## 5 Case Study  
In the section, the line maintenance of a wheel steering system installed on a regional jet's nose landing gear is used to illustrate the application of our proposed methods. The wheel steering system is given in Figure 6.  
The steering control motor, powered by the hydraulic source, is installed on the shock strut. The upside of the toque link is connected with the steering control motor, and its downside is connected with the wheel axle installed on the sliding tube. Hence the steering motor can turn the nose wheel via the sliding tube and torque link around the shock strut axle. We make the assumption that the system includes two line replaceable units, the steering control motor and the toque link, and their failure rates are \(1.25\times 10^{-4}\) per hour and \(3.12\times 10^{-5}\) per hour respectively. The line maintenance process of the wheel steering system includes two maintenance events, which are the replacement of the steering control motor and the replacement of the toque link. We assume that commonly used resources have already been available before the maintenance process, only the spare parts of the steering control motor are not available. The maintenance process of the wheel steering system based on the SCPN is shown in Figure 7.  
In Figure 7, \(s_{s1}\) to \(s_{s14}\) are state places, they only have one kind of token color \(c_{s}\). \(s_{c}\) is the consumable-resource place, and it has four kinds of token colors \(c_{c1}\), \(c_{c2}\), \(c_{c3}\) and \(c_{c4}\), which are used to denote spare parts of steering control motors, spare parts of toque links, washers and ctiter pins, respectively. \(s_{r}\) is the reusable-resource places, and it has six token colors \(c_{r1}\), \(c_{r2}\), \(c_{r3}\), \(c_{r4}\), \(c_{r5}\) and \(c_{r6}\), which are used to denote maintenance staffs, spanners, pliers, screwdrivers, lifting jacks and diagnostic equipment. \(t_{1}\) to \(t_{13}\) are transitions, among which, \(t_{3}\) and \(t_{10}\) are immediate transitions and others are timed transitions. Table 1 lists the maintenance tasks expressing by transitions in Figure 6 as well as the pertinent maintenance time distributions. The maintenance time distributions are obtained by statistical analysis of maintenance time samples collected from the line maintenance of a certain type of civil aircraft. Figure 6: The structure of the wheel steering system. Figure 7: Maintenance process of the wheel steering system. For all \(f\in S_{S}\times T\cup T\times S_{S}\), the weight function can be expressed as \(W(f)=c_{s}\). For other arcs, their weight functions \(w_{1}\sim w_{9}\) are expressed as \(2c_{r1}+c_{r5}\), \(2c_{r1}+c_{r5}+c_{r6}\), \(c_{r1}+c_{r5}\), \(2c_{r1}+2c_{r2}+2c_{r3}+c_{r5}\), \(2c_{r1}+2c_{r2}+2c_{r3}+2c_{r4}+c_{r5}\), \(c_{r1}+c_{r2}+c_{r3}+c_{r4}+c_{r5}\), \(c_{c1},4c_{c3}+2c_{c4}\) and \(c_{c2}+4c_{c3}+2c_{c4}\), respectively. By Equation (13), the incidence matrix C can be calculated as  
\[\mathbf{C}_{s\times t}=\left[ \begin{array}{ccccccccccccc}-c_{s}&0&0&0&0&0&0&0&0&0&0 &0\\ c_{s}&-c_{s}&0&0&0&0&0&0&0&0&0&0\\ 0&c_{s}&-c_{s}&0&0&0&0&0&-c_{s}&0&0\\ 0&0&c_{s}&-c_{s}&0&0&0&0&0&0&0\\ 0&0&0&c_{s}&0&-c_{s}&0&0&0&0&0&0\\ 0&0&0&0&c_{s}&-c_{s}&0&0&0&0&0&0\\ 0&0&0&0&c_{s}&-c_{s}&0&0&0&0&0&0\\ 0&0&0&0&0&c_{s}&-c_{s}&0&0&0&0&0\\ 0&0&0&0&0&0&c_{s}&-c_{s}&0&0&0&0\\ 0&0&0&0&0&0&0&c_{s}&-c_{s}&0&0&0&0\\ 0&0&0&0&0&0&0&0&c_{s}&-c_{s}&0&0&0\\ 0&0&0&0&0&0&0&0&0&c_{s}&-c_{s}&0\\ 0&0&0&0&0&0&0&0&c_{s}&0&0&c_{s}\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&w_{7}&0&0&-w_{7}&-w_{8}&-w_{8}&0&0&-w_{9}&0\end{array} \right] \tag{16}\]  
The initial markings of the maintenance process based on SCPN can be expressed as  
\[ \begin{split} M_{0}(s_{si})&=\left\{\begin{array}{ll}c_{s}&i=1 \\ 0&i\
eq 1\\ \end{array}\right.\\ M_{0}(s_{c})&=m_{0c1}c_{c1}+m_{0c2}c_{c2}+m_{0c3}c_{c3}+m_{0c4}c_{c4}\\ M_{0}(s_{r})&=m_{0r1}c_{r1}+m_{0r2}c_{r2}+m_{0r3}c_{r3}+m_{0r4}c_{r4}+m_{0r5}c _{r5}+m_{0r6}c_{r6}\end{split}