Partial synchronicity and the (max,+) semiring

Abstract

In this paper we illustrate how non-stochastic (max,+) techniques can be used to describe partial synchronization in a Discrete Event Dynamical System. Our work uses results from the spectral theory of dioids and analyses (max,+) equations describing various synchronization rules in a simple network. The network in question is a transport network consisting of two routes joined at a single point, and our Discrete Events are the departure times of transport units along these routes. We calculate the maximum frequency of circulation of these units as a function of the synchronization parameter. These functions allow us further to determine the waiting times on various routes, and here we find critical parameters (dependent on the fixed travel times on each route) which dictate the overall behavoiur. We give explicit equations for these parameters and state the rules which enable optimal performance in the network (corresponding to minimum waiting time).