An ODE for an Overloaded X Model Involving a Stochastic Averaging Principle

TL;DR

This paper derives an ODE model for a two-class, two-pool overloaded queueing system under a fixed-queue-ratio control, incorporating a stochastic averaging principle to describe system behavior during overloads.

Contribution

It introduces a novel ODE framework for the X model with stochastic averaging, capturing the dynamics under overload and control policies.

Findings

The ODE accurately models system performance during overloads.

The stochastic averaging principle simplifies the analysis by replacing fast-scale processes with their long-run averages.

The model provides insights into queue ratios and system stability under FQR-T control.

Abstract

We study an ordinary differential equation (ODE) arising as the many-server heavy-traffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the X model in the call-center literature, operates under 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 unanticipated overload. Each pool serves only its own class until a threshold is exceeded; then one-way sharing is activated with all customer-server assignments then driving the two queues toward a fixed ratio. For large systems, that fixed ratio is achieved approximately. The ODE describes system performance during an overload. The control is driven by a queue-difference stochastic process, which operates in a faster time scale than the queueing processes themselves, thus achieving a time-dependent steady state instantaneously in the limit. As a result, for the ODE, the driving process is replaced by its long-run average behavior at each instant of time; i.e., the ODE involves a heavy-traffic averaging principle (AP).