# Stochastic Analysis of a Priority Standby System under Preventive Maintenance
## 2 System Description
#### System States  
The configuration and states of the proposed system can be descried through the following notations:  
\(N_{O}/N_{OC}/N_{pf}\): \(\begin{array}{ll}N_{O}/N_{OC}/N_{pf}\): & the unit is in normal mode/in normal mode and is continued from the earlier state/in partial failure mode/in partial failure mode and is continued from the earlier state/in normal mode and is kept as cold standby. \(F_{r}/F_{R}/F_{ur}\): \(\begin{array}{ll}F_{r}/F_{R}/F_{ur}\): & the unit is in failure mode and under repair/the unit in failure mode and under repair is continued from the earlier state state/in failure mode and waiting for repair. \(N_{co}/N_{ch}\): \(\begin{array}{ll}N_{co}/N_{ch}\): & the unit is in normal mode and under costlier PM/in normal mode and under cheaper PM. \(N_{Co}/N_{Ch}\): \(\begin{array}{ll}N_{Co}/N_{Ch}\): & the unit in normal mode and under costlier PM is continued from the earlier state/the unit in normal mode and under cheaper PM is continued from the earlier state. By using these notations, the system can be presented in any one of the following states, where the first argument represents the state of the priority unit and the second argument represents the state of the ordinary unit:  
\( \begin{array}{ll}S_{0}=(N_{o},N_{\rm s})&S_{1}=(N_{pf},N_{\rm s})&S_{2}=(N _{co},N_{\rm o})&S_{3}=(F_{r},N_{\rm o})\\ S_{4}=(N_{Co},F_{ur})&S_{5}=(N_{o},F_{r})&S_{6}=(F_{R},F_{ur})&S_{7}=(N_{oc},N _{\rm s})\\ S_{8}=(N_{pf},F_{R})&S_{9}=(N_{pf},N_{\rm s})&S_{10}=(N_{ch},N_{\rm o})&S_{11}=( N_{Ch},F_{ur})\\ S_{12}=(F_{ur},F_{R}).&\end{array} \)The up states in this system are \(S_{0},S_{1},S_{2},S_{3},S_{5},S_{7},S_{8},S_{9}\), and \(S_{10}\); the down states are \(S_{4},S_{6},S_{11}\),and \(S_{12}\). Additionally, the states \(S_{0},S_{1},S_{2},S_{3},S_{5},S_{8},S_{10}\) and \(S_{12}\) are regenerative states, but \(S_{4},S_{6},S_{7},S_{9}\), and \(S_{11}\) are non-regenerative states; see Figure 1.  
In this this section, we calculate the transition probabilities between the different states of the system and the corresponding sojourn times for such states. From the system setup, it can be observed that the epochs of entry into any of the states \(S_{i}\in E\) are regenerative points. \(T_{0}(\equiv 0),T_{1},T_{2},\dots\) denotes the epochs at which the system enters any state \(S_{i}\in E\), and \(X_{n}\) denotes the state visited at epoch \(T_{n}+\), i.e., just after transition at \(T_{n}\). \(\{X_{n},T_{n}\}\) is the Markov renewal process with state space \(E\), and  
\[Q_{ij}(t)=P(X_{n+1}=j,T_{n+1}-T_{n}\leq t\mid X_{n}=i),\]  
is the semi-Markov kernel over \(E\). Thus, the transition probability matrix of embedded Markov chain is  
\[P=(p_{ij})=(Q_{ij}(\infty)=Q(\infty)),\]  
with non-zero elements \(p_{ij}\), which can be calculated by using the following probabilistic arguments:  
\[p_{01}=\int\tilde{A}(t)\tilde{F}_{2}(t)dF_{1}(t),\]  
where \(p_{01}\) means that the total failure of the priority unit does not occur until time \(t\); PM does is achieved until time \(t\); and partial failure of the priority unit occurs at time \(t\). All other transition probabilities listed below can be explained in the same manner. Figure 1: The Transitions diagram.