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A product form for the general stochastic matching model

Part of: Markov processes Special processes Graph theory

Published online by Cambridge University Press:  23 June 2021

Pascal Moyal Open the ORCID record for Pascal Moyal [Opens in a new window] ,
Ana Bušić  and
Jean Mairesse
Pascal Moyal*
Affiliation:
UTC/Université de Lorraine
Ana Bušić*
Affiliation:
Inria and DI ENS, PSL Research University
Jean Mairesse*
Affiliation:
CNRS, Sorbonne Université
*
*Postal address: LMAC, Université de Technologie de Compiègne, 60203 Compiègne Cedex, France and Institut Elie Cartan, Université de Lorraine, F-54506 Nancy Cedex, France. Email address: [email protected]
**Postal address: INRIA/Département d’Informatique (ENS), Université PSL, 2 rue Simone Iff, 75012 Paris, France. Email address: [email protected]
***Postal address: Sorbonne Université, CNRS, LIP6, F-75005, Paris, France. Email address: [email protected]
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Abstract

We consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

Keywords

Markov chainsreversibilitygraphs

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K25: Queueing theory
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05C63: Infinite graphs 05C80: Random graphs
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 449 - 468
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Adan, I. and Weiss, G. (2012). Exact FCFS matching rates for two infinite multi-type sequences. Operat. Res. 60, 475–489.CrossRefGoogle Scholar
Adan, I. and Weiss, G. (2014). A skill based parallel service system under FCFS-ALIS – steady state, overloads, and abandonments. Stoch. Sys. 4, 250–299.CrossRefGoogle Scholar
Adan, I., Bušić, A., Mairesse, J. and Weiss, G. (2018). Reversibility and further properties of the FCFM bipartite matching model. Math. Operat. Res. 43, 598–621.CrossRefGoogle Scholar
Adan, I., Kleiner, I., Righter, R., and Weiss, G. (2018). FCFS parallel service systems and matching models. Performance Evaluation 127–128, 253–272.10.1016/j.peva.2018.10.005CrossRefGoogle Scholar
Ayesta, U., Bodas, T., and Verloop, I. (2018). On a unifying product form framework for redundancy models. Performance Evaluation 127–128, 93–119.CrossRefGoogle Scholar
Bonald, T. and Comte, C. (2017). Balanced fair resource sharing in computer clusters. Performance Evaluation 116, 70–83.CrossRefGoogle Scholar
Boxma, O., David, I., Perry, D. and Stadje, W. (2011). A new look at organ transplantation models and double matching queues. Prob. Eng. Inf. Sci. 25, 135–155.10.1017/S0269964810000318CrossRefGoogle Scholar
Buke, B. and Chen, H. (2015). Stabilizing policies for probabilistic matching systems. Queueing Sys. Theor. Appl. 80, 35–69.CrossRefGoogle Scholar
Buke, B. and Chen, H. (2017). Fluid and diffusion approximations of probabilistic matching systems. Queueing Sys. Theor. Appl. 86, 1–33.CrossRefGoogle Scholar
Bušić, A. and Meyn, S. (2014). Approximate optimality with bounded regret in dynamic matching models. Preprint, arXiv:1411.1044 [math.PR].Google Scholar
Bušić, A., Gupta, V. and Mairesse, J. (2013). Stability of the bipartite matching model. Adv. Appl. Prob. 45, 351–378.10.1239/aap/1370870122CrossRefGoogle Scholar
Caldentey, R., Kaplan, E. H. and Weiss, G. (2009). FCFS infinite bipartite matching of servers and customers. Adv. Appl. Prob. 41, 695–730.CrossRefGoogle Scholar
Gardner, K. et al. (2016). Queueing with redundant requests: exact analysis. Queueing Sys. Theor. Appl. 83, 227–259.CrossRefGoogle Scholar
Gurvich, I. and Ward, A. On the dynamic control of matching queues. Stoch. Sys. 4, 1–45.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Mairesse, J. and Moyal, P. (2016). Stability of the stochastic matching model. J. Appl. Prob. 53, 1064–1077.10.1017/jpr.2016.65CrossRefGoogle Scholar
Moyal, P. and Perry, O. (2017). On the instability of matching queues. Ann. Appl. Prob. 27, 3385–3434.CrossRefGoogle Scholar
Moyal, P., Bušić, A. and Mairesse, J. (2018). Loynes construction for the extended bipartite matching. Preprint, arXiv:math.PR/1803.02788.Google Scholar
Nazari, M. and Stolyar, A. (2016). Reward maximization in general dynamic matching systems. Preprint, arXiv:math.PR/1608.01646.Google Scholar
Rahmé, Y. and Moyal, P. (2019). A stochastic matching model on hypergraphs. Preprint, arXiv:math.PR/1907.12711.Google Scholar
Talreja, R. and Whitt, W. Fluid models for overloaded multi-class many-service queueing systems with FCFS routing. Manag. Sci. 54, 1513–1527.10.1287/mnsc.1080.0868CrossRefGoogle Scholar