Appendix A
Proof of Theorem 1
An informal proof of this theorem is a straightforward algebra extension of [5, 19], and [20] with some differences caused by known fading coefficients. The coding scheme used to achieve this capacity is similar to the one used in [15, 19, 20]. The coding scheme and error probability analysis is straightforward and is not shown here due to a lack of space. We use definitions in Corollary 1 And 2 to compute the capacity of WC with knowing SI at the transmitter and fading coefficients. We have following assumptions:
$$ E\left[ {X^{2} } \right] = P, E\left[ {S^{2} } \right] = Q, E\left[ {\eta_{m}^{2} } \right] = N_{m} , E\left[ {\eta_{e}^{2} } \right] = N_{e} , $$
(13)
and \({U}_{m}\) and \({U}_{e}\) are generated by using generalized dirty paper coding presented in [23]. According to [5, 19, 20, 23] and knowing that \({Y}_{m}={h}_{m}X+{h}_{sm}S+{\eta }_{m}\), \({Y}_{e}={h}_{e}X+{h}_{sm}S+{\eta }_{e}\) we can write mutual information related to the term \({C}_{m}\) in corollary 1 as follows:
$$ \begin{aligned} I\left( {U_{m} ;Y_{m} } \right) & = H\left( {h_{m} X + h_{sm} S + \eta_{m} } \right) - H\left( {h_{m} X + h_{sm} S + \eta_{m} \left| {X + \alpha_{m} S} \right.} \right) \\ & = H\left( {h_{m} X + h_{sm} S + \eta_{m} } \right) + H\left( {X + \alpha_{m} S} \right) - H\left( {h_{m} X + h_{sm} S + \eta_{m} ,X + \alpha_{m} S} \right) \\ & = \log \left( {\left( {2\pi e} \right)^{2} \left( {\left| {h_{m} } \right|^{2} P + \left| {h_{sm} } \right|^{2} Q + N_{m} } \right)\left( {P + \alpha_{m}^{2} Q} \right)} \right) - \\ & \quad \log \left( {\left( {2\pi e} \right)^{2} \det \left( {cov\left( {h_{m} X + h_{sm} S + \eta_{m} ,X + \alpha_{m} S} \right)} \right)} \right) \\ & = \log \left( {\frac{{\left( {\left| {h_{m} } \right|^{2} P + \left| {h_{sm} } \right|^{2} Q + N_{m} } \right)\left( {P + \alpha_{m}^{2} Q} \right)}}{{PQ\left( {\left| {h_{sm} } \right| - \alpha_{m} \left| {h_{m} } \right|} \right)^{2} + N_{m} \left( {P + \alpha_{m}^{2} Q} \right)}}} \right), \\ \end{aligned} $$
(14)
and
$$ I\left( {U_{m} ;S} \right) = \log \left( {\frac{{P + \alpha_{m}^{2} Q}}{P}} \right). $$
(15)
Substituting (14) and (15) in \({C}_{m}\) with corollary 1, we obtain:
$$ C_{m} \left( {\alpha_{m} } \right) = \log \left( {\frac{{P\left( {\left| {h_{m} } \right|^{2} P + \left| {h_{sm} } \right|^{2} Q + N_{m} } \right)}}{{PQ\left( {\left| {h_{sm} } \right| - \alpha_{m} \left| {h_{m} } \right|} \right)^{2} + N_{m} \left( {P + \alpha_{m}^{2} Q} \right)}}} \right). $$
(16)
By maximizing \({C}_{m}\left({\alpha }_{m}\right)\) over \({\alpha }_{m}\), for corollary 1, we get
$$ C_{s} = \mathop {\max }\limits_{{\alpha_{m} }} C_{m} \left( {\alpha_{m} } \right) = \log \left( {1 + \frac{{\left| {h_{m} } \right|^{2} P}}{{N_{m} }}} \right) = \log \left( {1 + \gamma_{m} } \right). $$
(17)
So we have (9) for corollary 1. The proof of this theorem for corollary 2 is similar with negligible change and is omitted for the lack of space.
Appendix B
Proof of Theorem 2
First, we state some integrals that are needed to prove the theorems [26]. Therefore, we have.
$$ \smallint e^{ - \xi t} \log \left( {1 + \beta t} \right)dt = \frac{1}{\xi }\left[ {e^{{\frac{\xi }{\beta }}} Ei\left( { - \left( {\xi t + \frac{\xi }{\beta }} \right)} \right) - e^{ - \xi t} \log \left( {1 + \beta t} \right)} \right], $$
(18)
$$ \mathop \smallint \limits_{0}^{\infty } e^{ - \xi t} \log \left( {1 + \beta t} \right)dt = - \frac{{e^{{\frac{\xi }{\beta }}} }}{\xi }Ei\left( { - \frac{\xi }{\beta }} \right), $$
(19)
$$ \smallint e^{ - \nu t} Ei\left( { - \left( {\delta + \kappa t} \right)} \right)dt = \frac{1}{\nu }\left[ {e^{{\frac{\nu \delta }{\kappa }}} Ei\left( { - \frac{{\left( {\nu + \kappa } \right)\left( {\kappa t + \delta } \right)}}{k}} \right) - e^{vt} Ei\left( { - \left( {\kappa t + \delta } \right)} \right)} \right], $$
(20)
$$ \mathop \smallint \limits_{0}^{\infty } e^{ - \nu t} Ei\left( { - \left( {\delta + \kappa t} \right)} \right)dt = \frac{1}{\nu }\left[ {Ei\left( { - \delta } \right) - e^{{\frac{\nu \delta }{\kappa }}} Ei\left( { - \frac{{\left( {\nu + \kappa } \right)\delta }}{k}} \right)} \right], $$
(21)
where \(Ei\left(x\right)=-\underset{x}{\overset{\infty }{\int }}{t}^{-1}{e}^{-t}dt\). Also, \(Ei\left(-x\right)\) was approximated in [27] as follows:
$$ Ei\left( { - x} \right) = - 4\sqrt 2 a_{N} a_{I} \mathop \sum \limits_{n = 1}^{N + 1} \mathop \sum \limits_{i = 1}^{I + 1} \sqrt {b_{n} } e^{{ - 4b_{n} b_{i} x}} , $$
(22)
where \({\theta }_{0}=0<{\theta }_{1}<\dots <{\theta }_{N+1}=\frac{\pi }{2}\), \({a}_{N}=\frac{{\theta }_{n}-{\theta }_{n-1}}{\pi }\), \({b}_{n}=\frac{\mathrm{cos}\left({\theta }_{n-1}\right)-\mathrm{cos}({\theta }_{n})}{2({\theta }_{n}-{\theta }_{n-1})}\), and for \(N=I=1\) we have: \({a}_{N}={a}_{I}=\frac{1}{4}\), \({b}_{1}=\infty \) and \({b}_{2}=\frac{2}{\pi }\). Thus, \(Ei\left(-x\right)\) can be approximated as
$$ Ei\left( { - x} \right)\sim - \frac{\sqrt \pi }{2}e^{{ - \left( {\frac{16}{{\pi^{2} }}} \right)x}} . $$
(23)
For Corollary 1 First, we compute the distance \({d}_{m}\) between transmitter and legitimate receiver to ensure reliable transmission. Due to the secrecy outage probability definition, the reliable transmission between transmitter and legitimate receiver happens, when legitimate receiver can decode the confidential messages (i.e., \({C}_{m}>{R}_{s}\)) or
$$ R_{s} \le E_{{\gamma_{m} }} \left[ {\log \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)} \right] = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} }}{{\overline{{\gamma_{m} }} }}\log \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)d\gamma_{m} , $$
(24)
where by computing the above integral and substituting (23) into (24), the distance \({d}_{m}\) between transmitter and legitimate receiver for reliable transmission is obtained as below:
$$ d_{m} \le \left( {\frac{{\pi^{2} \overline{{\gamma_{m} }} }}{{\pi^{2} - 16}}\ln \left( {\frac{{2R_{s} }}{\sqrt \pi }} \right)} \right)^{{\frac{1}{\alpha }}} . $$
(25)
In this case we don’t have any limitation over \({d}_{e}\).
For Corollary 2 The distance \({d}_{m}\) for reliable transmission is obtained similar to (25). Now, using the derived distance \({d}_{m}\) between transmitter and legitimate receiver, we can find the SCR for secure transmission from definition (15) as follows
$$ R_{s} \le C_{s} \left( {d_{m} ,d_{e} } \right) = E_{{\gamma_{s} ,\gamma_{e} ,\gamma_{m} }} \left[ {\frac{{\log \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }} + \gamma_{sm} } \right)}}{{\log \left( {1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }} + \gamma_{se} } \right)}}} \right] = D_{1} - D_{2} . $$
(26)
We have
$$ D_{1} = \mathop \smallint \limits_{0}^{\infty } \mathop \smallint \limits_{0}^{{\gamma_{m} }} \mathop \smallint \limits_{0}^{\infty } \log \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }} + \overline{{\gamma_{sm} }} } \right)\frac{1}{{\overline{{\gamma_{sm} }} }}e^{{ - \frac{{\gamma_{sm} }}{{\overline{{\gamma_{sm} }} }}}} \frac{1}{{\overline{{\gamma_{m} }} }}e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} \frac{1}{{\overline{{\gamma_{e} }} }}e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} d\gamma_{sm} d\gamma_{e} d\gamma_{m} , $$
(27)
and
$$ D_{2} = \mathop \smallint \limits_{0}^{\infty } \mathop \smallint \limits_{0}^{{\gamma_{m} }} \mathop \smallint \limits_{0}^{\infty } \log \left( {1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }} + \gamma_{se} } \right)\frac{1}{{\overline{{\gamma_{se} }} }}e^{{ - \frac{{\gamma_{se} }}{{\overline{{\gamma_{se} }} }}}} \frac{1}{{\overline{{\gamma_{m} }} }}e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} \frac{1}{{\overline{{\gamma_{e} }} }}e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} d\gamma_{se} d\gamma_{e} d\gamma_{m} . $$
(28)
After some simplifications and by utilizing linear formulas of integration, for \({D}_{1}\), we receive to
$$ D_{1} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} }}{{\overline{{\gamma_{sm} }} .\overline{{\gamma_{e} }} .\overline{{\gamma_{m} }} }}(D_{1}^{^{\prime\prime}} )d\gamma_{m} , $$
(29)
where \({D}_{1}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}={\int }_{0}^{{\gamma }_{m}}{e}^{-\frac{{\gamma }_{e}}{\overline{{\gamma }_{e}}}}{D}_{1}\mathrm{^{\prime}}d{\gamma }_{e}\) and \({D}_{1}^{\mathrm{^{\prime}}}={\int }_{0}^{\infty }\mathrm{log}(1+\frac{{\gamma }_{m}}{{d}_{m}^{\alpha }}+{\gamma }_{sm}){e}^{-\frac{{\gamma }_{sm}}{\overline{{\gamma }_{sm}}}}d{\gamma }_{sm}=\overline{{\gamma }_{sm}}(\mathrm{ln}\left(1+\frac{{\gamma }_{m}}{{d}_{m}^{\alpha }}\right)+{e}^{\frac{1+\frac{{\gamma }_{m}}{{d}_{m}^{\alpha }}}{\overline{{\gamma }_{sm}}}}{E}_{1}(\frac{1+\frac{{\gamma }_{m}}{{d}_{m}^{\alpha }}}{\overline{{\gamma }_{sm}}})).\) So, we have
$$ D_{1}^{^{\prime\prime}} = j_{1} ^{\prime} + j_{2} ^{\prime}, $$
(30)
where
$$ j_{1}^{^{\prime}} = \mathop \smallint \limits_{0}^{{\gamma_{m} }} \overline{{\gamma_{sm} }} e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} \ln \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)d\gamma_{e} = \overline{{\gamma_{sm} }} \overline{{\gamma_{e} }} \ln \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)\left( {1 - e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{e} }} }}}} } \right), $$
(31)
and
$$ j_{2}^{^{\prime}} = \mathop \smallint \limits_{0}^{{\gamma_{m} }} \overline{{\gamma_{sm} }} e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} e^{{\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}}} E_{1} \left( {\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}} \right)d\gamma_{e} = \overline{{\gamma_{sm} }} \overline{{\gamma_{e} }} \left( {1 - e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{e} }} }}}} } \right)e^{{\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}}} E_{1} \left( {\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}} \right). $$
(32)
Thus, by replacing (31) and (32) in (30) and then (29), we have
$$ D_{1} = k_{1}^{^{\prime}} - k_{2}^{^{\prime}} + k_{3}^{^{\prime}} - k_{4}^{^{\prime}} , $$
(33)
where by exploiting (19) and (21), we have
$$ k_{1}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} }}{{\overline{{\gamma_{m} }} }}\ln \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)d\gamma_{m} = - e^{{\frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }}}} Ei\left( { - \frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }}} \right), $$
(34)
$$ k_{2} ^{\prime} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \gamma_{m} \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} }}{{\overline{{\gamma_{m} }} }}\ln \left( {1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}} \right)d\gamma_{m} = - \frac{{\overline{{\gamma_{e} }} .e^{{d_{m}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} }}{{\overline{{\gamma_{m} }} + \overline{{\gamma_{e} }} }}Ei\left( { - d_{m}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)} \right), $$
(35)
$$ k_{3}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}}} }}{{\overline{{\gamma_{m} }} }}E_{1} \left( {\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}} \right)d\gamma_{m} = \frac{{ - e^{{\frac{1}{{\overline{{\gamma_{sm} }} }}}} \left( {Ei\left( {\frac{ - 1}{{\overline{{\gamma_{sm} }} }}} \right) - e^{{\frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{sm} }} }}}} Ei\left( {\frac{{ - d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }}} \right)} \right)}}{{\overline{{\gamma_{m} }} \left( {\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } }}} \right)}}, $$
(36)
$$ \begin{aligned} k_{4}^{^{\prime}} & = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \gamma_{m} \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} e^{{\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}}} }}{{\overline{{\gamma_{m} }} }}E_{1} \left( {\frac{{1 + \frac{{\gamma_{m} }}{{d_{m}^{\alpha } }}}}{{\overline{{\gamma_{sm} }} }}} \right)d\gamma_{m} \\ & = \frac{{e^{{\frac{1}{{\overline{{\gamma_{sm} }} }}}} \left[ {Ei\left( {\frac{ - 1}{{\overline{{\gamma_{sm} }} }}} \right) - e^{{\frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }} + \frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{sm} }} }}}} Ei\left( { - \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)d_{m}^{\alpha } } \right)} \right]}}{{\frac{1}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } }} - \frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }}}}. \\ \end{aligned} $$
(37)
Similarly, for \({D}_{2}\), we have
$$ D_{2} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} }}{{\overline{{\gamma_{se} }} .\overline{{\gamma_{e} }} .\overline{{\gamma_{m} }} }}(D_{2}^{^{\prime\prime}} )d\gamma_{m} . $$
(38)
where \({D}_{2}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}={\int }_{0}^{{\gamma }_{m}}{e}^{-\frac{{\gamma }_{e}}{\overline{{\gamma }_{e}}}}({D}_{2}\mathrm{^{\prime}})d{\gamma }_{e}\) and \({D}_{2}\mathrm{^{\prime}}={\int }_{0}^{\infty }\mathrm{log}(1+\frac{{\gamma }_{e}}{{d}_{e}^{\alpha }}+{\gamma }_{se}){e}^{-\frac{{\gamma }_{se}}{\overline{{\gamma }_{se}}}}d{\gamma }_{se}=\overline{{\gamma }_{se}}(\mathrm{ln}\left(1+\frac{{\gamma }_{e}}{{d}_{e}^{\alpha }}\right)+{e}^{\frac{1+\frac{{\gamma }_{e}}{{d}_{e}^{\alpha }}}{\overline{{\gamma }_{se}}}}{E}_{1}(\frac{1+\frac{{\gamma }_{e}}{{d}_{e}^{\alpha }}}{\overline{{\gamma }_{se}}})).\) So, we have
$$ D_{2}^{^{\prime\prime}} = i_{1} ^{\prime} + i_{2} ^{\prime}, $$
(39)
where by exploiting (18) and (20), we have
$$ \begin{aligned} i_{1}^{^{\prime}} & = \mathop \smallint \limits_{0}^{{\gamma_{m} }} \overline{{\gamma_{se} }} e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} \ln \left( {1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }}} \right)d\gamma_{e} \\ & = \overline{{\gamma_{se} }} \overline{{\gamma_{e} }} \left[ {e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} Ei\left( { - \frac{{d_{e}^{\alpha } + \gamma_{m} }}{{\overline{{\gamma_{e} }} }}} \right) - e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{e} }} }}}} \ln \left( {1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }}} \right) - e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} Ei\left( {\frac{{ - d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right)} \right], \\ \end{aligned} $$
(40)
$$ \begin{aligned} i_{2} ^{\prime} = & \mathop \smallint \limits_{0}^{{\gamma_{m} }} \overline{{\gamma_{se} }} e^{{ - \frac{{\gamma_{e} }}{{\overline{{\gamma_{e} }} }}}} e^{{\frac{{1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }}}}{{\overline{{\gamma_{se} }} }}}} E_{1} \left( {\frac{{1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }}}}{{\overline{{\gamma_{se} }} }}} \right)d\gamma_{e} = - \frac{{\overline{{\gamma_{se} }} e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} }}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}} .[e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} }}}} Ei\left( { - \frac{{\gamma_{m} + d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right) \\ & - e^{{\gamma_{m} \left( {\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)}} Ei\left( { - \frac{{1 + \frac{{\gamma_{m} }}{{d_{e}^{\alpha } }}}}{{\overline{{\gamma_{se} }} }}} \right) - e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} }}}} Ei\left( { - \frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right) + Ei\left( { - \frac{1}{{\overline{{\gamma_{se} }} }}} \right)]. \\ \end{aligned} $$
(41)
Thus, by replacing (40) and (41) in (39) and then (38), we have
$$ D_{2} = d_{1} ^{\prime} - d_{2} ^{\prime} - d_{3} ^{\prime} - d_{4} ^{\prime} + d_{5} ^{\prime} + d_{6} ^{\prime} - d_{7} .^{\prime} $$
(42)
where by exploiting (19) and (21), we have
$$ d_{1}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} }}{{\overline{{\gamma_{m} }} }}Ei\left( { - \frac{{d_{e}^{\alpha } + \gamma_{m} }}{{\overline{{\gamma_{e} }} }}} \right)d\gamma_{m} = e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} \left[ {Ei\left( { - \frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right) - e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{m} }} }}}} Ei\left( { - \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)d_{e}^{\alpha } } \right)} \right], $$
(43)
$$ d_{2} ^{\prime} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{e} }} }}}} }}{{\overline{{\gamma_{m} }} }}\ln \left( {1 + \frac{{\gamma_{e} }}{{d_{e}^{\alpha } }}} \right)d\gamma_{m} = - \frac{{\overline{{\gamma_{e} }} }}{{\overline{{\gamma_{m} }} + \overline{{\gamma_{e} }} }}e^{{d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} Ei\left( { - d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)} \right), $$
(44)
$$ d_{3}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} }}{{\overline{{\gamma_{m} }} }}Ei\left( {\frac{{ - d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right)d\gamma_{m} = e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} Ei\left( {\frac{{ - d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right), $$
(45)
$$ \begin{aligned} d_{4}^{^{\prime}} & = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} }}{{\overline{{\gamma_{e} }} .\overline{{\gamma_{m} }} }}\left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)Ei\left( { - \frac{{\gamma_{m} + d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right)d\gamma_{m} \\ & = e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} \left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)\left( {Ei\left( { - \frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right) - e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} Ei\left( { - d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)} \right)} \right), \\ \end{aligned} $$
(46)
$$ \begin{aligned} d_{5}^{^{\prime}} & = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} }}{{\overline{{\gamma_{e} }} .\overline{{\gamma_{m} }} }}\left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)e^{{\gamma_{m} \left( {\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)}} Ei\left( { - \frac{{1 + \frac{{\gamma_{m} }}{{d_{e}^{\alpha } }}}}{{\overline{{\gamma_{se} }} }}} \right)d\gamma_{m} \\ & = \frac{1}{{\overline{{\gamma_{e} }} \overline{{\gamma_{m} }} }}e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} \left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}\frac{1}{{\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }} + \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)\left[ {Ei\left( { - \frac{1}{{\overline{{\gamma_{se} }} }}} \right) - e^{{\left( {d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }} + \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)} \right)}} Ei\left( { - d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }} + \frac{2}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)} \right)} \right], \\ \end{aligned} $$
(47)
$$ d_{6}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} }}{{\overline{{\gamma_{e} }} \overline{{\gamma_{m} }} }}\left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} }}}} Ei\left( { - \frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right)d\gamma_{m} = \frac{1}{{\overline{{\gamma_{e} }} }}\left( {\frac{{e^{{\frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}}} Ei\left( { - \frac{{d_{e}^{\alpha } }}{{\overline{{\gamma_{e} }} }}} \right)}}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right), $$
(48)
$$ d_{7}^{^{\prime}} = \mathop \smallint \limits_{0}^{\infty } \frac{{e^{{ - \frac{{\gamma_{m} }}{{\overline{{\gamma_{m} }} }}}} e^{{\frac{{N_{e} }}{{\overline{{\gamma_{s} }} }}}} }}{{\overline{{\gamma_{e} }} .\overline{{\gamma_{m} }} }}\left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)Ei\left( { - \frac{1}{{\overline{{\gamma_{se} }} }}} \right)d\gamma_{m} = \frac{1}{{\overline{{\gamma_{e} }} }}\left( {\frac{1}{{\frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} Ei\left( { - \frac{1}{{\overline{{\gamma_{se} }} }}} \right). $$
(49)
Therefore, from (26), we have
$$ \begin{aligned} R_{s} \le & k_{1}^{^{\prime}} - k_{2}^{^{\prime}} + k_{3}^{^{\prime}} - k_{4}^{^{\prime}} - d_{1}^{^{\prime}} - d_{2}^{^{\prime}} - d_{3}^{^{\prime}} - d_{4}^{^{\prime}} + d_{5}^{^{\prime}} + d_{6}^{^{\prime}} - d_{7}^{^{\prime}} \\ & = \frac{{\overline{{\gamma_{m} }} e^{{\frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }}}} }}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } - \overline{{\gamma_{m} }} }}Ei\left( { - \frac{{d_{m}^{\alpha } }}{{\overline{{\gamma_{m} }} }}} \right) - \left( {\frac{1}{{1 - \frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } }}}} - \frac{1}{{\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } }}}}} \right)e^{{\frac{1}{{\overline{{\gamma_{sm} }} }}}} Ei\left( {\frac{ - 1}{{\overline{{\gamma_{sm} }} }}} \right) + (\frac{{\overline{{\gamma_{e} }} }}{{\overline{{\gamma_{m} }} + \overline{{\gamma_{e} }} }} \\ & - \frac{1}{{\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }} - \frac{1}{{\overline{{\gamma_{sm} }} d_{m}^{\alpha } }}}})e^{{d_{m}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} Ei\left( { - d_{m}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)} \right) + \left( {\frac{1}{{1 - \frac{{\overline{{\gamma_{e} }} }}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)(1 - \frac{1}{{1 - \frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{e} }} }} + \frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{sm} }} d_{e}^{\alpha } }}}})e^{{\frac{1}{{\overline{{\gamma_{se} }} }}}} \\ & \cdot Ei\left( { - \frac{1}{{\overline{{\gamma_{se} }} }}} \right) + \left( {\frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{m} }} + \overline{{\gamma_{e} }} }} - \frac{1}{{1 - \frac{{\overline{{\gamma_{e} }} }}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)e^{{d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)}} Ei\left( { - d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} + \frac{1}{{\overline{{\gamma_{e} }} }}} \right)} \right) \\ & + \left( {\frac{1}{{1 - \frac{{\overline{{\gamma_{e} }} }}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right) \left( {\frac{1}{{1 - \frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{e} }} }} + \frac{{\overline{{\gamma_{m} }} }}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}}}} \right)e^{{d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }} + \frac{2}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)}} .Ei\left( { - d_{e}^{\alpha } \left( {\frac{1}{{\overline{{\gamma_{m} }} }} - \frac{1}{{\overline{{\gamma_{e} }} }} + \frac{2}{{\overline{{\gamma_{se} }} d_{e}^{\alpha } }}} \right)} \right). \\ \end{aligned} $$
(50)
Now, by considering the obtained approximation in (23), the proof is completed.
