Skip to main content

Heuristic Solution for the Optimal Thresholds in a Controllable Multi-server Heterogeneous Queueing System Without Preemption

Part of the Communications in Computer and Information Science book series (CCIS,volume 601)

Abstract

As it known the optimal policy which minimizes the long-run average cost per unit of time in a multi-server queueing system with heterogeneous servers without preemption has a threshold structure. It means that the slower server must be activated whenever all faster servers are busy and the number of customers in the queue exceeds some specified for this server threshold level. The optimal thresholds can be evaluated using the Howard iteration algorithm or by minimizing the function of the average cost which can be obtained in closed form as a function of unknown threshold levels. The both cases have sufficient restrictions on dimensionality of the model. In present paper we provide a heuristic method to derive expressions for the optimal threshold levels in explicit form as functions of system parameters like service intensities, usage and holding costs for an arbitrary number of servers. The proposed method is based on the fitting of the boundary planes between the areas where the optimal threshold takes a certain value.

Keywords

  • Controllable queueing system
  • Heterogeneous servers
  • Long-run average cost
  • Threshold policy
  • Optimal allocation

D. Efrosinin—This work was funded by the Russian Foundation for Basic Research (RFBR), Project No 15-08-08677-a.

This is a preview of subscription content, access via your institution.

Fig. 1.
Fig. 2.

References

  1. Aviv, Y., Federgruen, A.: The value-iteration method for countable state Markov decision processes. Oper. Res. Lett. 24(5), 223–234 (1999)

    MathSciNet  CrossRef  Google Scholar 

  2. Efrosinin, D.: Controlled Queueing Systems with Heterogeneous Servers. Dynamic Optimization and Monotonicity Properties. VDM Verlag, Saarbrücken (2008)

    MATH  Google Scholar 

  3. Efrosinin, D.: Queueing model of a hybrid channel with faster link subject to partial and complete failures. Ann. Oper. Res. 202(1), 75–102 (2013)

    MathSciNet  CrossRef  Google Scholar 

  4. Efrosinin, D., Rykov, V.: On performance characteristics for queueing systems with heterogeneous servers. Autom. Remote Control 69(1), 61–75 (2008)

    MathSciNet  CrossRef  Google Scholar 

  5. Howard, R.: Dynamic Programming and Markov Processes. Wiley Series, New York (1960)

    MATH  Google Scholar 

  6. Kumar, B.K., Arivudainambi, D.: Transient solution of an \(M/M/c\) queue with heterogeneous servers and balking. Inf. Manage. Sci. 12(3), 15–27 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Puterman, M.L.: Markov Decision Process. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1994)

    CrossRef  Google Scholar 

  8. Rykov, V., Efrosinin, D.: Optimal control of queueing systems with heterogeneous servers. Queueing Syst. 46, 389–407 (2004)

    MathSciNet  CrossRef  Google Scholar 

  9. Rykov, V., Efrosinin, D.: On the slow server problem. Autom. Remote Control 70(12), 2013–2023 (2009)

    MathSciNet  CrossRef  Google Scholar 

  10. Sennott, L.I.: Stochastic Dynamic Programming and the Control of Queueing Systems. Wiley, New York (1999)

    MATH  Google Scholar 

  11. Tijms, H.C.: Stochastic Models. An Algorithmic Approach. Wiley, New York (1994)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dmitry Efrosinin .

Editor information

Editors and Affiliations

Rights and permissions

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Reprints and Permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Efrosinin, D., Rykov, V. (2016). Heuristic Solution for the Optimal Thresholds in a Controllable Multi-server Heterogeneous Queueing System Without Preemption. In: Vishnevsky, V., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2015. Communications in Computer and Information Science, vol 601. Springer, Cham. https://doi.org/10.1007/978-3-319-30843-2_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30843-2_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30842-5

  • Online ISBN: 978-3-319-30843-2

  • eBook Packages: Computer ScienceComputer Science (R0)