Skip to main content
SpringerLink
Log in

Stability regions of systems with compatibilities and ubiquitous measures on graphs

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper addresses the ubiquity of remarkable measures on graphs and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supplies and demands, and so on. The stability region of such systems can then be seen as a set of measures on graphs, where the measures under consideration represent the arrival flows to the various classes of users, supplies, demands, etc., and the graph represents the compatibilities between those classes. In this paper, we show that these ‘stabilizing’ measures can always be easily constructed as a simple function of a family of weights on the edges of the graph. Second, we show that the latter measures always coincide with invariant measures of random walks on the graph under consideration. Some arguments in the proofs rely on the so-called matching rates of specific stochastic matching models. As a by-product of these arguments, we show that, in several cases, the matching rates are independent of the matching policy, that is, the rule for choosing a match between various compatible elements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Adan, I., Weiss, G.: Exact FCFS matching rates for two infinite multi-type sequences. Oper. Res. 60(2), 475–489 (2012)

    Article  Google Scholar 

  2. Adan, I., Weiss, G.: A skill based parallel service system under FCFS-ALIS—steady state, overloads, and abandonments. Stochastic Syst. 4(1), 250–299 (2014)

    Article  Google Scholar 

  3. Adan, I., Bu\(\check{\text{s}}\)ić, A., Mairesse, J., and Weiss, G.: Reversibility and further properties of the FCFM Bipartite matching model. Math. Oper. Res. 43(2), 598–621 (2018)

  4. Adan, I., Kleiner, I., Righter, R., Weiss, G.: FCFS parallel service systems and matching models. Perform. Eval. 127, 253–272 (2018)

    Article  Google Scholar 

  5. Anton, E., Ayesta, U., Jonckheere, M., Verloop, I. M.: A survey of stability results for redundancy systems. Modern Trends Controlled Stochastic Process. 266–283

  6. Aveklouris, A., DeValve, L., Ward, A.R., Wu, X.: Matching impatient and heterogeneous demand and supply. arXiv:2102.02710 [math.PR] (2021)

  7. Ayesta, U., Bodas, T., Dorsman, J.-P., Verloop, I.M.: A token-based central queue with order-independent service rates. Oper. Res. 70(1), 545–561 (2022)

    Article  Google Scholar 

  8. Begeot, J., Marcovici, I., Moyal, P., Rahmé, Y.: A general stochastic matching model on multigraphs. ALEA 18, 1325–1351 (2021)

    Article  Google Scholar 

  9. Boxma, O., David, I., Perry, D., Stadje, W.: A new look at organ transplantation models and double matching queues. Probab. Eng. Inf. Sci. 25, 135–155 (2011)

    Article  Google Scholar 

  10. Buke, B., Chen, H.: Stabilizing policies for probabilistic matching systems. Queueing Syst. Theor. Appl. 80(1–2), 35–69 (2015)

    Article  Google Scholar 

  11. Buke, B., Chen, H.: Fluid and diffusion approximations of probabilistic matching systems. Queueing Syst. Theor. Appl. 86(1–2), 1–33 (2017)

    Article  Google Scholar 

  12. Bu\(\check{\text{ s }}\)ić, A., Gupta, V., and Mairesse, J.: Stability of the bipartite matching model. Adv. Appl. Probab. 45(2), 351–378 (2013)

  13. Cadas, A., Bu\(\check{\text{ s }}\)ić, A., Doncel, J.: Optimal control of dynamic bipartite matching models. In: Proceedings of the 12th EAI International Conference on Performance Evaluation Methodologies and Tools, pp. 39–46 (2019)

  14. Caldentey, R., Kaplan, E.H., Weiss, G.: FCFS infinite bipartite matching of servers and customers. Adv. Appl. Probab. 41(3), 695–730 (2009)

    Article  Google Scholar 

  15. Chen, J., Dong, J., Shi, P.: A survey on skill-based routing with applications to service operations management. Queueing Syst. Theory Appl. 96, 53–82 (2020)

    Article  Google Scholar 

  16. Comte, C.: Stochastic non-bipartite matching models and order-independent loss queues. Stoch. Model. 38(1), 1–36 (2022)

    Article  Google Scholar 

  17. Comte, C., Dorsman, J.-P.: Performance Evaluation of Stochastic Bipartite Matching Models. Performance Engineering and Stochastic Modeling, Lecture in Computer Science, pp. 425–440. Springer, Berlin (2021)

  18. Farkas, J.: Theorie der einfachen Ungleichungen. Journal für die reine und angewandte Mathematik (Crelles J.) 124, 1–27 (1902)

    Google Scholar 

  19. Ford, L., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (2015)

    Google Scholar 

  20. Gardner, K., Righter, R.: Product forms for FCFS queueing models with arbitrary server-job compatibilities: an overview. Queueing Syst. Theory Appl. 96(1), 3–51 (2020)

    Article  Google Scholar 

  21. Gardner, K., Zbarsky, S., Doroudi, S., Harchol-Balter, M., Hyytia, E., Scheller-Wolf, A.: Queueing with redundant requests: exact analysis. Queueing Syst. Theory Appl. 83, 227–259 (2016)

    Article  Google Scholar 

  22. Gurvich, I., Ward, A.: On the dynamic control of matching queues. Stochastic Syst. 4(2), 1–45 (2014)

    Google Scholar 

  23. Hall, P.: On Representatives of Subsets. J. Lond. Math. Soc. 10(1), 26–30 (1935)

    Article  Google Scholar 

  24. Jonckheere, M., Moyal, P., Ramirez, C., Soprano-Loto, N.: Generalized Max-Weight policies in stochastic matching. Stochastic Syst. 1–19 (2022)

  25. Krzesinski, A.E.: Order independent queues. In: Boucherie, R.J., van Dijk, N.M. (eds.) Queueing Networks: A Fundamental Approach, pp. 85–120. Springer, Boston (2011)

    Chapter  Google Scholar 

  26. Mairesse, J., Moyal, P.: Stability of the stochastic matching model. J. Appl. Probab. 53(4), 1064–1077 (2017)

    Article  Google Scholar 

  27. Moyal, P., Bu\(\check{\text{ s }}\)ić, A., Mairesse, J.: A product form for the general stochastic matching model. J. Appl. Probab. 58(2), 449–468 (2021)

  28. Moyal, P., Bu\(\check{\text{ s }}\)ić, A., Mairesse, J.: Loynes construction for the Extended bipartite matching. arXiv:1803.02788 [math.PR] (2018)

  29. Moyal, P., Perry, O.: On the instability of matching queues. Ann. Appl. Probab. 27(6), 3385–3434 (2017)

    Article  Google Scholar 

  30. Nazari, M., Stolyar, A.L.: Reward maximization in general dynamic matching systems. Queueing Syst. Theory Appl. 91(1), 143–170 (2019)

    Article  Google Scholar 

  31. Rahme, Y., Moyal, P.: A stochastic matching model on hypergraphs. Adv. Appl. Probab. 53(4), 951–980 (2021)

    Article  Google Scholar 

  32. Tassiulas, L., Ephremides, A.: Dynamic server allocation to parallel queues with randomly varying connectivity. IEEE Trans. Inf. Theory 39(2), 466–478 (1993)

    Article  Google Scholar 

  33. Visschers, J., Adan, I., Weiss, G.: A product form solution to a system with multi-type jobs and multi-type servers. Queueing Syst. Theory Appl. 70(3), 269–298 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to warmly thank Emmanuel Jeandel, for suggesting the use of Farkas’ lemma in the proof of Theorem 1.

Author information

Authors and Affiliations

  1. Universite de Lorraine, Nancy, France

    Jocelyn Begeot, Irène Marcovici & Pascal Moyal

Authors
  1. Jocelyn Begeot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irène Marcovici
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pascal Moyal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pascal Moyal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Begeot, J., Marcovici, I. & Moyal, P. Stability regions of systems with compatibilities and ubiquitous measures on graphs. Queueing Syst 103, 275–312 (2023). https://doi.org/10.1007/s11134-023-09872-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-023-09872-0

Keywords

Mathematics Subject Classification

Search

Navigation