## 3 Related Definition
### Production Rule of FRCPN
#### 3.3.5 The Fifth Production Rule
In order to calculate the reliability of Web services and the reliability values of transitions, we make formal definitions of Web services and transitions in FRCPN. **Definition 8**. : _Formal definition of Web service in FRCPN._  
\[WS(IRS,PRTS,POTS,PIRS,ReliabilityValue)\]  
1. _WS_, Web Service represents a Web service in FRCPN. 2. _IRS_, Immediate Reachability Set. For a current Web service WS, if there exists a transition \(t\) satisfying some conditions, including \(WS\in I(t)\) and \(WS_{i}\in O(t)\), then we call \(WS_{i}\) the immediate reachability Web service of \(WS\). All of the immediate reachability Web services of \(WS\) are called an immediate reachability set of \(WS\), denoted as \(IRS\). 3. _PRTS_, PRepositional Transition Set. For a current Web service \(WS\), if there exists a transition t satisfying \(WS\in O(t)\), we call \(t\) the prepositional transition of \(WS\). All of the prepositional transitions of \(WS\) are called a prepositional transition set of \(WS\), denoted as \(PRTS\). 4. _POTS_, POstpositional Transition Set. For a current Web service WS, if there exists a transition t satisfying \(WS\in I(t)\), we call \(t\) the postpositional transition of \(WS\). All of the postpositional transitions of \(WS\) are called a postpositional transition set of \(WS\), denoted as \(POTS\). 5. _PIRS_, Prepositional Immediate Reachability Set. For a current Web service \(WS\), if there exists a transition \(t\) satisfying some conditions, including \(WS\in O(t)\) and \(WS_{i}\in I(t)\), we call \(WS_{i}\) the prepositional immediate reachability Web service of \(WS\). All of the prepositional immediate reachability Web services of \(WS\) are called a prepositional immediate reachability Web service set, denoted as \(PIRS\). 6. \(ReliabilityValue\), the reliability value of the current Web service. Before calculating the reliability value of all Web services in the graph, we set the initial Web service's reliability to 1, and the reliability value of other Web services to 0. **Definition 9**. : _Formal definition of transition in FRCPN._  
\[T(PRWSS,POWSS,Type,ReliabilityValue)\]  
1. T, Transition, represents a transition in FRCPN. 2. _PRWSS_, PRepositional Web Service Set. For a current transition \(t\), if there exists a Web service \(WS\) satisfying \(WS\in I(t)\), we call \(WS\) the prepositional Web service of \(t\). All of the prepositional Web services of \(t\) are called the prepositional Web service set of \(t\), denoted as \(PRWSS\). 3. _POWSS_, POstpositional Web Service Set. For a current transition \(t\), if there exists a Web service \(WS\) satisfying \(WS\in O(t)\), we call \(WS\) s the postpositional Web Service of \(t\). All of the postpositional Web Services of \(t\) are called the postpositional Web Service set of \(t\), denoted as \(POWSS\). 4. _Type_, the type of current transition. There are five types of transition. The first type of transition is shown in Figure 1, the second type of transition is shown in Figure 2, the third type of transition is shown in Figure 3, the fourth type of transition is shown in Figure 4, and the fifth type of transition is shown in Figure 5.
5. \(ReliabilityValue\), the reliability value of the current transition.