Fluid Limits for an ALOHA-type Model with Impatient Customers

TL;DR

This paper analyzes a time-slotted ALOHA network model with impatient users, demonstrating that the system's workload converges to a unique fluid limit described by an integral equation, regardless of initial conditions.

Contribution

It introduces a fluid limit analysis for an ALOHA model with impatience, establishing convergence to a unique solution and long-term stability.

Findings

Fluid limit coincides with a unique integral equation solution.

Workload converges to the same value over time from different initial states.

Model captures overload behavior with user impatience.

Abstract

Random multiple-access protocols of type ALOHA are used to regulate networks with a star configuration where client nodes talk to the hub node at the same frequency (finding a wide range of applications among telecommunication systems, including mobile telephone networks and WiFi networks). Such protocols control who talks at what time sharing the common idea "try to send your data and, if your message collides with another transmission, try resending later". In the present paper, we consider a time-slotted ALOHA model where users are allowed to renege before transmission completion. We focus on the scenario that leads to overload in the absence of impatience. Under mild assumptions, we show that the fluid (or law-of-large-numbers) limit of the system workload coincides a.s. with the unique solution to a certain integral equation. We also demonstrate that the fluid limits for distinct initial conditions converge to the same value as time tends to infinity.