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Applying a New Device in the Optimization of Exponential Queuing Systems

Author

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  • Steven A. Lippman

    (University of California, Los Angeles, California)

Abstract
We consider the problem of controlling M / M / c queuing systems. By providing a new definition of the time of transition, we enlarge the standard set of decision epochs and obtain a preferred version of the n -period problem in which the times between transitions are exponential random variables with constant parameter. Using this new device, we are able to utilize the inductive approach in a manner characteristic of inventory theory. The efficacy of the approach is demonstrated by successfully finding the form of an optimal policy for three distinct models that have appeared in the literature, namely, those of (i) Miller and Cramer, (ii) Crabill and Sabeti, and (iii) Low of particular note is our analysis of the Miller-Cramer model, in which we show that a policy optimal for all sufficiently small discount factors can be obtained from the usual average cost functional equation without recourse to further computation.

Suggested Citation

  • Steven A. Lippman, 1975. "Applying a New Device in the Optimization of Exponential Queuing Systems," Operations Research, INFORMS, vol. 23(4), pages 687-710, August.
    • Handle: RePEc:inm:oropre:v:23:y:1975:i:4:p:687-710
    DOI: 10.1287/opre.23.4.687
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    Cited by:

    1. De Munck, Thomas & Chevalier, Philippe & Tancrez, Jean-Sébastien, 2023. "Managing priorities on on-demand service platforms with waiting time differentiation," International Journal of Production Economics, Elsevier, vol. 266(C).
    2. Chatwin, Richard E., 2000. "Optimal dynamic pricing of perishable products with stochastic demand and a finite set of prices," European Journal of Operational Research, Elsevier, vol. 125(1), pages 149-174, August.
    3. van Wijk, A.C.C. & Adan, I.J.B.F. & van Houtum, G.J., 2019. "Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes," European Journal of Operational Research, Elsevier, vol. 272(2), pages 481-495.
    4. Papier, Felix & Thonemann, Ulrich W., 2011. "Capacity rationing in rental systems with two customer classes and batch arrivals," Omega, Elsevier, vol. 39(1), pages 73-85, January.
    5. Feng, Haofang & Zhang, Sheng Hao & Zhang, Yong, 2023. "Managing production-inventory-maintenance systems with condition monitoring," European Journal of Operational Research, Elsevier, vol. 310(2), pages 698-711.
    6. Izak Duenyas & Diwakar Gupta & Tava Lennon Olsen, 1998. "Control of a Single-Server Tandem Queueing System with Setups," Operations Research, INFORMS, vol. 46(2), pages 218-230, April.
    7. Satır, Benhür & Erenay, Fatih Safa & Bookbinder, James H., 2018. "Shipment consolidation with two demand classes: Rationing the dispatch capacity," European Journal of Operational Research, Elsevier, vol. 270(1), pages 171-184.
    8. Bhatnagar, Rohit & Lin, Bing, 2019. "The joint transshipment and production control policies for multi-location production/inventory systems," European Journal of Operational Research, Elsevier, vol. 275(3), pages 957-970.
    9. Kim, Hyunjung & Kim, Eungab, 2022. "A hybrid manufacturing system with demand for intermediate goods and controllable make-to-stock production rate," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1244-1257.
    10. Yu, Yimin & Shou, Biying & Ni, Yaodong & Chen, Li, 2017. "Optimal production, pricing, and substitution policies in continuous review production-inventory systems," European Journal of Operational Research, Elsevier, vol. 260(2), pages 631-649.
    11. Elhafsi, Mohsen & Hamouda, Essia, 2018. "Managing an integrated production and inventory system selling to a dual market: Long-term and walk-in," European Journal of Operational Research, Elsevier, vol. 268(1), pages 215-230.
    12. Cheng, T.C.E. & Gao, Chunyan & Shen, Houcai, 2011. "Production planning and inventory allocation of a single-product assemble-to-order system with failure-prone machines," International Journal of Production Economics, Elsevier, vol. 131(2), pages 604-617, June.
    13. ElHafsi, Mohsen & Fang, Jianxin & Hamouda, Essia, 2021. "Optimal production and inventory control of multi-class mixed backorder and lost sales demand class models," European Journal of Operational Research, Elsevier, vol. 291(1), pages 147-161.
    14. Huang, Boray & Wu, Andy, 2017. "Reduce shortage with self-reservation policy for a manufacturer paying both fixed and variable stockout expenditure," European Journal of Operational Research, Elsevier, vol. 262(3), pages 944-953.
    15. ElHafsi, Mohsen & Fang, Jianxin & Hamouda, Essia, 2020. "A novel decomposition-based method for solving general-product structure assemble-to-order systems," European Journal of Operational Research, Elsevier, vol. 286(1), pages 233-249.
    16. Xu, Jianjun & Serrano, Alejandro & Lin, Bing, 2017. "Optimal production and rationing policy of two-stage tandem production system," International Journal of Production Economics, Elsevier, vol. 185(C), pages 100-112.
    17. ElHafsi, Mohsen & Fang, Jianxin & Camus, Herve, 2018. "Optimal control of a continuous-time W-configuration assemble-to-order system," European Journal of Operational Research, Elsevier, vol. 267(3), pages 917-932.
    18. Janakiram Subramanian & Shaler Stidham & Conrad J. Lautenbacher, 1999. "Airline Yield Management with Overbooking, Cancellations, and No-Shows," Transportation Science, INFORMS, vol. 33(2), pages 147-167, May.
    19. Alavi Fard, Farzad & He, Jian & Ivanov, Dmitry & Jie, Ferry, 2019. "A utility adjusted newsvendor model with stochastic demand," International Journal of Production Economics, Elsevier, vol. 211(C), pages 154-165.
    20. Richard E. Chatwin, 1999. "Continuous-Time Airline Overbooking with Time-Dependent Fares and Refunds," Transportation Science, INFORMS, vol. 33(2), pages 182-191, May.
    21. Xiong, Yu & Li, Gendao & Zhou, Yu & Fernandes, Kiran & Harrison, Richard & Xiong, Zhongkai, 2014. "Dynamic pricing models for used products in remanufacturing with lost-sales and uncertain quality," International Journal of Production Economics, Elsevier, vol. 147(PC), pages 678-688.

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