Open Queueing Networks in Heavy Traffic

Abstract
This paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks. These limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity. The limit diffusion is reflected Brownian motion on the nonnegative orthant. This process behaves like Brownian motion on the interior of its state space, and reflects instantaneously on the boundaries. The reflection direction is a constant for each boundary hyperplane.