Optimization and Hierarchies for Lumped Distributed Networks

Abstract

The present paper is a generalization to lumped networks of our previous results [2, 5] on a stochastic model for performance evaluation of various distributed algorithms. In this model, a network is represented by N appropriate finite Markov chains, connected by relations R between entries of their transition matrices. “Solutions” ρ for a functioning of such a network are represented by vectors whose components are entries of the mentioned transition matrices satisfying R. The criterion of good functioning is based on the definition of a function F,named the guide function, to be optimized under constraints. The choice of F depends on the context. Two approaches are proposed: the first one uses mean recurrence tintes of ergodic states of the Markov chains; the other one uses mathematical expectations of random numbers of passages from states to others. These two approaches can lead to the same guide function F, which induces a hierarchy on “solutions” — the first order hierarchy, so named because conditional expectation and mathematical expectation are (probabilistic) first order moments. In this hierarchy, the best “solutions” ρ are optimal points of the optimization problem. When two “solutions” are equivalent in the first order hierarchy, we make use of variances for deciding between them. The better is the one with, globally, smaller variances. A function of variances is then defined, which generates a new hierarchy, named the second order hierarchy, because variance and conditional variance are (probabilistic) second order central moments. We show that the two mentioned approaches lead to two different ways for the second order hierarchy. As applications, we examine some particular problems: a problem of routing algorithms, the mutual exclusion problem and the dining philosophers’ problem. In addition to optimization, fuzzy sets are built which provide a comprehensive view of sets of “solutions”.