Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-19T19:22:25.475Z Has data issue: false hasContentIssue false
  • English
  • Français

A cyclic queueing network with dependent exponential service times

Published online by Cambridge University Press:  14 July 2016

Patricia A. Jacobs
Patricia A. Jacobs*
Affiliation:
Stanford University
Get access Rights & Permissions [Opens in a new window]

Abstract

A cyclic queueing network with two servers and a finite number of customers is studied. The service times for server 1 form an earma(1,1) process (exponential mixed autoregressive moving average process both of order 1) which is a sequence of positively correlated exponential random variables; the process in general is not Markovian. The service times for the other server are independent with a common exponential distribution. Limiting results for the number of customers in queue and the virtual waiting time at server 1 are obtained. Comparisons are made with the case of independent exponential service times for server 1.

Keywords

CYCLIC QUEUEDEPENDENT EXPONENTIAL SERVICE TIMESEARMA(1,1) SEQUENCEMULTIPROGRAMMED COMPUTER SYSTEMGENERAL STATE SPACE MARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 3 , September 1978 , pp. 573 - 589
Copyright
Copyright © Applied Probability Trust 1978 

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

Azéma, J., Duflo, M. and Revuz, D. (1969) Mesure invariante des processus de Markov recurrents. Séminaire Prob. Strasbourg III. Lecture Notes in Mathematics 88, Springer-Verlag, Berlin.Google Scholar
Çinlar, E. (1967) Time dependence of queues with semi-Markovian services. J. Appl. Prob. 4, 356–364.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, New Jersey.Google Scholar
Gaver, D. P. and Shedler, G. S. (1973) Processor utilization in multiprogrammed systems via diffusion approximations, Opns Res. 21, 569–576.Google Scholar
Gaver, D. P. and Thompson, G. L. (1973) Probability and Programming Models in Operations Research. Brooks/Cole, Monterey, Calif.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1977) A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1). Adv. Appl. Prob. 9, 87–104.Google Scholar
Jacod, J. (1973) Semi-groupes et mesures invariantes pour les processus semi-markoviens à espace d'état quelconque. Ann. Inst. H. Poincaré 9, 77–112.Google Scholar
Learmonth, G. P. and Lewis, P. A. W. (1973) Naval Postgraduate School random number generator package LLRANDOM. Naval Postgraduate School Report NPS55Lw73061A.Google Scholar
Learmonth, G. P. and Lewis, P. A. W. (1974) Statistical tests of some widely used and recently proposed uniform number generators. Proc. Seventh Conference on Computer Science and Statistics. Western Periodicals, North Hollywood, Calif. Google Scholar
Lewis, P. A. W., Goodman, A. S. and Miller, J. M. (1969) A pseudorandom number generator for the system/360. IBM Systems J. 8, 136–146.Google Scholar
Lewis, P. A. W. and Shedler, G. S. (1973) Empirically derived micromodels for sequences of page exceptions. IBM J. Res. Develop. 17, 86–100.Google Scholar
Loynes, R. M. (1962) Stationary waiting time distributions for single server queues. Ann. Math. Statist. 33, 1323–1339.Google Scholar
Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
Pearce, C. (1967) Queues with moving average service times. J. Appl. Prob. 4, 553–570.Google Scholar
Purdue, P. (1975) A queue with Poisson input and semi-Markov service times: busy period analysis. J. Appl. Prob. 12, 353–357.CrossRefGoogle Scholar