An application of the reflection principle to the transient analysis of the M/M/1 queue

Don Towsley

Naval Research Logistics (NRL), 1987, vol. 34, issue 3, 451-456

Abstract: This paper applies the well‐known reflection principle for random walks to the analysis of the transient M/M/1 queueing system. A closed‐form solution is obtained for the probability that exactly i arrivals and j departures occur over an interval of length t in an M/M/1 queueing system that contains n users at the beginning of the interval. The derivation of this probability is based on the calculation of the number of paths between two points in a two‐dimensional −y coordinate system that lie above the x axis and touch the x axis exactly r times. This calculation is readily performed through the application of the reflection principle.

Date: 1987
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https://doi.org/10.1002/1520-6750(198706)34:33.0.CO;2-8

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Persistent link: https://EconPapers.repec.org/RePEc:wly:navres:v:34:y:1987:i:3:p:451-456

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