Stochastic Petri Nets and Generalized Stochastic Petri Nets as well as other extensions to Stochastic Petri Nets have been widely applied as a model of asynchronous concurrent process, or as an aid to analyze or design concurrent systems. This paper presents an Extended Stochastic Petri Net model for a shift processing system in which three kinds of sink may occur and an arbitrary time distribution is incorporated, provides an analytical method based on a Markov renewal process with some non-regeneration points to clarify the probabilistic behavior of the system, and finally evaluates the performance of the system with numerical values.

Markov Renewal Process (MRP) is an extremely powerful analyzing tool for concurrent systems. But it is often difficult to determine the correct Markov model for even moderately complex systems. Petri net is very descriptive and flexible in modeling various kinds of systems of concurrency and asynchronization. This paper presents an aggregate approach of Markov Renewal Process and Extended Stochastic Petri Net (ESPN) model, which allows incorporation of arbitrary time distributions. Using a descriptive "language" like the ESPN for specifying a Markov model of a system, the analyzing power of MRP can be exploited more easily. Moreover, an Abstract Partial Reachability Graph (APRG) is introduced to simplify the Markov solution. In addition, the aggregate approach of MRP and ESPN is applied to analyze a parallel message system.

We develop a new class of stochastic Petri net: non-regenerative stochastic Petri net (NRSPN), which allows the firing time of its transitions with arbitrary distributions, and can automatically generate a bounded reachability graph that is equivalent to a generalization of the Markov renewal process in which some of the states may not constitute regeneration points. Thus, it can model and analyze behavior of a system whose states include some non-regeneration points. We show how to model a system by the NRSPN, and how to obtain numerical solutions for the NRSPN model. The probabilistic behavior of the modeled system can be clarified with the reliability measures such as the steady-state probability, the expected numbers of visits to each state per unit time, availability, unavailability and mean time between system failure. Finally, to demonstrate the modeling ability and analysis power of the NRSPN model, we present an example for a fault-tolerant system using the NRSPN and give numerical results for specific distributions.

We determine the optimum time TOPT to order a spare part for a system before the part in operation has failed. TOPT is a function of the part's failure-time distribution, the lead (delivery) time of the part, its inventory cost, and the cost of downtime while waiting delivery. The probabilities of the system's up and down states are obtained from a non-regenerative stochastic Petri net. TOPT is found by minimizing E[cost], the expected cost of inventory and downtime. Three cases are compared: 1) Exponential order and lead times, 2) Deterministic order time and exponential lead time, and 3) Deterministic order and lead times. In Case 1, it is shown analytically that, depending on the ratio of inventory to downtime costs, the optimum policy is one of three: order a spare part immediately at t = 0, wait until the part in operation fails, or order before failure at TOPT > 0. Numerical examples illustrate the three cases.