New CRT sequence sets for a collision channel without feedback

TL;DR

This paper introduces a new construction of user-irrepressible protocol sequences for collision channels without feedback, achieving shorter sequence periods and delays than previous methods, with proven optimality under certain conditions.

Contribution

The paper presents a CRTm construction for UI sequences with weight M+1, improving delay performance and establishing a tight lower bound on sequence period for equi-difference structures.

Findings

Shortest known sequence period for non-prime M ≥ 8

Better average delay than random access schemes

Proven optimality of the construction under certain conditions

Abstract

Protocol sequences are binary and periodic sequences used for deterministic multiple access in a collision channel without feedback. In this paper, we focus on user-irrepressible (UI) protocol sequences that can guarantee a positive individual throughput per sequence period with probability one for a slot-synchronous channel, regardless of the delay offsets among the users. As the sequence period has a fundamental impact on the worst-case channel access delay, a common objective of designing UI sequences is to make the sequence period as short as possible. Consider a communication channel that is shared by $M$ active users, and assume that each protocol sequence has a constant Hamming weight $w$. To attain a better delay performance than previously known UI sequences, this paper presents a CRTm construction of UI sequences with $w=M+1$, which is a variation of the previously known CRT construction. For all non-prime $M\geq 8$, our construction produces the shortest known sequence period and the shortest known worst-case delay of UI sequences. Numerical results show that the new construction enjoys a better average delay performance than the optimal random access scheme and other constructions with the same sequence period, in a variety of traffic conditions. In addition, we derive an asymptotic lower bound on the minimum sequence period for $w=M+1$ if the sequence structure satisfies some technical conditions, called equi-difference, and prove the tightness of this lower bound by using the CRTm construction.