Application of EM Algorithm to NHPP-Based Software Reliability Assessment with Ungrouped Failure Time Data

Abstract

This chapter presents computation procedures for maximum likelihood estimates (MLEs) of software reliability models (SRMs) based on nonhomogeneous Poisson processes (NHPPs). The idea behind our methods is to regard usual failure time data as incomplete data. This leads to quite simple computation procedures for NHPP-based SRMs based on the EM (expectation–maximization) algorithm, and these algorithms overcome a problem arising in practical use of SRMs. In this chapter, we discuss the algorithms for 10 types of NHPP-based SRMs. Numerical examples show that the proposed EM algorithms help us to reduce computational efforts in the parameter estimation of NHPP-based SRMs.

Appendix

1.1 Derivation of Eq. (40)

For notational simplification, $\mathrm{{E}}[\cdot | D_T; \omega , \varvec{\lambda }]$ is written by $\mathrm{{E}}[\cdot |D_T]$. From the left-hand side of Eq. (40), we have

$$\begin{aligned} \mathrm{{E}}\Bigg [\sum _{k=1}^N h(T_k) \Bigg | D_T \Bigg ]&= \mathrm{{E}}\Bigg [\sum _{k=1}^K h(T_k) \Bigg | D_T \Bigg ] + \mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&= \sum _{k=1}^K \text{ E }[h(T_k) | D_T ] + \mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&= \sum _{k=1}^K h(t_k) + \mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ]. \end{aligned}$$

(132)

Since $T_1, \ldots , T_N$ are IID samples, the first term of right-hand side of the above equation can be easily obtained. Then we focus on the derivation of the second term of right-hand side of Eq. (40). From the posterior distribution of $N$, we obtain

$$\begin{aligned}&\mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&= \Bigg ( \sum _{r = 0}^\infty e^{-\omega } \omega ^{K+r} \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \sum _{j=1}^r h(u_j) \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ) \nonumber \\&\quad \; \Bigg / \Bigg ( \sum _{r = 0}^\infty e^{-\omega } \omega ^{K+r} \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ), \end{aligned}$$

(133)

where $r$ corresponds to the number of failures experienced after time $T$. The integrals in Eq. (133) can be changed to

$$\begin{aligned}&\int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \sum _{j=1}^r h(u_j) \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \nonumber \\&\quad = \frac{r}{r!} \int _T^\infty h(u) f(u) {d}u \left( \int _T^\infty f(u) {d}u\right) ^{r-1}, \end{aligned}$$

(134)

and

$$\begin{aligned} \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 = \frac{1}{r!} \left( \int _T^\infty f(u) {d}u\right) ^r. \end{aligned}$$

(135)

By canceling the constants of numerator and denominator, Eq. (133) is reduced to

$$\begin{aligned}&\mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&= \Bigg ( \sum _{r = 1}^\infty \omega ^r \frac{1}{(r-1)!} \int _{T}^\infty h(u) f(u) {d}u \overline{F}(T)^{r-1} \Bigg ) \Bigg / \Bigg ( \sum _{r = 0}^\infty \omega ^r \frac{1}{r!} \overline{F}(T)^{r} \Bigg ) \nonumber \\&= \omega \int _T^\infty h(u) f(u) {d}u. \end{aligned}$$

(136)

1.2 Derivation of Eq. (56)

From the left-hand side of Eq. (56), we have

$$\begin{aligned}&\mathrm{{E}}\Bigg [\sum _{k=-\tilde{N}}^N h(T_k) \Bigg | D_T \Bigg ] = \mathrm{{E}}\Bigg [\sum _{k=1}^K h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&\qquad \; + \mathrm{{E}}\Bigg [\sum _{k=1}^{\tilde{N}} h(T_{-k}) \Bigg | D_T \Bigg ] + \mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&\quad = \sum _{k=1}^K h(t_k) + \mathrm{{E}}\Bigg [\sum _{k=1}^{\tilde{N}} h(T_{-k}) \Bigg | D_T \Bigg ] + \mathrm{{E}}\Bigg [\sum _{k=K+1}^N h(T_k) \Bigg | D_T \Bigg ]. \end{aligned}$$

(137)

Similar to the previous section, we consider the second and third terms of right-hand side of Eq. (137). Then we have

$$\begin{aligned}&\mathrm{{E}}\Bigg [\sum _{k=1}^{\tilde{N}} h(T_{-k}) \Bigg | D_T \Bigg ] \nonumber \\&= \Bigg ( \sum _{l = 0}^\infty \sum _{r = 0}^\infty e^{-\omega } \omega ^{K+l+r} \int _{-\infty }^0 \int _{-\infty }^{u_1} \cdots \int _{-\infty }^{u_{l-1}} \sum _{j=1}^l h(u_l) \prod _{j=1}^l f(u_j) \nonumber \\&\quad \times \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ) \nonumber \\&\quad \Bigg / \Bigg ( \sum _{l=0}^\infty \sum _{r=0}^\infty e^{-\omega } \omega ^{K+l+r} \int _{-\infty }^0 \int _{-\infty }^{u_1} \cdots \int _{-\infty }^{u_{l-1}} \prod _{j=1}^l f(u_j) \nonumber \\&\quad \times \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ) \nonumber \\&= \Bigg ( \sum _{l = 1}^\infty \omega ^l \frac{1}{(l-1)!} \int _{-\infty }^0 h(u) f(u) du F(0)^{l-1} \Bigg ) \Bigg / \Bigg ( \sum _{l = 0}^\infty \omega ^l \frac{1}{l!} F(0)^{l} \Bigg ) \nonumber \\&= \omega \int _{-\infty }^0 h(u) f(u) {d}u \end{aligned}$$

(138)

and

$$\begin{aligned}&\mathrm{{E}}\Bigg [\sum _{k=K+1}^{N} h(T_k) \Bigg | D_T \Bigg ] \nonumber \\&= \Bigg ( \sum _{l = 0}^\infty \sum _{r = 0}^\infty e^{-\omega } \omega ^{K+l+r} \int _{-\infty }^0 \int _{-\infty }^{u_1} \cdots \int _{-\infty }^{u_{l-1}} \prod _{j=1}^l f(u_j) \nonumber \\&\quad \times \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \sum _{j=1}^l h(u_r) \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ) \nonumber \\&\quad \Bigg / \Bigg ( \sum _{l=0}^\infty \sum _{r=0}^\infty e^{-\omega } \omega ^{K+l+r} \int _{-\infty }^0 \int _{-\infty }^{u_1} \cdots \int _{-\infty }^{u_{l-1}} \prod _{j=1}^l f(u_j) \nonumber \\&\quad \times \prod _{k=1}^K f(t_k) \int _{T}^\infty \int _{u_1}^\infty \cdots \int _{u_{r-1}}^\infty \prod _{j=1}^r f(u_j) {d}u_r \cdots {d}u_1 \Bigg ) \nonumber \\&= \Bigg ( \sum _{r = 1}^\infty \omega ^r \frac{1}{(r-1)!} \int _T^{\infty } h(u) f(u) {d}u \overline{F}(T)^{r-1} \Bigg ) \Bigg / \Bigg ( \sum _{r = 0}^\infty \omega ^r \frac{1}{r!} \overline{F}(T)^{r} \Bigg ) \nonumber \\&= \omega \int _T^{\infty } h(u) f(u) {d}u. \end{aligned}$$

(139)