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.

Stable schedules that repeat identical timing patterns for each work cycle have been important implication for cyclic manufacturing systems such as cluster tools for semiconductor manufacturing or flexible manufacturing systems. While it has been claimed that stable schedules have advantages including steady operation, predictable behavior, minimum cycle times, less work-in-progress inventory, recently stable schedules also play essential roles for meeting critical time window constraints on the operations such as wafer residency time constraint in a cluster tool or track equipment for some chemical vapor deposition processes or wet cleaning processes. However, when the process times or robot task times are subject to random variation of abrupt random disturbances, the stable schedule is disturbed and the perturbed wafer residency times may violate time window constraints. We prove that a cluster tool with dual arms, after any schedule disturbance, can be stabilized to the stable schedule. We characterize the condition for stabilization based on the event graph theory. We also present a control strategy that guarantees such stabilization and reduces the stabilization time.