The Availability of Quorum Systems

A quorum system is a collection of sets (quorums) every two of which have a nonempty intersection. Quorum systems have been used for a number of applications in the area of distributed systems. In this paper we study the fault-tolerance properties of quorum systems, and their implications on quorum based distributed protocols. For this study we adopt the following common model. The processors (elements of the system) are assumed to fail independently with a uniform probability p. The global failure probability F p (S) of a quorum system S is the probability that no complete quorum is functioning. This probability is a good measure of the (non) availability of protocols based on S. In this paper, the properties of F p are examined. It is shown that the most available quorum systems are the non-dominated coteries (NDC). Over the NDC's, the behavior of the failure probability F p depends on the exact value of p. It is shown that for any S 2 NDC, F p (S) is symmetric, i. From this, a new proof for a known lower bound on the number of sets in an NDC is derived. Furthermore, for 0 < p < 1 2 , the most available NDC is shown to be the \democracy" (namely, the minimal majority system), while the \monarchy" (singleton system) is the least available. Due to symmetry, the picture reverses for 1 2 < p < 1. The asymptotic behavior of F p is also considered, when the number of processors tends to innnity. It is shown that some NDC constructions enjoy the property that F p ! 0 when p < 1 2 , while others have a nonzero limit function.