# Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws
## 2 Assumptions and Cost Analysis of \(\{\tau,N\}\) Model
### Probability Analysis of \(\{\tau,N\}\) Model  
#### 2.3.1 Renewable Warranty Service Analysis of \(\{\tau,N\}\) Model  
Considering the possible causes of a warranty renewal, two cases, in which the number of visits to failure states set exceeds the threshold \(N\) and the time for one repair action is longer than threshold \(\tau\), are investigated and they are denoted Case 1-1 and Case 1-2. For Case 1-1, a product has gone through \(N\) repairs which take no more than \(\tau\) (short repair) and transit to failure states set \(D\) from the operating states set \(U\) within \((0,W_{1})\). Let \(\mathbf{T}^{0}\) be the time that the product has completed the above activities regardless of the threshold \(W_{1}\) and \([f_{ij}(t)]\) is the probability function of \(\mathbf{T}^{0}\) given the entry state \(i(i\in U)\) and the exit state \(j(j\in D)\). Assume that \(\mathbf{\varphi}^{0}(s)=[\mathbf{\varphi}^{0}_{ij}(s)](s\geq 0)\) is the Laplace transform matrix of \(\mathbf{T}^{0}(t)=[f^{0}_{ij}(t)]\) (\(i\in U,j\in D\)). **Proposition 1**. : _For a multi-state product with the failure process modeled by the homogeneous discrete-state continuous-time Markov chain \(\{X(t)\}\) and the warranty terms renewed according to the \(\{\tau,N\}\) regulation, the Laplace transform of the time that the product has gone through \(N\) short repairs and transited to failure states is:_  
\[ \begin{array}{rl}\varphi^{0}(s)=&(\mathbf{\mu}_{0}(s\mathbf{I}-\mathbf{Q}_{UU})^{- 1}\mathbf{Q}_{UD}\mathbf{P}_{SDU}\mathbf{e}_{U})\times\\ &(\mathbf{\mu}_{1}((s\mathbf{I}-\mathbf{Q}_{UU})^{-1}\mathbf{Q}_{UD}\mathbf{P}_{SDU})^{N-1}\bm {e}_{U})\mathbf{\mu}_{1}(s\mathbf{I}-\mathbf{Q}_{UU})^{-1}\mathbf{Q}_{UD}\mathbf{e}_{D},\end{array}  \tag{1}\]  
_where \(\mathbf{\mu}_{0}\) is the initial state probability vector, \(\mathbf{\mu}_{1}\) is the state probability vector after a repair action, and \(\mathbf{P}_{SDU}\) is the state transition probability matrix after a short repair._  
Figure 1: A possible product state evolution path under \(\{\tau,N\}\) model. **Proof. ** Given the entry state in \(U\) and the exit state in \(D\), the Laplace transform matrix of duration time in \(U\) is \({(s\mathbf{I}-\mathbf{Q}_{UU})}^{-1}\mathbf{Q}_{UD}\) (see [28]). According to the convolution property of the Laplace transform, \({(s\mathbf{I}-\mathbf{Q}_{UU})}^{-1}\mathbf{Q}_{UD}\mathbf{P}_{SDU}\) is the Laplace transform matrix of the time for a cycle under which the time for each repair is smaller than \(\tau\). Reusing the above results, Equation (1) can be obtained. \(\square\)  
Let \(T^{0}(t)\) be the inverse Laplace transform of \(\varphi^{0}(s)\). The probability that the warrantty term is renewed due to exceeding the threshold number of visits to the failure states sets is:  
\[p_{0}=\int_{0}^{W_{1}}T^{0}(t)\mathrm{d}tG(\tau)^{N} \tag{2}\]  
For Case 1-2, the first \(l-1(l=1,\cdots,N)\) short repairs and the \(l\)th long repair (a repair action which takes more than \(\tau\)) are completed within \(W_{1}\). Let \(\mathbf{T}^{l}\) be the time required to complete the above activities regardless of the threshold \(W_{1}\) and \([f_{ij}^{l}(t)]\) be the probability function of \(\mathbf{T}^{l}\) given the entry state \(i(i\in U)\) and the exit state \(j\in D\)). Assume that \(\mathbf{\varphi}^{l}(s)=[\varphi_{ij}^{l}(s)](s\geq 0)\) is the Laplace transform matrix of \(\mathbf{T}^{l}(t)=[f_{ij}^{l}(t)](i\in U,j\in D)\). **Proposition 2**. : _For a multi-state product with the failure process modeled by the homogeneous discrete-state continuous-time Markov chain \(\{X(t)\}\) and the warrantty terms renewed according to the \(\{\tau,N\}\) regulation, the Laplace transform of the time that the product has gone through \(l-1(l=1,\cdots,N)\) short repairs and a long repair is:_  
\[