Time-Dependent Solution of the Many-Server Poisson Queue

Abstract

In this paper we obtain the Laplace transform of the transient probabilities of the ordered queuing problem, with Poisson inputs, multiple channels, and exponential service times. Explicit expressions are derived for the two-channel case and known equilibrium conditions are shown to hold. The proof proceeds in two stages. The first obtains the Laplace transform of the generating function of the system and the second solves a first order linear partial differential equation in a restricted generating function introduced to determine the Laplace transform of the probability functions appearing in the first generating function. In the two-channel case the solution is decomposed into two components, one of which is immediately related to the well-known solution of the single-channel queue. We also study the problem with different service distributions for the two-channel case and compute the distribution of a busy period for that case.

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