**3. Background and mathematical fundamentals**

This section consists of background and mathematical fundamentals to carry out reliability analysis of instrumentation and control system: a case study of nuclear power plant.

#### **3.1 Petri net**

A Petri net (PN) is mathematically defined 5-tuple *PN* ¼ ð Þ *P*, *T*, *F*,*W*, *M***<sup>0</sup>** where *P* the finite is set places, *T* is a finite set of transitions, *F* is a finite set of arcs also referred to as flow relation, i.e., *F* ð Þ *P* � *T* ð Þ *T* � *P* , *W* : *F* f g **1**, **2**, **3**, … *:* is the

weight function, and *M***<sup>0</sup>** is the initial marking *M***<sup>0</sup>** : *P* f g **0**, **1**, **2**, **3**, … *:* . *P T* ¼ and *P T* 6¼ . If the Petri net does not have an initial marking, it is denoted as *N* ¼ ð Þ *P*, *T*, *F*,*W* with an initial marking denoted by ð Þ *N*, *M***<sup>0</sup>** . A simple example of the PN is shown in **Figure 1**.

The marking changes in the Petri net as per the transition firing are as follows:

