Validity of Steady-State Heavy Traffic
Approximations in Generalized Jackson Networks
David Gamarnik
Department of Math Sciences
IBM T.J.Watson Research Center
Wednesday, October 27, 2004
4:30 - 5:45 PM
Terman Engineering Center, Room 453
Abstract:
(Generalized) Jackson network is one of the oldest queueing network
models. The closed form type solution to this model is not achievable
in general, modulo some specific Markovian type cases. As a result, the
researchers focused on asymptotic approaches, and here the diffusion
approximation plays a very prominent role. In particular, a classical
theorem by Reiman asserts that a sequence of normalized queue length
processes converges weakly to an associated Reflected Brownian Motion
(RBM), as the network approaches the heavy traffic regime. However, the
result corresponds only to the transient behavior, and it is still not
known whether the stationary distribution of the RBM provides a valid
approximation of the underlying GJN in steady-state. We resolve this
open problem, thus validating a so-called ``interchange-of-limits'' for
this class of networks. The result has several useful implications. In
particular, in some cases, the asymptotic steady state distribution of
queue lengths and waiting times can be computed in closed form.
Joint work with Assaf Zeevi (Columbia University).
Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html