Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T04:51:18.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Regeneration in tandem queues with multiserver stations

Published online by Cambridge University Press:  14 July 2016

Karl Sigman
Karl Sigman*
Affiliation:
Columbia University
*
∗Postal address: Department of Industrial Engineering and Operations Research, Mudd Building, Columbia University, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.

Keywords

RANDOM ASSIGNMENTFIFOHARRIS ERGODIC MARKOV CHAIN
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 391 - 403
Copyright
Copyright © Applied Probability Trust 1988 

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

Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Asmussen, S. and Thorisson, H. (1987) A Markov chain approach to periodic queues. J. Appl. Prob. 24, 215–225.Google Scholar
Athreya, K. B. and Ney, P. (1978) A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493–501.Google Scholar
Charlot, F., Ghidouche, M. and Hamami, M. (1978) Irréducibilité et récurrence au sens de Harris des <temps d'attente> des files GI/GI/q. Z. Wahrscheinlichkeitsth. 43, 187–203. des files' href=https://scholar.google.com/scholar_lookup?title=Irr%C3%A9ducibilit%C3%A9+et+r%C3%A9currence+au+sens+de+Harris+des+%3Ctemps+d'attente%3E+des+files&author=Charlot+F.&author=Ghidouche+M.&author=Hamami+M.&publication+year=1978&journal=%0A++++++++++++++++++GI%2FGI%2Fq.+Z.+Wahrscheinlichkeitsth.%0A+++++++++++++++&volume=43&pages=187-203>Google Scholar
Daley, D. (1987) Certain optimality properties of the first-come first-served discipline for GI/G/s queues. Stoch. Proc. Appl. 25, 301–308.Google Scholar
Foss, S. G. (1980) Approximation of multi-channel queueing systems. Sib. Math. J. 21, 851–857.Google Scholar
Gnedenko, B. and Kovalenko, I. N. (1968) An Introduction to Queueing Theory. Israel Program for Scientific Translations, Jerusalem.Google Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Nummelin, E. (1978) A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309–318.Google Scholar
Nummelin, E. (1979) A conservation property for general GI/G/1 queues with application to tandem queues. Adv. Appl. Prob. 11, 660–672.Google Scholar
Nummelin, E. (1981) Regeneration in tandem queues. Adv. Appl. Prob. 13, 221–230.Google Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-Negative Operators. Cambridge Tracts in Mathematics.Google Scholar
Sigman, K. (1986) Applications of Harris Ergodic Markov Chains to the Regenerative Nature of Queueing Systems. Doctoral dissertation, Dept. of Operations Research, University of California, Berkeley.Google Scholar
Stoyan, D. (1976) A critical remark on a system approximation in queueing theory. Math. Operationsforsch. Statist. 7, 953–956.Google Scholar
Thorisson, H. (1983) The coupling of regenerative processes. Adv. Appl. Prob. 15, 531–561.Google Scholar
Whitt, W. (1982) Existence of limiting distributions in the GI/G/s queue. Math. Operat. Res. 7, 88–94; erratum, Math. Operat. Res. 9 (1984), 634.Google Scholar
Wolff, R. (1977) An upper bound for multichannel queues. J. Appl. Prob. 14, 884–888.Google Scholar
Wolff, R. (1984) Conditions for finite ladder height and delay moments. Operat. Res. 32, 909–916.Google Scholar
Wolff, R. (1987) Upper bounds on work in system for multichannel queues. J. Appl. Prob. 24, 547–551.Google Scholar
Wolfson, B. (1986) The uniqueness of stationary distribution for the GI/G/s queue. Math. Operat. Res. 11, 514–520.Google Scholar