*3.2. Incomplete Moments*

The *s*th incomplete moments can be listed as

$$
\omega\_s(t) = \sum\_{k,l=0}^{\infty} \varphi\_{k,l} \,\omega\_s^\*(t), \tag{9}
$$

ൌ

ǡୀ where *<sup>ω</sup>*<sup>∗</sup>*s* (*t*) = *t*0 *<sup>x</sup>sτk*+*l*+<sup>1</sup>(*x*)*dx*. Thus, the *s*th incomplete moments of the OEHLIEx model can be proposed as

$$\omega\_{\mathfrak{s}}(t) = \sum\_{k,l,n=0}^{\infty} \varphi\_{k,l} \frac{\{-\beta(k+l+1)\}^n}{n!(s-n-1)} t^{s-n-1}.\tag{10}$$

*3.3. Reliability Function of Linear Consecutive, Parallel, Series, and Bridge Type Network Systems*

If the random variable *X* has the OEHLIEx distribution, then the reliability function of the linear consecutive *k* − *out* − *of n* : *F* system can be expressed as

$$\begin{aligned} R\_L(t;k,n) &= \sum\_{\substack{j=0 \\ m}}^m N(j;k,n) \{ R^{n-j}(t) F(t) \}^j \\ &= \sum\_{j,l=0}^m (-1)^j \left\langle \begin{matrix} l \\ i \end{matrix} \right\rangle N(j;k,n) R^{n-j+l}(t) \\\\ \sum\_{j,l=0}^m (-1)^j \left\langle \begin{matrix} l \\ j \end{matrix} \right\rangle N(j;k,n) \left\{ 1 - \left\langle 1 - e^{-\lambda \left( \frac{\theta}{e^{\theta}-1} \right)^{-1}} \right\rangle^a \left\langle 1 + e^{-\lambda \left( \frac{\theta}{e^{\theta}-1} \right)^{-1}} \right\rangle^{-a} \right\}^{n-j+l}, \end{aligned} \tag{11}$$

for more details concerning the values of *m*, *<sup>N</sup>*(*j*; *k*, *<sup>n</sup>*), and *j*, see [26]. In the special case of the system *k* − *out* − *o f* − *n* : *F*, the parallel and series system when *k* = *n* and *k* = 1, respectively.

Consider two systems: one of them is parallel, whereas the other is a series with independent *n* components. Each component has the OEHLIEx model; thus, the reliability function in the case of the parallel system can be reported as

$$R\_{P-S}(\mathbf{x}) = 1 - \left[ \left\{ 1 - e^{-\lambda(e^{\frac{\beta}{\xi}} - 1)} \right\}^{a} \left\{ 1 + e^{-\lambda(e^{\frac{\beta}{\xi}} - 1)} \right\}^{-a} \right]^{n},\tag{12}$$

where as the reliability function in the case of the series system can be expressed as

$$R\_{S-S}(\mathbf{x}) = \left[ 1 - \left\{ 1 - e^{-\lambda(\varepsilon^{\frac{\beta}{\xi}} - 1)} \right\}^{n} \left\{ 1 + e^{-\lambda(\varepsilon^{\frac{\beta}{\xi}} - 1)} \right\}^{-n} \right]^{n}.\tag{13}$$

In reliability theory, there exists another type of engineering system in the so-called bridge-type network or a complex system, which has many applications in this field. Such systems as these can be evaluated by using many approaches such as conditional probability, a connection matrix, and tree diagrams, as well as cut and tie sets. Assume a bridge-type network consists of five components (A, B, C, D, and E) where each component has the OEHLIEx model, these components can be connected as follows:

Consider the previous network in which success requires that at least one of the paths AC, BD, AED, or BEC is good. To evaluate the reliability function of this network, the conditional probability approach has been utilized. The previous network in Figure 2 can be subdivided into two systems, one with E considered bad, i.e., it always failed, and one with E considered good, i.e., it cannot fail. Thus,

$$\begin{array}{lcl}R\_{NW} & R\_{NW} \text{(if E is good)} \ R\_E(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) + R\_{NW}(\text{if E is bad}) F\_E(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) \\ &= \{ (1 - F\_A(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) F\_B(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a})) (1 - F\_C(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) F\_D(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a})) \} R\_E(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) \\ &+ \{ 1 - (1 - R\_A(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) R\_C(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a})) (1 - R\_A(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a}) R\_D(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{a})) \} F\_E(\mathbf{x}; \lambda, \boldsymbol{\beta}, \boldsymbol{\beta}, \boldsymbol{a}), \end{array} \tag{14}$$

where *F*∗ = 1 − *R*∗ represents the unreliability function of a component (\*). Since *RA* = *RB* = *RC* = *RD* = *RE*, then the reliability *RNW* can be expressed as

$$R\_{NW} = 2\left\{ R(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \mathbf{a})^2 + R(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \mathbf{a})^3 + R(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \mathbf{a})^5 \right\} - 5R(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \mathbf{a})^4. \tag{15}$$

**Figure 2.** Bridge-type network.

Assume four different systems, namely, parallel, serious, linear consecutive, and bridge network; each system consists of 20 components, except the bridge network which consists of five components. Tables 4–6 list some numerical values of the reliability function for these systems using Maple software.

**Table 4.** Some numerical values of the reliability for different systems for various values of *α*.




**Table 6.** Some numerical values of the reliability for different systems for various values of *λ*.


Regarding Tables 4–6, it is clear that the reliability of parallel, series, linear consecutive, and bridge network systems increases in two cases, one of them for fixed values of *β* and *λ* with *α* → <sup>∞</sup>, and the other for fixed values of *α* and *λ* with *β* → ∞. Whereas, the reliability of these systems decreases for fixed values of *α* and *β* with *λ* → ∞.
