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Threshold properties of the M/M/1 queue under T-policy with applications

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  • Zhang, Xuelu
  • Wang, Jinting
  • Do, Tien Van

Abstract

In this paper, we consider a single server queueing system with a threshold control policy and its application for controlling the energy consumption of a computing server. If the number of customers in the system is less than a threshold, the service rate is set in a low value and it also can be switched to a high value once the number of customers reaches the threshold. We study the monotonicity, convexity or concavity properties of the key performance measures of the system such as the steady-state probability distribution, the expected number of customers in the system and in the queue, the expected sojourn time in the system and the waiting time in the queue, with respect to the threshold. Based on these properties, we further study a real-life problem regarding the energy consumption of a computing server. Numerical results show that the state-dependent service policy is a promising technique to balance the energy consumption and the quality of service of a computing server.

Suggested Citation

  • Zhang, Xuelu & Wang, Jinting & Do, Tien Van, 2015. "Threshold properties of the M/M/1 queue under T-policy with applications," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 284-301.
    • Handle: RePEc:eee:apmaco:v:261:y:2015:i:c:p:284-301
    DOI: 10.1016/j.amc.2015.03.109
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    References listed on IDEAS

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    1. Arie Harel, 1990. "Convexity Properties of the Erlang Loss Formula," Operations Research, INFORMS, vol. 38(3), pages 499-505, June.
    2. K. R. Balachandran, 1973. "Control Policies for a Single Server System," Management Science, INFORMS, vol. 19(9), pages 1013-1018, May.
    3. Daniel P. Heyman, 1977. "The T-Policy for the M/G/1 Queue," Management Science, INFORMS, vol. 23(7), pages 775-778, March.
    4. Arie Harel, 1988. "Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas," Management Science, INFORMS, vol. 34(8), pages 959-972, August.
    5. Arie Harel, 1988. "Erratum to: Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas," Management Science, INFORMS, vol. 34(10), pages 1277-1277, October.
    6. Alan J. Rolfe, 1971. "A Note on Marginal Allocation in Multiple-Server Service Systems," Management Science, INFORMS, vol. 17(9), pages 656-658, May.
    7. M. E. Dyer & L. G. Proll, 1977. "Note--On the Validity of Marginal Analysis for Allocating Servers in M/M/c Queues," Management Science, INFORMS, vol. 23(9), pages 1019-1022, May.
    8. Lotfi Tadj & Gautam Choudhury, 2005. "Optimal design and control of queues," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 359-412, December.
    9. Per Hokstad, 1978. "Approximations for the M / G / m Queue," Operations Research, INFORMS, vol. 26(3), pages 510-523, June.
    10. Pengfei Guo & Refael Hassin, 2011. "Strategic Behavior and Social Optimization in Markovian Vacation Queues," Operations Research, INFORMS, vol. 59(4), pages 986-997, August.
    11. Arie Harel & Paul H. Zipkin, 1987. "Strong Convexity Results for Queueing Systems," Operations Research, INFORMS, vol. 35(3), pages 405-418, June.
    12. Zhang, Zhe George & Tadj, Lotfi & Bounkhel, Messaoud, 2011. "Cost evaluation in M/G/1 queue with T-policy revisited, technical note," European Journal of Operational Research, Elsevier, vol. 214(3), pages 814-817, November.
    13. Arie Harel, 2011. "Convexity Results for the Erlang Delay and Loss Formulae When the Server Utilization Is Held Constant," Operations Research, INFORMS, vol. 59(6), pages 1420-1426, December.
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    Cited by:

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