Comparison Theorems and the Validity of Heavy Traffic Limit Distributions for Stochastic Networks

The issue of stability of stochastic networks continues to be one of the central issues in applications along with the computation of the stationary distribution. Under heavy traffic conditions, in recent works by Gamarnik and Zeevi (AAP'06) and Budhiraja and Lee (MOR'09) they showed that the steady state limits of networks with i.i.d inputs that satisfy a FCLT converge to the corresponding stationary distribution of the limit SRBM (Semi-martingale Reflecting Brownian Motion) or in other words the large time limit and the scaling limit can be interchanged. Their arguments were based on Lyapunov techniques.

In this talk we consider networks with Levy inputs and show that the corresponding interchange result holds via direct arguments without the need for Lyapunov techniques. Indeed the results follow in a direct manner from some well known results on stochastic networks with Levy inputs due to Kella and Whitt.

In the second part of the talk I will show some new comparison theorems for stochastic networks that yield useful results for showing conditions when monotonicity will hold. We conclude with a generalization of these results to general reflected diffusions with jumps even allowing for state-dependent reflections when the continuity of the Skorohod map cannot be shown (joint work with Dr. Francisco Piera (University of Chile) and Jean-Paul Haddad (Waterloo)).

Bio: The speaker was educated at the Indian Institute of Technology, Bombay (B.Tech, 1977), Imperial College, London (MSc, DIC, 1978) and UCLA (PhD, 1983). He is currently a University Research Chair Professor in the Dept. of ECE at the University of Waterloo, Ont., Canada where he has been since September 2004. Prior to this he was Professor of ECE at Purdue University, West Lafayette, USA where he continues to be an Adjunct Professor. He is a Fellow of the IEEE and the Royal Statistical Society. He is a recipient of INFOCOM 2006 Best Paper Award and was runner-up for the Best Paper Award at INFOCOM 1998. He has served as an editor of the IEEE/ACM Trans on Networking (2004-09) and as guest editor for a number of special issues of networking and applied probability related journals. He is also the author of a forthcoming monograph on performance modeling statistical multiplexing in networks to be published by Morgan and Claypool in early 2010. His research interests are in modeling, control, and performance analysis of both wireline and wireless networks, and in applied probability and stochastic analysis with applications to queueing, filtering, and control.