Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-17T10:18:53.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Some extensions of Takács's limit theorems

Published online by Cambridge University Press:  14 July 2016

D. N. Shanbhag
D. N. Shanbhag*
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

Keywords

GI/G/sRENEWAL THEOREMPOISSON ARRIVALSLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 752 - 761
Copyright
Copyright © Applied Probability Trust 1974 

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

Feller, W. (1965) An Introduction to Probability Theory and its Applications. Wiley, New York.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
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Shanbhag, D. N. (1965) On the asymptotic behaviour of a single server queue. Austral. J. Statist. 7, 48–53.Google Scholar
Shanbhag, D. N. (1968) A note on the queueing system GI/G/8. Proc. Camb. Phil. Soc. 64, 477–479.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232, 6–31.Google Scholar
Smith, W. L. (1958) Renewal theory and ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
Stidham, S. Jr. (1972) Regenerative processes in the theory of queues with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
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 times. Sankhya A 25, 91–100.Google Scholar
Takács, L. (1969) On Erlang's formula. Ann. Math. Statist. 40, 71–78.Google Scholar
Takács, L. (1974) A single-server queue with limited virtual waiting time. J. Appl. Prob. 11, 612–617.Google Scholar
Whitt, W. (1972) Embedded renewal processes in the GI/G/s queue. J. Appl. Prob. 9, 650–658.Google Scholar