## 2 IA for Two-Unit Series System with Three States
### Numerical Solution and Error Analysis  
When the failure time \(X_{i}\), the delay-repair time \(W_{i}\) and the repair time \(Y_{i}\) obey ordinary exponential distribution, it is difficult to get the analytic expression of \(A(t)\), so we will use numerical method to obtain \(A(t)\). In the following, we will introduce some definitions and lemmas. **Definition 1** ([20]). : _Divide the interval \([a,b]\) into \(n\) equivalent. Let \(x_{k}=a+kh,h=\dfrac{b-a}{n}(k=0,1,...,n)\), then apply trapezoidal formula on \(f(x)\) in the interval \([x_{k},x_{k}+1]\), \((k=0,1,...,n-1)\)_  
\[ \begin{split} I&=\int_{a}^{b}f(x)dx=\sum_{k=0}^{n-1 }\int_{x_{k}}^{x_{k+1}}f(x)dx\approx\sum_{k=0}^{n-1}\dfrac{h}{2}[f(x_{k})+f(x_ {k+1})]+R_{n}(f)\\ &=T_{n}+R_{n}(f),\end{split}  \tag{12}\]  
_where \(R_{n}(f)\) is reminder and_  
\[T_{n}=\dfrac{h}{2}[f(a)+2\sum_{k=0}^{n-1}f(x_{k})+f(b)] \tag{13}\]  
_is called the compound trapezoidal formula._  
**Lemma 1** ([20]). : _The reminder of the compound trapezoidal formula is_  
\[R_{n}(f)=-\dfrac{b-a}{12}h^{2}f^{\prime\prime}(\eta), \tag{14}\]  
_which means the error of compound trapezoidal formula is \(O(h^{2})\)._  
**Definition 2** ([20]). : _Divide the interval \([a,b]\) into \(n\) equivalent. Let \(x_{k}=a+kh,h=\dfrac{b-a}{n}(k=0,1,...,n)\), then apply [PERSON] formula to \(f(x)\) in the interval \([x_{k},x_{k}+1]\), \((k=0,1,...,n-1)\)_  
\[ \begin{split} I&=\int_{a}^{b}f(x)dx=\sum_{k=0}^{n-1 }\int_{x_{k}}^{x_{k+1}}f(x)dx\approx\sum_{k=0}^{n-1}\dfrac{h}{6}[f(x_{k})+4f (\dfrac{x_{k}+x_{k+1}}{2})+f(x_{k+1})]+R_{n}(f)\\ &=S_{n}+R_{n}(f),\end{split}  \tag{15}\]  
_where \(R_{n}(f)\) is reminder and_  
\[S_{n}=\dfrac{h}{6}[f(a)+4\sum_{k=0}^{n-1}f(\dfrac{x_{k}+x_{k+1}}{2})+\sum_{k= 1}^{n-1}f(x_{k})+f(b)] \tag{16}\]  
_is called the compound Simpson formula._  
**Lemma 2** ([20]). : _The reminder of the compound Simpson formula is_  
\[R_{n}(f)=-\dfrac{b-a}{2880}h^{4}f^{(4)}(\eta). \tag{17}\]_which means that the error of the compound Simpson formula is \(O(h^{4})\)._  
**Lemma 3** (discrete \(Gronwall\) inequation [21]). : _If discrete series \(\{w_{n}:n=0,...,N\}\) satisfy_  
\[w_{n}\leq A+h\sum_{k=1}^{n}B_{k}w_{k},\ n=0,...,N, \tag{18}\]  
_where \(A,B_{k}(k=1,...,N)\) are nonnegative constant; \(h=\dfrac{1}{N}\) and \(h\underset{1\leq k\leq N}{max}(B_{k})\leq\dfrac{1}{2}\). Then_  
\[\underset{0\leq n\leq N}{max}\left|w_{n}\right|\leq A\cdot\mathrm{e}^{2h \sum\limits_{k=1}^{N}B_{k}}. \tag{19}\]  
Next, according to the definitions and lemmas above, we will obtain \(A(t)\) for the two-unit series system with three states. For obtaining \(A(t)\), we will apply the compound \(Simpson\) formula and compound trapezoidal to the two integrals in the renewal Equations (8)-(11) respectively to transport the renewal equation into discrete equations. The specific steps are as follows:  
**STEP 1:**: Apply the compound \(Simpson\) formula to \(Q^{\prime}(t)\). Divide the interval \([0,T]\) into n equivalent. Let \(0=t_{0}<t_{1}<...<t_{n-1}<t_{n}=t,h=\dfrac{T}{n}\), then apply compound \(Simpson\) formula to \(Q^{\prime}(t)\). **STEP 2:**: Apply the compound trapezoidal formula to \(\int_{0}^{t_{i}}A(u)Q^{\prime}(t_{i}-u)du\).