Fluctuations of Telephone Traffic 



By V. E. BENES 



(Manuscript received November 9, 1956) 



The number of calls in progress in a simple telephone exchange model 

 characterized by unlimited call capacity, a general probability density of 

 holding-time, and randomly arriving calls is defined as N{t). A formula, 

 due to Riordan, for the generating function of the transition probabilities 

 of A^{t) is proved. From this generating function, expressions for the co- 

 variance function of Nif) and for the spectral density of N{t) are determined. 

 It is noted that the distributions of N(t) are completely specified by the co- 

 variance function. 



I INTRODUCTION 



The aim of this paper is to study the average fluctuations of telephone 

 traffic in an exchange, by means of a simple mathematical model to 

 which we apply concepts used in the theory of stochastic processes and 

 in the analysis of noise. 



The mathematical model we use is based on the following assumptions : 

 (1) requests for telephone service arise indi^'idually and collectively at 

 random at an average rate of a per second; (2) the holding-times of 

 calls are mutually independent random variables having the common 

 probability density function h{u); and (3) the capacity of the exchange 

 is effectively unlimited, and no call is blocked or delayed by lack of 

 equipment. This telephone exchange model has been described by J. 

 Riordan.^ 



As a measure of traffic, it is natural to use the number of calls in prog- 

 ress in the exchange. We are thus led to consider a random step-function 

 of time N(t), defined as the number of calls in progress at time t. iV(/) 

 fluctuates about an average in a manner depending on the calling-rate, 

 a, and the holding-time density, h{u). 



II PROOF OF RIORDAX'S FORMULA FOR TRANSITION PROBABILITIES 



Let Pm.n(t) be the probability that w calls are in progress at t if m 

 calls were in progress at 0. Define the generating function of these prob- 



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