Justifying Diffusion Approximations for Stochastic Processing Networks Under a Moment Condition

Abstract

Stochastic processing networks (SPN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original SPN. To validate such an approximation amounts to justifying the interchange of two limits, t→∞ and k→∞, with t being the time index and k, the scaling parameter. Here, we show this interchange of limits is justified for a broad class of SPN under a p*-th moment condition on the primitive data, interarrival and service times; and we provide an explicit characterization of the required order (p*), which depends naturally on the desired order of convergence of the workload process. To illustrate the generality of this moment condition, we first use it to establish the justification for resource-sharing networks, where, to be processed each job needs to concurrently occupy multiple resources (servers), whereas each resource is shared among different job classes following a so-called "proportional fair allocation'' scheme. We then show the same approach applies to the more traditional multi-class queueing networks that are known to have diffusion limits.

Keywords: stochastic processing network, resource sharing network, multiclass queueing network, diffusion limit, interchange of limits, uniform stability