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Transition Probabilities for Telephone Traffic

01 September 1960

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We shall study a stochastic process {iV(0, t ^ 0), which is a mathematical model for the occupancy of N service facilities, with no provisions for delays. For example, N(t) can he interpreted as the number of (fully accessible) telephone channels (trunks) out of a group of N such in use at time t, with lost calls cleared. Also, we can think of N(t) as the number of items on order at time t in an idealized inventory situation in which at most N items can be on order at one time (see Arrow, Karl in and Scarf1). Throughout the paper we use terminology appropriate to an application to telephone trnnking. The process N(t) is determined by the following assumptions: i. Holding times of trunks are independent, each with the same negative exponential distribution function, of mean 7' , 7 being the "hang-up rate." ii. Times between successive attempts to place a call (interarrival times) are independent ; each has the distribution function A{ -), where / ! ( · ) is arbitrary except for the condition A(0) = 0. This assumption covers Poisson arrivals as a special case. The mean of ^4(-)> when it exists, is denoted by MI · * This work was completed while the author was visiting lecturer at Dartmouth College, 1959-60. 1297