The large deviation principle for a general class of queueing systems. I

Authors:
Paul Dupuis and Richard S. Ellis

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2689-2751

MSC:
Primary 60F10; Secondary 60K25, 90B22

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290716-2

MathSciNet review:
1290716

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into . In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity--type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.

**[1]**Robert F. Anderson and Steven Orey,*Small random perturbation of dynamical systems with reflecting boundary*, Nagoya Math. J.**60**(1976), 189–216. MR**0397893****[2]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[3]**Jean-Dominique Deuschel and Daniel W. Stroock,*Large deviations*, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR**997938****[4]**Paul Dupuis,*Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets*, Stochastics**21**(1987), no. 1, 63–96. MR**899955**, https://doi.org/10.1080/17442508708833451**[5]**Paul Dupuis and Richard S. Ellis,*Large deviations for Markov processes with discontinuous statistics. II. Random walks*, Probab. Theory Related Fields**91**(1992), no. 2, 153–194. MR**1147614**, https://doi.org/10.1007/BF01291423**[6]**Paul Dupuis and Richard S. Ellis,*A weak convergence approach to the theory of large deviations*, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication. MR**1431744****[7]**Paul Dupuis, Richard S. Ellis, and Alan Weiss,*Large deviations for Markov processes with discontinuous statistics. I. General upper bounds*, Ann. Probab.**19**(1991), no. 3, 1280–1297. MR**1112416****[8]**Paul Dupuis and Hitoshi Ishii,*On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications*, Stochastics Stochastics Rep.**35**(1991), no. 1, 31–62. MR**1110990**, https://doi.org/10.1080/17442509108833688**[9]**Paul Dupuis, Hitoshi Ishii, and H. Mete Soner,*A viscosity solution approach to the asymptotic analysis of queueing systems*, Ann. Probab.**18**(1990), no. 1, 226–255. MR**1043946****[10]**G. Kieffer,*The large deviation principle for two-dimensional stable systems*, Ph. D. Thesis, Univ. of Massachusetts, 1995.**[11]**A. A. Mogul′skiĭ,*Large deviations for the trajectories of multidimensional random walks*, Teor. Verojatnost. i Primenen.**21**(1976), no. 2, 309–323 (Russian, with English summary). MR**0420798****[12]**Y. W. Park,*Large deviation theory for queueing systems*. Ph.D. thesis, Virginia Polytechnic Institute and State University, 1991.**[13]**D. W. Stroock,*An introduction to the theory of large deviations*, Universitext, Springer-Verlag, New York, 1984. MR**755154****[14]**Pantelis Tsoucas,*Rare events in series of queues*, J. Appl. Probab.**29**(1992), no. 1, 168–175. MR**1147776**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60F10,
60K25,
90B22

Retrieve articles in all journals with MSC: 60F10, 60K25, 90B22

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290716-2

Article copyright:
© Copyright 1995
American Mathematical Society