Stochastic Systems

On the dynamic control of matching queues

Itai Gurvich and Amy Ward

Full-text: Open access

Abstract

We consider the optimal control of matching queues with random arrivals. In this model, items arrive to dedicated queues, and wait to be matched with items from other (possibly multiple) queues. A match type corresponds to the set of item classes required for a match. Once a decision has been made to perform a match, the matching itself is instantaneous and the matched items depart from the system. We consider the problem of minimizing finite-horizon cumulative holding costs. The controller must decide which matchings to execute given multiple options. In principle, the controller may choose to wait until some “inventory” of items builds up to facilitate more profitable matches in the future.

We introduce a multi-dimensional imbalance process, that at each time $t$, is given by a linear function of the cumulative arrivals to each of the item classes. A non-zero value of the imbalance at time $t$ means that no control could have matched all the items that arrived by time $t$. A lower bound based on the imbalance process can be specified, at each time point, by a solution to an optimization problem with linear constraints. While not achievable in general, this lower bound can be asymptotically approached under a dedicated item condition (an analogue of the local traffic condition in bandwidth sharing networks). We devise a myopic discrete-review matching control that asymptotically–as the arrival rates become large–achieves the imbalance-based lower bound.

Article information

Source
Stoch. Syst., Volume 4, Number 2 (2014), 479-523.

Dates
First available in Project Euclid: 27 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1427462423

Digital Object Identifier
doi:10.1214/13-SSY097

Mathematical Reviews number (MathSciNet)
MR3353225

Zentralblatt MATH identifier
1327.60177

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60F17: Functional limit theorems; invariance principles

Keywords
Matching queues optimal control diffusion approximations

Citation

Gurvich, Itai; Ward, Amy. On the dynamic control of matching queues. Stoch. Syst. 4 (2014), no. 2, 479--523. doi:10.1214/13-SSY097. https://projecteuclid.org/euclid.ssy/1427462423


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References

  • [1] Ata, B. and Kumar, S., Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies. Ann. Appl. Prob., 2005.
  • [2] Bell, S. L. and Williams, R. J., Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy. Electronic Journal of Probability, 10(33):1044–1115, 2005.
  • [3] Dogru, M. K., Reiman, M. I., and Wang, Q., A stochastic programming based inventory policy for assemble-to-order systems with application to the W model. Operations Research, 58(4):849–864, 2010.
  • [4] Harrison, J. M., Assembly-like queues. Journal of Appl. Prob., 10:354–367, 1973.
  • [5] Harrison, J. M., The BIGSTEP approach to flow management in stochastic processing networks. In F. Kelly, S. Zachary, and I. Ziedins, editors, Stochastic Networks: Theory and Applications, pages 57–90. Oxford University Press, 1996.
  • [6] Harrison, J. M., Correction: Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Prob., 16(3):1703–1732, 2006.
  • [7] Harrison, J. M. and Lopez, M. J., Heavy traffic resource pooling in parallel-server systems. Queueing Systems, 33:339–368, 1999.
  • [8] Harrison, J. M. and Van Mieghem, J. A., Dynamic control of brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Prob., 7:747–771, 1996.
  • [9] Horvath, L., Strong approximation of renewal processes. Stochastic Processes and Their Applications, 18(1):127–138, 1984.
  • [10] Kang, W. N., Kelly, F. P., Lee, N. H., and Williams, R. J., State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Prob., 19(5):1719–1780, 2009.
  • [11] Mandelbaum, A. and Stolyar, S., Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized $c\mu$-rule. Operations Research, 52:836–855, 2004.
  • [12] Plambeck, E. L. and Ward, A. R., Optimal control of a high-volume assemble-to-order system. Mathematics of Operations Research, 31(3):453–477, 2006.
  • [13] Reiman, M. I. and Wang, Q., Asymptotically optimal inventory control for assemble-to-order systems with identical lead times, 2013. Working Paper.
  • [14] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, volume 293. Springer Verlag, 1999.
  • [15] Song, J. S. and Zipkin, P., Supply chain operations: Assemble-to-order and configure-to-order systems. In Handbooks in Operations Research and Management Science, volume XXX, pages 561–593, 2003.