Steady state analysis of a timed event graph with time window constraints

Abstract

A negative even graph, introduced by Lee et al. (2002) is a timed event graph that allows negative places and negative tokens for modeling time window constraints between any two transitions. Such time constrained discrete event systems are found in cluster tool scheduling for semiconductor manufacturing or microcircuit design. We examine the steady state behavior of the feasible firing schedules of a negative event graph that satisfy the time window constraints. We develop a recurrent equation for the feasible firing epochs based on the minimax algebra. By extending the steady state results of a conventional timed event graph based on the minimax algebra, we show that there are four classes of steady states that correspond to the earliest and latest feasible steady firing schedules for each of the minimum and maximum cycle times. We characterize how the cycle times and the steady schedules are computed through some matrix algebra and the associated graph algorithms.