Ascending Runs in Dependent Uniformly Distributed Random Variables: Application to Wireless Networks

Abstract

We analyze in this paper the longest increasing contiguous sequence or maximal ascending run of random variables with common uniform distribution but not independent. Their dependence is characterized by the fact that two successive random variables cannot take the same value. Using a Markov chain approach, we study the distribution of the maximal ascending run and we develop an algorithm to compute it. This problem comes from the analysis of several self-organizing protocols designed for large-scale wireless sensor networks, and we show how our results apply to this domain.

One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems

“…Under the assumption that the distribution F is discrete, the limit behaviour of MT depends strongly on the common law F, as in Reference [32] (see also References [33,34]) proved for the case of geometric and Poisson distribution. In Reference [35], the case of discrete uniform distribution is investigated, while in Reference [36], the authors study the asymptotic distribution of MT when the variables are uniformly distributed but not independent. In this section, we evaluate the distribution of the length of the longest increasing run using the methodology developed in Section 2.…”

Self-stabilizing philosophers with generic conflicts

“…Note that in order to satisfy nontrivial liveness properties we assume that our environment conforms to transmission fairness: if a process attempts to send infinitely many messages, all of its communication neighbors will receive infinitely many of them. This assumption is weaker than what is previously used for self-stabilizing algorithms in sensor networks [Herman and Tixeuil 2004;Mitton et al 2006;2008]: it is usually assumed that the expected message transmission time for one hop neighbors is constant. Our idea is to use the timeouts such that the lost messages are recovered.…”