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June, 1956 A Waiting Line Process of Markov Type
A. B. Clarke
Ann. Math. Statist. 27(2): 452-459 (June, 1956). DOI: 10.1214/aoms/1177728268


Waiting-line or queuing processes of the Markov type are studied, the incoming traffic being of Poisson type and having negative-exponential holding time. The parameters are allowed to depend on time. The problem of finding an exact solution for the probability distribution of the waiting-line length as a function of time is reduced to the solution of an integral equation of the Volterra type. When the ratio of the parameters for the incoming and out-going traffic is constant, this equation can be solved explicitly and the required distribution obtained. Using this solution, the behavior of the process for large values of $t$ is studied, particularly for the unstable case with traffic intensity $\geqq 1$.


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A. B. Clarke. "A Waiting Line Process of Markov Type." Ann. Math. Statist. 27 (2) 452 - 459, June, 1956.


Published: June, 1956
First available in Project Euclid: 28 April 2007

zbMATH: 0073.13002
MathSciNet: MR79857
Digital Object Identifier: 10.1214/aoms/1177728268

Rights: Copyright © 1956 Institute of Mathematical Statistics

Vol.27 • No. 2 • June, 1956
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