Diffusion Limits for Measure-Valued Queueing Models, With Application to General Random Order of Service Queues

Abstract

Currently, there is no general theory for deriving diffusion approximations of queueing systems with high- or infinite-dimensional state descriptors. In this paper, we explore one path for deriving diffusion limit equations of queueing models. The method hinges on a martingale decomposition of dynamics driven by time-changed renewal processes, which are a common feature of many queueing models. We then prove a central limit theorem for models decomposed in this way, which gives the form of stochastic differential equations (SDEs) that will be satisfied in the diffusion limit of such a system. We demonstrate the approach on a multiclass, multi-server random order of service queue with reneging and generally distributed interarrival, service, and patience times. In this setting, the state descriptor is measure-valued and the workload is nonlinear in the fluid limit. We prove tightness and uniqueness of limit points, and derive SDEs satisfied by the diffusion limit. Our results offer a broadly applicable method for diffusion approximations in complex queueing systems without relying on dimension reductions or model-specific structure.