Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-15T01:54:16.951Z Has data issue: false hasContentIssue false
  • English
  • Français

A conservation property for general GI/G/1 queues with an application to tandem queues

Published online by Cambridge University Press:  01 July 2016

E. Nummelin
E. Nummelin*
Affiliation:
Helsinki University of Technology
*
∗ Postal address: Institute of Mathematics, Helsinki University of Technology, 02150 Espoo 15, Finland.
Get accessRights & Permissions [Opens in a new window]

Abstract

We show that, if the input process of a general GI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

Keywords

QUEUEINPUT PROCESSOUTPUT PROCESSMARKOV RENEWAL PROCESSMARKOV CHAINHARRIS RECURRENT

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979, pp. 660 - 672
Copyright
Copyright © Applied Probability Trust 1979 

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.)

Article purchase

Temporarily unavailable

References

Arjas, E. (1972) On the use of a fundamental identity in the theory of semi-Markov queues. Adv. Appl. Prob. 4, 271–284.Google Scholar
Arjas, E. and Speed, T. P. (1973) Topics in Markov additive processes. Math. Scand. 33, 171–192.CrossRefGoogle Scholar
Barbour, A. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 584–591.Google Scholar
Çinlar, E. (1969) On semi-Markov processes on arbitrary spaces. Proc. Comb. Phil. Soc. 66, 381–392.Google Scholar
Çinlar, E. (1972) Markov additive processes. I. Z. Wahrscheinlichkeitsth. 24, 85–93.Google Scholar
Daley, D. J. (1968) The correlation structure of the output process of some single server queueing systems. Ann. Math. Statist. 39, 1007–1019.CrossRefGoogle Scholar
Daley, D. J. (1974) Notes on queueing output processes. In Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 98, Springer-Verlag, Berlin, 351–358.Google Scholar
Daley, D. J. (1976) Queueing output processes. Adv. Appl. Prob. 8, 395–415.CrossRefGoogle Scholar
Disney, R. L. and Cherry, W. P. (1974) Some topics in queueing network theory. In Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 98, Springer-Verlag, Berlin, 23–44.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Jacod, J. (1971) Théorème de renouvellement et classification pour les chaînes semi-markoviennes. Ann. Inst. H. Poincaré B VII, 83–129.Google Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
Kesten, H. (1974) Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386.Google Scholar
McDonald, D. (1976) On semi-Markov and semi-regenerative processes. To appear.Google Scholar
Neuts, M. F. (1970) Two servers in series, studied in terms of a Markov renewal branching process. Adv. Appl. Prob. 2, 110–149.Google Scholar
Nummelin, E. (1978) Uniform and ratio limit theorems for Markov renewal and semi-regenerative processes on a general state space. Ann. Inst. H. Poincaré XIV, 119–143.Google Scholar
Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand, New York.Google Scholar
Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
Vlach, T. L. and Disney, R. L. (1969) The departure process from the GI/G/1 queue. J. Appl. Prob. 6, 704–707.CrossRefGoogle Scholar