Advertisement

Near-Optimal Switching Strategies for a Tandem Queue

  • Daphne van LeeuwenEmail author
  • Rudesindo Núñez-Queija
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 248)

Abstract

Motivated by various applications in logistics, road traffic and production management, we investigate two versions of a tandem queueing model in which the service rate of the first queue can be controlled. The objective is to keep the mean number of jobs in the second queue as low as possible, without compromising the total system delay (i.e. avoiding starvation of the second queue). The balance between these objectives is governed by a linear cost function of the queue lengths. In the first model, the server in the first queue can be either switched on or off, depending on the queue lengths of both queues. This model has been studied extensively in the literature. Obtaining the optimal control is known to be computationally intensive and time consuming. We are particularly interested in the scenario that the first queue can operate at larger service speed than the second queue. This scenario has received less attention in literature. We propose an approximation using an efficient mathematical analysis of a near-optimal threshold policy based on a matrix-geometric solution of the stationary probabilities that enables us to compute the relevant stationary measures more efficiently and determine an optimal choice for the threshold value.

In some of our target applications, it is more realistic to see the first queue as a (controllable) batch-server system. We follow a similar approach as for the first model and obtain the structure of the optimal policy as well as an efficiently computable near-optimal threshold policy.

We illustrate the appropriateness of our approximations using simulations of both models.

Keywords

Queue Length Service Rate Distribution Center Threshold Strategy Tandem Queue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    F. Avram, Optimal Control of Fluid Limits of Queueing Networks and Stochasticity Corrections. Lectures in Applied Mathematics, vol. 33 (Springer, New York, 1997), pp. 1–36Google Scholar
  2. 2.
    R.E. Bellman, Dynamic Programming. (Princeton University Press, Princeton, NJ, 2003) Republished 2003: DoverGoogle Scholar
  3. 3.
    G.L. Curry, R.M. Feldman, An M/M/1 queue with a general bulk service rule. Nav. Res. Logist. Q. 32 (4), 595–603 (1985)CrossRefGoogle Scholar
  4. 4.
    A. El-Rayes, M. Kwiatkowska, G. Norman, Solving infinite stochastic process algebra models through matrix-geometric methods. School of Computer Science Research Reports, University of Birmingham (1999)Google Scholar
  5. 5.
    A. Gajrat, A. Hordijk, A. Ridder, Large-deviations analysis of the fluid approximation for a controllable tandem queue. Ann. Appl. Probab. 13, 1423–1448 (2003)CrossRefGoogle Scholar
  6. 6.
    G. Koole, Convexity in tandem queues. Probab. Eng. Inf. Sci. 18 (1), 13–31 (2004)CrossRefGoogle Scholar
  7. 7.
    G. Latouche, M. Neuts, Efficient algorithmic solutions to exponential tandem queues with blocking. SIAM J. Algebr. Discr. Meth. 1, 93–106 (1980)CrossRefGoogle Scholar
  8. 8.
    G. Latouche, V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-and-death processes. J. Appl. Probab. 30, 650–674 (1993)CrossRefGoogle Scholar
  9. 9.
    S.A. Lippman, Applying a new device in the optimisation of exponential queueing systems. Oper. Res. 23 (4), 687–710 (1975)CrossRefGoogle Scholar
  10. 10.
    S.P. Meyn, Control Techniques for Complex Networks (Cambridge University Press, Cambridge, 2008). ISBN 978-0-521-88441-9 hardbackGoogle Scholar
  11. 11.
    M. Neuts, A general class of bulk queues with Poisson input. Ann. Math. Stat. 38 (3), 759–770 (1967)CrossRefGoogle Scholar
  12. 12.
    M. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. (Courier Dover Publications, Mineola, 1981)Google Scholar
  13. 13.
    M.L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming (Wiley, New York, 1994)CrossRefGoogle Scholar
  14. 14.
    Z. Rosberg, P.P. Varaiya, J. Walrand, Optimal control of service in tandem queues. IEEE Trans. Autom. Control 27 (3), 600-610 (1982)CrossRefGoogle Scholar
  15. 15.
    R.R. Weber, S. Stidham, Optimal control of service rates in networks of queues. Adv. Appl. Probab. 19, 202–218 (1987)CrossRefGoogle Scholar
  16. 16.
    J. Weiss, The computation of optimal control limits for a queue with batch services. Manag. Sci. 25 (4), 320–328 (1979)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CWIAmsterdamThe Netherlands

Personalised recommendations