Balanced Allocations and Global Clock in Population Protocols: An Accurate Analysis

The context of this paper is the two-choice paradigm which is deeply used in balanced online resource allocation, priority scheduling, load balancing and more recently in population protocols. The model governing the evolution of these systems consists in throwing balls one by one and independently of each others into n bins, which represent the number of agents in the system. At each discrete instant, a ball is placed in the least filled bin among two bins randomly chosen among the n ones. A natural question is the evaluation of the difference between the number of balls in the most loaded bin and the one in the least loaded. At time t, this difference is denoted by \(\text {Gap}(t)\). A lot of work has been devoted to the derivation of asymptotic approximations of this gap for large values of n. In this paper we go a step further by showing that for all \(t \ge 0\), \(n \ge 2\) and \(\sigma > 0\), the variable \(\text {Gap}(t)\) is less than \(a(1+\sigma )\ln (n) + b\) with probability greater than \(1-1/n^\sigma \), where the constants a and b, which are independent of t, \(\sigma \) and n, are optimized and given explicitly, which to the best of our knowledge has never been done before.