940 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



The original work on this particular model for telephone traffic is in 

 Palm,^ and Palm's results have been reported by Feller^ and Jensen.* 

 The results have been extended heuristically to arbitrar}^ absolutely con- 

 tinuous holding-time distributions by Riordan," following some ideas of 

 Newland* suggested by S. 0. Rice. 



Let Pijit) be the probability that there are j trunks busy at t if there 

 were i busy at 0. And let Pi{t, x) be the generating function of these 

 probabilities, defined by 



PAUx) = T^x^PiM)- 



Then Palm^ has shown (pp. 56 et seq.) that 



PiiU x) = [1 + (.r - 1) e-''X exp {(.r - \)ah (1 - e"^')}. 



This is formula (12) of Riordan" with his g replaced by e""^*. It can be 

 verified that the random variable N{t) is jMarkovian; the limit of Piit, x) 

 as ^ ^ 2c is 



exp !(.r - 1) ah], 



so that the equilibrium distribution of the numl)er of trunks in use is a 

 Poisson distribution with mean h = ah. The shifted random variable 

 [N{t) — h] then has mean zero, and covariance function 6e~^'. 



For additional work on this model the reader is referred to F. W. 

 Rabe,^° and to H. Stormer.^' 



Ill DISCUSSION OF THE MODEL 



Let us envisage the operation of the model we have described by con- 

 sidering the random variable N{t) ecjual to the number of trunks busy 

 at time t. As a random function of time, N{t) jumps up one unit step each 

 time a demand for service occurs, and it jumps down one vmit step each 

 time a con^-ersation ends'; If N{t) reaches zero, it stays there until there 

 is another demand for service. If N{t) = ?t, the probabilit}' that a con- 

 versation ends in the next small time-interval M is 



h-yM + o(A0, 



because the n conversations are mutually independent. A graph of a 

 sample of A^(0 is shown in Fig. 1. 



The model we described departs from realitj^ in several important 

 ways, which it is well to discuss. First, the assumption that the number 

 of trunks is infinite is not realistic, and is justified only by the mathe- 

 matical complication which results when we assume the number of trunks 



