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Controlling the workload of M/G/1 queues via the q-policy

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  • Fajardo, Val Andrei
  • Drekic, Steve

Abstract

We consider a single-server queueing system with Poisson arrivals and generally distributed service times. To systematically control the workload of the queue, we define for each busy period an associated timer process, {R(t), t ≥ 0}, where R(t) represents the time remaining before the system is closed to potential arrivals. The process {R(t), t ≥ 0} is similar to the well-known workload process, in that it decreases at unit rate and consists of up-jumps at the arrival instants of admitted customers. However, if X represents the service requirement of an admitted customer, then the magnitude of the up-jump for the timer process occurring at the arrival instant of this customer is (1 − q)X for a fixed q ∈ [0, 1]. Consequently, there will be an instant in time within the busy period when the timer process hits level zero, at which point the system immediately closes and will remain closed until the end of the current busy period. We refer to this particular blocking policy as the q-policy. In this paper, we employ a level crossing analysis to derive the Laplace–Stieltjes transform (LST) of the steady-state waiting time distribution of serviceable customers. We conclude the paper with a numerical example which shows that controlling arrivals in this fashion can be beneficial.

Suggested Citation

  • Fajardo, Val Andrei & Drekic, Steve, 2015. "Controlling the workload of M/G/1 queues via the q-policy," European Journal of Operational Research, Elsevier, vol. 243(2), pages 607-617.
    • Handle: RePEc:eee:ejores:v:243:y:2015:i:2:p:607-617
    DOI: 10.1016/j.ejor.2014.12.036
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    References listed on IDEAS

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    1. Uri Yechiali, 1971. "On Optimal Balking Rules and Toll Charges in the GI / M /1 Queuing Process," Operations Research, INFORMS, vol. 19(2), pages 349-370, April.
    2. Doshi, Bharat T., 1977. "Continuous time control of the arrival process in an m/g/1 queue," Stochastic Processes and their Applications, Elsevier, vol. 5(3), pages 265-284, July.
    3. Shaler Stidham, 2002. "Analysis, Design, and Control of Queueing Systems," Operations Research, INFORMS, vol. 50(1), pages 197-216, February.
    4. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
    5. Knudsen, Niels Chr, 1972. "Individual and Social Optimization in a Multiserver Queue with a General Cost-Benefit Structure," Econometrica, Econometric Society, vol. 40(3), pages 515-528, May.
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    Cited by:

    1. Val Andrei Fajardo & Steve Drekic, 2017. "Waiting Time Distributions in the Preemptive Accumulating Priority Queue," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 255-284, March.

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