Skip to main content
SpringerLink
Log in

A Markov-modulated M/G/1 queue I: Stationary distribution

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.

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. S. Halfin and M. Segal, A priority queueing model for a mixture of two types of customers, SIAM J. Appl. Math. 23(1972)369.

    Google Scholar 

  2. S. Halfin, Steady-state distribution for the buffer content of an M/G/1 queue with varying service rate, SIAM J. Appl. Math. 23(1972)356.

    Google Scholar 

  3. D.P. Gaver and J.P. Lehoczky, Channels that cooperatively service a data stream and voice messages, Technical Report, Department of Operations Research, Naval Postgraduate School (1979).

  4. G.J.K. Regterschot and J.H.A. de Smit, The queue M/G/1 with Markov-modulated arrivals and service, Math. Oper. Res. 11(1986).

  5. D.P. Gaver and J.P. Lehoczky, Diffusion approximation for the cooperative service of voice and data messages, J. Appl. Prob. 18(1981)660.

    Google Scholar 

  6. D.P. Gaver and J.P. Lehoczky, Channels that cooperatively service a data stream and voice messages, IEEE Trans. Comm., Vol. Com-30, No. 5(1982)1153.

    Google Scholar 

  7. D.Y. Burman and D.R. Smith, Asymptotic analysis of a queueing model with bursty traffic, Bell Syst. Tech. J. 62 No. 6(1983)1433.

    Google Scholar 

  8. D.Y. Burman and D.R. Smith, An asymptotic analysis of a queueing system with Markov-modulated arrivals, Opns. Res. 34(1986)105.

    Google Scholar 

  9. L. Kleinrock,Queueing Systems, Vol. 1 (John Wiley, New York, 1975).

    Google Scholar 

  10. D.Y. Burman, An analytic approach to diffusion approximations in queueing, Ph.D. dissertation, Courant Institute of Mathematical Sciences, New York University (1979).

  11. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I, Adv. Appl. Prob. 2(1970)150.

    Google Scholar 

  12. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic II: Sequences, networks and batches, Adv. Appl. Prob. 2(1970)335.

    Google Scholar 

  13. W. Whitt, Multiple channel queues in heavy traffic III: Random server selection, Adv. Appl. Prob. 2(1970)370.

    Google Scholar 

  14. R.E. O'Malley, Jr.,Introduction to Singular Perturbations (Academic Press, New York, 1974).

    Google Scholar 

  15. J. Kevorkian and J.D. Cole,Perturbation Methods in Applied Mathematics (Springer-Verlag, New York, 1981).

    Google Scholar 

  16. C.M. Bender and S.A. Orszag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

    Google Scholar 

  17. N.G. de Bruijn,Asymptotic Methods in Analysis (North-Holland, Amsterdam, 1958).

    Google Scholar 

  18. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Asymptotic analysis of a state-dependent M/G/1 queueing system, SIAM J. Appl. Math. 46(1986)483.

    Google Scholar 

  19. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, On the performance of state-dependent single server queues, SIAM J. Appl. Math. 46(1986)657.

    Google Scholar 

  20. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, System crash in a finite capacity M/G/1 queue, Comm. Statist.: Stochastic Models 2(1986)171.

    Google Scholar 

  21. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, A finite capacity single server queue with customer loss, Comm. Statist.: Stochastic Models 2(1986)97.

    Google Scholar 

  22. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Asymptotic expansions for a closed multiple access system, SIAM J. Comp., to appear.

  23. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Two parallel queues with dynamic routing, IEEE Trans. on Comm., Vol. Com-34(1986)1170.

    Google Scholar 

  24. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, An M/G/1 queue with Markov-modulated arrival rates and service time densities I: Stationary distribution, Applied Mathematics Technical Report No. 8423, Northwestern University (1986).

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Sciences and Applied Mathematics, Northwestern University, 60201, Evanston, IL, USA

    C. Knessl & B. J. Matkowsky

  2. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Z. Schuss

  3. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 60680, Chicago, IL, USA

    C. Tier

Authors
  1. C. Knessl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. J. Matkowsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Schuss
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Tier
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knessl, C., Matkowsky, B.J., Schuss, Z. et al. A Markov-modulated M/G/1 queue I: Stationary distribution. Queueing Syst 1, 355–374 (1987). https://doi.org/10.1007/BF01150670

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01150670

Keywords

Search

Navigation