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This article presents a new way of generating a generalized state equation that is useful for analyzing the token flow of the Petri Net (PN) with priority. The transition values in the firing vector as used in the conventional state equation are replaced with transition variables, which are generated by multiplying a series of firing condition functions taking the weighted inhibitor arc into account. The actual value of a transition variable is determined by taking priority and the present marking into account. The proposed state equation generalizes the conventional one by using the transition variable form and by containing the formulation of priority. Given the initial marking, the subsequent marking evolution can be determined successively from the generalized state equation as the simultaneous firings occur. A PN with deadlock is analyzed as an example to establish the validity of the generalized state equation.

This article introduces a new way of generating a state equation which is useful for analyzing token flows in a Petri Net (PN). Transition values as used in the conventional state equation are replaced with transition variables, which are generated by multiplying a series of firing condition functions. The proposed state equation generalizes the conventional one in that it is represented with the transition variable form. A token flow with deadlock is analyzed as an example to establish the validity of the proposed state equation.

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This paper presents the evaluation technique of the mean time to failure (MTTF) and reliability corresponding to a programmable logic controller (PLC) structures such as a single controller, a dual controller and a triple modular redundancy (TMR) controller. The failure and repair period of PLCs are surveyed, and the failure rate of each unit composing a PLC is calculated. The MTTF of a single PLC system and a dual PLC system is computed with failure rate and compared. Reliability and MTTF with respect to a dual controller and a TMR controller are calculated and analyzed by using the Markov model in order to present the operation standards of a PLC.