# Induced idleness leads to deterministic heavy traffic limits for queue-based random-access algorithms

## TL;DR

This paper studies a queue-based random-access algorithm's heavy traffic limits, revealing nonstandard deterministic behavior caused by induced idleness, and introduces a new stochastic averaging method for analysis.

## Contribution

It establishes the deterministic heavy traffic limit for a queue-based algorithm on a complete interference graph, highlighting the impact of induced idleness and developing a novel analytical approach.

## Key findings

- Heavy traffic limit is deterministic.
- Scaling depends on algorithm parameters, not standard.
- Induced idleness causes nonstandard behavior.

## Abstract

We examine a queue-based random-access algorithm where activation and deactivation rates are adapted as functions of queue lengths. We establish its heavy traffic behavior on a complete interference graph, which turns out to be highly nonstandard in two respects: (1) the scaling depends on some parameter of the algorithm and is not the $N/N^2$ scaling usually found in functional central limit theorems; (2) the heavy traffic limit is deterministic. We discuss how this nonstandard behavior arises from the idleness induced by the distributed nature of the algorithm. In order to prove our main result, we developed a new method for obtaining a fully coupled stochastic averaging principle.