Fluctuations of telephone traffic

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 N(t) is proved. From this generating function, expressions for the co-variance function of N(t) 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.