# 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
#### 3.2.2 Nonrenewable Warranty Service Analysis of \(\{\tau,N_{1},1\}\) Model  
According to the modeling assumptions, if warranty terms are not renewed under the \(\{\tau,N_{1},1\}\) model, the product has undergone \(m\)\((l=0,\cdots,N_{1})\) short repairs due to minor failures and \(0\) or \(1\) catastrophic failure with short repairs over \((0,\ W_{1})\).  
The Laplace transform for the probability density function of the time that the product has gone without catastrophic failure, \(m(m=1,\cdots,N_{1})\) times in the working state set, and \(m(m=1,\cdots,N_{1})\) short repairs due to minor failures is:  
\[\varphi_{f}^{0m}(s)=(\mu_{0}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})(\mathbf{ \mu}_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{m-1}. \tag{45}\]  
The associated probability density functions are denoted by \(f_{m}^{0}(t)\).  
Laplace transforms for the probability density functions of the times that the product has experienced \(1\) short repair due to catastrophic failure at the \(k\)th cycle and \(m(m=1,\cdots,N_{1})\) short repairs due to minor failure are given as:  
\[\begin{array}{rl}\varphi_{kf}^{1m}(s)=&(\mu_{0}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD _{1}U}\mathbf{e}_{U})^{k-1}(\mathbf{\mu}_{1}\mathbf{G}_{UD_{2}}(s)\mathbf{P}_{SD_{2}U}\mathbf{e}_{ U})\times\\ &(\mu_{1}\mathbf{G}_{UD_{1}}(s)\mathbf{P}_{SD_{1}U}\mathbf{e}_{U})^{m-(k-1)},\ k=1,\cdots,m +1,\ m=1,2,\cdots,N_{1}.\end{array} \tag{46}\]  
\[\varphi_{1f}^{10}(s)=\mu_{0}\mathbf{G}_{UD_{2}}(s)\mathbf{P}_{SD_{2}U}\mathbf{e}_{U}. \tag{47}\]  
The associated probability density functions are denoted by \(f_{mk}^{1}(t)\) and \(f_{01}^{1}(t)\), respectively.  
Using the same method as Proposition 3, we can obtain the following two propositions.  
Proposition 7: _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 probability that the product has undergone \(m(m=1,\cdots,N_{1})\) short repairs due to minor failures and the warranty terms are not renewed is:_  
\[\widetilde{p}_{m}^{0}=\left(\int_{0}^{N_{1}}\int_{W_{1}-t_{1}}^{+\infty}f_{m} ^{0}(t_{1})f(t_{2})\mathrm{d}t_{1}\mathrm{d}t_{2}\right)G(\tau),\ \ \ m=1,\cdots,N_{1}. \tag{48}\]  
_where \(f(t_{2})\) is the inverse Laplace transform of \(\mu_{1}(s\mathbf{I}-\mathbf{Q}_{UU})^{-1}\mathbf{Q}_{UD}\mathbf{e}_{D}\)._  
Proposition 8: _For a multi-state product with the failure process modelled by the homogeneous discrete-state continuous-time Markov chain \(\{X(t)\}\) and warranty terms renewed according to the \(\{\tau,N_{1},1\}\) regulation, the probability that the product has undergone one short repair due to catastrophic failure at the \(k\)th cycle, \(m(m=1,\cdots,N_{1})\) short repairs due to minor failures, and the warranty terms are not renewed is:_  
\[\widetilde{p}_{mk}^{1}=\left(\sum_{k=1}^{m+1}\int_{0}^{W_{1}}\int_{W_{1}-t_{1 }}^{+\infty}f_{mk}^{1}(t_{1})\ f(t_{2})\mathrm{d}t_{1}\mathrm{d}t_{2}\right)G (\tau)^{m+1}, \tag{49}\]  
\[\widetilde{p}_{01}^{1}=\left(\int_{0}^{W_{1}}\int_{W_{1}-t_{1}}^{+\infty}f_{01 }^{1}(t_{1})\ f(t_{2})\mathrm{d}t_{1}\mathrm{d}t_{2}\right)G(\tau). \tag{50}\]