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Transient characteristics of an M/M/∞ system

Part of: Computer system organization Miscellaneous topics of analysis in the complex domain Markov processes

Published online by Cambridge University Press:  01 July 2016

Fabrice Guillemin  and
Alain Simonian
Fabrice Guillemin*
Affiliation:
France Telecom
Alain Simonian*
Affiliation:
France Telecom
*
* Postal address: France Telecom, CNET Lannion-A, 2 Ave. P. Marzin, 22300 Lannion, France.
** Postal address: France Telecom, CNET Paris-A. 38–40 rue du Général Leclerc, 92131 Issy-les-Mlx, France.
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Abstract

Convergence results are given for transient characteristics of an M/M/∞ system such as the period of time the occupation process remains above a given state, the area swept by this process above this state and the number of customers arriving during this period. These results are precise in contrast to approximations derived in the framework of the Poisson clumping heuristic introduced by Aldous.

Keywords

TRANSIENT ANALYSISPOINT PROCESS THEORYANALYTIC FUNCTION THEORYSPECIAL FUNCTIONS

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 30E15: Asymptotic representations in the complex domain 68M20: Performance evaluation; queueing; scheduling

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 862 - 888
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Aldous, D. (1989) Probability Approximations via the Clumping Poisson Heuristic. Springer-Verlag, Berlin.Google Scholar
[2] Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, Berlin.Google Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Borovkov, A. A. (1984) Asymptotic Methods in Queueing Theory. Wiley, New York.Google Scholar
[5] Erdélyi, A. (1953) Higher Transcendental Functions. MacGraw-Hill, New York.Google Scholar
[6] Henrici, P. Applied and Computational Complex Analysis, Vol.2. Wiley, New York.Google Scholar
[7] Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.Google Scholar
[8] Keilson, J. and Servi, L. D. (1993) The matrix M/M/8 system: retrial models and Markov modulated sources. Technical Report CIS 93-05, Computer and Information Systems, Graduate School of Business Administration, University of Rochester.Google Scholar
[9] Kleinrock, L. (1975) Queueing Systems, Vol. I: Theory. Wiley, New York.Google Scholar
[10] Lebedev, N. (1965) Special Functions and their Applications. Prentice-hall, Englewood Cliffs, NJ.CrossRefGoogle Scholar
[11] Morrison, J. Shepp, L. and Van Wyk, C. (1987) A queueing analysis of hashing with lazy deletion. SIAM J. Computing 16, 6, 1155–1164.Google Scholar
[12] Murray, J. D. (1984) Asymptotic Analysis. Springer-Verlag. Berlin.Google Scholar
[13] Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744–756.Google Scholar
[14] Simonian, , Guillemin, A., Louvion, F., Louvion, J.R. and Romoeuf, L. (1992) Transient analysis of statistical multiplexing on an ATM link. Globecom'92, Orlando, FL.Google Scholar
[15] Syski, R. (1986) Introduction to Congestion Theory in Telephone Systems. Elsevier, Amsterdam.Google Scholar
[16] Veech, W. A. (1967) A Second Course of Complex Analysis. W. A. Benjamin, New York.Google Scholar