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Fluid limit of generalized Jackson queueing networks with stationary and ergodic arrivals and service times

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Marc Lelarge
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France. Email address: marc.lelarge@ens.fr
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Abstract

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We use a sample-path technique to derive asymptotics of generalized Jackson queueing networks in the fluid scale; that is, when space and time are scaled by the same factor n. The analysis only presupposes the existence of long-run averages and is based on some monotonicity and concavity arguments for the fluid processes. The results provide a functional strong law of large numbers for stochastic Jackson queueing networks, since they apply to their sample paths with probability 1. The fluid processes are shown to be piecewise linear and an explicit formulation of the different drifts is computed. A few applications of this fluid limit are given. In particular, a new computation of the constant that appears in the stability condition for such networks is given. In a certain context of a rare event, the fluid limit of the network is also derived explicitly.

Keywords

Generalized Jackson networksample-path techniquefixed pointfluid limit

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 491 - 512
Copyright
© Applied Probability Trust 2005 

References

Baccelli, F. and Foss, S. (1994). Ergodicity of Jackson-type queueing networks. Queuing Systems Theory Appl. 17, 5–72.CrossRefGoogle 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., Foss, S. and Lelarge, M. (2005). Tails in generalized Jackson networks with subexponential service-time distributions. J. Appl. Prob. 42, 513–530.CrossRefGoogle Scholar
Billingsley, P. (1979). Probability and Measure. John Wiley, New York.Google Scholar
Chen, H. (1995). Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines. Ann. Appl. Prob. 5, 637–665.Google Scholar
Chen, H. and Mandelbaum, A. (1991). Discrete flow networks: bottleneck analysis and fluid approximations. Math. Operat. Res. 16, 408–446.Google Scholar
Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Prob. 5, 49–77.Google Scholar
Dai, J. G. (1996). A fluid limit model criterion for the instability of multiclass queueing networks. Ann. Appl. Prob. 6, 751–757.Google Scholar
Foss, S. (1991). Ergodicity of queueing networks. Siberian Math. J. 32, 184–203.Google Scholar
Gordon, W. J. and Newell, G. F. (1967). Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.Google Scholar
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.Google Scholar
Jackson, J. R. (1963). Jobshop-like queueing systems. Manag. Sci. 10, 518–527.Google Scholar
Majewski, K. (2000). Single class queueing networks with discrete and fluid customers on the time interval {R}. Queueing Systems 36, 405–435.Google Scholar
Massey, W. A. (1981). Non-stationary queues. , Department of Mathematics, Stanford University.Google Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Skorokhod, A. V. (1961). Stochastic equations for diffusions in a bounded region. Theory Prob. Appl. 6, 264–274.Google Scholar
Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5, 285–309.Google Scholar