Calculation of traversing time distributions in semi-Markov chains with application on Petri Nets

This paper shows how to obtain probability distribution of traversing time between initial and final states in Markov Chains underlying Petri Nets. The exact closed form solution is obtained for the negative exponential transitions (firing times) with or without one deterministic transition, and the approximate solution for the mix of negative exponential with more than one deterministic transitions. Then the known distribution enables to find the percentile estimates. We apply our method to obtain the percentile of a packet delay in the network. This approach can be applied to any performance tool which reduces to Markov chains, such as Finite State Machines as well as Queuing Networks.