Petri net modeling and wafer sojourn time analysis of single-arm cluster tools with residency time constraints and activity time variation

With wafer residency time constraints, it is crucial to schedule a cluster tool in semiconductor fabrication such that the wafer sojourn time in a processing module is in a given range. However, because of the activity time variation in wafer fabrication by cluster tools, a feasible schedule obtained under the assumption of deterministic activity times may become infeasible. To solve this problem, it is critically important to reveal the wafer sojourn time fluctuations with bounded activity time variation. This paper targets at single-arm cluster tools. They are modeled by a Petri net to describe the fabrication processes. Based on the net, a real-time control policy is proposed such that its use offsets the effect of the activity time variation as much as possible. Then, the wafer sojourn time delay is analyzed and analytical expressions are given to calculate the upper bound. With the proposed method, we can check if a given schedule is feasible under bounded activity time variation. Examples are given to show the applications of the research results.