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Exact FCFS Matching Rates for Two Infinite Multitype Sequences

Author

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  • Ivo Adan

    (Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands)

  • Gideon Weiss

    (Department of Statistics, The University of Haifa, 31905 Haifa, Israel)

Abstract

Motivated by queues with multitype servers and multitype customers, we consider an infinite sequence of items of types (C-script) = { c 1 ,..., c I }, and another infinite sequence of items of types (S-script) = { s 1 ,..., s J }, and a bipartite graph G of allowable matches between the types. We assume that the types of items in the two sequences are independent and identically distributed (i.i.d.) with given probability vectors (alpha), (beta). Matching the two sequences on a first-come, first-served basis defines a unique infinite matching between the sequences. For ( c i , s j ) (in) G we define the matching rate r c i , s j as the long-term fraction of ( c i , s j ) matches in the infinite matching, if it exists. We describe this system by a multidimensional countable Markov chain, obtain conditions for ergodicity, and derive its stationary distribution, which is, most surprisingly, of product form. We show that if the chain is ergodic, then the matching rates exist almost surely, and we give a closed-form formula to calculate them. We point out the connection of this model to some queueing models.

Suggested Citation

  • Ivo Adan & Gideon Weiss, 2012. "Exact FCFS Matching Rates for Two Infinite Multitype Sequences," Operations Research, INFORMS, vol. 60(2), pages 475-489, April.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:2:p:475-489
    DOI: 10.1287/opre.1110.1027
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    References listed on IDEAS

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    1. Noah Gans & Ger Koole & Avishai Mandelbaum, 2003. "Telephone Call Centers: Tutorial, Review, and Research Prospects," Manufacturing & Service Operations Management, INFORMS, vol. 5(2), pages 79-141, September.
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    Cited by:

    1. Francisco Castro & Hamid Nazerzadeh & Chiwei Yan, 2020. "Matching queues with reneging: a product form solution," Queueing Systems: Theory and Applications, Springer, vol. 96(3), pages 359-385, December.
    2. Ross Anderson & Itai Ashlagi & David Gamarnik & Yash Kanoria, 2017. "Efficient Dynamic Barter Exchange," Operations Research, INFORMS, vol. 65(6), pages 1446-1459, December.
    3. Itai Ashlagi & Maximilien Burq & Patrick Jaillet & Vahideh Manshadi, 2019. "On Matching and Thickness in Heterogeneous Dynamic Markets," Operations Research, INFORMS, vol. 67(4), pages 927-949, July.
    4. Ivo Adan & Ana Bušić & Jean Mairesse & Gideon Weiss, 2018. "Reversibility and Further Properties of FCFS Infinite Bipartite Matching," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 598-621, May.
    5. Dongyuan Zhan & Gideon Weiss, 2018. "Many-server scaling of the N-system under FCFS–ALIS," Queueing Systems: Theory and Applications, Springer, vol. 88(1), pages 27-71, February.
    6. Jingui Xie & Yiming Fan & Mabel C. Chou, 2017. "Flexibility design in loss and queueing systems: efficiency of k-chain configuration," Flexible Services and Manufacturing Journal, Springer, vol. 29(2), pages 286-308, June.
    7. Burak Büke & Hanyi Chen, 2017. "Fluid and diffusion approximations of probabilistic matching systems," Queueing Systems: Theory and Applications, Springer, vol. 86(1), pages 1-33, June.
    8. Frank Kelly & Elena Yudovina, 2015. "A Markov model of a limit order book: thresholds, recurrence, and trading strategies," Papers 1504.00579, arXiv.org, revised Mar 2017.
    9. Jean Mairesse & Pascal Moyal, 2022. "New frontiers for stochastic matching," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 473-475, April.
    10. Jose H. Blanchet & Martin I. Reiman & Viragh Shah & Lawrence M. Wein & Linjia Wu, 2020. "Asymptotically Optimal Control of a Centralized Dynamic Matching Market with General Utilities," Papers 2002.03205, arXiv.org, revised Jun 2021.
    11. Gideon Weiss, 2020. "Directed FCFS infinite bipartite matching," Queueing Systems: Theory and Applications, Springer, vol. 96(3), pages 387-418, December.
    12. Frank Kelly & Elena Yudovina, 2018. "A Markov Model of a Limit Order Book: Thresholds, Recurrence, and Trading Strategies," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 181-203, February.
    13. Adan, Ivo & Foss, Sergey & Shneer, Seva & Weiss, Gideon, 2020. "Local stability in a transient Markov chain," Statistics & Probability Letters, Elsevier, vol. 165(C).
    14. Jocelyn Begeot & Irène Marcovici & Pascal Moyal, 2023. "Stability regions of systems with compatibilities and ubiquitous measures on graphs," Queueing Systems: Theory and Applications, Springer, vol. 103(3), pages 275-312, April.
    15. Kristen Gardner & Rhonda Righter, 2020. "Product forms for FCFS queueing models with arbitrary server-job compatibilities: an overview," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 3-51, October.
    16. Mohammadreza Nazari & Alexander L. Stolyar, 2019. "Reward maximization in general dynamic matching systems," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 143-170, February.

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