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Transient analysis of the Erlang A model

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  • Charles Knessl
  • Johan Leeuwaarden

Abstract

We consider the Erlang A model, or $$M/M/m+M$$ M / M / m + M queue, with Poisson arrivals, exponential service times, and m parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth–death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green’s functions and contour integrals related to hypergeometric functions. Our results are specialized to the $$M/M/\infty $$ M / M / ∞ queue, the M / M / m queue, and the M / M / m / m loss model. We also obtain some corresponding results for diffusion approximations to these models. Copyright The Author(s) 2015

Suggested Citation

  • Charles Knessl & Johan Leeuwaarden, 2015. "Transient analysis of the Erlang A model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 143-173, October.
    • Handle: RePEc:spr:mathme:v:82:y:2015:i:2:p:143-173
    DOI: 10.1007/s00186-015-0498-9
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    References listed on IDEAS

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    1. Ward Whitt, 2005. "Engineering Solution of a Basic Call-Center Model," Management Science, INFORMS, vol. 51(2), pages 221-235, February.
    2. Ward Whitt, 2006. "Sensitivity of Performance in the Erlang-A Queueing Model to Changes in the Model Parameters," Operations Research, INFORMS, vol. 54(2), pages 247-260, April.
    3. Ward Whitt, 2004. "Efficiency-Driven Heavy-Traffic Approximations for Many-Server Queues with Abandonments," Management Science, INFORMS, vol. 50(10), pages 1449-1461, October.
    4. Thomas L. Saaty, 1960. "Time-Dependent Solution of the Many-Server Poisson Queue," Operations Research, INFORMS, vol. 8(6), pages 755-772, December.
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