Sharpness: A Tight Condition for Scalability

• Authors: Augustin Chaintreau
• Venue: SIROCCO '08 Proceedings of the 15th international colloquium on Structural Information and Communication Complexity
• Year: 2008

A distributed system is scalable if the rate at which it completes its computation and communication tasks does not depend on its size. As an example, the scalability of a peer-to-peer application that transmits data among a large group depends on the topology and the synchronization implemented between the peers. This work describes a model designed to shed light on the conditions that enable scalability. Formally, we model here a collection of tasks, each requiring a random amount of time, which are related by precedence constraints. We assume that the tasks are organized along an euclidean lattice of dimension d. Our main assumption is that the precedence relation between these tasks is invariant by translation along any of these dimensions, so that the evolution of the system follows Uniform Recurrence Equations (UREs). Our main result is that scalability may be shown under two general conditions: (1) a criterion called "sharpness" satisfied by the precedence relation and (2) a condition on the distribution of each task completion time, which only depends on the dimension d. These conditions are shown to be tight. This result offers a universal technique to prove scalability which can be useful to design new systems deployed among an unlimited number of collaborative nodes.