Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T03:26:34.090Z Has data issue: false hasContentIssue false
  • English
  • Français

The cyclic queue with one general and one exponential server

Published online by Cambridge University Press:  01 July 2016

O. J. Boxma
O. J. Boxma*
Affiliation:
University of Utrecht
*
∗Postal address: Mathematical Institute, State University of Utrecht, Budapestlaan 6, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue.

Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.

Keywords

M/G/1 QUEUERESPONSE TIMEJOINT DISTRIBUTIONCYCLE TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 857 - 873
Copyright
Copyright © Applied Probability Trust 1983 

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] Boxma, O. J. (1983) Joint distribution of sojourn time and queue length in the M/G/1 queue with (in)finite capacity. European J. Operat. Res. CrossRefGoogle Scholar
[2] Boxma, O. J. (1983) Response time distributions in cyclic queues. Bull. ISI 50.Google Scholar
[3] Boxma, O. J. and Donk, P. (1982) On response time and cycle time distributions in a two-stage cyclic queue. Performance Evaluation 2, 181–194.CrossRefGoogle Scholar
[4] Boxma, O. J., Kelly, F. P. and Konheim, A. G. (1983) The product form for sojourn time distributions in cyclic exponential queues. J. Assoc. Comput. Mach. CrossRefGoogle Scholar
[5] Chow, W. M. (1980) The cycle time distribution of exponential cyclic queues. J. Assoc. Comput. Mach. 27, 281–286.Google Scholar
[6] Cohen, J. W. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[7] Daduna, H. (1982) Passage times for overtake-free paths in Gordon-Newell networks. Adv. Appl. Prob. 14, 672–686.Google Scholar
[8] Daduna, H. (1982) The cycle time distribution of cyclic two-stage queues with a non-exponential server. Report, Technical University of Berlin.Google Scholar
[9] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[10] Kelly, F. P. and Pollett, P. K. (1983) Sojourn times in closed queueing networks. Adv. Appl. Prob. 15, 638–656.CrossRefGoogle Scholar
[11] Lavenberg, S. S. (1975) The steady-state queueing time distribution for the M/G/1 finite capacity queue. Management Sci. 21, 501–506.CrossRefGoogle Scholar
[12] Reiser, M. (1981) Calculation of response time distributions in cyclic exponential queues. Performance Evaluation 1, 331–333.Google Scholar
[13] Schassberger, R. and Daduna, H. (1983). The time for a round-trip in a cycle of exponential queues. J. Assoc. Comput. Mach. 30, 146–150.CrossRefGoogle Scholar
[14] Takács, L. (1963) Introduction to the Theory of Queues. Wiley, New York.Google Scholar