An extended event graph with negative places and negative tokens for time window constraints

We introduce negative places and negative tokens into timed event graphs in order to model and analyze time window constraints. A negative token moves backward at an associated transition firing. We develop a necessary and sufficient condition, based on the circuits, for the event graph to be live. A sequence of feasible infinite firing epochs exists for each transition. We present a linear programming model for finding a stable firing schedule. We also characterize the minimal cycle time as a revised critical circuit ratio and the range of the feasible cycle times.