**3.2 Stochastic Petri net**

A stochastic Petri net (SPN) is the extension of Petri net. In SPN, each transition is associated with a time delay that is an exponentially distributed random variable that expresses delay denoted by *SPN* ¼ ð Þ *P*, *T*, *F*,*W*,*M***<sup>0</sup>** .

## **3.3 Reachability**

Reachability is the fundamental study of the dynamic property of the system. A marking *Mn* is said to be reachable from another marking *M***<sup>1</sup>** if there exists a firing sequence that transforms *Mn* to *<sup>M</sup>***<sup>1</sup>** such that*<sup>∂</sup>* <sup>¼</sup> f g *<sup>M</sup>***1***t***0***M***2***t***1***M***<sup>3</sup>** … *:tnMn* .

## **3.4 Reachability graph and Markov chain (MC)**

A marking *M* is reachable from the initial marking *M***<sup>0</sup>** if there exists a firing *∂* that brings back from the initial state of PN to a state that corresponds to *M***0**.

The Markov chain (MC) is the Markov process with discrete state space. The MC is obtained from the reachability graph of the SPN. Let SPN be the reversible, i.e., *M***<sup>0</sup> ∈** *R M*ð Þ*<sup>i</sup>* for every *Mi* in *R M*ð Þ**<sup>0</sup>** , then the SPN generates an ergodic continuous time Markov chain (CTMC) and it is possible to compute the steady-state probability distribution Q by solving the following (Eq. (1)) and (Eq. (2)).

**Figure 1.** *Simple Petri net.*

*Reliability Analysis of Instrumentation and Control System: A Case Study of Nuclear Power… DOI: http://dx.doi.org/10.5772/intechopen.101099*

$$\sum\_{i=1}^{s} \pi\_i = \mathbf{1} \tag{2}$$

Where, *<sup>π</sup><sup>i</sup>* is the probability being in the state *Mi* and <sup>Q</sup> <sup>¼</sup> ð Þ *<sup>π</sup>***1**, *<sup>π</sup>***2**, … *<sup>π</sup><sup>s</sup>* .
