# Stochastic Analysis of an Industrial System with Preparation Time Repair under Warranty Policy
## 5 Numerical Results
### Numerical Results for \(R(t)\), MTSF, and \(Av(\infty)\)  
In this part of this section, we use numerical examples to study how \(R(t)\), MTSF, and \(Av(\infty)\) are affected by changing various system parameters. The numerical results for \(R(t)\) are offered first. We fix \(\lambda_{2}=0.6\) and \(\alpha=0.4\) for each value of \(\lambda_{1}\) and \(t\), where the parameter \(\lambda_{1}\) takes the values 0.1, 0.4, and 0.8 and the time \(t\) ranges from 1 to 10 with a constant step of 1; we obtain Table 1. Table 2 explains the changes that occur in the reliability of the system \(R(t)\) when \(\lambda_{2}\) takes the values 0.1, 0.5, and 0.8, respectively, and the time from 1 to 10, fixing the other parameters that affect the reliability function. Finally, in Table 3, we examine the effect of the warranty expiration rate \(\alpha\) on the reliability of the system \(R(t)\), where the warranty expiration rate \(\alpha\) takes the values 0.1, 0.5, and 0.9, the time changes from 1 to 10, and the other system parameters remain unchanged. From all these considerations on the study of the effects of system parameters on system reliability, the following conclusions can be drawn:  
Table 1: the influence of parameter \(\lambda_{1}\) and \(t\) on \(R(t)\) is shown as follows. 1. In the first column of this table, the value of \(R(t)\) decreases when \(\lambda_{1}=0.1\), \(\lambda_{2}=0.6\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. 2. In the second column of same table, the value of \(R(t)\) decreases when \(\lambda_{1}=0.4\), \(\lambda_{2}=0.6\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. 3. Finally, in the third column, we show that the value of \(R(t)\) decreases when \(\lambda_{1}=0.8\), \(\lambda_{2}=0.6\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. From (1), (2) and (3) we can conclude that the reliability of the system \(R(t)\) decreases when both \(\lambda_{1}\) and \(t\) increases and \(\alpha=0.4\), \(\lambda_{2}=0.6\). Table 2 shows the effect of system parameter \(\lambda_{2}\) and \(t\) on \(R(t)\) as follows. 1. In the first column of this table, we notice that the value of \(R(t)\) decreases when \(\lambda_{1}=0.5\), \(\lambda_{2}=0.1\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. 2. In the second column shows that the value of \(R(t)\) decreases when \(\lambda_{1}=0.5\), \(\lambda_{2}=0.5\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. 3. In the third column, we show that the value of \(R(t)\) decreases when \(\lambda_{1}=0.5\), \(\lambda_{2}=0.8\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. From the previous three points, we conclude that the value of \(R(t)\) decreases when \(\lambda_{1}=0.8\), \(\lambda_{2}=0.6\), \(\alpha=0.4\), and \(t\) varies from 1 to 10. From the previous three points, we conclude that the value of \(R(t)\) decreases when the value of \(\lambda_{2}\) increases, and the value of \(t\) varies from 1 to 10 with constant values \(\lambda_{1}\) and \(\alpha\). Table 3 also illustrates the effect of \(\alpha\) when it has the values of \(\{0.2,0.5,0.9\}\) and t varies from 1 to 10 on \(R(t)\) when \(\lambda_{1}=0.5\) and \(\lambda_{2}=0.6\). Therefore, we note that the reliability of the system \(R(t)\) of the system decreases when both \(\alpha\) and \(t\) increases, and \(\lambda_{1}\) and \(\lambda_{2}\) take the constant values of 0.5 and 0.6, respectively. Second, we examine the effect of the system parameters \(\lambda_{1}\), \(\alpha\), and \(\lambda_{2}\) on the mean to system failure, which can be seen in Figures 2 and 3. The variations of the mean time to system failure are summarized as follows: Figure 2 shows that the mean time to failure of the system in the interval \(\lambda_{1}\in[0,0.4]\) increases when the parameter \(\alpha\) decreases from 0.9 to 0.4 to 0.2 at \(\lambda_{2}=0.4\). In the same figure, the mean time to system failure in the interval \(\lambda_{1}\in[0.4,1]\) increases by increasing the value of parameter \(\alpha\) from 0.2 to 0.4 to 0.  
at \(\lambda_{2}=0.4\).