The availability of crumbling wall quorum systems

Abstract

A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information. Crumbling walls are a general class of quorum systems. The elements (processors) of a wall are logically arranged in rows of varying widths. A quorum in a wall is the union of one full row and a representative from every row below the full row. This class considerably generalizes a number of known quorum system constructions. In this paper we study the availability of crumbling wall quorum systems. We show that if the row width is bounded, or if the number of rows is bounded, then the wall's failure probability F p does not vanish as the number of elements tends to infinity (i.e., F p is not Condorcet). If the wall may grow in both the row number and row width, we show that the behavior depends on the rate of growth of the row width. We establish a sharp threshold rate: when the row width n i ⩽ ⌊ log 2 2 i⌋ then F p is Condorcet, and when n i ⩾ (1 + ε) log 2 i then F p is not Condorcet.