Large deviations for a simple closed queueing model

We consider a simple queueing model with one service station. The arrival and service processes have intensitiesa(N−Qt) andNf(N−1Qt), where Qt is the queue length,N is a large integer,a>0 andf(x) is a positive continuous function. We establish the large deviation principle for the sequence of the normalized queue length processqNt =N−1Qt,N⩾1 for both light (a<f(0)) and heavy (a⩾f(0)) traffic and use this result for an investigation of ergodic properties ofqNt,N⩾ 1.

[1]  P. Dupuis,et al.  Large deviations for Markov processes with discontinuous statistics, II: random walks , 1992 .

[2]  P. Dupuis,et al.  Large deviations for Markov processes with discontinuous statistics , 1991 .

[3]  A. A. Mogul'skii.  Large Deviations for Trajectories of Multi-Dimensional Random Walks , 1977 .

[4]  A. Skorokhod.  Stochastic Equations for Diffusion Processes in a Bounded Region , 1961 .

[5]  P. Baldi,et al.  An extension of ventsel-freidlin estimates , 1988 .

[6]  A. A. Pukhal'skii,et al.  Limit theorems on large deviations for semimartingales , 1992, math/0510028.

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

[8]  Mikami Toshio.  Some generalizations of wentzell's lower estimates on large deviations , 1988 .

[9]  R. Azencott,et al.  Grandes deviations et applications , 1980 .

[10]  D. Stroock.  An Introduction to the Theory of Large Deviations , 1984 .

[11]  Hiroshi Tanaka.  Stochastic differential equations with reflecting boundary condition in convex regions , 1979 .

[12]  S. Leonov,et al.  Action functional for diffusion process with discontiuous drift , 1991 .

[13]  J. Jacod.  Shiryaev : Limit theorems for stochastic processes , 1987 .

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

[15]  A. Borovkov.  Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces , 1967 .

[16]  S. Varadhan.  Large Deviations and Applications , 1984 .

[17]  A. Friedman.  Stochastic Differential Equations and Applications , 1975 .

[18]  Kiyomasa Narita.  Large deviation principle for diffusion processes , 1988 .

[19]  A. A. Pukhal'skii,et al.  On functional principle of large deviations , 1991 .

[20]  J. Lynch,et al.  Large Deviations for Processes with Independent Increments , 1987 .

[21]  Srinivasa R. S. Varadhan,et al.  Asymptotic probabilities and differential equations , 1966 .

[22]  F. Chorlton,et al.  Differential Equations and Applications , 1966 .

[23]  Nigel J. Cutland.  An extension of the Ventcel-Freidlin large deviation principle , 1988 .

[24]  Sergei Leonov,et al.  Action functional for diffusions in discontinuous media , 1993 .

[25]  S. V. Anulova,et al.  Diffusional Approximation for Processes with the Normal Reflection , 1991 .