# Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws
## 2 Assumptions and Cost Analysis of \(\{\tau,N\}\) Model
### Cost Analysis of \(\{\tau,N\}\) Model  
In this section, the cost rate of the \(\{\tau,N\}\) model will be given. According to the assumptions in Section 2.2, the warranty terms would be renewed if the number of visits to the failure states set \(D\) is more than \(N\) or the repair time of a failure (regardless of the failure type) is more than \(\tau\) within the warranty period; otherwise the warranty policy would not be renewed and the warranty period will end with the warranty services over \((0,W_{1})\). After the last repair of a renewable warranty, the old product has residual value \(c_{s}\). The cost of updating a product is \(c_{nw}(c_{nw}>c_{s})\). Let \(W_{\{\tau,N\}}\) denote the length of the warranty under the \(\{\tau,N\}\) regulation. It is clear that \(W_{\{\tau,N\}}\) is a random variable depending on the total number of warranty renewals, the inter-arrival times between two successive renewals of the warranty terms, and the length of the original warranty period \(W_{1}\). Hence,  
\[W_{\{\tau,N\}}=T_{R_{1}}+T_{R_{2}}+\cdots+T_{R_{N_{R}}}+W_{1}, \tag{11}\]  
where \(N_{R}\) is the number of renewals during the warranty period, \(T_{R_{i}}(i=1,\cdots,N_{R})\) are the time intervals between two subsequent warranty renewals, and they are independent and identically distributed non-negative random variables. **Theorem 1**. : _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 \(\{\tau,N\}\) regulation, the expected length of the warranty service is:_  
\[E(W_{\tau,N})=\frac{\sum\limits_{l=0}^{N}p_{l}L_{i}}{1-p_{\tau N}}+W_{1}. \tag{12}\]  
**Proof. ** According to the assumptions in Section 2.2, the probability that the warranty terms are renewed is \(p_{\tau N}=\sum_{l=0}^{N}p_{l}\). If the product is renewed, the expected inter-arrival times between two successive renewals is:  
\[E(T_{R_{1}})=\frac{1}{p_{\tau N}}\sum\limits_{l=0}^{N}p_{l}L_{l}. \tag{13}\]  
\(\square\)  
The probability that the warranty terms are renewed \(s\) times is:  
\[P(N_{R}=s)=p_{\tau N}s^{(}1-p_{\tau N}),s=0,\cdots. \tag{14}\]  
Given \(N_{R}=s\), the expected length of the warranty service is:  
\[E(W_{\{\tau,N\}}|N_{R}=s)=E(T_{R_{1}}+T_{R_{2}}+\cdots+T_{R_{s}}+W_{1})=\frac {s}{p_{\tau N}}\sum\limits_{l=0}^{N}p_{l}L_{l}+W_{1}. \tag{15}\]  
By taking the expectation for the conditional expectation of Equation (15) with respect to \(N_{R}\),  
\[ \begin{array}{rl}E(W_{\{\tau,N\}})&=E(E(W_{\{\tau,N\}}|N_{R})\\ &=\sum\limits_{s=0}^{+\infty}E(W_{W_{\{\tau,N\}}}|N_{R}=s)P(N_{R}=s)\\ &=\sum\limits_{s=0}^{+\infty}(\frac{s}{p_{\tau N}}\sum\limits_{l=0}^{N}p_{l}L _{l}+W_{1})p_{\tau N}s^{(}1-p_{\tau N})\\ &=\frac{\sum\limits_{l=0}^{N}p_{l}L_{l}}{1-p_{\tau N}}+W_{1}.\end{array}  \tag{16}\]  
Let \(C_{\{\tau,N\}}\) denote the total cost of the warranty service under the \(\{\tau,N\}\) regulation. Then,  
\[C_{\{\tau,N\}}=C_{R_{1}}+C_{R_{2}}+\cdots+C_{R_{N_{R}}}+C_{W_{1}}, \tag{17}\]  
where \(C_{R_{i}}(i=1,\cdots,N_{R})\) is the cost of the \(i\)th warranty renewal and they are independent and identically distributed non-negative random variables. \(C_{W_{1}}\) is the warranty service cost during the nonrenewable warranty period \(W_{1}\). **Theorem 2**. : _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 \(\{\tau,N\}\) regulation, the expected renewable warranty service cost over the warranty period is:_  
\[E(C_{\{\tau,N\}}^{sw})=\frac{p_{0}Nc_{nr}^{s}+\sum\limits_{l=1}^{N}p_{l}((l- 1)c_{nr}^{s}+c_{nr}^{l})+p_{\tau N}(c_{nw}-c_{s})}{1-p_{\tau N}}, \tag{18}\]  
_where \(c_{nr}^{s}\) and \(c_{nr}^{l}\) are the cost for one short repair and long repair, respectively._  
**Proof.