Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T14:20:41.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Tails in generalized Jackson networks with subexponential service-time distributions

Part of: Special processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

François Baccelli ,
Serguei Foss  and
Marc Lelarge
François Baccelli*
Affiliation:
INRIA-ENS
Serguei Foss*
Affiliation:
Heriot-Watt University and Institute of Mathematics, Novosibirsk
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France.
∗∗∗Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@ma.hw.ac.uk
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

Keywords

Generalized Jackson networksubexponential random variableheavy tailintegrated tailVeraverbeke's theoremfluid limit

MSC classification

Primary: 60K25: Queueing theory
Secondary: 62G20: Asymptotic properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 513 - 530
Copyright
© Applied Probability Trust 2005 

Footnotes

***

Supported by INTAS grant 265 and by EPSRC grant R58765/01.

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.Google Scholar
Baccelli, F. and Foss, S. (1994). Ergodicity of Jackson-type queueing networks. Queuing Systems Theory Appl. 17, 5–72.Google Scholar
Baccelli, F. and Foss, S. (1995). On the saturation rule for the stability of queues. J. Appl. Prob. 32, 494–507.Google Scholar
Baccelli, F. and Foss, S. (2004). Moments and tails in monotone-separable stochastic networks. Ann. Appl. Prob. 14, 612–650.Google Scholar
Embrechts, P., Goldie, C. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails, eds Adler, R. J., Feldman, R. E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 435–459.Google Scholar
Lelarge, M. (2005). Fluid limit of generalized Jackson queueing networks with stationary and ergodic arrivals and service times. J. Appl. Prob. 42, 491–512.CrossRefGoogle Scholar