The throughput in multi-channel (slotted) ALOHA: large deviations and analysis of bad events

TL;DR

This paper analyzes the throughput of multi-channel slotted ALOHA protocols using large deviations theory, providing probabilistic formulas, optimization, and insights into system performance under various conditions.

Contribution

It introduces large deviation principles for multi-channel ALOHA, deriving formulas and laws of large numbers for throughput and analyzing the causes of low delivery success.

Findings

Derived large deviation principles for message delivery quantities

Optimized throughput over transmission probability

Provided insights into causes of low success events

Abstract

We consider ALOHA and slotted ALOHA protocols as medium access rules for a multi-channel message delivery system. Users decide randomly and independently with a minimal amount of knowledge about the system at random times to make a message emission attempt. We consider the two cases that the system has a fixed number of independent available channels, and that interference constraints make the delivery of too many messages at a time impossible. We derive probabilistic formulas for the most important quantities like the number of successfully delivered messages and the number of emission attempts, and we derive large-deviation principles for these quantities in the limit of many participants and many emission attempts. We analyse the rate functions and their minimizers and derive laws of large numbers for the throughput. We optimize it over the probability parameter. Furthermore, we are interested in questions like ``if the number of successfully delivered messages is significantly lower than the expectation, was the reason that too many or too few sending attempts were made?''. Our main tools are basic tools from probability and the theory of (the probabilities of) large deviations.