Initial requirements to the optimal performance of systems modeled by timed place Petri nets

The initial token value required to the optimal performance of discrete event systems can be decided by sum of delay time and synchronic time ratio, which are new synchronic variables in timed place Petri nets. For the system consisting of two live-and-bounded circuits (LB-circuits) fused in common transition-transition-path or common place-place-path, we grove that the synchronic time ratio is the initial token ratio between two LB-circuits to optimally perform system functions. These results are generalized and formulated as a theorem. The initial tokens of a specific place can imply shared resources. Using the theorem, we can decide the minimum number of the shared resources to obtain the optimal performance, and minimize the idling time of resources. As an example, an automated assembly system is modeled by timed place Petri net, and the initial tokens to achieve the optimal system performance are identified. All the values are verified by simulation