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A Markov-modulated M/G/1 queue I: Stationary distribution

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Abstract

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.

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Authors and Affiliations

  1. Department of Engineering Sciences and Applied Mathematics, Northwestern University, 60201, Evanston, IL, USA

    C. Knessl & B. J. Matkowsky

  2. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Z. Schuss

  3. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 60680, Chicago, IL, USA

    C. Tier

Authors
  1. C. Knessl
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  2. B. J. Matkowsky
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  3. Z. Schuss
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  4. C. Tier
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Additional information

This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.

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Knessl, C., Matkowsky, B.J., Schuss, Z. et al. A Markov-modulated M/G/1 queue I: Stationary distribution. Queueing Syst 1, 355–374 (1987). https://doi.org/10.1007/BF01150670

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  • DOI: https://doi.org/10.1007/BF01150670

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