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The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure

Author

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  • Warren B. Powell

    (Princeton University, Princeton, New Jersey)

  • Pierre Humblet

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

Abstract

This paper develops a general framework for analyzing a wide class of vehicle dispatching strategies for bulk arrival, bulk service queues. We provide a simple derivation of the queue length transform for the embedded Markov chain, and present a new computational procedure for finding the moments of the queue length distribution. Extensive computational tests demonstrate that the new procedure is significantly faster and more stable than the standard method, from the literature, that requires solving a set of simultaneous linear equations. We give formulas for the mean and variance of the length of the queue for the general case of compound Poisson arrivals, random batch capacities, general service times and a general control strategy.

Suggested Citation

  • Warren B. Powell & Pierre Humblet, 1986. "The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure," Operations Research, INFORMS, vol. 34(2), pages 267-275, April.
    • Handle: RePEc:inm:oropre:v:34:y:1986:i:2:p:267-275
    DOI: 10.1287/opre.34.2.267
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    Citations

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    Cited by:

    1. K. Sikdar & S. K. Samanta, 2016. "Analysis of a finite buffer variable batch service queue with batch Markovian arrival process and server’s vacation," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 553-583, September.
    2. Dieter Claeys & Koenraad Laevens & Joris Walraevens & Herwig Bruneel, 2010. "Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(1), pages 1-23, August.
    3. Nam K. Kim & Kyung C. Chae & Mohan L. Chaudhry, 2004. "An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions," Operations Research, INFORMS, vol. 52(5), pages 756-764, October.
    4. Dall'Orto, Leonardo Campo & Crainic, Teodor Gabriel & Leal, Jose Eugenio & Powell, Warren B., 2006. "The single-node dynamic service scheduling and dispatching problem," European Journal of Operational Research, Elsevier, vol. 170(1), pages 1-23, April.
    5. 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.
    6. Jens Baetens & Bart Steyaert & Dieter Claeys & Herwig Bruneel, 2018. "Delay analysis of a two-class batch-service queue with class-dependent variable server capacity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(1), pages 37-57, August.
    7. Papadaki, Katerina P. & Powell, Warren B., 2002. "Exploiting structure in adaptive dynamic programming algorithms for a stochastic batch service problem," European Journal of Operational Research, Elsevier, vol. 142(1), pages 108-127, October.
    8. Warren B. Powell, 1987. "Waiting‐time distributions for bulk arrival, bulk service queues with vehicle‐holding and cancellation strategies," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(2), pages 207-227, April.
    9. Noah Gans & Garrett van Ryzin, 1999. "Dynamic Vehicle Dispatching: Optimal Heavy Traffic Performance and Practical Insights," Operations Research, INFORMS, vol. 47(5), pages 675-692, October.
    10. Seok Ho Chang & Dae Won Choi, 2006. "Modeling and Performance Analysis of a Finite-Buffer Queue with Batch Arrivals, Batch Services, and Setup Times: The M X /G Y /1/K + B Queue with Setup Times," INFORMS Journal on Computing, INFORMS, vol. 18(2), pages 218-228, May.
    11. Katerina P. Papadaki & Warren B. Powell, 2003. "An adaptive dynamic programming algorithm for a stochastic multiproduct batch dispatch problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 742-769, October.
    12. 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.
    13. Yi, Xeung W. & Kim, Nam K. & Yoon, Bong K. & Chae, Kyung C., 2007. "Analysis of the queue-length distribution for the discrete-time batch-service Geo/Ga,Y/1/K queue," European Journal of Operational Research, Elsevier, vol. 181(2), pages 787-792, September.
    14. Çetinkaya, SIla & Bookbinder, James H., 2003. "Stochastic models for the dispatch of consolidated shipments," Transportation Research Part B: Methodological, Elsevier, vol. 37(8), pages 747-768, September.
    15. Apoorv Saxena & Dieter Claeys & Bo Zhang & Joris Walraevens, 2020. "Cloud data storage: a queueing model with thresholds," Annals of Operations Research, Springer, vol. 293(1), pages 295-315, October.
    16. Srinivas R. Chakravarthy & Shruti & Alexander Rumyantsev, 2021. "Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1551-1579, December.
    17. Gopinath Panda & Veena Goswami, 2023. "Analysis of a Discrete-time Queue with Modified Batch Service Policy and Batch-size-dependent Service," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-18, March.
    18. Chakravarthy, Srinivas R., 2016. "Queueing models with optional cooperative services," European Journal of Operational Research, Elsevier, vol. 248(3), pages 997-1008.
    19. S. R. Chakravarthy & Arunava Maity & U. C. Gupta, 2017. "An ‘(s, S)’ inventory in a queueing system with batch service facility," Annals of Operations Research, Springer, vol. 258(2), pages 263-283, November.
    20. M. A. A. Boon & A. J. E. M. Janssen & J. S. H. Leeuwaarden & R. W. Timmerman, 2019. "Pollaczek contour integrals for the fixed-cycle traffic-light queue," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 89-111, February.
    21. Sergei Dudin & Olga Dudina, 2023. "Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System," Mathematics, MDPI, vol. 11(22), pages 1-20, November.

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