The Heavy Traffic Diffusion Approximation for Sojourn Times in Jackson Networks

Abstract

Using a heavy traffic limit theorem for open queueing networks, we find the correct diffusion approximation (D.A.) for sojourn times in Jackson networks with single server stations. The D.A. for sojourn times is a function of the D.A. for the queue length process, which is reflected Brownian motion on the nonnegative orthant.

We display a partial differential equation with boundary conditions which is satisfied by the stationary density of the D.A. for the queue length process. The solution of this equation is a product of exponentials when the diffusion is obtained as a limit of Jackson networks. Finally, we derive an expression for the stationary distribution of the D.A. for sojourn times based on the relationship between the D.A.’s for sojourn times and queue lengths. For many special cases, the stationary distribution of the D.A. for sojourn times precisely matches that of the Jackson network.