An Algebraic Model For Quorum Systems

Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturing trust assumptions. A quorum system is a collection of subsets of all processes, called quorums, with the property that each pair of quorums have a non-empty intersection. They can be found at the core of many reliable distributed systems, such as cloud computing platforms, distributed storage systems and blockchains. In this paper we give a new interpretation of quorum systems, starting with classical majority-based quorum systems and extending this to Byzantine quorum systems. We propose an algebraic representation of the theory underlying quorum systems making use of multivariate polynomial ideals, incorporating properties of these systems, and studying their algebraic varieties. To achieve this goal we will exploit properties of Boolean Groebner bases. The nice nature of Boolean Groebner bases allows us to avoid part of the combinatorial computations required to check consistency and availability of quorum systems. Our results provide a novel approach to test quorum systems properties from both algebraic and algorithmic perspectives.Comment: 15 pages, 3 algorithm

Open Heterogeneous Quorum Systems

In contrast to proof-of-work replication, Byzantine replicated systems maintain consistency with higher throughput, modest energy consumption, and deterministic liveness guarantees. If complemented with open membership and heterogeneous trust, they have the potential to serve as a global financial infrastructure. This paper presents a general model of heterogeneous quorum systems, where each participant can declare its own quorums, and captures the consistency, availability, and inclusion properties of these systems. In order to support open membership, it then presents reconfiguration protocols for heterogeneous quorum systems: joining and leaving of a process, and adding and removing of a quorum. It presents trade-offs for the properties that reconfigurations can preserve, and accordingly, presents reconfiguration protocols and proves their correctness. It further presents a graph characterization of heterogeneous quorum systems, and its application for reconfiguration optimization

Well-formed Properties of Heterogeneous Quorum Systems

Byzantine quorum systems provide higher throughput than proofof-work and incur modest energy consumption. Further, their modern incarnations incorporate personalized and heterogeneous trust. Thus, they are emerging as an appealing candidate for global financial infrastructure. However, since their quorums are not uniform across processes anymore, the properties that they should maintain to support abstractions such as reliable broadcast and consensus are not well-understood. In this paper, we first see a general model of heterogeneous quorum systems where each participant can declare its own quorums, and capture their properties. It has been shown that the two properties quorum intersection and availability are necessary. In this paper, we prove that they are not sufficient. We then define the notion of quorum inclusion, and show that the three conditions together are sufficient: we present reliable broadcast and consensus protocols, and prove their correctness for quorum systems that provide the three properties

Finding Optimal Quorum Assigments for Distributed Databases

Replication has been studied as a method of increasing the availability of a data item in a distributed database subject to component failures and consequent partitioning. The potential for partitioning requires that a protocol be employed which guarantees that any access to a data item is aware of the most recent update to that data item. By minimizing the number of access requests denied due to this constraint, we maximize availability. In the event that all access requests are reads, placing one copy of the data item at each site clearly leads to maximum availability. The other extreme, all access requests are write requests or are treated as such, has been studied extensively in the literature. In this paper we investigate the performance of systems with both read and write requests. We describe a distributed on-line algorithm for determining the optimal parameters, or optimal quorum assignments, for a commonly studied protocol, the quorum consensus protocol[9]. We also show how to incorporate these optimization techniques into a dynamic quorum reassignment protocol. In addition, we demonstrate via simulation both the value of this algorithm and the effect of various read-write rations on availability. This simulation, on 101 sites and up to 5050 links(fully- connected), demonstrates that the techniques described here can greatly increase data availability, and that the best quorum assignments are frequently realized at the extreme values of the quorum parameters

A Performance Evaluation of a Quorum-Based State-Machine Replication Algorithm for Computing Grids

Quorum systems are well-known tools that improve the performance and the availability of distributed systems. In this report we explore their use as a means to achieve low response time for network services that are replicated and accessed over computing grids. To that end, we propose both a quorum construction and a quorum-based state-machine replication algorithm that tolerates crash failures in a partially synchronous model. We show through the evaluation of a real implementation that although simple, this quorum construction and replication algorithm exhibit a response time 20% lower than that of a regular active replication algorithm in appropriate conditions