Token Loss Detection for Random Walk based Algorithm

Abstract

Self-stabilizing token circulation algorithms are not always adapted for dynamic networks. Random walks are well known to play a crucial role in the design of randomized algorithms. The combination of these two concepts makes it possible to design a solution that is adaptive to topology changes and is tolerant to transient faults. Our purpose in this paper is to study the behavior of such algorithms with possible transient failures. We provide a probabilistic analysis of the waiting time. More precisely, we give two bounds on the probability for a processor to wait for the token more than a certain amount of time. The first bound is based on the token return time (the expected time for the token to visit again a processor) and the second one, a tighter upper bound, is based on its variance. Next, we characterize a local ¿criterion of suspicion¿ for each processor to be in a faulty global configuration; in fact a token loss detection. Thanks to this criterion, we propose to refine the timeout procedure used in [6, 1]. Thus, an improved version of an adaptive and self-stabilizing random walk token circulation algorithm can be designed.