Invariant states and rates of convergence for a critical fluid model of a processor sharing queue

This paper contains an asymptotic analysis of a fluid model for a heavily loaded processor sharing queue. Specifically, we consider the behavior of solutions of critical fluid models as time approaches1. The main theorems of the paper provide sufficient conditions f or a fluid model solution to converge to an invariant state and, under slightly more restrictive assumptions, provide a rate of convergence. These results are used in a related work by Gromoll for establishing a heavy traffic diffusion approximation for a processor sharing queue.

[1]  H. C. Gromoll.  Diffusion approximation for a processor sharing queue in heavy traffic , 2004, math/0405298.

[2]  Ruth J. Williams,et al.  Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse , 1998, Queueing Syst. Theory Appl..

[3]  Maury Bramson,et al.  Convergence to equilibria for fluid models of FIFO queueing networks , 1996, Queueing Syst. Theory Appl..

[4]  Hong Chen,et al.  Fluid approximations for a processor-sharing queue , 1997, Queueing Syst. Theory Appl..

[5]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[6]  Gennady Samorodnitsky,et al.  Heavy Tails and Long Range Dependence in On/Off Processes and Associated Fluid Models , 1998, Math. Oper. Res..

[7]  R. Durrett.  Probability: Theory and Examples , 1993 .

[8]  Kai Lai Chung,et al.  Book Review — An Introduction to Probability Theory and its Applications 2, 2nd ed. , 2004 .

[9]  Maury Bramson,et al.  Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks , 1996, Queueing Syst. Theory Appl..

[10]  Maury Bramson,et al.  State space collapse with application to heavy traffic limits for multiclass queueing networks , 1998, Queueing Syst. Theory Appl..

[11]  E. Perkins,et al.  9. Markov Processes, Characterization and Convergence , 1988 .

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[13]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[14]  K. Chung.  Review: William Feller, An Introduction to Probability Theory and its Applications 2 , 1973 .

[15]  J. Wendelberger.  Adventures in Stochastic Processes , 1993 .

[16]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

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

[18]  Amber L. Puha,et al.  THE FLUID LIMIT OF A HEAVILY LOADED PROCESSOR SHARING QUEUE , 2002 .