*4.8. Stress-Strength Model*

The stress–strength model defines the life of an element which has a random strength *Y* that is subjected to an accidental stress *X*. The component fails at the instant that the stress applied to it exceeds the strength, and the component will function suitably whenever *X* < *Y*. Hence, *R* = *P*(*X* < *Y*) is a measure of component reliability (Kotz et al., 2003 [16]). It has many applications, especially in reliability engineering. We derive the reliability *R* when *Y* and *X* are two independent continuous random variables from the GOLE-F (*a*1, *b*1, *c*1, *φ*1) and GOLE-F (*a*2, *b*2, *c*2, *φ*2) distributions, respectively. The reliability is defined by

$$R = \int\_0^\infty g\_Y(\mathbf{x}) G\_X(\mathbf{x}) d\mathbf{x}.\tag{37}$$

Using the PDF in (19) and the CDF in (18), we obtain

$$R = \sum\_{m,t=0}^{\infty} \delta\_{m+1} \delta\_t R\_{m+1,t} \, \text{s} \tag{38}$$

where

*Rm*+1,*<sup>t</sup>* <sup>=</sup> <sup>∞</sup> <sup>0</sup> *πm*+1(*x*, *φ*1)Π*t*(*x*, *φ*2)*dx*, and *πm*+1(*x*, *φ*1) = (*m* + 1)*f*(*x*, *φ*1)*F*(*x*, *φ*1) *<sup>m</sup>*, Π*t*(*x*, *φ*2) = *F*(*x*, *φ*2) *t* . The constants *δt*, *δm*+<sup>1</sup> are defined as:

$$\delta\_{l} = \sum\_{l=t}^{\infty} \sum\_{i=0}^{\infty} \sum\_{j=0}^{i} \sum\_{k=0}^{\infty} \frac{(-1)^{i+l+t+1} b\_2^j a\_2^i \binom{i}{j} \binom{l}{t} \binom{i+j+k-1}{k} \binom{c\_2(i+j+k)}{l}}{i! 2^j}$$

for *t* ≥ 1. For *t* = 0, then

$$\delta\_0 = 1 - \sum\_{l=0}^{\infty} \sum\_{i=0}^{\infty} \sum\_{j=0}^{i} \sum\_{k=0}^{\infty} \frac{(-1)^{i+l+m} b\_2^j a\_2^i}{l} \binom{i}{j} \binom{i+j+k-1}{k} \binom{c\_2(i+j+k)}{l}$$

and

$$\delta\_{m+1} = \sum\_{l=m+1}^{\infty} \sum\_{i=0}^{\infty} \sum\_{j=0}^{i} \sum\_{k=0}^{\infty} \frac{(-1)^{i+l+m+2} b\_1^j a\_1^i \binom{i}{j} \binom{l}{m+1} \binom{i+j+k-1}{k} \binom{c\_1(i+j+k)}{l}}{i! 2^l}$$

for *m* ≥ 0.

If *φ*<sup>1</sup> = *φ*2, then the model reduces to

$$R = \sum\_{m,t=0}^{\infty} \frac{\delta\_{m+1} \delta\_t (m+1)}{m+t+1}.\tag{39}$$

#### **5. Estimation and Simulation**
