Skip to main content
SpringerLink
Log in

On the slow server problem

  • Classical Models of the Queuing Theory and Generalizations
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

In this paper the problem of optimal control over a Markov queueing system with heterogeneous servers and a joint queue is considered, which is also known in the literature as “the slow server problem.” The classical model is generalized here to the case with delay and call serving penalties. It is proved here that the optimal control policy for servers’ activating is of monotonic and threshold nature.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kitaev, M.Yu. and Rykov, V.V., Controlled Queueing Systems, New York: CRC Press, 1995.

    MATH  Google Scholar 

  2. Rykov, V.V., On Monotonicity Conditions for Optimal Policies for Controlling Systems, Autom. Remote Control, 1999, no. 9, pp. 1290–1302.

  3. Topkis, D., Minimizing a Submodular Function on a Lattice, Oper. Res. 1978, vol. 26, no. 2, pp. 305–321.

    Article  MATH  MathSciNet  Google Scholar 

  4. Vishnevskii, V.M. and Semenova, O.V., Sistemy pollinga: teoriya i primenenie v shirokopolosnykh besprovodnykh setyakh (Polling Systems: Theory and Application in the Broadband Networks), Moscow: Technosphera, 2007.

    Google Scholar 

  5. Pedro, T., One Size Does Not Fit All: A Case for Heterogeneous Multiprocessor Systems, IADIS Int. Conf. Appl. Comput., 2005.

  6. Krishnamoorthy, B., On Poisson Queue with Two Heterogeneous Servers, Oper. Res., 1963, vol. 11, pp. 321–330.

    Article  Google Scholar 

  7. Hajek, B., Optimal Control of Two Interacting Service Stations, IEEE Trans. Automat. Control, 1984, vol. 29, pp. 491–499.

    Article  MATH  MathSciNet  Google Scholar 

  8. Lin, W. and Kumar, P.R., Optimal Control of a Queueing System with Two Heterogeneous Servers, IEEE Trans. Automat. Control, 1984, vol. 29, pp. 696–703.

    Article  MATH  MathSciNet  Google Scholar 

  9. Koole, G., A Simple Proof of the Optimality of a Threshold Policy in Two-server Queueing System, Syst. & Control Lett., 1995, vol. 26, pp. 301–303.

    Article  MATH  MathSciNet  Google Scholar 

  10. Weber, R., On a Conjecture about Assigning Jobs to Processors of Different Speeds, IEEE Trans. Automat. Control, 1993, vol. 38, no. 1, pp. 166–170.

    Article  MathSciNet  Google Scholar 

  11. Rykov, V.V., Monotone Control of Queueing System with Heterogeneous Servers, QUESTA, 2001, vol. 37, pp. 391–403.

    MATH  MathSciNet  Google Scholar 

  12. Vericourt, F. and Zhou, Y.P., On the Incomplete Results for the Heterogeneous Server Problem, Queueing Syst., 2006, vol. 52, pp. 189–191.

    Article  MATH  MathSciNet  Google Scholar 

  13. Efrosinin, D.V. and Rykov, V.V., Numerical Study of the Optimal Control of a System with Heterogeneous Servers, Autom. Remote Control, 2003, no. 2, pp. 302–309.

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

    Google Scholar 

  15. Efrosinin, D. and Breuer, L., Threshold Policies for Controlled Retrial Queues with Heterogeneous Servers, Ann. Oper. Res., 2006, vol. 141, pp. 139–162.

    Article  MathSciNet  Google Scholar 

  16. Ghoneim, H.A. and Stidham, S., Control of Arrivals to Two Queues in Series, Eur. J. Oper. Res., 1985, vol. 21, pp. 399–409.

    Article  MATH  MathSciNet  Google Scholar 

  17. Sennott, L.I., Average Cost Optimal Stationary Policies in Infinite State Markov Decision Processes with Unbounded Costs, Oper. Res., 1989, vol. 37, pp. 626–633.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Peoples Friendship University, Moscow, Russia

    V. V. Rykov & D. V. Efrosinin

Authors
  1. V. V. Rykov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. V. Efrosinin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.V. Rykov, D.V. Efrosinin, 2009, published in Avtomatika i Telemekhanika, 2009, No. 12, pp. 81–91.

This work was supported by the Russian Foundation for Basic Research projects nos. 07-07-00088_a and 08-07-90102-Mol_a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rykov, V.V., Efrosinin, D.V. On the slow server problem. Autom Remote Control 70, 2013–2023 (2009). https://doi.org/10.1134/S0005117909120091

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117909120091

Keywords

Search

Navigation