# Stochastic Analysis of a Priority Standby System under Preventive Maintenance
## 7 Effect of PM on System Reliability  
In this section, we discuss the effect of PM on the reliability measures of the system (MSTF), availability, and cost function. Practically, PM performs after time \(t_{0}\) from the beginning of operation of the system, that is  
\[A(t)=\begin{cases}1,&\text{if }t\geq t_{0},\\ 0,&\text{if }t<t_{0}.\end{cases} \tag{40}\]Then, using (40), some of the transition probabilities, mean sojourn times, and reliability measures can be reformulated as follows:  
\[p_{01}(t_{0}) = \int_{0}^{t_{0}}\bar{F}_{2}(t)dF_{1}(t),\qquad p_{03}(t_{0})=\int_{0 }^{t_{0}}\bar{F}_{1}(t)dF_{2}(t),\] \[p_{02}(t_{0}) = p\bar{P}_{1}(t_{0})\bar{F}_{2}(t_{0}),\ \ p_{010}(t_{0})=(1-p)\bar{F}_{1}(t_{0})\bar{F}_{2}(t_{0}),\] \[p_{12}(t_{0}) = 1-F_{3}(t_{0}),\ \ P_{13}=F_{3}(t_{0}),\] \[p_{53}^{(7)}(t_{0}) = \int_{0}^{\infty}\bar{F}_{1}(t)(1-e^{-\delta t})dF_{2}(t)-(1-e^{- \delta t_{0}})\int_{t_{0}}^{\infty}\bar{F}_{1}(t)dF_{2}(t),\] \[p_{51}^{(7)}(t_{0}) = \int_{0}^{\infty}\bar{F}_{2}(t)(1-e^{-\delta t})dF_{1}(t)-(1-e^{- \delta t_{0}})\int_{t_{0}}^{\infty}\bar{F}_{2}(t)dF_{1}(t),\] \[p_{52}^{(7)}(t_{0}) = p(1-e^{-\delta t_{0}})\bar{F}_{1}(t_{0})\bar{F}_{2}(t_{0}),\ \ p_{510}^{(7)}(t_{0})=(1-p)(1-e^{-\delta t_{0}})\bar{F}_{1}(t_{0})\bar{F}_{2}( t_{0}),\] \[p_{83}^{(9)}(t_{0}) = \int_{0}^{\infty}(1-e^{-\delta t})dF_{3}(t)-(1-e^{-\delta t_{0}} )\bar{F}_{3}(t_{0}),\ \ p_{82}^{(9)}(t_{0})=(1-e^{-\delta t_{0}})\bar{F}_{3}(t_{0}),\] \[\mu_{0}(t_{0}) = \int_{0}^{t_{0}}t\bar{F}_{1}(t)\bar{F}_{2}(t)dt,\qquad\mu_{1}(t_{ 0})=\int_{0}^{t_{0}}t\bar{F}_{3}(t)dt_{,},\] \[\mu_{01}(t_{0}) = \int_{0}^{t_{0}}t\bar{F}_{2}(t)dF_{1}(t),\qquad\mu_{03}(t_{0})= \int_{0}^{t_{0}}t\bar{F}_{1}(t)dF_{2}(t),,\] \[\mu_{02}(t_{0}) = pt_{0}\bar{F}_{1}(t_{0})\bar{F}_{2}(t_{0}),\ \ \mu_{010}(t_{0})=(1-p)t_{0}\bar{F}_{1}(t_{0})\bar{F}_{2}(t_{0}),\] \[\mu_{12}(t_{0}) = t_{0}\bar{F}_{3}(t_{0}),\ \ \mu_{13}(t_{0})=t_{0}\bar{F}_{3}(t_{0})-\int_{0}^{t_{0}} \bar{F}_{3}(t)dt,\] \[\mu_{53}^{(7)}(t_{0}) = \int_{0}^{\infty}t\bar{F}_{1}(t)(1-e^{-\delta t})dF_{2}(t)-(1-e^ {-\delta t_{0}})\int_{t_{0}}^{\infty}t\bar{F}_{1}(t)dF_{2}(t),\] \[\mu_{51}^{(7)}(t_{0}) = \int_{0}^{\infty}t\bar{F}_{2}(t)(1-e^{-\delta t})dF_{1}(t)-(1-e^{ -\delta t_{0}})\int_{t_{0}}^{\infty}t\bar{F}_{2}(t)dF_{1}(t),\] \[\mu_{52}^{(7)}(t_{0}) = pt_{0}(1-e^{-\delta t_{0}})\bar{F}_{1}(t_{0})\bar{F}_{2}(t_{0}),\ \ \mu_{510}^{(7)}(t_{0})=t_{0}(1-p)(1-e^{-\delta t_{0}})\bar{F}_{1}(t_{0})\bar{F}_{ 2}(t_{0}),\] \[\mu_{83}^{(9)}(t_{0}) = \int_{0}^{\infty}t(1-e^{-\delta t})dF_{3}(t)-t_{0}(1-e^{-\delta t _{0}})\bar{F}_{3}(t_{0}),\ \ \mu_{82}^{(9)}(t_{0})=t_{0}(1-e^{-\delta t_{0}})\bar{F}_{3}(t_{0}),\] \[MTSF(t_{0}) = \frac{N_{0}(t_{0})}{D_{0}(t_{0})}, \tag{41}\] \[AV_{0}(t_{0}) = \frac{N(t_{0})}{D(t_{0})}, \tag{42}\]  
\[R_{0}^{1}(t_{0})=\frac{N_{1}(t_{0})}{D(t_{0})},\ \ R_{0}^{2}(t_{0})= \frac{N_{2}(t_{0})}{D(t_{0})},\ \ R_{0}^{3}(t_{0})=\frac{N_{3}(t_{0})}{D(t_{0})},\ \ R_{0}^{4}(t_{0})= \frac{N_{4}(t_{0})}{D(t_{0})}, \tag{43}\] \[C(t_{0})=\{\alpha N(t_{0})-k_{1}N_{1}(t_{0})-k_{2}N_{2}(t_{0})-k _{3}N_{3}(t_{0})-k_{4}N_{4}(t_{0})\}/D(t_{0}), \tag{44}\]  
where \(N_{0}(t_{0})\), \(D_{0}(t_{0})\), \(N(t_{0})\), \(D(t_{0})\), and \(N_{i}(t_{0}),i=1,2,3,4\) are given in (7), (8), (18), (19), (29), (31), (33), and (35), respectively, after replacing \(p_{ij},p_{ij}^{(k)},\mu_{i},\mu_{ij},\mu_{ij}^{(k)}\) by \(p_{ij}(t_{0}),p_{ij}^{(k)}(t_{0})\), \(\mu_{i}(t_{0}),\mu_{ij}(t_{0}),\mu_{ij}^{(k)}(t_{0})\).