# Stochastic Analysis of an Industrial System with Preparation Time Repair under Warranty Policy
## 6 Conclusions  
This study discusses the reliability and sensitivity analyses of a system with a single unit with a preparation time for repair under product warranty policy conditions. Using the state-transition rate diagram of the system, the Laplace transform of the probability of each state was derived, and reliability measures of the system were successfully obtained. Sensitivity and relative sensitivity analyses of two system measures were also derived. Finally, some numerical results were extracted to draw the following conclusions:  
1. The values presented in Tables 1-3 show that the reliability function (\(R(t)\)) of the system decreases with the increase of the failure rate (\(\lambda_{1}\)), (\(\lambda_{2}\)), and the rate of termination of the warranty (\(\alpha\)) in relation to time (\(t\)) and with constant values of other parameters. It is also analyzed that there are no sudden jumps in the values of the reliability function and that the system remains reliable over a long period of time.
2. The behavior of the steady-state availability \(Av(\infty)\) is shown in Tables 4 and 5. 1. Table 4 shows that the steady-state availability \(Av(\infty)\) decreases with the increase of the failure rate (\(\lambda_{2}\)) in all cases of rates of repair (\(\mu_{3}>\mu_{4}\), \(\mu_{3}<\mu_{4}\), \(\mu_{3}=\mu_{4}\)) and for fixed values of other parameters.
3. Table 5 shows that the steady-state availability \(Av(\infty)\) increases with the increase of the preparation of repair rate (\(\kappa_{2}\)) in all cases of rates of repair (\(\mu_{3}>\mu_{4}\), \(\mu_{3}<\mu_{4}\), \(\mu_{3}=\mu_{4}\)).
4. Figures 2 and 3 explain the behavior of MTSF for the proposed system as follows: 1. Figure 2 shows that the mean time to system failure MTSF increases when the rate of termination of the warranty (\(\alpha\)) decreases in the interval \(\lambda_{1}\in[0,\ 0.4)\) and conversely MTSF increases when the rate of termination of the warranty (\(\alpha\)) increases in the interval \(\lambda_{1}\in(0.4,\ 1]\). 2. Figure 3 also shows that the mean time to system failure MTSF increases when the rate of termination of the warranty (\(\alpha\)) increases in the interval \(\lambda_{2}\in[0,\ 0.5)\),  
Figure 4: Sensitivity analysis with different parameters.  
and conversely, MTSF increases when the rate of termination of the warranty (\(\alpha\)) decreases in the interval \(\lambda_{2}\in(0.5,\,1]\).  
According to Figure 4, only \(\lambda_{1}\) and \(\lambda_{2}\) have significant sensitivities for \(R(t)\). Finally, the sensitivity and relative sensitivity of the MTSF to the system parameters are arranged as follows:  
1. \(\lambda_{2}>\lambda_{1}>\alpha\) when \(\lambda_{1}=\lambda_{2}<0.5\)
2. \(\lambda_{1}>\lambda_{2}>\alpha\) when \(\lambda_{1}=\lambda_{2}\geq 0.5\)  
From prior relationships, it can be observed that the parameters \(\lambda_{1}\) and \(\lambda_{2}\) are the most important parameters that affect both performance metrics; therefore, it is necessary to monitor their values.  
Conceptualization, M.S.E.-S.; Methodology, Z.M.H.; Software, M.S.E.-S.; Formal analysis, Z.M.H.; Investigation, Z.M.H. and M.S.E.-S.; Writing--original draft, Z.M.H.; Writing--review & editing, M.S.E.-S.; Supervision, M.S.E.-S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability Statement: Data are contained with the article.  
The authors declare