# OSS Project Stability Assessment Support Tool Considering EVM Based on Wiener Process Models
## 2 Wiener Process Models for Effort Estimation  
Considering the characteristic of the operation phase in OSS projects, the time-dependent effort expenditure phenomenon of operation phase keeps an irregular state, because there is variability among the levels of developer and skill through version-upgrade, and the OSS are developed and maintained by several developers and users. Then, the time-dependent effort expenditure phenomenon of maintenance phase becomes unstable. Thus, the operation phases of many OSS projects are influenced from external factors by triggers such as the difference of skill of developers and users, time lags of development and maintenance activities. From above points, we discus the stochastic differential equation modeling for the OSS project. Then, let \(\Psi\left(t\right)\) be the cumulative maintenance effort up to operational time \(t\left(t\geq 0\right)\) in the OSS project. Suppose that \(\Psi\left(t\right)\) takes on continuous real values. Since the estimated maintenance effort are observed during the operational phase of the OSS project, \(\Psi\left(t\right)\) gradually increases as the operational procedures go on. Based on the software reliability growth modeling approach [5; 6; 7; 8], the following linear differential equation in terms of maintenance effort can be formulated:  
\[\frac{d\Psi\left(t\right)}{dt}=\beta\left(t\right)\{\alpha-\Psi\left(t\right)\}, \tag{1}\]  
Figure 1: The characteristics of the proposed method. where \(\beta\left(t\right)\) is the increase rate of maintenance effort at operational time \(t\) and a non-negative function, and \(\alpha\) means the estimated maintenance effort required until the end of operation. Therefore, we extend Equation (1) to the following stochastic differential equation with Brownian motion [17]:  
\[\frac{d\Psi\left(t\right)}{dt}=\{\beta\left(t\right)+\sigma\upsilon\left(t \right)\}\{\alpha-\Psi\left(t\right)\}, \tag{2}\]  
where \(\sigma\) is a positive constant representing a magnitude of the irregular fluctuation, and \(\upsilon\left(t\right)\) a standardized Gaussian white noise. By using [PERSON]'s formula [18], we can obtain the solution of Equation (2) under the initial condition \(\Psi\left(0\right)=0\) as follows:  
\[\Psi\left(t\right)=\alpha[1-\exp\{-\int_{0}^{t}\beta(s)ds-\sigma\omega(t)\}], \tag{3}\]  
where \(\omega\left(t\right)\) is one-dimensional Wiener process which is formally defined as an integration of the white noise \(\upsilon\left(t\right)\) with respect to time \(t\). Moreover, we define the increase rate of maintenance effort in case of \(\beta\left(t\right)\) defined as [19]:  
\[\int_{0}^{t}\beta(s)ds\doteq\frac{\frac{d\Gamma_{e}\left(t\right)}{dt}}{ \alpha-F_{*}\left(t\right)}. \tag{4}\]  
In this paper, we assume the following equations based on software reliability models \(F_{*}\left(t\right)\) as the cumulative maintenance effort function of the proposed model:  
\[F_{e}(t) \equiv \alpha(1-e^{-\beta t}), \tag{5}\] \[F_{s}(t) \equiv \alpha\left\{1-(1+\beta t)e^{-\beta t}\right\}, \tag{6}\]  
where \(\Psi_{e}\left(t\right)\) means the cumulative maintenance effort for the exponential software reliability growth model with \(F_{e}\left(t\right)\). Similarly, \(\Psi_{s}\left(t\right)\) is the cumulative maintenance effort for the delayed S-shaped software reliability growth model with \(F_{s}\left(t\right)\). Therefore, the cumulative maintenance effort up to time \(t\) is obtained as follows:  
\[\Psi_{e}\left(t\right) = \alpha[1-\exp\{-\beta t-\sigma\omega\left(t\right)\}], \tag{7}\] \[\Psi_{s}\left(t\right) = \alpha[1-(1+\beta t)\exp\{-\beta t-\sigma\omega\left(t\right)\}]. \tag{8}\]  
In this model, we assume that the parameter \(\sigma\) depends on several noises by external factors from several triggers in OSS projects.