Diffusion Limits for Measure-Valued Queueing Models Without State Space Collapse, With Application to General Random Order of Service Queues

TL;DR

The paper addresses diffusion-approximation for queueing systems with high- or infinite-dimensional state descriptors, where reneging and general primitives preclude state-space collapse. It develops a martingale-decomposition framework for time-changed renewal dynamics, derives a central limit theorem for renewal-driven systems, and obtains diffusion-limit SDEs in the space $D([0,ty), \mathscr{S}')$, applicable to complex, non-Markovian queues. The authors demonstrate the method on a multiclass, multi-server random order of service queue with reneging, producing a tightness result and a diffusion-limit description via a system of SDEs, without relying on workload-based dimension reductions. This framework provides a broadly applicable, model-agnostic approach to diffusion approximations in modern stochastic networks where traditional state-space-collapse techniques fail or are intractable, with potential applications to enzymatic processing and other non-HL queueing systems.

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 address this by developing a novel framework 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. Unlike existing approaches, our framework does not require state space collapse or tractable workload representations, enabling its application to systems with features such as reneging. 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 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.

Theorems & Definitions (56)

• Remark 2.1
• Definition 3.1: Fluid model parameters
• Definition 3.2: Fluid Model Solution
• Theorem 3.1: Loeser--Williams
• Theorem 4.1
• Remark 4.1
• Remark 5.1
• Remark 5.2
• Definition 5.1: Fluid-scaled Renewal Process
• Definition 5.2: Diffusion-scaled Renewal Process