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Large deviations of the long term distribution of a non Markov process. (English) Zbl 1441.60025
The author continues investigations in the field of large deviation theory by means of idempotent measure theory. The paper considers the long-term distribution of the non-Markov process of queue length in an ergodic generalized Jackson networks and establishes a large deviation principle. The deviation function is given by the quasipotential related to the stationary idempotent distribution of the limit idempotent process. The latter distribution is also a unique long-term idempotent distribution. The geometric ergodicity of the queue length process implies that the long-term idempotent distribution is the large deviation limit of the long-term queue length distributions.
MSC:
60F10 Large deviations
60K25 Queueing theory (aspects of probability theory)
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[1] S. L. Bell and R. J. Williams, Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy, Ann. Appl. Probab. 11 (2001), no. 3, 608-649. · Zbl 1015.60080
[2] M. Bramson, Stability of queueing networks, Lecture Notes in Mathematics, vol. 1950, Springer, Berlin, 2008, Lectures from the 36th Probability Summer School held in Saint-Flour, July 2-15, 2006. · Zbl 1189.60005
[3] M. Bramson, Stability of queueing networks, Probab. Surv. 5 (2008), 169-345. · Zbl 1189.60005
[4] H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations, Math. Oper. Res. 16 (1991), no. 2, 408-446. · Zbl 0735.60095
[5] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Nauka, 1979, In Russian, English translation: Springer, 1984.
[6] S. P. Meyn and D. Down, Stability of generalized Jackson networks, Ann. Appl. Probab. 4 (1994), no. 1, 124-148. · Zbl 0807.68015
[7] A. Puhalskii, Large deviations and idempotent probability, Chapman & Hall/CRC, 2001. · Zbl 0983.60003
[8] A. Puhalskii, On large deviation convergence of invariant measures, J. Theoret. Probab. 16 (2003), no. 3, 689-724. · Zbl 1028.60023
[9] A. A. Puhalskii, The action functional for the Jackson network, Markov Processes and Related Fields 13 (2007), 99-136. · Zbl 1132.60068
[10] A. Shwartz and A. Weiss, Large deviations for performance analysis, Stochastic Modeling Series, Chapman & Hall, London, 1995, Queues, communication, and computing, with an appendix by R. J. Vanderbei.
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