The Action Functional for the Jackson Network

The large deviation principle is established for the queue length process in a general queueing network with a homogeneous customer population. The action functional has an intuitive form and is expressed in terms of solutions to certain mathematical programming problems. The key to identifying the action functional is to show uniqueness of a weak solution to the system of idempotent equations that describes the large deviation limit of the queueing processes.

[1]  F. Smithies,et al.  Convex Functions and Orlicz Spaces , 1962, The Mathematical Gazette.

[2]  Anatolii A. Puhalskii,et al.  Large deviation analysis of the single server queue , 1996 .

[3]  P. Dupuis,et al.  The large deviation principle for a general class of queueing systems. I , 1995 .

[4]  Paul Dupuis,et al.  Large deviations and queueing networks: Methods for rate function identification , 1998 .

[5]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[6]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[7]  R. Dobrushin,et al.  Large deviations for tandem queueing systems , 1994 .

[8]  Robert Sh. Liptser.  Large deviations for a simple closed queueing model , 1993, Queueing Syst. Theory Appl..

[9]  J. García.  An Extension of the Contraction Principle , 2004 .

[10]  David D. Yao,et al.  Fundamentals of Queueing Networks , 2001 .

[11]  Anatolii A. Puhalskii,et al.  Large Deviations and Idempotent Probability , 2001 .

[12]  Irina Ignatiouk-Robert,et al.  Large deviations of Jackson networks , 2000 .

[13]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[14]  Anatolii A. Puhalskii.  Stochastic processes in random graphs , 2004 .

[15]  A. A. Pukhal'skii.  On the Theory of Large Deviations , 1994 .

[16]  Kurt Majewski,et al.  Large Deviation Bounds for Single Class Queueing Networks and Their Calculation , 2004, Queueing Syst. Theory Appl..

[17]  A. Puhalskii.  On Some Degenerate Large Deviation Problems , 2004 .

[18]  Ward Whitt,et al.  Functional Large Deviation Principles for Waiting and Departure Processes , 1998, Probability in the Engineering and Informational Sciences.

[19]  Venkat Anantharam,et al.  Stationary tail probabilities in exponential server tandems with renewal arrivals , 1996, Queueing Syst. Theory Appl..

[20]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[21]  P. Dupuis,et al.  A VISCOSITY SOLUTION APPROACH TO THE ASYMPTOTIC ANALYSIS OF QUEUEING SYSTEMS , 1990 .

[22]  Hong Chen,et al.  Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations , 1991, Math. Oper. Res..

[23]  Anatolii A. Puhalskii,et al.  Moderate deviations for queues in critical loading , 1999, Queueing Syst. Theory Appl..