Steady state analysis of timed event graphs with time window constraints

Abstract

For discrete event systems, time window constraints between two events are often required to control their behavior. Such time constrained discrete event systems are found in various industrial systems such as chemical vapor deposition processes for wafer fabrication, electroplating lines for printed circuit board fabrication, microcircuit design etc. A negative event graph (NEG) which is an extension of the timed event graph (TEG) has been proposed by Lee and Park (2005) to model and analyze discrete event systems with time window constraints. An NEG can model time window constraints between any two transitions by introducing negative places and negative tokens. This study examines the steady state behavior of an NEG that satisfies the time window constraints. We develop a recurrent equation for the feasible steady firing epochs based on the minimax algebra. In addition, we identify 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 by extending the steady state results of a conventional TEG based on the minimax algebra. Besides, we characterize how the cycle times and the steady schedules are computed through the matrix algebra and the associated graph algorithms.