Secrecy Analysis of Overlay Mechanism in Radio Frequency Energy Harvesting Networks with Jamming under Nakagami-m fading

Abstract

Overlay mechanism offers secondary users (SUs) chances to access licensed spectrum of primary users (PUs) to improve spectrum utilization efficiency. However, these chances impose serious security challenges when wire-tappers emulate legal users (SUs/PUs) to eavesdrop secret information. Moreover, energy sources available for SUs to transmit its private information as well as assist PUs in the overlay mechanism are problematic in green communication age. This paper proposes SUs who are able to scavenge radio frequency energy in signals of PUs to meet green communication requirement and jam wire-tappers to secure both secondary and primary information. Important secrecy performance indicators of overlay mechanism in radio frequency energy harvesting networks with jamming under Nakagami-m fading are evaluated by proposed precise expressions. Various results validate these expressions and show that secrecy performances of SUs and PUs are flexibly controlled, compromised and optimized with proper selections of system parameters as well as are dramatically affected by fading severity.

Appendices

Appendix A: Special Pdfs

Let

$$\begin{aligned} {Z_{cb}} = 1 + \frac{{U{g_{cb}}}}{{{J_{cb}}{g_{cb}} + {\epsilon _b}}}, \end{aligned}$$

(74)

and

$$\begin{aligned} {Z_{cw}} = 1 + \frac{{U{g_{cw}}}}{{{J_{cw}}{g_{cw}} + {\epsilon _w}}}, \end{aligned}$$

(75)

where U, $J_{cw}$, and $J_{cb}$ are defined in (29), (30), and (33), respectively.

This appendix derives the pdfs of ${Z_{cb}}$ and ${Z_{cw}}$.

From (74), one obtains

$$\begin{aligned} {g_{cb}} = \frac{{\left( {{Z_{cb}} - 1} \right) {\epsilon _b}}}{{U + {J_{cb}} - {J_{cb}}{Z_{cb}}}}. \end{aligned}$$

(76)

Because $g_{cb} \ge 0$, ${Z_{cb}}$ is constrained in $\left[ {\left. {1,\frac{U}{{{J_{cb}}}} + 1} \right) } \right.$.

The Jacobian coefficient is computed as

$$\begin{aligned} \frac{{d{g_{cb}}}}{{d{Z_{cb}}}} = \frac{{U{\epsilon _b}}}{{{{\left( {U + {J_{cb}} - {J_{cb}}{Z_{cb}}} \right) }^2}}}. \end{aligned}$$

(77)

Given the variable transform in (74), the pdf of $Z_{cb}$ can be obtained from the pdf of $g_{cb}$ and the Jacobian coefficient as

$$\begin{aligned} {f_{{Z_{cb}}}}\left( z \right) = {f_{{g_{cb}}}}\left( {\frac{{\left( {z - 1} \right) U}}{{U + {J_{cb}} - {J_{cb}}z}}} \right) {\left| {\frac{{d{g_{cb}}}}{{d{Z_{cb}}}}} \right| _{{Z_{cb}} = z}}. \end{aligned}$$

(78)

Substituting ${f_{{g_{cb}}}}\left( z \right) = \frac{{\xi _{cb}^{{\alpha _{cb}}}}}{{\Gamma \left( {{\alpha _{cb}}} \right) }}{z^{{\alpha _{cb}} - 1}}{e^{ - {\xi _{cb}}z}}$ and the Jacobian coefficient into (78), the pdf of $Z_{cb}$ is obtained as

$$\begin{aligned} {f_{{Z_{cb}}}}\left( z \right) = \frac{{Q_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}U}}{{{J_{cb}}\Gamma \left( {{\alpha _{cb}}} \right) }}{e^{\frac{{{W_{cb}}}}{{z - {L_{cb}}}}}}\frac{{{{\left( {z - 1} \right) }^{{\alpha _{cb}} - 1}}}}{{{{\left( {{L_{cb}} - z} \right) }^{{\alpha _{cb}} + 1}}}}, \ \ \ 1 \le z < {L_{cb}}, \end{aligned}$$

(79)

where $L_{cb}$, $Q_{cb}$, and $W_{cb}$ in (34), (36), and (39), respectively.

Similarly, the pdf of $Z_{cw}$ is represented as

$$\begin{aligned} {f_{{Z_{cw}}}}\left( t \right) = \frac{{Q_{cw}^{{\alpha _{cw}}}{e^{{Q_{cw}}}}U}}{{{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{\frac{{{W_{cw}}}}{{t - {L_{cw}}}}}}\frac{{{{\left( {t - 1} \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {{L_{cw}} - t} \right) }^{{\alpha _{cw}} + 1}}}}, \ \ \ 1 \le t < {L_{cw}}, \end{aligned}$$

(80)

where $L_{cw}$, $Q_{cw}$, and $W_{cw}$ in (31), (41), and (42), respectively.

Appendix B: Special Integral

This appendix solves the following integral

$$\begin{aligned} \mathcal{M}\left( {m,n,{S_{cb}}} \right) = \int \limits _m^n {{f_{{Z_{cb}}}}\left( z \right) dz}, \end{aligned}$$

(81)

where ${S_{cb}} = \left\{ {U,J_{cb},{\xi _{cb}},{\alpha _{cb}},{\epsilon _b}} \right\}$ denotes the set of parameters related to the $\texttt {C}\rightarrow \texttt {B}$ transmission and $1 \le m < n \le L_{cb}$.

Inserting ${{f_{Z_{cb}}}\left( z \right) }$ in (79) into (81) and after changing the variable, (81) is rewritten as

$$\begin{aligned} \mathcal{M}\left( {m,n,{S_{cb}}} \right)&= \int \limits _m^n {\frac{{Q_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}U}}{{{J_{cb}}\Gamma \left( {{\alpha _{cb}}} \right) }}{e^{\frac{{{W_{cb}}}}{{z - {L_{cb}}}}}}\frac{{{{\left( {z - 1} \right) }^{{\alpha _{cb}} - 1}}}}{{{{\left( {{L_{cb}} - z} \right) }^{{\alpha _{cb}} + 1}}}}dz} \nonumber \\&= ^{t = \frac{1}{{z - {L_{cb}}}}} \frac{{Q_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}U}}{{{J_{cb}}\Gamma \left( {{\alpha _{cb}}} \right) }}{\left( { - \frac{U}{{{J_{cb}}}}} \right) ^{{\alpha _{cb}} - 1}}\int \limits _{\frac{1}{{n - {L_{cb}}}}}^{\frac{1}{{m - {L_{cb}}}}} {{e^{{W_{cb}}t}}{{\left( {t + \frac{{{J_{cb}}}}{U}} \right) }^{{\alpha _{cb}} - 1}}dt}. \end{aligned}$$

(82)

Executing the binomial expansion to ${{{\left( {t + \frac{{{J_{cb}}}}{U}} \right) }^{{\alpha _{cb}} - 1}}}$ and after some simplifications, (82) is reduced to

$$\begin{aligned} \mathcal{M}\left( {m,n,{S_{cb}}} \right)&= {\left( { - 1} \right) ^{{\alpha _{cb}} + 1}}\frac{{W_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}}}{{\Gamma \left( {{\alpha _{cb}}} \right) }}\int \limits _{\frac{1}{{n - {L_{cb}}}}}^{\frac{1}{{m - {L_{cb}}}}} {{e^{{W_{cb}}t}}\left\{ {\sum \limits _{l = 0}^{{\alpha _{cb}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cb}} - 1}\\ l \end{array}} \right) } {t^l}{{\left( {\frac{{{J_{cb}}}}{U}} \right) }^{{\alpha _{cb}} - 1 - l}}} \right\} dt} \nonumber \\&= {\left( { - 1} \right) ^{{\alpha _{cb}} + 1}}\frac{{Q_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}U}}{{{J_{cb}}\Gamma \left( {{\alpha _{cb}}} \right) }}\sum \limits _{l = 0}^{{\alpha _{cb}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cb}} - 1}\\ l \end{array}} \right) } {\left( {\frac{U}{{{J_{cb}}}}} \right) ^l}\int \limits _{\frac{1}{{n - {L_{cb}}}}}^{\frac{1}{{m - {L_{cb}}}}} {{t^l}{e^{{W_{cb}}t}}dt}. \end{aligned}$$

(83)

By invoking [31, eq. (2.321.2)], the last integral in (83) is computed in closed-form, reducing (83) to

$$\begin{aligned} \mathcal{M}\left( {m,n,{S_{cb}}} \right)&= {\left( { - 1} \right) ^{{\alpha _{cb}} + 1}}\frac{{Q_{cb}^{{\alpha _{cb}}}{e^{{Q_{cb}}}}U}}{{{J_{cb}}\Gamma \left( {{\alpha _{cb}}} \right) }}\sum \limits _{l = 0}^{{\alpha _{cb}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cb}} - 1}\\ l \end{array}} \right) } {\left( {\frac{U}{{{J_{cb}}}}} \right) ^l} \nonumber \\&\quad \times \left\{ {\mathcal{R}\left( {{{\left[ {m - {L_{cb}}} \right] }^{ - 1}},{W_{cb}},l} \right) - \mathcal{R}\left( {{{\left[ {n - {L_{cb}}} \right] }^{ - 1}},{W_{cb}},l} \right) } \right\} , \end{aligned}$$

(84)

where

$$\begin{aligned} \mathcal{R}\left( {s,u,p} \right) = \frac{{{s^p}}}{u}{e^{us}}\left( {1 + \sum \limits _{k = 1}^p {{{\left( { - us} \right) }^{ - k}}\prod \limits _{i = 0}^{k - 1} {\left( {p - i} \right) } } } \right) . \end{aligned}$$

(85)

Appendix C: Proof of Lemma 1

Substituting ${{\hat{C}_a}}$ in (22) into $\Delta$ in (25), one obtains

$$\begin{aligned} \Delta = \Pr \left\{ {\left. {1 + {\Theta _b} < K\left( {1 + {\Theta _{Wa}}} \right) } \right| {C_c} \ge \bar{C}} \right\} , \end{aligned}$$

(86)

where K is defined in (32).

Invoking (14) and (17) for the case of ${{C_c} \ge {\bar{C}}}$, (86) is rewritten as

$$\begin{aligned} \Delta = \Pr \left\{ {{Z_{cb}} < K{Z_{cw}}} \right\} , \end{aligned}$$

(87)

where $Z_{cb}$ and $Z_{cw}$ are defined in (74) and (75), correspondingly.

Due to statistical independence of $Z_{cb}$ in (74) and $Z_{cw}$ in (75), their joint pdf is the product of their individual pdfs. Consequently, $\Delta$ in (87) can be computed as

$$\begin{aligned} \Delta = \iint \limits _{z < Kt} {{f_{{Z_{cb}}}}\left( z \right) {f_{{Z_{cw}}}}\left( t \right) dzdt}. \end{aligned}$$

(88)

Since $f_{Z_{cb}}\left( z\right)$ in (79) is non-zero for $z \in [1, L_{cb})$, the double integral in (88) can be solved by considering three cases of Kt in correlation to $[1, L_{cb})$: (Case 1: $Kt < 1$), (Case 2: $1 \le Kt < L_{cb}$), (Case 3: $Kt > L_{cb}$). “Case 1” means $t< K^{-1} < 1$ and hence, $\Delta$ for this case is $\int \limits _{t < {K^{ - 1}}} {\left[ {\int \limits _0^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt}=0$ because ${{f_{{Z_{cw}}}}\left( t \right) }$ in (80) is zero for $t < 1$. Meanwhile, “Case 2” and “Case 3” respectively mean $K^{-1} \le t<V$ and $t>V$ where V is given in (35). Therefore, (88) is rewritten for “Case 2” and “Case 3” as

$$\begin{aligned} \Delta = \int \limits _{{K^{ - 1}} \le t < V} {\left[ {\int \limits _{z = 1}^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt} + \int \limits _{t > V} {\left[ {\int \limits _{z = 1}^{{L_{cb}}} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(89)

Since ${{f_{{Z_{cw}}}}\left( t \right) }$ is non-zero for $t \in [1, L_{cw})$, in order to determine integral limits for t in (89), one must consider whether V falls in $[1, L_{cw})$ or not. Consequently, three contexts for V are investigated as follows.

Context 1: $L_{cw}<V$

Under this context, the second term in (89) is zero since ${{f_{{Z_{cw}}}}\left( t \right) }=0$ as $t=V>L_{cw}$. Consequently, (89) is simplified as

$$\begin{aligned} {\Delta _1} = \int \limits _{t = 1}^{{L_{cw}}} {\left[ {\int \limits _{z = 1}^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(90)

The inner integral in (90) is solved by using (81) and hence,

$$\begin{aligned} {\Delta _1}&= \int \limits _1^{{L_{cw}}} {\mathcal{M}\left( {1,Kt,{S_{cb}}} \right) {f_{{Z_{cw}}}}\left( t \right) dt} \nonumber \\&= \int \limits _1^{{L_{cw}}} {\frac{{{{\left( { - {Q_{cb}}} \right) }^{{\alpha _{cb}} - 1}}}}{{{e^{ - {Q_{cb}}}}\Gamma \left( {{\alpha _{cb}}} \right) }}\sum \limits _{l = 0}^{{\alpha _{cb}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cb}} - 1}\\ l \end{array}} \right) } {{\left( {\frac{U}{{{J_{cb}}}}} \right) }^l} } \nonumber \\&\qquad \times \left\{ {\frac{{{e^{{W_{cb}}/\left( {1 - {L_{cb}}} \right) }}}}{{{{\left( {1 - {L_{cb}}} \right) }^l}}}\left( {1 + \sum \limits _{k = 1}^l {{{\left( {\frac{{{L_{cb}} - 1}}{{{W_{cb}}}}} \right) }^k}\prod \limits _{i = 0}^{k - 1} {\left( {l - i} \right) } } } \right) } \right. \nonumber \\&\quad - \frac{{{e^{{W_{cb}}/\left( {Kt - {L_{cb}}} \right) }}}}{{{{\left( {Kt - {L_{cb}}} \right) }^l}}}\left. {\left( {1 + \sum \limits _{k = 1}^l {{{\left( {\frac{{{L_{cb}} - Kt}}{{{W_{cb}}}}} \right) }^k}\prod \limits _{i = 0}^{k - 1} {\left( {l - i} \right) } } } \right) } \right\} {f_{{Z_{cw}}}}\left( t \right) dt. \end{aligned}$$

(91)

Because $\int \limits _1^{{L_{cw}}} {{f_{{Z_{cw}}}}\left( t \right) dt}=1$, (91) is simplified to (40) where

$$\begin{aligned} \mathcal{L}\left( {l,v} \right) = \int \limits _1^{{L_{cw}}} {\frac{{{e^{v/\left( {t - V} \right) }}}}{{{{\left( {t - V} \right) }^l}}}{f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(92)

Consequently, the next step of the proof is to prove that (92) coincides (44). To this end, inserting ${{f_{Z_{cw}}}\left( t \right) }$ in (80) into (92) and after some simplications, one reduces (92) to

$$\begin{aligned} \mathcal{L}\left( {l,v} \right)&= \int \limits _1^{{L_{cw}}} {\frac{{{e^{v/\left( {t - V} \right) }}}}{{{{\left( {t - V} \right) }^l}}}\frac{{Q_{cw}^{{\alpha _{cw}}}{e^{{Q_{cw}}}}U}}{{{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{\frac{{{W_{cw}}}}{{t - {L_{cw}}}}}}\frac{{{{\left( {t - 1} \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {{L_{cw}} - t} \right) }^{{\alpha _{cw}} + 1}}}}dt} \nonumber \\&= \frac{{Q_{cw}^{{\alpha _{cw}}}{e^{{Q_{cw}}}}U}}{{{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}\int \limits _1^{{L_{cw}}} {\frac{{{{\left( {t - 1} \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {t - V} \right) }^l}{{\left( {{L_{cw}} - t} \right) }^{{\alpha _{cw}} + 1}}}}{e^{\frac{v}{{t - V}} + \frac{{{W_{cw}}}}{{t - {L_{cw}}}}}}dt}. \end{aligned}$$

(93)

Executing consecutive variable transforms (${z = 1/\left( {t - L_{cw}} \right) }$ and then ${t = - z}$), one simplifies (93) to

$$\begin{aligned} \mathcal{L}\left( {l,v} \right)&= \frac{{{{\left( {{L_{cw}} - 1} \right) }^{{\alpha _{cw}} - 1}}Q_{cw}^{{\alpha _{cw}}}U}}{{{{\left( {{L_{cw}} - V} \right) }^l}{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{Q_{cw}} + \frac{v}{{{L_{cw}} - V}}}} \nonumber \\&\quad \times \int \limits _{\frac{1}{{{L_{cw}} - 1}}}^\infty {{t^l}\frac{{{{\left( {t - 1/\left[ {{L_{cw}} - 1} \right] } \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {t + 1/\left[ {V - {L_{cw}}} \right] } \right) }^l}}}{e^{\frac{{v/{{\left( {{L_{cw}} - V} \right) }^2}}}{{t + 1/\left( {V - {L_{cw}}} \right) }} - {W_{cw}}t}}dt}. \end{aligned}$$

(94)

Performing the series expansion ${e^x} = \sum \limits _{n = 0}^\infty {\frac{{{x^n}}}{{n!}}}$ for the term ${e^{\frac{{v/{{\left( {{L_{cw}} - V} \right) }^2}}}{{t + 1/\left( {V - {L_{cw}}} \right) }}}}$ in (94) and then executing the binomial expansion, one achieves

$$\begin{aligned} \mathcal{L}\left( {l,v} \right)&= \frac{{{{\left( { - 1} \right) }^{{\alpha _{cw}} + 1}}Q_{cw}^{{\alpha _{cw}}}U}}{{{{\left( {{L_{cw}} - V} \right) }^l}{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{Q_{cw}} + \frac{v}{{{L_{cw}} - V}}}} \nonumber \\&\quad \times \sum \limits _{q = 0}^\infty {\sum \limits _{u = 0}^{{\alpha _{cw}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cw}} - 1}\\ u \end{array}} \right) } } \frac{{{{\left( {1 - {L_{cw}}} \right) }^u}{v^q}}}{{{{\left( {{L_{cw}} - V} \right) }^{2q}}q!}}\int \limits _{\frac{1}{{{L_{cw}} - 1}}}^\infty {\frac{{{t^{l + u}}{e^{ - {W_{cw}}t}}}}{{{{\left( {t + 1/\left[ {V - {L_{cw}}} \right] } \right) }^{l + q}}}}dt}. \end{aligned}$$

(95)

Executing the variable change $z = t + {\left( {V - L_{cw}} \right) ^{ - 1}}$ and performing the binomial expansion, one simplifies (95) to (44) where

$$\begin{aligned} \eta&= \int \limits _I^\infty {\frac{{{z^t}}}{{{z^{l + q}}}}{e^{ - {W_{cw}}z}}dz} \nonumber \\&= \left\{ {\begin{array}{*{20}{c}} {\int \limits _I^\infty {{z^{t - l - q}}{e^{ - {W_{cw}}z}}dz} }&{}{,t > l + q}\\ {\int \limits _I^\infty {{e^{ - {W_{cw}}z}}dz} }&{}{,t = l + q}\\ {\int \limits _I^\infty {\frac{{{e^{ - {W_{cw}}z}}}}{{{z^{l + q - t}}}}dz} }&{}{,t < l + q} \end{array}} \right. \end{aligned}$$

(96)

with I being defined in (43). As such, the last step of the proof is to show that (96) coincides (45). To this end, the following notes should be paid attention. First, it is straightforward to solve the second integral in the last equality of (96). Second, the first integral has a common form as $\mathcal {G}\left( {g,v,b} \right) = \int \limits _g^\infty {{t^{v - 1}}{e^{ - bt}}dt}$, which is analytically evaluated as (46) by invoking [31, eq. (3.381.3)]. Finally, the third integral has a general form as $\mathcal {Y}\left( {v,g,n} \right) = \int \limits _v^\infty {\frac{{{e^{ - gt}}}}{{{t^{n + 1}}}}dt}$, which is also analytically evaluated as (47) by invoking [31, eq. (358.4)]. After solving these three integrals, (96) is simplified to (45), completing the proof of $\Delta _1$.

Context 2: $1\le V<L_{cw}$

This condition reduces (89) to

$$\begin{aligned} {\Delta _2} = \int \limits _{t = {K^{ - 1}}}^V {\left[ {\int \limits _{z = 1}^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt} + \int \limits _{t = V}^{{L_{cw}}} {\left[ {\int \limits _{z = 1}^{{L_{cb}}} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(97)

Before solving (97), it should be reminded that ${{f_{{Z_{cb}}}}\left( z \right) }$ is non-zero for $z\in \left[ 1, L_{cb}\right)$ and thus, ${\int \limits _{z = 1}^{{L_{cb}}} {{f_{{Z_{cb}}}}\left( z \right) dz} }=1$. Furthermore, $\int \limits _1^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz}$ and $\int \limits _V^{{L_{cw}}} {{f_{{Z_{cw}}}}\left( t \right) dt}$ are represented in closed-form as $\mathcal{M}\left( {1,Kt,{S_{cb}}} \right)$ and $\mathcal{M}\left( {V,{L_{cw}},{S_{cw}}} \right)$ according to (81), correspondingly, where ${S_{cb}} = \left\{ {U,{J_{cb}},{\xi _{cb}},{\alpha _{cb}},{\epsilon _b}} \right\}$ and ${S_{cw}} = \left\{ {U,{L_{cw}},{\xi _{cw}},{\alpha _{cw}},{\epsilon _w}} \right\}$ are the sets of parameters related to the $\texttt {C}\rightarrow \texttt {B}$ and $\texttt {C} \rightarrow \texttt {W}$ transmission. Accordingly, (97) is reduced to

$$\begin{aligned} {\Delta _2} = \int \limits _1^V {\mathcal{M}\left( {1,Kt,{S_{cb}}} \right) {f_{{Z_{cw}}}}\left( t \right) dt} + \mathcal{M}\left( {V,{L_{cw}},{S_{cw}}} \right) . \end{aligned}$$

(98)

Substituting ${\mathcal{M}\left( {1,Kt,{S_{cb}}} \right) }$ in (84) into (98) and after some manipulations, (98) is reduced to (48) where

$$\begin{aligned} \mathcal{Q}\left( {l,v} \right) = \int \limits _1^V {\frac{{{e^{v/\left( {t - V} \right) }}}}{{{{\left( {t - V} \right) }^l}}}{f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(99)

Accordingly, the next step of the proof of $\Delta _2$ is to prove that (99) is expressed as (49). To this end, inserting ${{f_{{Z_{cw}}}}\left( t \right) }$ in (80) into (99) and then executing consecutive variable transforms ($z=1/\left( t-V\right)$ and then $t=-z$), one simplifies (99) to

$$\begin{aligned} \mathcal{Q}\left( {l,v} \right)&= \frac{{{{\left( { - 1} \right) }^l}Q_{cw}^{{\alpha _{cw}}}U}}{{{{\left( {V - {L_{cw}}} \right) }^{{\alpha _{cw}} + 1}}{J_{cw}}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{Q_{cw}} + \frac{{{W_{cw}}}}{{V - {L_{cw}}}}}}\sum \limits _{u = 0}^{{\alpha _{cw}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cw}} - 1}\\ u \end{array}} \right) } {\left( {1 - V} \right) ^u} \nonumber \\&\quad \times \int \limits _{1/\left( {V - 1} \right) }^\infty {\frac{{{t^{l + u}}}}{{{{\left( {t + 1/\left[ {{L_{cw}} - V} \right] } \right) }^{{\alpha _{cw}} + 1}}}}{e^{\frac{{{W_{cw}}/{{\left( {V - {L_{cw}}} \right) }^2}}}{{t + 1/\left( {{L_{cw}} - V} \right) }} - vt}}dt}. \end{aligned}$$

(100)

Performing some similar (series and binomial expansions) manipulations as (92), one obtains (49) where

$$\begin{aligned} \theta = \int \limits _O^\infty {\frac{{{z^t}{e^{ - vz}}}}{{{z^{{\alpha _{cw}} + 1 + q}}}}dz}, \end{aligned}$$

(101)

which is further decomposed into three sub-cases as

$$\begin{aligned} \theta = \left\{ {\begin{array}{*{20}{c}} {\int \limits _O^\infty {{z^{t - {\alpha _{cw}} - 1 - q}}{e^{ - vz}}dz} }&{}{,t > {\alpha _{cw}} + 1 + q}\\ {\int \limits _O^\infty {{e^{ - vz}}dz} }&{}{,t = {\alpha _{cw}} + 1 + q}\\ {\int \limits _O^\infty {\frac{{{e^{ - vz}}}}{{{z^{{\alpha _{cw}} + 1 + q - t}}}}dz} }&{}{,t < {\alpha _{cw}} + 1 + q} \end{array}} \right. , \end{aligned}$$

(102)

where O is defined in (51). Consequently, the last step of the proof of $\Delta _2$ is to show that (102) is reduced to (50). To this end, the following notes are helpful. First, it is easy to solve the second integral in the last equality of (102). Second, the first and the third integrals are represented in terms of $\mathcal {G}\left( \cdot ,\cdot ,\cdot \right)$ and $\mathcal {Y}\left( \cdot ,\cdot ,\cdot \right)$ in (46) and (47), correspondingly. After solving these three integrals, one reduces (102) to (50), finishing the proof of $\Delta _2$.

Context 3: $V<1$

This condition reduces (89) to

$$\begin{aligned} \Delta = \int \limits _{t = {K^{ - 1}}}^V {\left[ {\int \limits _{z = 1}^{Kt} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt} + \int \limits _{t = 1}^{{L_{cw}}} {\left[ {\int \limits _{z = 1}^{{L_{cb}}} {{f_{{Z_{cb}}}}\left( z \right) dz} } \right] {f_{{Z_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(103)

Since ${{f_{{Z_{cw}}}}\left( t \right) }=0$ for $t=V<1$, the first term in (103) is zero. Moreover, ${{f_{{Z_{cb}}}}\left( z \right) }$ is non-zero for $z\in \left[ 1,L_{cb}\right)$ and ${{f_{{Z_{cw}}}}\left( t \right) }$ is non-zero for $t\in \left[ 1,L_{cw}\right)$ and therefore, the second term in (103) is one. Plugging these results into (103), one infers $\Delta = 1$, which agrees with (28) for $V<1$. This completes the proof of $\Delta$ for $V<1$.

By combining three above contexts, $\Delta$ is proved to match (28), completing the proof of Lemma 1.

Appendix D: Proof of Lemma 2

Using (13) and (18) for the case of ${{C_c} \ge {\bar{C}}}$ to rewrite $\Omega _1$ in (55) as

$$\begin{aligned} {\Omega _1} = \Pr \left\{ {1 + \frac{{\phi \left( {1 - \tau } \right) {P_c}{g_{cd}}}}{{\phi \tau {P_c}{g_{cd}} + {\epsilon _d}}} < K\left( {1 + \frac{{\phi \left( {1 - \tau } \right) {P_c}{g_{cw}}}}{{\left( {\phi \tau + 1 - \phi } \right) {P_c}{g_{cw}} + {\epsilon _w}}}} \right) } \right\} . \end{aligned}$$

(104)

It is seen from (87) and (104) that $\Delta$ and $\Omega _1$ have the same form. Consequently, the precise closed-form expression of $\Omega _1$ can be taken from that of $\Delta$ with appropriate variable substitutions as shown in Lemma 2, completing the proof.

Appendix E: Proof of Lemma 3

Let ${Z_{cd}} = 1 + \bar{K}{g_{cd}}$ and ${{\bar{Z}}_{cw}} = 1 + \frac{{\bar{H}{g_{cw}}}}{{\bar{M}{g_{cw}} + {\epsilon _w}}}$ where $\bar{K}$, $\bar{H}$, and $\bar{M}$ are presented in (61), (59), and (58), respectively. Given the variable change, the pdf of $Z_{cd}$ is taken from the pdf of $g_{cd}$, ${f_{{g_{cd}}}}\left( z \right)$, as

$$\begin{aligned} {f_{{g_{cd}}}}\left( z \right) = \frac{1}{{\bar{K}}}{f_{{g_{cd}}}}\left( {\frac{{z - 1}}{{\bar{K}}}} \right) ,\mathrm{{ }}z \ge 1. \end{aligned}$$

(105)

Moreover, the pdf of ${\bar{Z}}_{cw}$ is given by

$$\begin{aligned} {f_{{{\bar{Z}}_{cw}}}}\left( t \right) = \frac{{\bar{Q}_{cw}^{{\alpha _{cw}}}{e^{{{\bar{Q}}_{cw}}}}\bar{H}}}{{\bar{M}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{\frac{{{{\bar{W}}_{cw}}}}{{t - \bar{U}}}}}\frac{{{{\left( {t - 1} \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {\bar{U} - t} \right) }^{{\alpha _{cw}} + 1}}}},\ \ \ 1 \le t < \bar{U}, \end{aligned}$$

(106)

where $\bar{U}$, ${{{\bar{Q}}_{cw}}}$ and ${{{\bar{W}}_{cw}}}$ are presented in (60), (63) and (64), respectively. Equation (106) is derived directly from the results in Appendix A.

Using (13) and (18) for the case of ${{C_c} < {\bar{C}}}$ to rewrite $\Omega _2$ in (55) as

$$\begin{aligned} {\Omega _2} = \Pr \left\{ {1 + \frac{{\rho {P_c}{g_{cd}}}}{{{\epsilon _d}}} < K\left( {1 + \frac{{\rho {P_c}{g_{cw}}}}{{\left( {1 - \rho } \right) {P_c}{g_{cw}} + {\epsilon _w}}}} \right) } \right\} , \end{aligned}$$

(107)

which is also expressed in terms of $Z_{cd}$ and ${\bar{Z}}_{cw}$ as

$$\begin{aligned} {\Omega _2} = \Pr \left\{ {{Z_{cd}} < K{{\bar{Z}}_{cw}}} \right\} . \end{aligned}$$

(108)

Since $Z_{cd}$ and ${\bar{Z}}_{cw}$ are statistically independent, (108) can be expressed in terms of the individual pdfs of $Z_{cd}$ and ${\bar{Z}}_{cw}$ as

$$\begin{aligned} {\Omega _2} = \iint \limits _{z < Kt} {{f_{{Z_{cd}}}}\left( z \right) {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dzdt}. \end{aligned}$$

(109)

Since ${{f_{{Z_{cd}}}}\left( z \right) }$ is non-zero for $z \ge 1$, the double integral in (109) can be solved by considering two cases of Kt in correlation to 1: (Case 1: $Kt<1$) and (Case 2: $Kt>1$). Therefore, (109) is rewritten in a compact form as

$$\begin{aligned} {\Omega _2} = \int \limits _{Kt< 1} {\left[ {\int \limits _{z < Kt} {{f_{{Z_{cd}}}}\left( z \right) dz} } \right] {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} + \int \limits _{Kt > 1} {\left[ {\int \limits _{z = 1}^{Kt} {{f_{{Z_{cd}}}}\left( z \right) dz} } \right] {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(110)

The first term in (110) is zero since $\int \limits _{z < Kt} {{f_{{Z_{cd}}}}\left( z \right) dz} = 0$ owing to ${{f_{{Z_{cd}}}}\left( z \right) }=0$ for $z=Kt < 1$. As such, by substituting (105) into the second term in (110) and after solving the inner integral in (110), one obtains

$$\begin{aligned} {\Omega _2}&= \int \limits _1^{\bar{U}} {\left[ {\int \limits _{z = 1}^{Kt} {\frac{1}{{\bar{K}}}{f_{{g_{cd}}}}\left( {\frac{{z - 1}}{{\bar{K}}}} \right) dz} } \right] {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} \nonumber \\&= ^{x = \frac{{z - 1}}{{\bar{K}}}} \int \limits _1^{\bar{U}} {\left[ {\int \limits _0^{\frac{{Kt - 1}}{{\bar{K}}}} {{f_{{g_{cd}}}}\left( x \right) dx} } \right] {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} \nonumber \\&= \int \limits _1^{\bar{U}} {{F_{{g_{cd}}}}\left( {\frac{{Kt - 1}}{{\bar{K}}}} \right) {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} \nonumber \\&= \int \limits _1^{\bar{U}} {\frac{{\gamma \left( {{\alpha _{cd}},\left( {Kt - 1} \right) {\xi _{cd}}/\bar{K}} \right) }}{{\Gamma \left( {{\alpha _{cd}}} \right) }}{f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt}. \end{aligned}$$

(111)

Executing the series expansion of $\gamma \left( \cdot ,\cdot \right)$ in [31, eq. (8.352.1)] and then applying the binomial expansion, (111) reduces to

$$\begin{aligned} {\Omega _2}&= \int \limits _1^{\bar{U}} {\left[ {1 - {e^{ - \frac{{{\xi _{cd}}\left( {Kt - 1} \right) }}{{\bar{K}}}}}\sum \limits _{m = 0}^{{\alpha _{cd}} - 1} {\frac{1}{{m!}}{{\left\{ {\frac{{{\xi _{cd}}\left( {Kt - 1} \right) }}{{\bar{K}}}} \right\} }^m}} } \right] {f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} \nonumber \\&= 1 - \sum \limits _{m = 0}^{{\alpha _{cd}} - 1} {\sum \limits _{l = 0}^m {\left( {\begin{array}{*{20}{c}} m \\ l \end{array}} \right) } \frac{{{{\left( { - 1} \right) }^{m - l}}}}{{m!{K^{ - l}}}}{e^{\frac{{{\xi _{cd}}}}{{\bar{K}}}}}{{\left( {\frac{{{\xi _{cd}}}}{{\bar{K}}}} \right) }^m}} \underbrace{\int \limits _1^{\bar{U}} {{t^l}{e^{ - \bar{J}t}}{f_{{{\bar{Z}}_{cw}}}}\left( t \right) dt} }_\zeta \end{aligned}$$

(112)

where $\bar{J}$ is defined in (62).

Inserting ${{f_{{{\bar{Z}}_{cw}}}}\left( t \right) }$ in (106) into (112), one obtains

$$\begin{aligned} \zeta&= \int \limits _1^{\bar{U}} {{t^l}{e^{ - \bar{J}t}}\frac{{\bar{Q}_{cw}^{{\alpha _{cw}}}{e^{{{\bar{Q}}_{cw}}}}\bar{H}}}{{\bar{M}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{\frac{{{{\bar{W}}_{cw}}}}{{t - \bar{U}}}}}\frac{{{{\left( {t - 1} \right) }^{{\alpha _{cw}} - 1}}}}{{{{\left( {\bar{U} - t} \right) }^{{\alpha _{cw}} + 1}}}}dt} \nonumber \\&= ^{z = \frac{1}{{t - \bar{U}}}} - \frac{{\bar{Q}_{cw}^{{\alpha _{cw}}}\bar{H}}}{{\bar{M}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{{\bar{Q}}_{cw}} - \bar{U}\bar{J}}}\int \limits _{\frac{1}{{1 - \bar{U}}}}^{ - \infty } {{{\left( {\frac{1}{z} + \bar{U}} \right) }^l}{e^{{{\bar{W}}_{cw}}z - \frac{{\bar{J}}}{z}}}{{\left( {\left[ {1 - \bar{U}} \right] z - 1} \right) }^{{\alpha _{cw}} - 1}}dz} \nonumber \\&= ^{t = - z} \frac{{\bar{Q}_{cw}^{{\alpha _{cw}}}\bar{H}}}{{\bar{M}\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{{\bar{Q}}_{cw}} - \bar{U}\bar{J}}}\int \limits _{\frac{1}{{\bar{U} - 1}}}^\infty {{{\left( {\bar{U} - \frac{1}{t}} \right) }^l}{e^{\frac{{\bar{J}}}{t} - {{\bar{W}}_{cw}}t}}{{\left( {\left[ {\bar{U} - 1} \right] t - 1} \right) }^{{\alpha _{cw}} - 1}}dt}. \end{aligned}$$

(113)

Executing the binomial expansion to reduce (113) to

$$\begin{aligned} \zeta = \frac{{\bar{W}_{cw}^{{\alpha _{cw}}}}}{{\Gamma \left( {{\alpha _{cw}}} \right) }}{e^{{{\bar{Q}}_{cw}} - \bar{U}\bar{J}}}\sum \limits _{j = 0}^l {\sum \limits _{g = 0}^{{\alpha _{cw}} - 1} {\left( {\begin{array}{*{20}{c}} {{\alpha _{cw}} - 1} \\ g \end{array}} \right) \left( {\begin{array}{*{20}{c}} l \\ j \end{array}} \right) {{\left( {\frac{{\bar{M}}}{{\bar{H}}}} \right) }^g}} {{\bar{U}}^j}{{\left( { - 1} \right) }^{l - g - j}}} \underbrace{\int \limits _{\bar{M}/\bar{H}}^\infty {{z^{\bar{m}}}{e^{\frac{{\bar{J}}}{z} - {{\bar{W}}_{cw}}z}}dz} }_{\hat{\zeta }}, \end{aligned}$$

(114)

where $\bar{m}$ is defined in (65).

Applying the series expansion to ${e^{\bar{J}/x}}$, the last integral in (114) reduces to

$$\begin{aligned} \hat{\zeta } = \sum \limits _{u = 0}^\infty {\frac{{{{\bar{J}}^u}}}{{u!}}} \underbrace{\int \limits _{\bar{M}/\bar{H}}^\infty {\frac{{{z^{\bar{m}}}}}{{{z^u}}}{e^{ - {{\bar{W}}_{cw}}z}}dz} }_\beta . \end{aligned}$$

(115)

Dependent on the difference between $\bar{m}$ and u, $\beta$ in (115) has different values

$$\begin{aligned} \beta = \left\{ {\begin{array}{*{20}{c}} {\int \limits _{\bar{M}/\bar{H}}^\infty {{z^{\bar{m} - u}}{e^{ - {{\bar{W}}_{cw}}z}}dz} }&{}{,\bar{m} > u}\\ {\int \limits _{\bar{M}/\bar{H}}^\infty {{e^{ - {{\bar{W}}_{cw}}z}}dz} }&{}{,\bar{m} = u}\\ {\int \limits _{\bar{M}/\bar{H}}^\infty {\frac{{{e^{ - {{\bar{W}}_{cw}}z}}}}{{{z^{u - \bar{m}}}}}dz} }&{}{,\bar{m} < u} \end{array}} \right. \end{aligned}$$

(116)

Three integrals in (116) are solved as follows. The second integral is easy to solve while the first and the third integrals are represented in terms of $\mathcal {G}\left( \cdot ,\cdot ,\cdot \right)$ and $\mathcal {Y}\left( \cdot ,\cdot ,\cdot \right)$ in (46) and (47), correspondingly. Given solutions of these three integrals, one reduces (116) to (66). Moreover, inserting (116) into (115), then plugging (115) into (114) and finally substituting (114) into (112), one reduces (112) to (57), finishing the proof of Lemma 3.