Repeated Games, Optimal Channel Capture, and Open Problems for Slotted Multiple Access

TL;DR

This paper analyzes a classical slotted multiple access problem, reports on a 2-user game with a winning algorithm, and proposes an efficient, conjectured optimal algorithm for n-user channel capture with proven cases.

Contribution

It introduces the 4-State algorithm for 2-user access, analyzes its optimality, and develops a new algorithm for n-user scenarios with proven optimality in specific cases.

Findings

The 4-State algorithm is a consistent winner in 2-user games.

An efficient algorithm for n-user channel capture is proposed and conjectured to be optimal.

Optimality is proven for specific small cases using a novel analytical technique.

Abstract

This paper revisits a classical problem of slotted multiple access with success, idle, and collision events on each slot. First, results of a 2-user multiple access game are reported. The game was conducted at the University of Southern California over multiple semesters and involved competitions between student-designed algorithms. An algorithm called 4-State was a consistent winner. This algorithm is analyzed and shown to have an optimal expected score when competing against an independent version of itself. The structure of 4-State motivates exploration of the open question of how to minimize the expected time to capture the channel for a $n$-user situation. It is assumed that the system delivers perfect feedback on the number of users who transmitted at the end of each slot. An efficient algorithm is developed and conjectured to have an optimal expected capture time for all positive integers $n$. Optimality is proven in the special cases $n \in \{1, 2, 3, 4, 6\}$ using a novel analytical technique that introduces virtual users with enhanced capabilities.