Mathematical Finance and Probability Seminars (Since covid these events are taking place online.)

Diffusion Approximations for Multiscale Stochastic Networks in Heavy traffic

Tuesday, April 03, 2012 at 11:00am - 12:00pm

Speaker: Xin Liu, Institute for Mathematics and its Applications, University of Minnesota

We study a sequence of nearly critically loaded queueing networks, with time varying arrival and service rates and routing structure. The nth network is described in terms of two independent finite state Markov processes {Xn(t): t ? 0} and {Yn(t) : t ? 0} which can be interpreted as the random environment in which the system is operating. The process Xn changes states at a much higher rate than the typical arrival and service times in the system, while the reverse is true for Yn. The variations in the routing mechanism of the network are governed by Xn, whereas the arrival and service rates at various stations depend on the state process (i.e. queue length process) and both Xn and Yn. Denoting by Zn the suitably normalized queue length process, it is shown that, under appropriate heavy traffic conditions, the pair Markov process (Zn, Yn) converges weakly to the Markov process (Z, Y), where Y is a finite state continuous time Markov process and the process Z is a reflected diffusion with drift and diffusion coefficients depending on (Z, Y) and the stationary distribution of Xn. We also study stability properties of the limit process (Z, Y).

Speaker: Xin Liu, Institute for Mathematics and its Applications, University of Minnesota

Location   Hill 705