We consider a scheduling problem of a robotized cluster tool for semiconductor manufacturing, which should control wafer delays within a chamber so as not to exceed a specified limit under time variation. The prior research has proposed a   ${\rm p}^{+}$  -time event graph, an extension of Petri nets, for modeling the scheduling problem and developed a method of verifying whether a cyclic   ${\rm p}^{+}$  -time event graph or a cluster tool, which repeats identical work cycles, can satisfy the time constraints under time variation. In this paper, we extend and simplify the schedulability analysis method in the prior research for a noncyclic event graph or a cluster tool which performs start-up and close-down operation for a lot or lot switching, and obtain specialized results. We assume that a robot task sequence or firing sequence of transitions is given. Based on the schedulability analysis, we also propose a way of modifying a not always schedulable noncyclic   ${\rm p}^{+}$  -time event graph with some qualifications to be always schedulable, that is, we prolong token holding times at some places or equivalently delay firings of some transitions so as to be always feasible.
['Hyun-Jung Kim', 'Jun-Ho Lee', 'Tae-Eog Lee']
Schedulability Analysis for Noncyclic Operation of Time-Constrained Cluster Tools With Time Variation