Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T08:20:24.729Z Has data issue: false hasContentIssue false
  • English
  • Français

A new informative embedded Markov renewal process for the PH/G/1 queue

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
University of Delaware
*
∗Present address: Department of Systems and Industrial Engineering, University of Arizona, Tucson AZ 85721, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

Keywords

WAITING TIMESQUEUE LENGTH: COMPUTATIONAL PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 533 - 557
Copyright
Copyright © Applied Probability Trust 1986 

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. Ali, O. M. E. and Neuts, M. F. (1986) A queue with service times dependent on their order within the busy periods. Stochastic Models 2. To appear.Google Scholar
2. Daley, D. J. (1977) Inequalities for moments of tails of random variables with a queueing application. Z Wahrscheinlichkeitsth. 41, 139–143.Google Scholar
3. Lucantoni, D. M. (1983) An Algorithmic Analysis of a Communication Model with Retransmission of Flawed Messages. Pitman Research Notes in Mathematics No. 81.Google Scholar
4. Neuts, M. F. (1976) Moment formulas for the Markov renewal branching process. Adv. Appl. Prob. 8, 690–711.Google Scholar
5. Neuts, M. F. (1977) Some explicit formulas for the steady-state behavior of the queue with semi-Markovian service times. Adv. Appl. Prob. 9, 141–157.Google Scholar
6. Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
7. Neuts, M. F. (1986) Partitioned Stochastic Matrices of M/G/1 Type and their Applications. In preparation.Google Scholar
8. Ramaswami, V. (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261.Google Scholar