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Optimal Average-Cost Policy for a Queue with Start-Up and Shut-Down Costs

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  • Matthew J. Sobel

    (Yale University, New Haven, Connecticut)

Abstract

This paper examines pure stationary policies for starting and stopping service in a single-server queue with respect to the state space consisting of queue length and whether or not the service facility is operating. Almost any such policy is shown to be equivalent to one that depends on two parameters. For a wide class of cost functions, the “almost any” is shown to be without loss of optimality in the sense of average cost per unit time over an infinite horizon. The results are largely distribution-free and are obtained with random-walk arguments. They apply also to certain repair problems and to production smoothing problems in which demand is a renewal process and goods are made sequentially.

Suggested Citation

  • Matthew J. Sobel, 1969. "Optimal Average-Cost Policy for a Queue with Start-Up and Shut-Down Costs," Operations Research, INFORMS, vol. 17(1), pages 145-162, February.
    • Handle: RePEc:inm:oropre:v:17:y:1969:i:1:p:145-162
    DOI: 10.1287/opre.17.1.145
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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. Xiuli Chao & Frank Y. Chen, 2005. "An Optimal Production and Shutdown Strategy when a Supplier Offers an Incentive Program," Manufacturing & Service Operations Management, INFORMS, vol. 7(2), pages 130-143, March.
    3. Foreest, N. D. van & Wijngaard, J., 2010. "On the Optimal Policy for the Single-product Inventory Problem with Set-up Cost and a Restricted Production Capacity," Research Report 10005, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    4. 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.
    5. Ying Shi & Zhaotong Lian, 2016. "Equilibrium Strategies and Optimal Control for a Double-Ended Queue," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(03), pages 1-18, June.
    6. Cigdem Gurgur, 2013. "Optimal configuration of a decentralized, market-driven production/inventory system," Annals of Operations Research, Springer, vol. 209(1), pages 139-157, October.
    7. 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.
    8. 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.
    9. 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.
    10. repec:dgr:rugsom:10005 is not listed on IDEAS
    11. Jau-Chuan Ke, 2006. "An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(2), pages 357-369, May.
    12. Apurva Jain & Kamran Moinzadeh, 2005. "A Supply Chain Model with Reverse Information Exchange," Manufacturing & Service Operations Management, INFORMS, vol. 7(4), pages 360-378, November.
    13. Gullu, Refik, 1998. "Base stock policies for production/inventory problems with uncertain capacity levels," European Journal of Operational Research, Elsevier, vol. 105(1), pages 43-51, February.
    14. Ananth V. Iyer & Apurva Jain, 2004. "Modeling the Impact of Merging Capacity in Production-Inventory Systems," Management Science, INFORMS, vol. 50(8), pages 1082-1094, August.
    15. 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.
    16. Bruno Gaujal, 2022. "Learning in queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 521-523, April.
    17. Nicky D. Van Foreest & Jacob Wijngaard, 2014. "On Optimal Policies for Production-Inventory Systems with Compound Poisson Demand and Setup Costs," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 517-532, May.
    18. 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.
    19. Jain, Apurva, 2007. "Value of capacity pooling in supply chains with heterogeneous customers," European Journal of Operational Research, Elsevier, vol. 177(1), pages 239-260, February.
    20. 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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