An infinitesimal generator approach on weak convergence of regulated multi-class matching systems

TL;DR

This paper develops a diffusion approximation for a complex multi-class matching system with reneging, using an infinitesimal generator approach to analyze its heavy traffic behavior and connect it with existing models.

Contribution

It introduces a novel generator-based method to derive diffusion limits for regulated multi-class matching systems under heavy traffic, addressing structural challenges.

Findings

Established a tractable diffusion approximation under heavy traffic.

Connected generator-based limits with existing stochastic integral equation models.

Provided insights into the dynamics of regulated coupled matching systems.

Abstract

We consider a regulated multi-class instantaneous matching system with reneging, in which each event requires $K \geq 2$ distinct impatient agents who wait in their respective queues. Each agent class is subject to a buffer capacity, allowing for the special case without buffers. Due to the instantaneous matching behavior, at any give time, at least one category has an empty queue. Under the Markovian assumption, the system dynamics are described by a Markov chain with innovative rate matrices that capture all possible queue configurations across all classes. To effectively circumvent the structural challenges introduced by instantaneous matching, we establish a non-trivial yet tractable diffusion approximation under heavy traffic conditions by leveraging the infinitesimal generator in conjunction with appropriate regulation and boundary conditions. This asymptotic analysis offers a direct explanation of the dynamics of the regulated coupled heavy-traffic limiting process. Furthermore, we demonstrate the connection between the diffusion-scaled limit derived from the generator approach and the one established in the literature. The latter is typically described by a regulated coupled stochastic integral equation.