Model of Open Source Software Reliability with Fault Introduction Obeying the Generalized Pareto Distribution

Abstract

Open source software reliability is an important factor affecting the quality of open source software. The developed reliability models of open source software cannot meet the actual evaluation of open source software reliability because of the complexity, dynamics, and uncertainty of open source software development. Considering the dynamic changes of fault introduction in open source software development, we propose an open source software reliability model with fault introduction based on the generalized Pareto distribution. We use three Apache open source software projects to validate the proposed model. Least squares estimation is used to estimate the model parameter values. Experimental results indicate that the proposed model has better fitting and predictive performance than other existing models. The generalized Pareto distribution of the fault introduction is consistent with that in actual open source software development. Thus, the proposed model can assist developers and managers in evaluating the reliability of open source software in the actual process of open source software development.

Appendix A

$$ \left\{ {\begin{array}{*{20}l} {\frac{{dM(t)}}{{dt}} = B(t)(A(t) - M(t))} \hfill \\ {A(t) = AF(t) + C} \hfill \\ {F(t) = 1 - \left( {1 - \frac{{\beta t}}{\gamma }} \right)^{{\frac{1}{\beta }}} \quad \beta \ne 0} \hfill \\ \end{array} } \right.$$

(A.1)

Given G(t) = $\int_{0}^{t} {B(x){\text{d}}x}$ as B(x) = $\frac{b}{{1 + \mu \exp ( - bx)}}$, then

 $\exp [G(t)] = \frac{{\mu + \exp (bt)}}{{1 + \mu }}$.

Equation (A.1) can be transferred as follows:

$$ \begin{gathered} \exp [G(t)]{\text{d}}M(t) + B(t)\exp [G(t)]M(t){\text{d}}t = B(t)\exp [G(t)]A(t){\text{d}}t \hfill \\ \int {{\text{d}}(\exp [G(t)]M(t))} = \int {B(t)\exp [G(t)]A(t){\text{d}}t} \hfill \\ \exp [G(t)]M(t) = \int {A(t){\text{d}}(\exp [G(t)])} \hfill \\ \end{gathered} $$

$$ \begin{aligned} M(t) & = (\exp [G(t)]A(t) - \int {\exp [G(t)]{\text{d}}(A(t))} )/\exp [G(t)] \\ & = A(t) - \int {\exp [G(t)]{\text{d}}(A(t))} /\exp [G(t)] \\ & = A(t) - \left( {\frac{{1 + \mu }}{{\mu + \exp (bt)}}} \right)\int {\left( {\frac{{\mu + \exp (bt)}}{{1 + \mu }}} \right){\text{d}}(A(t))} \\ & = A(t) - \left( {\frac{1}{{\mu + \exp (bt)}}} \right)\int {(\mu + \exp (bt)){\text{d}}(A(t))} \\ & = A(t) - \frac{{\mu A(t)}}{{\mu + \exp (bt)}} - \frac{1}{{\mu + \exp (bt)}}\int {\exp (bt){\text{d}}(A(t))} \\ & = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{1}{{\mu + \exp (bt)}}\int {\exp (bt){\text{d}}(A(t))} \\ \end{aligned} $$

$$ \begin{array}{*{20}l} {{\text{d}}(A(t)) = A\gamma ^{{ - 1}} \left( {1 - \frac{{\beta t}}{\gamma }} \right)^{{\frac{1}{\beta } - 1}} {\text{d}}t} & {\beta \ne 0} \\ \end{array} $$

_{Expansion according to Taylor's formula}

$$ \exp (bt) = \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{i} }}{{i!}}} $$

$$ \begin{aligned} \left( {1 - \frac{{\beta t}}{\gamma }} \right)^{{\frac{1}{\beta } - 1}} & = 1 - (1 - \beta )\gamma ^{{ - 1}} t + (1 - \beta )(1 - 2\beta )\gamma ^{{ - 2}} t^{2} \\ & \quad + \cdots + ( - 1)^{n} (1 - \beta )(1 - 2\beta ) \cdots (1 - n\beta )\gamma ^{{ - n}} t^{n} \\ & = 1 - (1 - \beta )\gamma ^{{ - 1}} t + (1 - \beta )(1 - 2\beta )\gamma ^{{ - 2}} t^{2} \\ & \quad + \cdots + ( - 1)^{n} \left( {\prod\limits_{{j = 0}}^{n} {(1 - j\beta )} } \right)\gamma ^{{ - n}} t^{n} \\ \end{aligned} $$

then

$$ M(t) = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\int {\left( {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{i} }}{{i!}}} } \right)} \left( {1 - \frac{{\beta t}}{\gamma }} \right)^{{\frac{1}{\beta } - 1}} {\text{d}}t $$

$$ \begin{aligned} M(t) & = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\int {\left( {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{i} }}{{i!}}} } \right)} \left( {1 - {\text{(1}} - \beta {\text{)}}\gamma ^{{ - 1}} t} \right. \\ & \quad + (1 - \beta )(1 - 2\beta )\gamma ^{{ - 2}} t^{2} + \cdots + \\ & \quad ( - 1)^{n} (1 - \beta )(1 - 2\beta ) \cdots (1 - n\beta )\gamma ^{{ - n}} t^{n} ){\text{d}}t \\ \end{aligned} $$

$$ \begin{aligned} M(t) & = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\left( {\int {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{i} }}{{i!}}{\text{d}}t - \int {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - 1}} (1 - \beta )t^{{i + 1}} }}{{i!}}{\text{d}}t} } } } } \right. \\ & \quad + \cdots + ( - 1)^{n} \int {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - n}} (\prod\limits_{{j = 0}}^{n} {(1 - j\beta ))} t^{{i + n}} }}{{i!}}{\text{d}}t} } ) \\ \end{aligned} $$

$$ \begin{gathered} M(t) = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\left( {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{{i + 1}} }}{{i!(i + 1)}}} - \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - 1}} (1 - \beta )t^{{i + 2}} }}{{i!(i + 2)}}} {\text{ + }}} \right. \hfill \\ \quad \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - 2}} (1 - \beta )(1 - 2\beta )t^{{i + 3}} }}{{i!(i + 3)}}} \hfill \\ \left. {\quad + \cdots + ( - 1)^{n} \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - n}} \prod\limits_{{j = 0}}^{n} {(1 - j\beta )} t^{{i + n + 1}} }}{{i!(i + n + 1)}}} + C_{1} } \right) \hfill \\ \end{gathered} $$

When t = 0, M(t) = 0.

$$ C_{1} = A^{{ - 1}} C\gamma $$

$$ \begin{aligned} M(t) = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\left( {\sum\limits_{{i = 0}}^{n} {\frac{{b^{i} t^{{i + 1}} }}{{i!(i + 1)}}} - \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - 1}} (1 - \beta )t^{{i + 2}} }}{{i!(i + 2)}}} } \right. \\ \quad + \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - 2}} (1 - \beta )(1 - 2\beta )t^{{i + 3}} }}{{i!(i + 3)}}} + \cdots \\ \left. { \quad + ( - 1)^{n} \sum\limits_{{i = 0}}^{n} {\frac{{b^{i} \gamma ^{{ - n}} \left( {\prod\limits_{{j = 0}}^{n} {(1 - j\beta )} } \right)t^{{i + n + 1}} }}{{i!(i + n + 1)}}} + A^{{ - 1}} C\gamma } \right) \\ = \frac{{A(t)}}{{1 + \mu \exp ( - bt)}} \\ \quad - \frac{{A\gamma ^{{ - 1}} }}{{\mu + \exp (bt)}}\left( {\sum\limits_{{k = 0}}^{n} {\sum\limits_{{i = 0}}^{n} {\frac{{( - 1)^{k} b^{i} \gamma ^{{ - k}} \left( {\prod\limits_{{j = 0}}^{k} {(1 - j\beta )} } \right)t^{{i + k + 1}} }}{{i!(i + k + 1)}} + A^{{ - 1}} C\gamma } } } \right) \\ = \frac{{A\left( {1 - \left( {1 - \frac{{\beta t}}{\gamma }} \right)^{{\frac{1}{\beta }}} } \right) + C}}{{1 + \mu \exp ( - bt)}} \\ \quad - \frac{A}{{\mu + \exp (bt)}}\left( {\sum\limits_{{k = 0}}^{n} {\sum\limits_{{i = 0}}^{n} {\frac{{( - 1)^{k} b^{i} \gamma ^{{ - k - 1}} \left( {\prod\limits_{{j = 0}}^{k} {(1 - j\beta )} } \right)t^{{i + k + 1}} }}{{i!(i + k + 1)}}} } + A^{{ - 1}} C} \right) \\ \end{aligned} $$