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M/M/c Queue with Two Priority Classes

Author

Listed:
  • Jianfu Wang

    (Nanyang Business School, Nanyang Technological University, Singapore, 639798)

  • Opher Baron

    (Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada)

  • Alan Scheller-Wolf

    (Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

Abstract

This paper provides the first exact analysis of a preemptive M/M/c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the two-dimensionally infinite Markov chain (MC), representing the two class state space, into a one-dimensionally infinite MC, from which the generating function (GF) of the number of low-priority jobs can be derived in closed form . (The high-priority jobs form a simple M/M/c system and are thus easy to solve.) We demonstrate our methodology for the c = 1, 2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c ≥ 2. Numerical examples demonstrate the accuracy of our algorithm and generate insights on (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compared with one having fewer fast servers; and (iii) the validity of the square root staffing rule in maintaining a fixed service level for the low-priority class. Finally, we demonstrate the potential of our methodology to solve other problems using the M/M/c queue with two priority classes, where the high-priority class is completely impatient.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:63:y:2015:i:3:p:733-749
    DOI: 10.1287/opre.2015.1375
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    References listed on IDEAS

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    9. 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.
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    Cited by:

    1. Gabor, Adriana F. & van Vianen, Lars A. & Yang, Guangyuan & Axsäter, Sven, 2018. "A base-stock inventory model with service differentiation and response time guarantees," European Journal of Operational Research, Elsevier, vol. 269(3), pages 900-908.
    2. van Vianen, L.A. & Gabor, A.F. & van Ommeren, J.C.W., 2016. "Waiting times in classical priority queues via elementary lattice path counting," Econometric Institute Research Papers EI2016-17, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Opher Baron & Antonis Economou & Athanasia Manou, 2018. "The state-dependent M / G / 1 queue with orbit," Queueing Systems: Theory and Applications, Springer, vol. 90(1), pages 89-123, October.
    4. 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.
    5. Gabor, A.F. & van Vianen, L.A. & Yang, G. & Axsäter, S., 2016. "Enabling customer satisfaction and stock reduction through service differentiation with response time guarantees," Econometric Institute Research Papers EI2016-13, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.

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