Diffusion Approximation for an Overloaded X Model Via a Stochastic Averaging Principle

TL;DR

This paper develops a refined diffusion approximation for an overloaded two-class, two-pool queueing system with a fixed-queue-ratio control, extending previous fluid models using a stochastic averaging principle.

Contribution

It introduces a diffusion approximation for the X model under FQR-T control, based on a heavy-traffic FCLT, advancing the theoretical understanding of such systems.

Findings

Established a diffusion limit for the X model under overload conditions

Extended the stochastic averaging principle to a diffusion scale

Provided a more precise approximation for system performance in heavy traffic

Abstract

In previous papers we developed a deterministic fluid approximation for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. The control aims to keep the two queues at a pre-specified fixed ratio. We supported the fluid approximation by establishing a many-server heavy-traffic functional weak law of large numbers (FWLLN) involving an averaging principle. In this paper we develop a refined diffusion approximation for the same model based on a many-server heavy-traffic functional central limit theorem (FCLT).