compared with the expected values obtained by calculations through the mathematical model.

The general variables available from the results of the simulated model are:

real length of the simulation (calculation time),

number of simulated calls,

average service time per one call (1/μ),

average number of calls in the system (K),

average time spent by user in the system (T),



number calls placed in the waiting queue,

probability of blocking PB, respectively probability of waiting PC,

average utilization of agents η.

3. Calculations using the Erlang C model Erlang C model works in the basic form with 3 input

parameters (A, N, PC). By use of relation (2) it is possible to divide the value of load A (generated by incoming calls) into 2 components: λ and 1/μ. Calculator thus always works with 4 values, while 3 of them act as input parameters and the last parameter is the output parameter. By adding of the waiting queue we can obtain a set of new parameters that can be calculated and then compared with the simulation results:

average number of requests in the waiting queue Q (using (7)),

average number of requests in the call center K (using (9)),

average waiting time in the waiting queue W (using (6)),

average time spent in the call center T (using (10)),

value of GoS for AWT=20 s (using (8)),

average utilization of agents η (using (11)).

Probability of insertion into waiting queue PC can be entered in three different variants:

direct entry of value PC,

entry of average waiting time W,

entry of GoS.

When using a PC as input variable, this variable is considered to the upper limit.

3.1 Calculation of the parameter Pc

The calculation of the unknown value of probability of waiting can be easily realized by basic relation for Erlang C model (1). The dividing of two large number could lead to numerical errors and the obtained result might not be accurate. Therefore we derived an alternative relation, that is identical to the original (1) (in terms of result):

N

i

i

k

C

A

kN

N

ANANP

1

1

0

1

1, . (17)

Similarly, we can use Horner scheme [8] also for more efficient calculations.

3.2 Calculation of the parameter N

Analytical calculation according to (1) respectively (17) would be very difficult. The solution is therefore through a numerical method. The easiest technique is to gradually increase of the number of agents and continuous checking the stop conditions. This depends on the form, in which the input value PC is inserted (thus direct, or as W, or as GoS). At the moment, where the current value in the calculation is less or equal to the required value, the necessary number of agents is found. The implementation uses this idea, but applying of the relation (17) in every step of the calculation it is possible to use the current result obtained from the previous iteration cycle for the lower value of N. So, this is very quickly method to find the unknown value N.

3.3 Calculation of the parameter λ or 1/μ

The calculation of one of these unknown quantities in terms of Erlang C model means the find out of the load A, that can the system process at the specified parameters N and PC (respectively W and GoS parameters). Consequently, we can calculate the second one by applying the formula (2) and one of the value λ and 1/μ.

In terms of implementation, the finding of the unknown value A is the most difficult of all three combinations. As the solution we can use the feature, that allows to search the value of unknown x, for which it holds f(x) = 0. In this case, the function f(x) for input PC is:

0),()( _ INPUTCC PNxPxf . (18)

If the input value is defined as the average waiting time W, respectively GoS, we can use the following substitutions by (6) and (10):

ANWPC , (19)

AWTANCe

GoSP

1. (20)

In both cases we have the value μ. Therefore, if we need the solution for the unknown value 1/ μ, we must find the solution for the value μ and only then to calculate the load A by (2). The functions f(x) for input W (by use of substitution (19)) are:

0),()( xNWNxPxf C , (21)

respectively, if μ is unknown, then:

03600

,3600

)(

xN

x

WN

xPxf C

. (22)

If the input parameter is GoS, the substitution (20) is used and the function for parameter λ is:

01

,)(

AWTxNC

e

GoSNxPxf , (23)

respectively, if μ is unknown, then:



01

,3600

)(

3600

AWT

xNx

C

e

GoSN

xPxf

. (24)

4. Simulations The whole process of simulation for Erlang C model

is shown in the following fig. 1:

Fig. 1. Process of simulation for Erlang C model

4.1 Simulation results

The parameters of simulated model are:

667 incoming calls per 1 hour,

average call service time is 150 seconds,

simulation time of the call center work is 30 hours with 1 second step (run to steady state is 1 hour).

The calls that cannot be processed immediately are placed into the waiting queue (according to Erlang C formula). For the system stability it is necessary to satisfy the condition A < N. Therefore the calculations and simulations are realized for the number of agents in the range of 28 to 37. In this range are the most notable changes observed in output parameters. The table 1 and table 2 shows the results obtained by calculation using the

mathematical model and results also obtained by simulations.

Tab. 1. The results of calculations

N Pc

[%] K T [s] Q

W [s]

GoS [%]

η [%]

28 95,4 155 836,6 127,2 686,6 7,2 99,3 29 75,3 45,1 243,5 17,3 93,5 35,9 95,8 30 58,7 35,2 189,9 7,4 39,9 56,3 92,6 31 45,1 31,7 171,1 3,9 21,1 70,6 89,7 32 34,1 30 162,1 2,2 12,1 80,6 86,8 33 25,3 29,1 157,3 1,4 7,3 87,3 84,2 34 18,5 28,6 154,5 0,8 4,5 91,9 81,7 35 13,3 28,3 152,8 0,5 2,8 94,9 79,4 36 9,4 28,1 151,7 0,3 1,7 96,8 77,2 37 6,5 28 151,1 0,2 1,1 98,1 75,1

Tab. 2. The results of Erlang C model simulations

N Pc

[%] K T [s] Q

W [s]

GoS [%]

η [%]

28 91,9 88,5 478 60,9 328 12,3 98,8 29 74 43,2 233 15,4 83,4 38 95,6 30 59,3 35,1 189 7,2 38,8 55,9 93 31 46,2 32,4 174 4,5 24,1 69,3 90 32 33,7 30 162 2,2 11,9 81,3 86,9 33 25,1 29,2 158 1,4 7,3 87,5 84,4 34 19,2 28,8 155 0,9 4,7 91,5 82,1 35 13 28,3 153 0,5 2,5 95,5 79,4 36 8,7 28 151 0,3 1,5 97,4 77,1 37 6,4 28,1 151 0,2 1 98,2 75,3

Obtained simulation results and calculations are very similar. The differences exist only in the case of the minimum number of agents. However, it is probably that any company will not carry on the call center with extremely poor quality of service [9, 10, 11, 12] delivering (mainly the very long waiting time).

From the caller point of view, the most important parameters are the average waiting time in the waiting queue W and parameter GoS (this case is evaluated for AWT=20 seconds). The caller expects the lowest value of W and also the value GoS, which is close to 100%. From the call center point of view, the most important parameters are the number of agents N and their utilization η, because these two variables significantly affect the financial demands of service. The aim of the operator is to minimize the number of agents and to maximize their utilization. The aim of the analysis is therefore to find such a minimum number of agents N, when the operation parameters are yet on the sufficient level. As suitable we can consider the GoS parameter on level 80% and the average waiting time about 10seconds.



Fig. 2. Simulation results

The fig. 2 shows the characteristic curve of the important call center simulation results according to Erlang C model assumptions in relation with the number of agents N. The characteristic curve of the average waiting time has a very strong exponential character and thus a slight increase of the number of agent (about 1 to 2 agents) can bring a significant improvement of this parameter. A similar, even though less aggressive, is the characteristic curve of the probability PC. GoS parameter also exponentially converges to the level 100% and we can see that a small change in the number of agents can bring significant improvement. The characteristic curve of agents load η is in the displayed range almost linear.

According to the above mentioned requirements on the provided quality of service by the call center is in this case possible to consider 32 agents as sufficient. By adding two agents it is possible to shorten the average waiting time by half and to increase the value of GoS parameter at 10% on very decent level (90%). The utilization of agents does not decrease below 80% and therefore it does not create unnecessarily long pauses, when the agents were redundant.

5. Conclusion Based on calculations and simulations it can be stated,

that in term of simplicity and accuracy of obtained results Erlang C formula is applicable for call center simulations. However, its shortness is the possibility of calculations only for one service group, and also the need to define for all agents the same service time.

Despite these limitations, it is possible to use the basic Erlang C formula also for calculations for call center with several service groups. In this case when it is possible to determine the probability that calls are routed to the individual service groups, then it is possible to use the basic Erlang C formula for calculations for each service group individually. The number of incoming calls per time unit is the aliquot portion of the total number of incoming calls into the call center. The Erlang C formula calculation

can be used in the case of different performance of agents. It is possible to determine the value of the average call processing time or average number of the processed calls per time unit by dividing the total number of processed calls per time unit of all agents and the total number of agents in service group.

For the purpose of further study it would be interesting to expand the simulations by more independent service groups at the same time and the random distribution of call between them according to defined probabilities. Another interesting possibility could be different average service time for individual agents. Finally there is possibility to simulate the impact of unequal performance of agents on the results of the whole call center.

Acknowledgments

This work is a part of research activities conducted at Slovak University of Technology Bratislava, Faculty of Electrical Engineering and Information Technology, Department of Telecommunications, within the scope of the projects VEGA No. 1/0565/09 „Modeling of traffic parameters in NGN telecommunication networks and services“ and ITMS 26240120029 “Support for Building of Centre of Excellence for SMART technologies, systems and services II”.

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About Authors ... Erik CHROMÝ was born in Veľký Krtíš, Slovakia, in 1981. He received the Master degree in telecommunications in 2005 from Faculty of Electrical Engineering and Information Technology of Slovak University of Technology (FEI STU) Bratislava. In 2007 he submitted PhD work

from the field of Observation of statistical properties of input flow of traffic sources on virtual paths dimensioning and his scientific research is focused on optimizing of processes in convergent networks. Nowadays he works as assistant professor at the Department of Telecommunications of FEI STU Bratislava.

Tibor MIŠUTH is a student of PhD. study at Department of Telecommunications, Faculty of Electrical Engineering and Information Technology of Slovak University of Technology Bratislava. He focuses on application of Erlangs' equations both in

classic telecommunication networks and modern IP networks.

Matej KAVACKÝ was born in Nitra, Slovakia, in 1979. He received the Master degree in telecommunications in 2004 from Faculty of Electrical Engineering and Information Technology of Slovak University of Technology (FEI STU) Bratislava. In 2006 he submitted PhD

work “Quality of Service in Broadband Networks”. Nowadays he works as assistant professor at the Department of Telecommunications of FEI STU Bratislava and his scientific research is focused on the field of quality of service and private telecommunication networks.

probabilidad de erlang

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