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

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  • Don Towsley

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.

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  • Don Towsley, 1987. "An application of the reflection principle to the transient analysis of the M/M/1 queue," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(3), pages 451-456, June.
    • Handle: RePEc:wly:navres:v:34:y:1987:i:3:p:451-456
    DOI: 10.1002/1520-6750(198706)34:33.0.CO;2-8
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    References listed on IDEAS

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    1. Claude Dennis Pegden & Matthew Rosenshine, 1982. "Some New Results for the M/M/1 Queue," Management Science, INFORMS, vol. 28(7), pages 821-828, July.
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    Cited by:

    1. Seki, Yoichi & Hoshino, Naoto, 1999. "Transient behavior of a single-stage kanban system based on the queueing model," International Journal of Production Economics, Elsevier, vol. 60(1), pages 369-374, April.

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