A queueing system with on-demand servers: local stability of fluid limits

TL;DR

This paper analyzes a queueing system with on-demand servers, establishing local stability of its fluid limits using switched linear systems and Lyapunov functions, and explores conditions for global stability and system performance.

Contribution

It introduces a stability analysis framework for a queueing system with on-demand servers using fluid limits and switched linear systems, providing new stability conditions.

Findings

Derived sufficient local stability conditions for the fluid limits.

Used Lyapunov functions to analyze stability of the system.

Numerical experiments support the conjecture that local stability implies global stability.

Abstract

We study a system, where a random flow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some fixed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic objective is to minimize both customer and agent waiting times. We establish the system process fluid limits in the asymptotic regime where the customer arrival rate goes to infinity. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of fluid limits at the desired equilibrium point, and derive a variety of sufficient local stability conditions. For our model, we conjecture that local stability is in fact sufficient for global stability of fluid limits; the validity of this conjecture is supported by numerical and simulation experiments. When local stability conditions do hold, simulations show good overall performance of the scheme.