## 2 Nhpp Srm  
### Nhpp Srm  
A repairable system means a system capable of switching from an active state to a faulty state and back to an active state. Statistical analysis of repairable systems is a very important part of the reliability field because there are many more repairable products compared to products that are virtually non-repairable. Such a repairable system includes a dummy model, a maintenance model,and a repair model, and the selection of the model may vary depending on how the failure is handled. In this case, if the time required for replacement, maintenance, and repair is negligible, that is, if it can be assumed to be zero, the failure time can be adapted to a point process. The Poisson process, which is one of the most important processes in the count process, shows probabilistic characteristics when the number of occurrences in a given interval follows the Poisson distribution and the number of occurrences of the events are independent of each other. This Poisson process can be classified as homogeneous Poisson process (HPP) and NHPP, which deals only with the NHPP.  
If the counting process \(\{\text{N}(\text{t}),\ \text{t}\geq 0\}\) satisfies three conditions: \(\text{N}(0)=0\), with independent increments, and the average of the number of failures in the interval \([\text{t}_{1},\ \text{t}_{2}]\), it is called an NHPP with an intensity function \(\lambda(\text{t})\). \(\text{N}(\text{t})\) (\(\text{t}\geq 0\)) follows a Poisson distribution with parameter m(t):  
\[\text{Pr}\{\text{N}(\text{t})=\text{n}\}=\frac{\{\text{m}(\text{t})\}^{\text{n }}}{\text{n}!}\exp\{-\text{m}(\text{t})\},\ \text{n}=0,\ 1,\ 2,\ 3,\ldots.\]  
where m(t) is the mean value function of the NHPP.  
The intensity function \(\lambda(\text{t})\) is as follows.  
\[\frac{\text{dm}(\text{t})}{\text{dt}}=\lambda(\text{t}).\]  
[PERSON] et al. [22] formalized the general framework for the software reliability based on NHPP and provided numerical expressions for the mean value function m(t) using differential equations. The mean value function m(t) of the general NHPP SRM with different values for a(t) and b(t), which reflect various assumptions, can be obtained with the initial condition N(0) = 0.  
\[\frac{\text{d}\,\text{m}(\text{t})}{\text{dt}}=\text{b}(\text{t})[\text{a}( \text{t})-\text{m}(\text{t})]. \tag{1}\]  
The general solution of (1) is  
\[\text{m}(\text{t})=\text{e}^{-\text{B}(\text{t})}[\text{m}_{0}+\int_{\text{t} _{0}}^{\text{t}}\text{a}(\text{s})\text{b}(\text{s})\text{e}^{\text{B}(\text {s})}\text{b}\text{s}], \tag{2}\]  
where \(\text{B}(\text{t})=\int_{\text{t}_{0}}^{\text{t}}\text{b}(\text{s})\text{d}\)s, and \(\text{m}(\text{t}_{0})=\text{m}_{0}\) is the marginal condition of (2).  
A generalized NHPP SRM that incorporates uncertainty in the operating environment is formulated as follows [23]:  
\[\frac{\text{d}\ \text{m}(\text{t})}{\text{dt}}=\eta[\text{b}(\text{t})][\text{N}- \text{m}(\text{t})], \tag{3}\]  
where \(\eta\) is a random variable that represents the uncertainty of the system fault detection rate in the operating environment with a probability density function g; b(t) is the fault detection rate function, which also represents the average failure rate caused by faults; N is the expected number of faults that exist in the software before testing.  
Thus, a generalized mean value function, m(t), where the initial condition m(0) = 0, is given by  
\[\text{m}(\text{t})=\int_{\eta}\text{N}(1-\text{e}^{-\eta\int_{0}^{\text{t}} \text{b}(\text{s})\text{d}\text{x}})\text{d}\text{g}(\eta). \tag{4}\]  
The mean value function [24] from (4) using the random variable \(\eta\) has a generalized probability density function g with two parameters \(\alpha\geq 0\) and \(\beta\geq 0\), and is given by  
\[\text{m}(\text{t})=\text{N}(1-\frac{\beta}{\beta+\int_{0}^{\text{t}}\text{b}( \text{s})\text{d}\text{s}})^{\alpha}, \tag{5}\]  
where b(t) is the fault detection rate per fault per unit time.