## 3 Software Cost Model Considering Operating Environment  
### Cost Model  
A software cost model was proposed to determine the optimal time for releasing a software product under certain circumstances. The software cost model is shown in Figure 2. As subsequently discussed, the model considers the following [19]:  
Additional considerations and aims:  
* The cost of removing faults during the operating period is proportional to the total time required to remove all the faults detected during the time interval, \([T,\ T+x]\). * The time required for removing faults during the operating period is much longer than that for removing faults during the testing period. * Time is necessary to remove errors, and we assume that the time to remove each error follows a truncated exponential distribution. * The aim is to apply the SRM to the cost model to determine whether there is a change in the optimal release time depending on the length of the software operation period. * The optimal software release time, \(\mathrm{T}^{*}\), with the expected minimum in the environment, is determined as follows: \[\mathrm{Minimize}\ \mathrm{C}(\mathrm{T})\]  
The software reliability function \(\mathrm{R}(\mathrm{t})\), which represents the probability that a software error will not occur within the interval \([0,\mathrm{t}]\), is expressed as  
\[\mathrm{R}(\mathrm{t})=\mathrm{e}^{-\mathrm{m}(\mathrm{t})}=\mathrm{e}^{- \int_{0}^{\mathrm{t}}\lambda(\mathrm{s})\mathrm{d}\mathrm{s}}\]  
If \(\mathrm{t}+\mathrm{x}\) is given, then the software reliability can be expressed as a conditional probability \(\mathrm{R}(\mathrm{x}\,\mathrm{t})\), as expressed below:  
\[\mathrm{R}(\mathrm{x}|\mathrm{t})=\mathrm{e}^{-\left[\mathrm{m}(\mathrm{t}+ \mathrm{x})-\mathrm{m}(\mathrm{t})\right]}\]  
Many existing cost models propose models without assuming the time it takes to eliminate errors, but in reality, it takes time to eliminate errors, and often, it takes much time. The costs associated with this may be significant. Therefore, the time and cost of eliminating errors and the number of failures must be included. Let a random variable \(\mathrm{Y}\) denote the time required to remove an error in the testing environment. The time to remove each error is assumed to follow a truncated exponential distribution. The probability density distribution of \(\mathrm{Y}\) is given by  
\[\mathrm{s}(\mathrm{y})=\frac{\lambda\mathrm{e}^{-\lambda\mathrm{y}}}{\int_{0} ^{\mathrm{T}_{0}}\lambda\mathrm{e}^{-\lambda\mathrm{z}}\mathrm{d}\mathrm{z}} =\frac{\lambda\mathrm{e}^{-\lambda\mathrm{y}}}{1-\mathrm{e}^{-\lambda \mathrm{T}_{0}}},\ \mathrm{for}\ \ 0\leq\mathrm{y}\leq\mathrm{T}_{0}.\]  
The expected time to remove each fault is expressed as  
\[\mathrm{E}(\mathrm{y})=\mu_{\mathrm{y}}=\int_{0}^{\mathrm{T}_{0}}\mathrm{y} \ \mathrm{s}(\mathrm{y})\mathrm{d}\mathrm{y}\ =\int_{0}^{\mathrm{T}_{0}}\mathrm{y}\frac{\lambda\mathrm{e}^{- \lambda\mathrm{y}}}{\int_{0}^{\mathrm{T}_{0}}\lambda\mathrm{e}^{-\lambda \mathrm{z}}\mathrm{d}\mathrm{z}}\mathrm{d}\mathrm{y}\ =\int_{0}^{\mathrm{T}_{0}}\frac{\mathrm{y}\lambda\mathrm{e}^{- \lambda\mathrm{y}}}{1-\mathrm{e}^{-\lambda\mathrm{T}_{0}}}\mathrm{d}\mathrm{y} =\frac{1-(\lambda\mathrm{T}_{0}+1)\mathrm{e}^{-\lambda\mathrm{T}_{0}}}{\lambda( 1-\mathrm{e}^{-\lambda\mathrm{T}_{0}})}.\]  
Figure 2: Software cost model structure. Additionally, \(Z\) is a random variable denoting the time required to remove an error in the operating environment. The probability density function of \(Z\) and the expected time to remove each fault are expressed as follows:  
\[\mathrm{s(z)}=\frac{\lambda\mathrm{e}^{-\lambda z}}{1-\mathrm{e}^{-\lambda T_{1} }},\ \mathrm{E(z)}=\mu_{z}=\frac{1-(\lambda T_{1}+1)\mathrm{e}^{-\lambda T_{1}}}{ \lambda(1-\mathrm{e}^{-\lambda T_{1}})}.\]  
The expected total software cost, \(\mathrm{C(T)}=\sum\mathrm{E}_{i}(\mathrm{T})\), is expressed in terms of the following costs:  
1. Testing phase cost, \[\mathrm{E}_{1}(\mathrm{T})=\mathrm{C}_{1}\cdot\mathrm{T}^{\theta},\]
2.