# Stochastic Petri net

Stochastic Petri nets are a form of Petri net where the transitions fire after a probabilistic delay determined by a random variable.

## Definition

A stochastic Petri net is a five-tuple SPN = (P, T, F, M0, Λ) where:

1. P is a set of states, called places.
2. T is a set of transitions.
3. F where F (P × T) (T × P) is a set of flow relations called "arcs" between places and transitions (and between transitions and places).
4. M0 is the initial marking.
5. Λ = is the array of firing rates λ associated with the transitions. The firing rate, a random variable, can also be a function λ(M) of the current marking.

## Correspondence to Markov process

The reachability graph of stochastic Petri nets can be mapped directly to a Markov process. It satisfies the Markov property, since its states depend only on the current marking. Each state in the reachability graph is mapped to a state in the Markov process, and the firing of a transition with firing rate λ corresponds to a Markov state transition with probability λ.

## References

1. ^ Dingle, N. J.; Knottenbelt, W. J.; Suto, T. (2009). "PIPE2". ACM SIGMETRICS Performance Evaluation Review. 36 (4): 34. doi:10.1145/1530873.1530881. S2CID 3265173.
2. ^ Carnevali, L.; Ridi, L.; Vicario, E. (2013). "A Quantitative Approach to Input Generation in Real-Time Testing of Stochastic Systems". IEEE Transactions on Software Engineering. 39 (3): 292. doi:10.1109/TSE.2012.42. S2CID 8064028.
3. ^ Amparore, E. G. (2014). "A New GreatSPN GUI for GSPN Editing and CSLTA Model Checking". Quantitative Evaluation of Systems. Lecture Notes in Computer Science. Vol. 8657. pp. 170–173. doi:10.1007/978-3-319-10696-0_13. ISBN 978-3-319-10695-3.