# 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
** For \(k=1\), during the first cycle, a catastrophic failure occurs and the repair time is not more than \(\tau\); hence the Laplace transform for the probability density function of the first cycle's length is \(\boldsymbol{\mu}_{0}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{P}_{SD_{2} \boldsymbol{\mu}}\boldsymbol{e}_{U}\). During the following \(m\) cycles, the product has goes through \(m\) times of short duration in \(D_{1}\), which gives the term \(\left(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{1}}(s)\boldsymbol{P}_{SD_{1} \boldsymbol{\mu}}\boldsymbol{e}_{U}\right)^{m}\). The last term \(\boldsymbol{\mu}_{1}\boldsymbol{G}_{UD_{2}}(s)\boldsymbol{e}_{D_{2}}\) is the Laplace transform for the probability density function of the time that passes from the working state to the catastrophic failure set. Equations (23) and (24) can be proved in a similar manner. \(\square\)  
Let \(h^{mk}(t)\) be the inverse Laplace transform of \(\boldsymbol{\phi}^{mk}(s)\). Given the number of visits to the minor failures set \(m(m=0,\cdots,N_{1})\) and the serial number of visits to the catastrophic failures set \(k(k=1,\cdots,m+1)\), the probability that the warranty terms will be renewed is:  
\[p_{mk}=\int_{0}^{W_{1}}h^{mk}(t)\mathrm{d}tG(\tau)^{m+1}. \tag{25}\]For Case 2-2, the product experiences 0 or 1 times of short duration in \(D_{2}\), \(N_{1}\) times of short duration in \(D_{1}\), and transits to the minor failure set \(D_{1}\) from the operating states set \(U\) within \(W_{1}\) times. Let \(T^{0}_{ij}\) and \(T^{1}_{kij}\) be the total times of the aforementioned process for the case that the product visits the catastrophic failure set 0 and 1 times, respectively, given the entry state \(i(i\in U)\) and the exit state \(j(j\in D)\). Assume that \(h^{0}_{ij}(t)\) and \(h^{1}_{kij}(t)\) are the probability density functions of \(T^{0}_{ij}\) and \(T^{1}_{kij}\), respectively, and \(\mathbf{\phi}^{0}(s)=[\phi^{0}_{ij}(s)]\) and \(\mathbf{\phi}^{1}_{k}(s)=[\phi^{1}_{k}(s)](s\geq 0)\) are the Laplace transform matrices of \(\mathbf{h}^{0}(t)=[h^{1}_{ij}(t)]\) and \(\mathbf{h}^{1}_{k}(t)=[h^{1}_{kij}(t)]\), respectively. **Proposition 5**. : _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 undergone \(N_{1}\) short repairs due to minor failure, no or one catastrophic failure in the \(kth(k=1,\cdots,m+1)\) cycle, and a long repair due to a minor failure is:_  
\[\phi^{0}(s)=(\mathbf{\mu}_{0}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})(\mathbf{\mu }_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{N_{1}-1}\mathbf{\mu}_{1}\mathbf{G} _{UD_{1}}(s)\mathbf{e}_{D_{1}}. \tag{26}\]  
\[\phi^{1}_{k}(s)= (\mathbf{\mu}_{0}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})(\mathbf{\mu }_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{k-2}\times \tag{27}\] \[(\mathbf{\mu}_{1}\mathbf{G}_{UD_{2}}(s)\mathbf{P}_{SD_{2}U}\mathbf{e}_{U})(\mathbf{\mu }_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{N_{1}-(k-1)}\mathbf{\mu}_{1} \mathbf{G}_{UD_{1}}(s)\mathbf{e}_{D_{1}},\] \[k=2,\cdots,N_{1},\]  
\[\phi^{1}_{1}(s)=(\mathbf{\mu}_{0}\mathbf{G}_{UD_{2}}(s)\mathbf{P}_{SD_{2}U}\mathbf{e}_{U})(\bm {\mu}_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{N_{1}-1}\mathbf{\mu}_{1} \mathbf{G}_{UD_{1}}(s)\mathbf{e}_{D_{1}}. \tag{28}\]  
\[\phi^{1}_{N_{1}+1}(s)= (\mathbf{\mu}_{0}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})(\mathbf{\mu }_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{N_{1}-1}\times \tag{29}\] \[(\mathbf{\mu}_{1}\mathbf{G}_{UD_{2}}(s)\mathbf{P}_{SD_{2}U}\mathbf{e}_{U})\mathbf{\mu }_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{e}_{D_{1}}.\]  
Equations (26)-(29) can be proved in a similar manner for Equation (22).