Level Crossings in Point Processes Applied to Queues: Single-Server Case

P. H. Brill and M. J. M. Posner
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P. H. Brill: University of Toronto, Toronto, Ontario
M. J. M. Posner: University of Toronto, Toronto, Ontario

Operations Research, 1977, vol. 25, issue 4, 662-674

Abstract: This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

Date: 1977
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:25:y:1977:i:4:p:662-674

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