Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T18:45:34.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic inequalities for an overflow model

Published online by Cambridge University Press:  14 July 2016

Arie Hordijk  and
Ad Ridder
Arie Hordijk*
Affiliation:
University of Leiden
Ad Ridder*
Affiliation:
University of Leiden
*
∗Postal address: University of Leiden, Department of Mathematics and Computer Science, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
∗Postal address: University of Leiden, Department of Mathematics and Computer Science, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.

Keywords

QUEUEING NETWORKSSTATIONARY DISTRIBUTIONINSENSITIVE BOUNDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 3 , September 1987 , pp. 696 - 708
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Cohen, J. W. (1980) Sensitivity and insensitivity. Delft Progr. Rep. 5, 159–173.Google Scholar
[2] Cooper, R. B. (1981) Introduction to Queueing Theory, 2nd edn., North-Holland, Amsterdam.Google Scholar
[3] Hordijk, A. (1983) Insensitivity for stochastic networks. In Mathematical Computer Performance and Reliability, ed. Iazeolla, G., Courtois, P. J. and Hordijk, A., 77–94.Google Scholar
[4] Hordijk, A. and Ridder, A. (1986) Insensitive bounds for the stationary distribution of non-reversible Markov chains. J. Appl. Prob. To appear.Google Scholar
[5] Hordijk, A. and Schassberger, R. (1982) Weak convergence for generalized semi-Markov proceses. Stoch. Proc. Appl. 12, 271–291.Google Scholar
[6] Hordijk, A. and Van Dijk, N. (1983) Networks of queues. Proc. Internat. Seminar on Modelling and Performance Evaluation Methodology, Lecture Notes in Control and Information Sciences 60, Springer-Verlag, Berlin, 151–205.Google Scholar
[7] Hordijk, A. and Van Dijk, N. (1983) Adjoint processes, job local balance and insensitivity for stochastic networks. Proc. 44th Session ISI, 776–788.Google Scholar
[8] Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[9] Keilson, J. (1971) Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.Google Scholar
[10] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[11] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models; an Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
[12] Schassberger, R. (1973) Warteschlangen. Springer-Verlag, Wien.Google Scholar
[13] Schassberger, R. (1977) Insensitivity of steady-state distributions of generalized semi-Markov processes, Part I. Ann. Prob. 5, 87–99; Part II. Ann. Prob. 6, 85–93.Google Scholar
[14] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[15] Dijk, N. Van (1986) Simple and insensitive bounds for a grading and an overflow model. Operat. Res. Letters. To appear.Google Scholar
[16] Dijk, N. Van and Lamond, B. F. (1985) Simple bounds for finite single-server exponential tandem queues. Operat. Res. Google Scholar
[17] Marion, E. W. B. Van (1968) Influence of holding time distributions on blocking probabilities of a grading. TELE 20, 17–12.Google Scholar
[18] Whitt, W. (1980) Continuity of generalized semi-Markov processes. Math. Operat. Res. 5, 494–501.Google Scholar
[19] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.CrossRefGoogle Scholar
[20] Whitt, W. (1985) Blocking when service is required from several facilities simultaneously. AT&T Tech. J. 64, 1807–1856.Google Scholar
[21] Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.Google Scholar