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Optimal Operating Policies for M / G /1 Queuing Systems

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  • Daniel P. Heyman

    (Bell Telephone Laboratories, Holmdel, New Jersey)

Abstract

We consider the economic behavior of a M / G /1 queuing system operating with the following cost structure: a server start-up cost, a server shut-down cost, a cost per unit time when the server is turned on, and a holding cost per unit time spent in the system for each customer. We prove that for the single server queue there is a stationary optimal policy of the form: Turn the server on when n customers are present, and turn it off when the system is empty. For the undiscounted, infinite horizon problem, an exact expression for the cost rate as a function of n and a closed form expression for the optimal value of n are derived. When future costs are discounted, we obtain an equation for the expected discounted cost as a function of n and the interest rate, and prove that for small interest rates the optimal discounted policy is approximately the optimal undiscounted policy. We conclude by establishing the recursion relation to find the optimal (nonstationary) policy for finite horizon problems.

Suggested Citation

  • Daniel P. Heyman, 1968. "Optimal Operating Policies for M / G /1 Queuing Systems," Operations Research, INFORMS, vol. 16(2), pages 362-382, April.
    • Handle: RePEc:inm:oropre:v:16:y:1968:i:2:p:362-382
    DOI: 10.1287/opre.16.2.362
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    Citations

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    Cited by:

    1. R. E. Lillo, 2000. "Optimal Operating Policy for an M/G/1 Exhaustive Server-Vacation Model," Methodology and Computing in Applied Probability, Springer, vol. 2(2), pages 153-167, August.
    2. Mirchevski, Stefan & Bakeva, Verica, 2024. "Cost function analysis of a single-server queueing system with Poisson input stream and Erlang-k service time," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    3. Dong-Yuh Yang & Po-Kai Chang, 2015. "A parametric programming solution to the -policy queue with fuzzy parameters," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(4), pages 590-598, March.
    4. Douglas G. Down, 2022. "Optimal control of energy-aware queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 417-419, April.
    5. Tirdad, Ali & Grassmann, Winfried K. & Tavakoli, Javad, 2016. "Optimal policies of M(t)/M/c/c queues with two different levels of servers," European Journal of Operational Research, Elsevier, vol. 249(3), pages 1124-1130.
    6. 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.
    7. Jae‐Dong Son & Yaghoub Khojasteh Ghamari, 2008. "Optimal admission and pricing control problems in service industries with multiple servers and sideline profit," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(4), pages 325-342, July.
    8. Jing-Sheng Song & Li Xiao & Hanqin Zhang & Paul Zipkin, 2017. "Optimal Policies for a Dual-Sourcing Inventory Problem with Endogenous Stochastic Lead Times," Operations Research, INFORMS, vol. 65(2), pages 379-395, April.
    9. Tayfur Altiok & Goang An Shiue, 1995. "Single‐stage, multi‐product production/inventory systems with lost sales," Naval Research Logistics (NRL), John Wiley & Sons, vol. 42(6), pages 889-913, September.
    10. Wei Li & Attahiru Sule Alfa, 2000. "Optimal policies for M/M/m queue with two different kinds of (N, T)‐policies," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(3), pages 240-258, April.
    11. Cree S. Dawson & Charles J. McCallum & R. Bradford Murphy & Eric Wolman, 2000. "Operations Research at Bell Laboratories through the 1970s: Part II," Operations Research, INFORMS, vol. 48(3), pages 351-361, June.
    12. Jim (Junmin) Shi & Michael N. Katehakis & Benjamin Melamed & Yusen Xia, 2014. "Production-Inventory Systems with Lost Sales and Compound Poisson Demands," Operations Research, INFORMS, vol. 62(5), pages 1048-1063, October.
    13. Vasiliki Kostami, 2020. "Price and Lead time Disclosure Strategies in Inventory Systems," Production and Operations Management, Production and Operations Management Society, vol. 29(12), pages 2760-2788, December.
    14. Won Seok Yang & Nam K. Kim & Sungjune Park & Chandrasekar Subramaniam, 2020. "Improving service in the presence of surge traffic: a K-policy voluntary flushing queueing system," Annals of Operations Research, Springer, vol. 295(1), pages 411-423, December.
    15. Boronico, Jess S. & Siegel, Philip H., 1998. "Capacity planning for toll roadways incorporating consumer wait time costs," Transportation Research Part A: Policy and Practice, Elsevier, vol. 32(4), pages 297-310, May.
    16. Li Xia & Zhe George Zhang & Quan‐Lin Li, 2022. "A c/μ‐Rule for Job Assignment in Heterogeneous Group‐Server Queues," Production and Operations Management, Production and Operations Management Society, vol. 31(3), pages 1191-1215, March.
    17. Erim Kardeş & Fernando Ordóñez & Randolph W. Hall, 2011. "Discounted Robust Stochastic Games and an Application to Queueing Control," Operations Research, INFORMS, vol. 59(2), pages 365-382, April.

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