Table 1 m(t) i.e., mean value function of selected SRGMs.

1

Delayed S-Shaped⁴¹

\(\:m\left(t\right)=a(1-(1+bt){e}^{-bt}\)

2

GO model⁴²

\(\:m\left(t\right)=a(1-{e}^{-bt})\)

3

Inflection S-shaped⁴³

\(\:m\left(t\right)=\frac{a(1-{e}^{-bt})}{a(1-{\beta\:e}^{-bt})}\)

4

S-shaped change-point⁹

\(\:m\left(t\right)\:\:\:=\left\{\begin{array}{c}a(1-{b}_{1}t){e}^{-{b}_{1}t}),\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\:0\le\:t\le\:\tau\:\\\:a\left(1-\frac{\left(1+{b}_{1}\tau\:\right)}{\left(1+{b}_{2}\tau\:\right)}(1+{b}_{2}t){e}^{-{b}_{1}\tau\:-{b}_{2}(t-\tau\:)}\right)\:\:\:\tau\:<t\end{array}\right.\)

5

Pham-Zhang⁴⁴

\(\:m\left(t\right)=\frac{1}{\left(1-{\beta\:e}^{-bt}\right)}(\left(c+a\right)\left(1-{e}^{-bt}\right)-\frac{ab}{b-\alpha\:}({e}^{-\beta\:t}-{e}^{-bt})\)

6

Pure error generation¹⁰

\(\:m\left(t\right)=\left(\frac{a}{1-\alpha\:}\right)(1-{e}^{-b\left(1-\alpha\:\right)t})\)

7

Roy-Mahapatra-Dey model⁴⁵

\(\:m\left(t\right)=a\alpha\:(\)1\(\:-{e}^{-bt})-\frac{ab}{b-\beta\:}({e}^{-\beta\:t}-{e}^{-bt}\left)\right)\)

8

Yamada Exponential⁴¹

\(\:m\left(t\right)=a(1-{e}^{-\gamma\:\alpha\:\left(1-{e}^{-\beta\:t}\right)})\)

9

Yamada Imperfect 1⁴⁶

\(\:m\left(t\right)=\frac{ab}{\alpha\:+b}({e}^{\alpha\:t}-{e}^{-bt})\)

10

Yamada Imperfect 2⁴¹

\(\:m\left(t\right)=a\left(1-{e}^{-bt}\right)\left(1-\frac{\alpha\:}{b}\right)+\alpha\:at\)

11

Yamada Rayleigh model⁴⁶

\(\:m\left(t\right)=a(1-{e}^{-\gamma\:\alpha\:\left(1-{e}^{-\beta\:{t}^{2}/2}\right)})\)

12

Proposed Model

\(\:\varvec{S}\left(\varvec{t}\right)=\left\{\begin{array}{c}N-Nexp\left(\frac{-{\varvec{b}}_{1}\varvec{t}}{1-{\varvec{\alpha\:}}_{1}}\right)+\:{\varvec{\sigma\:}}_{1}exp\left(\frac{{-\varvec{b}}_{1}\varvec{t}}{1-{\varvec{\alpha\:}}_{1}}\right){\int\:}_{0}^{\varvec{t}}\begin{array}{c}{\varvec{\sigma\:}}_{1}exp\left(\frac{{\varvec{b}}_{1}\varvec{l}}{1-{\varvec{\alpha\:}}_{1}}\right)d{\varvec{C}}_{\varvec{l}},\:\:\:\:\:\:\:t\le\:\tau\:\:\:\\\:\:\end{array}\\\:{\varvec{U}}_{2}\left(\varvec{t}\right)\left[\varvec{S}\left(\varvec{\tau\:}\right)+\varvec{N}\varvec{e}\varvec{x}\varvec{p}\left(\frac{{\varvec{b}}_{2}(\varvec{t}-\varvec{\tau\:})}{1-{\varvec{\alpha\:}}_{2}}\right)-\varvec{N}+{\int\:}_{\varvec{\tau\:}}^{\varvec{t}}\frac{{\varvec{\sigma\:}}_{2}}{{\varvec{U}}_{2}\left(\varvec{l}\right)}\varvec{d}{\varvec{C}}_{\varvec{l}}\:\:\right],\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:t>\tau\:\:\:\:\:\:\end{array}\right.\)