Schedulability analysis of time-constrained cluster tools with bounded time variation by an extended Petri net

Abstract

Cluster tools for some wafer fabrication processes such as low-pressure chemical vapor deposition have strict wafer delay constraints. A wafer that completes processing in a processing chamber should leave the chamber within a specified time limit. Otherwise, the wafer suffers from severe quality troubles due to residual gases and heat within the chamber. An important engineering problem is to verify whether for given task times there exists a tool operation schedule that satisfies the wafer delay limit. There have been studies on the problem, which all assume deterministic task times. However, in reality, the task times are subject to random variation. In this paper, we develop a systematic method of determining schedulability of time-constrained decision-free discrete-event systems, where time variation can be confined within finite intervals. To do this, we propose an extended Petri net for modeling such systems. We then develop a necessary and sufficient condition for which there always exists a feasible schedule and one for which there never exists any feasible schedule. We develop a graph-based computational procedure for verifying the schedulability conditions and determining the worst-case task delay. We demonstrate how the procedure can be used for cluster tool engineering to control wafer delays against wafer alignment failures and time variation. Note to Practitioners-Many automated systems such as cluster tools for semiconductor manufacturing repeat predetermined cyclic work cycles. However, they may have strict time window constraints between a pair of events and also their activity times are subject to random variation. However, the random variation can be confined within a finite range with some confidence level. Our proposed method can verify whether such systems can satisfy the time constraints. By using our proposed method, engineers can quickly examine validity of their engineering decisions such as tool configuration and process time choices.