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On heavy traffic approximations for transient characteristics of M/M/∞ queues

Part of: Limit theorems Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Fabrice M. Guillemin ,
Ravi R. Mazumdar  and
Alain D. Simonian
Fabrice M. Guillemin*
Affiliation:
France Telecom
Ravi R. Mazumdar*
Affiliation:
INRS Télécom
Alain D. Simonian*
Affiliation:
France Telecom
*
∗ Postal address: France Telecom, CNET Lannion-A, Route de Trégastel, 22300 Lannion, France.
∗∗ Postal address: INRS Télécom, 16 P1. du Commerce, Ile des Soeurs, Verdun, Québec, Canada H3E 1H6.
∗∗∗ Postal address: France Telecom, CNET Paris-A, 38–40 Rue du Général Leclerc, 92131 Issy-les-Mlx, France.
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Abstract

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

Keywords

M/M/∞QUEUESTRANSIENT BEHAVIOURHEAVY TRAFFIC APPROXIMATIONSORNSTEIN–UHLENBECK PROCESSCONVERGENCE OF DISTRIBUTION

MSC classification

Primary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60F05: Central limit and other weak theorems 60J60: Diffusion processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 490 - 506
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Aldous, D. (1989) Probability Approximations via the Clumping Poisson Heuristic. Springer, Berlin.Google Scholar
[2] Baccelli, F. and Bremaud, P. (1980) Palm Probabilities and Stationary Queues. (Lecture Notes in Statistics 41). Springer, Berlin.Google Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Doetsch, G. (1955) Handbuch der Laplace-Transformation. Vol. II. Birkhaüser, Basel.Google Scholar
[5] El Karoui, N. (1978) Sur les montées des semi-martingales. In Temps locaux , ed. Azéma, J. and Yor, M. Astérisques 52-53. Société Mathémathique de France.Google Scholar
[6] Friedman, A. (1975) Stochastic Differential Equations and Applications. Vol. 1. Academic Press, New York.Google Scholar
[7] Guillemin, F. and Simonian, A. (1995) Transient characteristics of an M/M/8 system. Adv. Appl. Prob. 27, 862–888.CrossRefGoogle Scholar
[8] Guillemin, F., Rubino, G., Sericola, B. and Simonian, A. (1994) Transient analysis of statistical multiplexing of data connections on an ATM link. Submitted.Google Scholar
[9] Iscoe, I., Mcdonald, D. and Qian, K. (1993) Capacity of ATM switches. Ann. Appl. Prob. 3, 277–295.CrossRefGoogle Scholar
[10] Karatzas, I. and Shreve, S. (1987) Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
[11] Kleinrock, L. (1975) Queueing Systems. Vol. I: Theory. Wiley, New York.Google Scholar
[12] Lebedev, N. (1965) Special Functions and their Applications. Prentice Hall, New York.Google Scholar
[13] Morrison, J., Shepp, L. and Van Wyk, C. (1987) A queueing analysis of hashing with lazy deletion. SIAM J. Comput. 16, 1155–1164.Google Scholar
[14] Murray, J. D. (1984) Asymptotic Analysis. Springer, Berlin.CrossRefGoogle Scholar
[15] Nobile, A., Ricciardi, L. and Sacerdote, L. (1985) Exponential trends of the Ornstein-Uhlenbeck first passage time densities. J. Appl. Prob. 22, 360–369.Google Scholar
[16] Oberhettinger, O. and Badii, A. (1973) Tables of Laplace Transforms. Springer, Berlin.Google Scholar
[17] Ricciardi, L. M. and Sato, S. (1988) First-passage time density and moments of the Ornstein-Uhlenbeck process. J. Appl. Prob. 25, 43–57.Google Scholar
[18] Simonian, A., Guillemin, F., Louvion, J. R. and Romoeuf, L. (1992) Transient analysis of statistical multiplexing on an ATM link. Globecom '92. Orlando, December 1992.Google Scholar
[19] Wasow, W. (1965) Asymptotic Expansions of Ordinary Differential Equations. Interscience, New York.Google Scholar
[20] Watson, G. N. (1966) A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge.Google Scholar
[21] Whitt, W. (1982) On the heavy traffic limit theorem for G/G/8 queues. Adv. Appl. Prob. 14, 171–190.Google Scholar