# Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws
## 3 Assumptions and Probability Analysis of \(\{\tau,N_{1},1\}\) Model
### Probability Analysis of \(\{\tau,N_{1},1\}\) Model  
#### 3.2.1 Renewable Warranty Analysis of \(\{\tau,N_{1},1\}\) Model  
According to the modeling assumptions, three cases--renewal due to the over-limit of catastrophic failures, minor failures, and repair time--are considered and they are denoted Case 2-1, Case 2-2, and Case 2-3, respectively. For Case 2-1, the product has gone through one short duration in \(D_{2},m(m=1,\cdots,N_{1})\) times short duration in \(D_{1}\), and transits to the catastrophic failure states set \(D_{2}\) from the operating states set \(U\) within \(W_{1}\). Let \(T_{ij}^{mk}\) be the time to complete the above activities regardless of the threshold \(W_{1}\), given the total number of visits to the minor failures set \(m(m=0,\cdots,N_{1})\), the serial number of visits to the catastrophic failures set \(k(k=1,\cdots,m+1)\), the entry state \(i(i\in U)\), and the exit state \(j(j\in D_{2})\). \(h_{ij}^{mk}\) is the probability function of \(T_{ij}^{mk}\). Assume that \(\boldsymbol{\phi}^{mk}(s)=[\boldsymbol{\phi}_{ij}^{mk}(s)](s\geq 0)\) is the Laplace transform matrix of \(\boldsymbol{h}^{mk}(t)=[h_{ij}^{mk}(t)](i\in U,j\in D_{2})\). **Proposition 4**. : _For a multi-state product with the failure process modeled by the homogeneous discrete-state continuous-time Markov chain \(\{X(t)\}\) and warranty terms renewed according to the \(\{\tau,\,N_{1},1\}\) regulation, the Laplace transform of the time that the product has gone through \(m(m=1,\cdots,N_{1})\) short repairs due to minor failures, one catastrophic failure in the \(kth(k=1,\cdots,m+1)\) cycle, and transits to catastrophic failure states is:_  
\[ \begin{array}{ll}\boldsymbol{\phi}^{m1}(s)=&\left(\boldsymbol{\mu}_{0} \boldsymbol{G}_{UD_{2}}(s)\boldsymbol{P}_{SD_{2}\boldsymbol{\mu}}\boldsymbol{ e}_{U}\right)\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{1}}(s)\boldsymbol{P}_{SD _{1}\boldsymbol{\mu}}\boldsymbol{e}_{U}\right)^{m}\times\\ &\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{e}_{D_{2}} \right),\quad m=0,\cdots,N_{1}.\end{array}  \tag{22}\]  
\[ \begin{array}{ll}\boldsymbol{\phi}^{mk}(s)=&\left(\boldsymbol{\mu}_{0} \boldsymbol{G}_{UD_{1}}(s)\boldsymbol{P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol {e}_{U}\right)\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{1}}(s)\boldsymbol {P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol{e}_{U}\right)^{k-2}\times\\ &\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{P}_{SD_{2} \boldsymbol{\mu}}\boldsymbol{e}_{U}\right)\left(\boldsymbol{\mu}_{1} \boldsymbol{G}_{UD_{1}}(s)\boldsymbol{P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol {e}_{U}\right)^{m-(k-1)}\times\\ &\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{e}_{D_{2}} \right),\quad k=2,\cdots,m,\ m=2,\cdots,N_{1}.\end{array}  \tag{23}\]  
\[ \begin{array}{ll}\boldsymbol{\phi}^{mm+1}(s)=&\left(\boldsymbol{\mu}_{0} \boldsymbol{G}_{UD_{1}}(s)\boldsymbol{P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol {e}_{U}\right)\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{1}}(s)\boldsymbol {P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol{e}_{U}\right)^{m-1}\times\\ &\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{P}_{SD_{2} \boldsymbol{\mu}}\boldsymbol{e}_{U}\right)\boldsymbol{\mu}_{1}\boldsymbol{ \mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{e}_{D_{2}},\ m=1,\cdots,N_{1}. \end{array}  \tag{24}\]  
_where \(\boldsymbol{G}_{UD_{1}}(s)=\left(s\mathbf{I}-\boldsymbol{Q}_{UU}\right)^{-1} \boldsymbol{Q}_{UD_{1}},\boldsymbol{G}_{UD_{2}}(s)=\left(s\mathbf{I}-\boldsymbol {Q}_{UU}\right)^{-1}\boldsymbol{Q}_{UD_{2}},\)\(\boldsymbol{P}_{SD_{1}\boldsymbol{\mu}}\boldsymbol{e}_{U}\), and \(\boldsymbol{P}_{SD_{2}\boldsymbol{U}}\) are the state transition probability matrixes after a short repair due to a minor failure and a catastrophic failure, respectively._  
**Proof.