Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T15:09:40.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient analysis of the M/M/1 queue

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

P. Leguesdron ,
J. Pellaumail ,
G. Rubino  and
B. Sericola
P. Leguesdron*
Affiliation:
INSA, Rennes
J. Pellaumail*
Affiliation:
INSA, Rennes
G. Rubino*
Affiliation:
IRISA-INRIA, Rennes
B. Sericola*
Affiliation:
IRISA-INRIA, Rennes
*
∗ Postal address: INSA, 20 avenue des Buttes de Coësmes, 35043 Rennes Cédex, France.
∗ Postal address: INSA, 20 avenue des Buttes de Coësmes, 35043 Rennes Cédex, France.
∗∗ Postal address: IRISA-INRIA, Campus Universitaire de Beaulieu, 35042 Rennes Cédex, France.
∗∗ Postal address: IRISA-INRIA, Campus Universitaire de Beaulieu, 35042 Rennes Cédex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new approach is used to obtain the transient probabilities of the M/M/1 queueing system. The first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this generating function. A new analytical expression of the transient probabilities of the M/M/1 queue is then obtained.

Keywords

UNIFORMIZARONGENERATING FUNCTION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 702 - 713
Copyright
Copyright © Applied Probability Trust 1993 

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] Abate, J. and Whitt, W. (1988) Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Prob. 20, 145–178.CrossRefGoogle Scholar
[2] Abate, J. and Whitt, W. (1989) Calculating time-dependent performance measures for the M/M/1 queue. IEEE Trans. Commun. 37, 1102–1104.CrossRefGoogle Scholar
[3] Cohen, J. W. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[4] Kijima, M., Abate, J. and Whitt, W. (1991) Decompositions of the M/M/1 transition function. QUESTA 9, 323–336.Google Scholar
[5] Le Ny, L. M., Rubino, G. and Sericola, B. (1991) Calculating the busy period distribution of the M/M/1 queue. Technical Report 1501, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France.Google Scholar
[6] Parthasarathy, P. R. (1987) A transient soluton to an M/M/1 queue: a new simple approach. Adv. Appl. Prob. 19, 997–998.CrossRefGoogle Scholar
[7] Pellaumail, J. (1991) Systèmes markoviens discrets stationnaires et applications. Technical Report 1447, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France.Google Scholar
[8] Riordan, J. (1968) Combinatorial Identities. Wiley, New York.Google Scholar
[9] Riordan, J. (1962) Stochastic Service Systems. Wiley, New York.Google Scholar
[10] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[11] Saaty, T. L. (1961) Elements of Queueing Theory with Applications. McGraw-Hill, New York.Google Scholar
[12] Singh, H. and Gupta, R. D. (1992) On the probability that the kth customer finds an M/M/1 queue empty. Adv. Appl. Prob. 24, 238–239.CrossRefGoogle Scholar
[13] Sourouri, Y. and Krinik, A. (1990) Taylor series solutions of classical queueing systems. Abstr. Amer. Math. Soc. 11(5), October.Google Scholar
[14] Syski, R. (1986) Introduction to Congestion Theory in Telephone Systems. North-Holland, Amsterdam.Google Scholar