## 5 Experimental Results
### Examples  
**Example 1**. : _Firstly, please refer to Figure 2, there are two idle places, six operation places and three resource places, which are denoted as \(P^{0}=\left\{p_{1},p_{8}\right\}\), \(P_{A}=\left\{p_{2},p_{3},p_{4},p_{5},p_{6},p_{7}\right\}\) and \(P_{R}=\left\{p_{9},p_{10},p_{11}\right\}\). Since there are two prebuilt process competing resources in this example, deadlock states hence occur. Further, according to reachability graph analysis, the system's reachability graph consists of 20 reachable states in total, shown in Figure 3, including fifteen legal and five illegal markings (three quasi-dead markings, i.e., \(M_{5}\), \(M_{9}\) and \(M_{10}\), andtwo deadlock markings, i.e., \(M_{14}\) and \(M_{15}\)). Please notice that under our proposed transition-based recovery control policy, only deadlock and legal markings are needed to be considered, and are adopted to build its GCAM and SGCAM matrixes. Therefore, we show both of them in Tables 2 and 3, respectively. Furthermore, for easy tracking we show the GCAM matrix in PN places form. Based on Equation (3), each leading (control) transition can then be obtained under two deadlock markings and fifteen legal ones. For instance, \(t_{14,1}\) can be built by the two markings, \(\left[M_{14},M_{1}\right]\), shown in Equation (5) as follows:_  
\[t_{14,1}=M_{1}-M_{14} \tag{5}\]  
_In Equation (5), it is worthy note that the marking \(M_{1}\) belongs to \(\mathcal{M}_{\mathrm{L}}\) and the marking \(\mathrm{M}_{14}\) belongs to \(\mathcal{M}_{\mathrm{D}}\). Further, Equation (5) can then be Equation (6) as follows:_  
\[t_{14,1}=M_{1}-M_{14}=[3p_{1}+3p_{8}+p_{9}+p_{10}+p_{11}]-[p_{1}+p_{2}+p_{3}+p_ {5}+2p_{8}] \tag{6}\]  
_After the computation in Equation (6), the leading transition of \(t_{14,1}\) is equal to \([2p_{1}-p_{2}-p_{3}-p_{5}+p_{8}+p_{9}+p_{10}+p_{11}]\). It is also present in Table 2 (please refer to bold face). Note that all leading transitions in GCAM can be viewed as control transitions since they can recover their corresponding (relative) dead markings simultaneously. Furthermore, it means that they can recover the same dead markings once two sets or more of markings (one legal with one dead markings) in GCAM have the same leading transition._  
_Accordingly, one can recognize that the two sets of markings \([M_{14},M_{4}]\) and \([M_{15},M_{6}]\) have the same leading transition since \(t_{14,4}=t_{15,6}=[p_{1}-p_{2}-p_{5}+p_{8}+p_{9}+p_{11}]\). In other words, it means that the set of leading transition can make the two deadlock markings \(M_{14}\) and \(M_{15}\) live and make this system recovery. Further, this system will be optimal controlled once the control transition \(t_{c}\) is equal to \([p_{1}-p_{2}-p_{5}+p_{8}+p_{9}+p_{11}]\). Figure 4 shows that the control transition is added into the system. Further, all illegal (i.e., quasi-dead and deadlock) markings are recovered and become live as shown in Figure 5. In summary, the system has real maximal permissive states since all original 20 reachable markings are held totally._  
Figure 4: The controller for Figure 2.  
**Example 2**. : _Here is another classical example used to evaluate our policy and presented in Figure 6, with totally 19 places and 14 transitions. In this PN model, there are two idle places, eleven operation places and six resource places, denoted as \(P^{0}=\{p_{1},p_{8}\}\), \(P_{A}=\{p_{2}-p_{7},p_{9}-p_{13}\}\) and \(P_{R}=\{p_{14}-p_{19}\}\). After reachability graph analysis, there are 282 markings identified totally, including 16 deadlock markings (\(M_{89}\), \(M_{107}\), \(M_{154}\), \(M_{162}\), \(M_{170}\), \(M_{174}\), \(M_{198}\), \(M_{202}\), \(M_{205}\), \(M_{211}\), \(M_{227}\), \(M_{249}\), \(M_{256}\), \(M_{257}\), \(M_{261}\), and \(M_{277}\)) and 205 legal ones based on PN-tools [18]. Please note that due to the space limitation of this paper, we will not present the detailed information of CGAM, SCGAM, and all detailed process of seeking leading transitions. We just show the key information.