Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-05T00:37:35.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary waiting-time distributions in the GI/PH/1 queue

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
University of Delaware
*
∗Postal address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

Keywords

MATRIX-GEOMETRIC SOLUTIONSGI/PH/1 QUEUEWAITING-TIME DISTRIBUTIONSLINDLEY'S EQUATIONCOMPUTATIONAL PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 4 , December 1981 , pp. 901 - 912
Copyright
Copyright © Applied Probability Trust 1981 

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.)

Footnotes

Research supported by the National Science Foundation under Grant No. ENG–7908351 and by the Air Force Office of Scientific Research under Grant No. AFOSR–77–3236.

References

[1] Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
[2] Geiszler, C. (1980) A numerical procedure for the selection of the constant interarrival time to a single server queue. Technical Report No. 63B, Applied Mathematics Institute, University of Delaware.Google Scholar
[3] Neuts, M. F. (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185–212.CrossRefGoogle Scholar
[4] Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.CrossRefGoogle Scholar
[5] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models — An Algorithmic Approach. The Johns Hopkins Press, Baltimore, MD.Google Scholar
[6] Neuts, M. F. and Chakravarthy, S. (1981) A single server queue with platooned arrivals and phase type services. European J. Operat. Res .CrossRefGoogle Scholar
[7] Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service time. Sankhya 25, 91–100.Google Scholar