Hostname: page-component-7f64f4797f-tldsr Total loading time: 0.001 Render date: 2025-11-04T06:39:02.182Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rates of convergence of Erlang's model

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Christine Fricker ,
Philippe Robert  and
Danielle Tibi
Christine Fricker*
Affiliation:
INRIA
Philippe Robert*
Affiliation:
INRIA
Danielle Tibi*
Affiliation:
Université de Paris 7
*
∗ Postal address: INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France.
∗ Postal address: INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France.
∗∗∗ Postal address: Université de Paris 7, URA 1321, 2 Place Jussieu, 75251 Paris Cedex 05, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

Keywords

The M/M/N/N queueconvergence to equilibriumcut-off propertyhitting times of the M/M/∞ queueOrnstein–Ühlenbeck processesGamma functions

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1167 - 1184
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Article purchase

Temporarily unavailable

References

Abate, J., and Whitt, W. (1998). Calculating transient characteristics of the Erlang loss model by numerical transform inversion. Commun. Statist. Stoch. Models 14, 663–680.Google Scholar
Aldous, D. J. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Séminaire de Probabilités de Strasbourg XVII (Lecture Notes in Math. 986). Springer, Berlin, pp. 243–297.Google Scholar
Aldous, D. J., and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97.Google Scholar
Anantharam, V. (1988). The settling time of a closed Jackson network. Unpublished manuscript.Google Scholar
Barbour, A., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford Science Publications, Oxford.Google Scholar
Beneš, V. E. (1961). The covariance function of a simple truck group, with applications to traffic measurement. Bell System Technical Journal 40, 117–148.Google Scholar
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Math. Appl. 13 Gordon and Breach Science, New York.Google Scholar
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sciences USA 93, 1659–1664.Google Scholar
Ethier, S., and Kurtz, T. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
Flajolet, P. (1980). Combinatorial aspects of continued fractions. Discrete Math. 32, 125–161.Google Scholar
Fricker, C., Robert, P., and Tibi, D. (1999). On the spectral gap of the M/M/N/N queue. Unpublished manuscript. In preparation.Google Scholar
Goulden, I., and Jackson, D. (1983). Combinatorial Enumeration. Wiley, New York.Google Scholar
Guillemin, F., and Simonian, A. (1995). Transient characteristics of an M/M/∞ system. Adv. Appl. Prob. 27, 862–888.Google Scholar
Hadjiev, D. I. (1985). The first passage problem for generalized Ornstein–Ühlenbeck processes with nonpositive jumps. In Séminaire de Probabilités, XIX (Lecture Notes in Math. 1123). Springer, Berlin, pp. 80–90.Google Scholar
Hunt, P. (1995). Loss networks under diverse routing: the symmetric star network. Adv. Appl. Prob. 25, 255–272.Google Scholar
Hunt, P., and Kurtz, T. (1994). Large loss networks. Stoch. Proc. Appl. 53, 363–378.Google Scholar
Karlin, S., and McGregor, J. (1957). The classification of birth and death processes. Trans. AMS 86, 366–400.CrossRefGoogle Scholar
Kelly, F. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.CrossRefGoogle Scholar
Knessl, C. (1990). On the transient behavior of the Mz/M/m/m loss model. Commun. Statist. Stoch. Models 6, 749–776.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley-Interscience, New York.Google Scholar
Mitra, D., and Weiss, A. (1988). The transient behavior in Erlang's model for large trunk groups and various traffic conditions. Technical Report, AT&T Bell Laboratories. In Proc. 12th Internat. Teletraffic Congress, Torino, Italy.Google Scholar
Newell, G. F. (1984). The M/M/∞ Service System with Ranked Servers in Heavy Traffic Lecture Notes in Economics and Mathematical Systems 231. Springer, Berlin.Google Scholar
Rogers, L., and Williams, D. (1987). Diffusions, Markov Processes and Martingales Vol. 2: Itô Calculus. Wiley, New York.Google Scholar
Rogers, L., and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 1: Foundations, 2nd edn. Wiley, New York.Google Scholar
Shwartz, A., and Weiss, A. (1995). Large Deviations for Performance Analysis. Stochastic Modeling Series. Chapman & Hall, London.Google Scholar
Srikant, R., and Whitt, W. (1996). Simulation run lengths to estimate blocking probabilities. ACM Trans. Modeling and Computer Simulation 6, 7–52.Google Scholar
Takacs, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Whitt, W. (1982). On the heavy-traffic limit theorem for GI/G/∞ queues. Adv. Appl. Prob. 14, 171–190.Google Scholar
Whittaker, E. T., and Watson, G. N. (1996). A Course of Modern Analysis. Cambridge Mathematical Library. CUP, Cambridge. (Reprint of the 4th edn (1927).)Google Scholar