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Time-dependent analysis of an M / M / c preemptive priority system with two priority classes

Author

Listed:
  • Jori Selen

    (Eindhoven University of Technology
    Eindhoven University of Technology)

  • Brian Fralix

    (Clemson University)

Abstract

We analyze the time-dependent behavior of an M / M / c priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least c high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $$c - 1$$ c - 1 high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami’s formula from the theory of M / G / 1-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution; these results seem to yield the most explicit expressions known to date.

Suggested Citation

  • Jori Selen & Brian Fralix, 2017. "Time-dependent analysis of an M / M / c preemptive priority system with two priority classes," Queueing Systems: Theory and Applications, Springer, vol. 87(3), pages 379-415, December.
    • Handle: RePEc:spr:queues:v:87:y:2017:i:3:d:10.1007_s11134-017-9541-2
    DOI: 10.1007/s11134-017-9541-2
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    References listed on IDEAS

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    1. Douglas R. Miller, 1981. "Computation of Steady-State Probabilities for M / M /1 Priority Queues," Operations Research, INFORMS, vol. 29(5), pages 945-958, October.
    2. N. K. Jaiswal, 1961. "Preemptive Resume Priority Queue," Operations Research, INFORMS, vol. 9(5), pages 732-742, October.
    3. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    4. Fralix, Brian, 2015. "When are two Markov chains similar?," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 199-203.
    5. Alan Cobham, 1954. "Priority Assignment in Waiting Line Problems," Operations Research, INFORMS, vol. 2(1), pages 70-76, February.
    6. Jianfu Wang & Opher Baron & Alan Scheller-Wolf, 2015. "M/M/c Queue with Two Priority Classes," Operations Research, INFORMS, vol. 63(3), pages 733-749, June.
    7. Richard H. Davis, 1966. "Waiting-Time Distribution of a Multi-Server, Priority Queuing System," Operations Research, INFORMS, vol. 14(1), pages 133-136, February.
    8. Mor Harchol-Balter & Takayuki Osogami & Alan Scheller-Wolf & Adam Wierman, 2005. "Multi-Server Queueing Systems with Multiple Priority Classes," Queueing Systems: Theory and Applications, Springer, vol. 51(3), pages 331-360, December.
    9. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
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    Cited by:

    1. Brian Fralix, 2018. "A new look at a smart polling model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 339-367, December.
    2. Wenqing Wu & Yuanyuan Zhang, 2020. "Analysis of a Markovian queue with customer interjections and finite buffer," OPSEARCH, Springer;Operational Research Society of India, vol. 57(2), pages 301-319, June.

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