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With wafer residency time constraints for some wafer fabrication processes, such as low pressure chemical-vapor deposition, the schedulability and scheduling problems are still open. This paper aims to solve both problems. A Petri net (PN) model is developed for the system. This model describes when the robot should wait and a robot wait is modeled as an event in an explicit way. Thus, to schedule a single-arm cluster tool with wafer residency time constraint is to decide how long a robot wait should be. Based on this model, for the first time, we present the necessary and sufficient conditions under which a single-arm cluster tool with residency time constraints is schedulable, which can be checked analytically. Meanwhile, a closed form scheduling algorithm is developed to find an optimal periodic schedule if it is schedulable. Also, a simple method is presented for the implementation of the periodic schedule for steady state, which is not seen in any previous work.