Markov Processes Representing Traffic in Connecting Networks

A class of Markov stochastic processes xt , suitable as models for random traffic in connecting networks with blocked calls cleared, is described and analyzed. These models take into account the structure of the connecting network, the set S of its permitted states, the random epochs at which new calls are attempted and calls in progress are ended, and the method used for routing calls. The probability of blocking, or the fraction of blocked attempts, is defined in a rigorous way as the stochastic limit of a ratio of counter readings, and a formula for it is given in terms of the stationary probability vector p of xt . This formula is ( p , P) 2 or xtS PxQz PxOLx where (3Z is the number of blocked idle inlet-outlet pairs in state x, and ax is the number of idle inlet-outlet pairs in state x. On the basis of this formula, it is shown that in some cases a simple algebraic relationship exists between the blocking probability b, the traffic parameter X (the calling rate per idle inlet-outlet pair), the mean m of the load, carried, and the variance a 1 of the load carried. For a one-sided connecting network of T inlets (= outlets), this relation is 1 r 1 -- o = 2m -- (T -- 2m) + 4a 2' X (T - 2m) 2 for a two-sided network with N inlets on one side and M outlets on the other, it is (N - m) (M 2795 - m) + a 2' 2796 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1963 The problem of calculating the vector p of stationary state probabilities is fully resolved in principle by three explicit formulas for the components of p: a determinant formula, a sum of products along paths on S, and an expansion in a power series around any point X ^ 0.