Stationary deterministic flows in discrete systems 1

Abstract We consider a deterministic system whose state space is the n -dimensional first orthant. It may be considered as a network of (deterministic) queues, a Karp-Miller vector addition system, a Petri net, a complex computer system, etc. Weak assumptions are then made concerning the asymptotic or limiting behaviour of the instants at which events are observed across a cut in the system: these instants may be considered as ‘arrival’ or ‘departure’ instants. Thus, like in operational analysis, we deal with deterministic and observable properties and we need no stochastic assumptions or restrictions (such as independence, identical distributions, etc.). We consider however asymptotic or stationary properties, as in conventional queuing analysis. Under our assumptions a set of standard theorems are proved: concerning arrival and departure instant measures, concerning ‘birth and death’ type equations, and concerning Little's formula. Our intention is to set the framework for a new approach to performance modelling of computer systems in a context close to that used in actual measurements, but taking into account infinite time behaviour in order to take advantage of the useful mathematical properties of asymptotic results.