# Stochastic Analysis of an Industrial System with Preparation Time Repair under Warranty Policy
## 2 Description of the System  
The system in question has a single active unit with a single repairer who performs detection of the causes of failure, preparation for repair, repair of the failed unit, and post-repair inspection to ensure that the failed unit is repaired or re-repaired again, so we assume the following:  
1. The system has single active unit to work.
2. A single repairman is available for inspection of the causes of failure, repair preparation, repair of the failed unit, and post-repair inspection.
3. Inspection time, preparation for repair time, repair time, and re-repair time are independent and perfect.
4. After the failure of the active unit (within the warranty period), the unit goes for inspection to determine whether or not this failure was caused by the user.
5. After the discovery of the defect by the inspection and the determination of whether the unit is still subject to the warranty of the company or if the repair is at the expense of the user, the preparation of the repair begins, and after the completion of the repair preparation, the repair is started directly.
6. After the repair (within the warranty period/after the warranty period has expired), the unit will be inspected to determine whether the repair is satisfactory or not. If the repaired unit is not satisfactory, it will be sent for re-repair.
7. The manufacturer will cover the cost of repairing the defective unit during the warranty period, provided that the user is not the one who causes the malfunction of the unit.
8. The distribution of all times in this system follows negative exponential distribution \(f(x)=\eta\exp(-\eta x),x>0,\eta>0\). The selection of all system times for this distribution is due to the following reasons: The exponential distribution is a simple distribution with only one parameter and is commonly used to model reliability data. The exponential distribution is actually a special case of the Weibull distribution with \(\beta=1\). The exponential distribution provides a good model for the phase of a product or item's life when it is just as likely to fail at any time, regardless of whether it is brand new, a year old, or several years old. The exponential distribution is frequently used to model electronic components that usually do not wear out until long after the expected life of the product in which they are installed. Examples include components of high-quality integrated circuits, such as diodes, transistors, resistors, and capacitors.
9. The system works like new after repair or re-repair.  
State-Specification.  
To study the behavioral changes of this system at any virtual time \(t\), we can describe the system states as follows:  
\begin{tabular}{l l} \hline S\({}_{0}\)/S\({}_{8}\) & the unit is active during/within the warranty period/after the warranty \\  & period has expired; \\ S\({}_{1}\) & the failed unit under inspection, to determine whether or not the system is under warranty; \\ S\({}_{2}\)/S\({}_{6}\) & the failed unit under preparation for repair during/within the warranty period/after the warranty period has expired; \\ S\({}_{3}\)/S\({}_{7}\) & the failed unit under repair during/within the warranty period/after the warranty period has expired; \\ S\({}_{4}\)/S\({}_{9}\) & the unit under inspection after repair, to determine whether the repair is satisfactory or not (during/within the warranty period/after the warranty period has expired); \\ S\({}_{5}\)/S\({}_{10}\) & the unit under re-repair after inspection during/within the warranty period/after the warranty period has expired. \\ \hline \end{tabular}  
With these symbols, the possible states of the system under study are as follows: