Skip to main content
SpringerLink
Log in

Simple bounds and monotonicity results for finite multi-server exponential tandem queues

  • Contributed Papers
  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

Simple and computationally attractive lower and upper bounds are presented for the call congestion such as those representing multi-server loss or delay stations. Numerical computations indicate a potential usefulness of the bounds for quick engineering purposes. The bounds correspond to product-form modifications and are intuitively appealing. A formal proof of the bounds and related monotonicity results will be presented. The technique of this proof, which is based on Markov reward theory, is of interest in itself and seems promising for further application. The extension to the non-exponential case is discussed. For multiserver loss stations the bounds are argued to be insensitive.

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. I.J.B.F Adan and J. van der Wal, Monotonicity of the throughput of a closed queueing network in the number of jobs, Memorandum 87-03, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1987. To appear: Opns. Res.

  2. I.J.B.F. Adan and J. van der Wal, Monotonicity of the throughput in single server production and assembly networks with respect to the buffer sizes. To appear: Proc. 1th Int. workshop on queuing systems with blocking.

  3. U.N. Bhat, Finite capacity assembly-like queues, Queueing Systems 1 (1986) 85–101.

    Google Scholar 

  4. N.M. van Dijk, Simple bounds for queueing systems with breakdowns, Perf. Evaluation 8 (1988) 117–128.

    Google Scholar 

  5. N.M. van Dijk, A formal proof for the insensitivity of simple bounds for finite multi-server non-exponential tandem queues, Stochastic Processes 27 (1988) 261–277.

    Google Scholar 

  6. N.M. van Dijk and B.F. Lamond, Bounds for the call congestion of finite single-server exponential tandem queues, Opns. Res. 36 (1988) 470–477.

    Google Scholar 

  7. N.M. van Dijk, P. Tsoucas and J. Walrand, Simple bounds and monotonicity of the call congestion of finite multiserver delay systems, Probability in the Engineering and Informational Sciences 2 (1988), 129–138.

    Google Scholar 

  8. A. Hordijk and N. van Dijk, Networks of queues with blocking, Performance '81 (North-Holland, 1981) 51–65.

  9. A. Hordijk and N. van Dijk, Adjoint process, job-local-balance and insensitivity for stochastic networks Bull. 44-th Session Int.Stat.Inst., 50 (1983) 776–788.

    Google Scholar 

  10. A. Hordijk and A. Ridder, Stochastic inequalities for an overflow model, J. Appl. Probability 24 (1987) 696–708.

    Google Scholar 

  11. E.H. Lipper and B. Sengupta, Assembly-like queues with finite capacity: Bounds, asymptotics and approximations, Queueing Systems 1 (1986) 67–83.

    Google Scholar 

  12. T.S. Robertazzi and A.A. Lazar. On the modelling and optimal flow control of the Jacksonian network Perf. Evaluation 5 (1985) 29–43.

    Google Scholar 

  13. J.G. Shanthikumar and D.D. Yao, Stochastic monotonicity of the queue lengths in closed queueing networks, Research Report, University of California, Berkeley, Opns. Res. 35 (1987), 583–588.

    Google Scholar 

  14. J.G. Shanthikumar and D.D. Yao, General queueing networks: Representation and stochastic monotonicity, Proc. of 26th IEEE Conference on Decision and Control, (1987) 1084–1087.

  15. D. Stoyan,Comparison Method for Queues and other Stochastic models (Wiley, New York, 1983).

    Google Scholar 

  16. R. Suri, A concept of monotonicity and its characterization for closed queueing networks, Opns. Res. 33 (1985) 606–624.

    Google Scholar 

  17. P. Tsoucas and J. Walrand, Monotonicity of throughput in non-Markovian networks, To appear: J. Appl. Probability.

  18. J. van der Wal. Monotonicity of the throughput of a closed exponential network in the number of jobs, Research report COSOR 85-21, Eindhoven University of Technology, 1985.

  19. W. Whitt, Comparing counting processes and queues, Adv. Appl. Probability 13 (1981) 207–220.

    Google Scholar 

  20. R.W. Wolff, Poisson arrivals see time averages, Opns. Res. 30 (1982) 223–231.

    Google Scholar 

  21. D.D. Yao, Some properties of the throughput function of closed networks of queues, Oper. Res. Letters 3 (1985) 313–317.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, Free University, Postbox 7161, 1007, MC Amsterdam, The Netherlands

    Nico M. van Dijk

  2. Department of Mathematics and Computing Science, Eindhoven University of Technology, Postbox513, 5600, MB Eindhoven, The Netherlands

    Jan van der Wal

Authors
  1. Nico M. van Dijk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan van der Wal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Dijk, N.M., van der Wal, J. Simple bounds and monotonicity results for finite multi-server exponential tandem queues. Queueing Syst 4, 1–15 (1989). https://doi.org/10.1007/BF01150852

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01150852

Keywords

Search

Navigation