Advertisement

Queueing Systems

, Volume 91, Issue 1–2, pp 143–170 | Cite as

Reward maximization in general dynamic matching systems

  • Mohammadreza NazariEmail author
  • Alexander L. Stolyar
Article

Abstract

We consider a matching system with random arrivals of items of different types. The items wait in queues—one per item type—until they are “matched.” Each matching requires certain quantities of items of different types; after a matching is activated, the associated items leave the system. There exists a finite set of possible matchings, each producing a certain amount of “reward.” This model has a broad range of important applications, including assemble-to-order systems, Internet advertising, and matching web portals. We propose an optimal matching scheme in the sense that it asymptotically maximizes the long-term average matching reward, while keeping the queues stable. The scheme makes matching decisions in a specially constructed virtual system, which in turn controls decisions in the physical system. The key feature of the virtual system is that, unlike the physical one, it allows the queues to become negative. The matchings in the virtual system are controlled by an extended version of the greedy primal–dual (GPD) algorithm, which we prove to be asymptotically optimal—this in turn implies the asymptotic optimality of the entire scheme. The scheme is real time; at any time, it uses simple rules based on the current state of the virtual and physical queues. It is very robust in that it does not require any knowledge of the item arrival rates and automatically adapts to changing rates. The extended GPD algorithm and its asymptotic optimality apply to a quite general queueing network framework, not limited to matching problems, and therefore are of independent interest.

Keywords

Dynamic matching EGPD algorithm Virtual queues Optimal control Utility maximization Stability 

Mathematics Subject Classification

60K25 68M20 90B22 

References

  1. 1.
    Adan, I., Bušić, A., Mairesse, J., Weiss, G.: Reversibility and further properties of FCFS infinite bipartite matching. Math. Oper. Res. 43, 598–621 (2017)CrossRefGoogle Scholar
  2. 2.
    Adan, I., Weiss, G.: Exact FCFS matching rates for two infinite multitype sequences. Oper. Res. 60(2), 475–489 (2012)CrossRefGoogle Scholar
  3. 3.
    Büke, B., Chen, H.: Stabilizing policies for probabilistic matching systems. Queueing Syst. 80(1–2), 35–69 (2015)CrossRefGoogle Scholar
  4. 4.
    Bušić, A., Gupta, V., Mairesse, J.: Stability of the bipartite matching model. Adv. Appl. Probab. 45(2), 351–378 (2013)CrossRefGoogle Scholar
  5. 5.
    Bušić, A., Meyn, S.: Optimization of dynamic matching models. (2014). arXiv preprint arXiv:1411.1044
  6. 6.
    Caldentey, R., Kaplan, E.H., Weiss, G.: FCFS infinite bipartite matching of servers and customers. Adv. Appl. Probab. 41(3), 695–730 (2009)CrossRefGoogle Scholar
  7. 7.
    Gurvich, I., Ward, A.: On the dynamic control of matching queues. Stoch. Syst. 4(2), 479–523 (2014)CrossRefGoogle Scholar
  8. 8.
    Kashyap, B.: The double-ended queue with bulk service and limited waiting space. Oper. Res. 14(5), 822–834 (1966)CrossRefGoogle Scholar
  9. 9.
    Lovász, L., Plummer, M.D.: Matching Theory, vol. 367. American Mathematical Society, Providence (2009)Google Scholar
  10. 10.
    Mairesse, J., Moyal, P.: Stability of the stochastic matching model. J. Appl. Probab. 53(4), 1064–1077 (2016)CrossRefGoogle Scholar
  11. 11.
    Mehta, A.: Online matching and ad allocation. Theor. Comput. Sci. 8(4), 265–368 (2012)Google Scholar
  12. 12.
    Plambeck, E.L., Ward, A.R.: Optimal control of a high-volume assemble-to-order system with maximum leadtime quotation and expediting. Queueing Syst. 60(1), 1–69 (2008)CrossRefGoogle Scholar
  13. 13.
    Stolyar, A.L.: Maximizing queueing network utility subject to stability: greedy primal-dual algorithm. Queueing Syst. 50(4), 401–457 (2005)CrossRefGoogle Scholar
  14. 14.
    Stolyar, A.L.: Greedy primal-dual algorithm for dynamic resource allocation in complex networks. Queueing Syst. 54(3), 203–220 (2006)CrossRefGoogle Scholar
  15. 15.
    Stolyar, A.L., Tezcan, T.: Control of systems with flexible multi-server pools: a shadow routing approach. Queueing Syst. 66(1), 1–51 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lehigh UniversityBethlehemUSA
  2. 2.University of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations