Multiscale Diffusion Approximations for Open Queueing Networks in Heavy Traffic

• Activity Number: 133
• Type: Contributed
• Date/Time: Monday, August 2, 2010 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Xin Liu*+
• Companies: The University of North Carolina at Chapel Hill
• Keywords: Diffusion approximations ; Queueing networks in random environment ; Heavy traffic ; Multiscale analysis ; Reflected Markov modulated diffusions ; Constrained martingale problems

Abstract

Open queueing networks with time varying arrival and service rates and routing structure are studied. Time variations are governed, in addition to the state of the system, by two independent Markov processes X and Y. Transition times of X are significantly smaller than typical interarrival and service times whereas the reverse is true for the Markov process Y. By a suitable martingale problem, diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. It is shown that properly normalized queue length processes converge weakly to a reflected diffusion with drift and diffusion coefficients that, in addition to depending on the state process, are functions of the invariant distributions of X. Additionally, the coefficients are modulated by a finite state Markov process which is independent of the driving Brownian motion.