Performance analysis and optimization of a hybrid TSR–PSR protocol for AF, DF and hybrid AF–DF relaying under Weibull fading

Abstract

In this paper, a generalized Interference-aided radio frequency energy harvesting (EH) cooperative network has been presented under Weibull faded environment. The performance analysis of the system in terms of Throughput and Outage Probability for various cooperation protocols, Decode-and-Forward, Amplify-and-Forward and Hybrid-Decode-Amplify-Forward has been shown. The throughput of the network has been optimized with respect to the EH fractions of the hybrid TSR–PSR protocol. Integral-form expressions for Outage probability and Throughput are derived for the given protocols. In addition, the throughput of the proposed network has been evaluated under different channel conditions which show the superiority of different EH fractions in different relays. Analytical expressions have been well substantiated through extensive simulation results. The influence of system parameters such as time switching and power splitting fractions, number of relay nodes, channel parameters on outage probability and throughput performance of the proposed network has been shown. Finally, the throughput of the proposed model has been compared with various benchmark techniques.

Appendices

Appendix: A

1.1 A.1 Outage probability for DF protocol

For deriving the outage probability for DF protocol, the SINR of DF is put into the expression of outage probability.

By using Eqs. (7)–(11), the resultant expression of outage probability for ${j}{\text{th}}$ relay is obtained which is shown in Eq. (48).

$$ \begin{aligned} P_{outj} & = 1 - \Pr \left( {P_{j} \left| {h_{j} } \right|^{2} > \gamma_{th} ,Q_{j} \left( {\left| {h_{j} } \right|^{2} + U_{j} } \right)\left| {g_{j} } \right|^{2} > \gamma_{th} } \right) \\ & = 1 - \Pr \left( {\left| {h_{j} } \right|^{2} > \frac{{\gamma_{th} }}{{P_{j} }} ,\left| {g_{j} } \right|^{2} > \frac{{\gamma_{th} }}{{Q_{j} \left( {\left| {h_{j} } \right|^{2} + U_{j} } \right)}}} \right) \\ & = 1 - \mathop \int \limits_{{\frac{{\gamma_{th} }}{{P_{j} }}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( z \right)\left[ {{\text{Pr}}(\left| {g_{j} } \right|^{2} > \frac{{\gamma_{th} }}{{Q_{j} \left( {z_{j} + U_{j} } \right)}})} \right]dz \\ & = 1 - \mathop \int \limits_{{\frac{{\gamma_{th} }}{{P_{j} }}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( z \right)\left[ {1 - {\text{Pr}}\left( {\left| {g_{j} } \right|^{2} \le \frac{{\gamma_{th} }}{{Q_{j} \left( {z_{j} + U_{j} } \right)}}} \right)} \right]dz \\ \end{aligned} $$

(48)

Putting the p.d.f. and c.d.f. of the Weibull distribution Eq. (49) is obtained.

$$ P_{outj} = 1 - \frac{{K_{hj} }}{{\lambda_{hj}^{{k_{hj} }} }}\mathop \int \limits_{{\frac{{\gamma_{th} }}{{P_{j} }}}}^{\infty } z_{j}^{{k_{hj} - 1}} e^{{ - \left\{ {\left( {\frac{{z_{j} }}{{\lambda_{hj} }}} \right)^{{k_{hj} }} + \left( {\frac{{\gamma_{th} }}{{Q_{j} \lambda_{gj} \left( {z_{j} + U_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} dz_{j} $$

(49)

Substitution of $p_{j} = \left( {z_{j} /\lambda _{{hj}} } \right)^{{k_{{hj}} }}$ into Eq. (50) can be derived.

$${P}_{outj}=1-{\int }_{{\left(\frac{{\gamma }_{th}}{{P}_{j}{\lambda }_{hj}}\right)}^{{k}_{hj}}}^{\infty}{ e}^{-\left\{{p}_{j}+{\left(\frac{{\gamma }_{th}}{{Q}_{j}{\lambda }_{gj}{\lambda }_{hj}}.\frac{1}{\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}+\frac{{U}_{j}}{{\lambda }_{hj}}\right)}\right)}^{{k}_{gj}}\right\}}d{p}_{j}$$

(50)

Now, using the Eqs. (6), (8), (11) and (33)–(35), the Eqs. (51)–(53) are formulated.

$$\frac{{\gamma }_{th}}{{P}_{j}{\lambda }_{hj}}=\frac{{\gamma }_{th}\left((1-{\rho }_{j}){I}_{Rj}+{\sigma }_{Rj}^{2}\right)}{{{L}_{1j}}^{2}{ P}_{s}.(1-{\rho }_{j}){\lambda }_{hj}}={M}_{j}$$

(51)

$$\frac{{\gamma }_{th}}{{Q}_{j}{\lambda }_{gj}{\lambda }_{hj}}=\frac{{\gamma }_{th}\left({I}_{D}+{\sigma }_{D}^{2}\right)(1-{\alpha }_{j}-{\beta }_{j})}{{\lambda }_{hj}{\lambda }_{gj}({{\eta }_{Tj}\alpha }_{j}+{\eta }_{Pj}{\rho }_{j}{\beta }_{j}){{L}_{1j}}^{2}{{L}_{2j}}^{2}{ P}_{s}}=N_{j}$$

(52)

$$\frac{{U}_{j}}{{\lambda }_{hj}}=\frac{{I}_{Rj}}{{L}_{1j}^{2}{P}_{s }{\lambda }_{hj}}={X}_{j}$$

(53)

Equations (50)–(53) lead to Eq. (54).

$${P}_{outj}=1-{\int }_{{{M}_{j}}^{{k}_{hj}}}^{\infty}{e}^{-\left\{{p}_{j}+{\left(\frac{{N}_{j}}{\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}+ {X}_{j}\right)}\right)}^{{k}_{gj}}\right\}}d{p}_{j}$$

(54)

Equation (54) shows the outage probability of ${j}{\text{th}}$ relay in DF protocol.

1.2 A.2: Outage probability for AF protocol

For deriving the outage probability for DF protocol, the SINR of DF is put into the expression of outage probability. The alternative expression of outage probability for ${j}{\text{th}}$ relay is shown in Eq. (55).

$${P}_{outj}=\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th}\right)+\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj}\le {\gamma }_{th}\right)$$

(55)

Putting the Eq. (7) in $\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{thj}\right)$ Eq. (56) is obtained.

$$ \begin{aligned} \Pr \left( {\gamma_{SRj} \le \gamma_{th} } \right) & = \Pr \left( {\frac{{\left( {1 - \rho_{j} } \right)P_{s} }}{{L_{1j}^{2} \left( {\sigma_{jR}^{2} + \left( {1 - \rho_{j} } \right)I_{jR} } \right)}}|h_{j} |^{2} \le \gamma_{th} } \right) \\ & = {\text{Pr}}\left( {|h_{j} |^{2} \le \gamma_{th} .\frac{{L_{1j}^{2} \left[ {\left( {1 - \rho_{j} } \right)\sigma_{jR}^{2} + I_{jR} } \right]}}{{\left( {1 - \rho_{j} } \right)P_{s} }} } \right) \\ \end{aligned} $$

(56)

The constants in Eqs. (19)–(22) are modified into Eqs. (57)–(60).

$$c=c{^{\prime}}{\gamma }_{th}$$

(57)

$$d=d{^{\prime}}{\gamma }_{th}$$

(58)

$$e=e{^{\prime}}{\gamma }_{th}$$

(59)

$$f=f{^{\prime}}{\gamma }_{th}$$

(60)

Therefore, Eq. (61) is obtained.

$$ \Pr \left( {\gamma _{{SRj}} \le \gamma _{{th}} } \right) = \Pr \left( {|h_{j} |^{2} \le \frac{d}{a}} \right) = \mathop {\mathop \int \limits^{{\frac{d}{a}}} }\limits_{{z = 0}} f_{{\left| {h_{j} } \right|^{2} }} \left( z \right)dz $$

(61)

The Eq. (62) shows the expression for $\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th},{ \gamma }_{RDj}\le {\gamma }_{th}\right)$.

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj} \le \gamma_{th} } \right) \\ & = \Pr \left( \frac{{\left({1 - \rho_{j} } \right)P_{s} }}{{L_{1j}^{2} \left[ {\sigma_{jR}^{2} + \left( {1 - \rho_{j} } \right)I_{jR} } \right]}}|h_{j} |^{2}\right.\\ &> \left.\gamma_{th} , \frac{{a\left| {g_{j} } \right|^{2} |h_{j} |^{4} + b\left| {g_{j} } \right|^{2} |h_{j} |^{2} }}{{c^{\prime}|h_{j} |^{2} + d^{^{\prime}} \left| {g_{j} } \right|^{2} |h_{j} |^{2} + e^{\prime}\left| {g_{j} } \right|^{2} + f^{\prime}}} \le \gamma_{th} \right) \\ & = \Pr \left(|h_{j} |^{2} > \gamma_{th} .\frac{{L_{1j}^{2} \left[ {\sigma_{jR}^{2} +\left( {1 - \rho_{j} } \right) I_{jR} } \right]}}{{\left( {1 - \rho_{j} } \right)P_{s} }},\left| {g_{j} } \right|^{2}\right.\\ &\quad \left.\left\{ {a|h_{j} |^{4} + b\left| {h_{j} } \right|^{2} - \gamma_{th} d^{\prime}|h_{j} |^{2} - \gamma_{th} e^{\prime}} \right\} \le \gamma_{th} \left( {c^{\prime} |h_{j} |^{2} + f^{\prime}} \right)\right) \\ & = \Pr \left(|h_{j} |^{2} > \frac{d}{a} ,\left| {g_{j} } \right|^{2} \left\{ {a|h_{j} |^{4} + b\left| {h_{j} } \right|^{2} - d|h_{j} |^{2} - e} \right\}\right.\\ &\le \left.\left( {c |h_{j} |^{2} + f} \right) \right) \\ \end{aligned} $$

(62)

From Eqs. (17), (18), (20), (21), (58), (59), $\frac{b}{a}=\frac{e{^{\prime}}}{d{^{\prime}}}=\frac{e}{d}$. Thus,

$$\left\{a{{|h}_{j}|}^{4}+b{\left|{h}_{j}\right|}^{2}-d{{|h}_{j}|}^{2}-e\right\}=a\left({{|h}_{j}|}^{2}-\frac{d}{a}\right)\left({{|h}_{j}|}^{2}+\frac{b}{a}\right)$$

(63)

If,${{|h}_{j}|}^{2}>\frac{d}{a}$, then the right-hand side of the Eq. (63) will be positive. The Eq. (62) leads to Eq. (64).

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj} \le \gamma_{th} } \right) \\ & = \Pr \left( {|h_{j} |^{2} > \frac{d}{a} ,\left| {g_{j} } \right|^{2} \le \frac{{\left( {c |h_{j} |^{2} + f} \right)}}{{a\left( {|h_{j} |^{2} - \frac{d}{a}} \right)\left( {|h_{j} |^{2} + \frac{b}{a}} \right)}}} \right) \\ & = \mathop \int \limits_{{z_{j} = \frac{d}{a}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( {z_{j} } \right)Pr\left( {\left| {g_{j} } \right|^{2} \le \frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}} \right)dz_{j} \\ \end{aligned} $$

(64)

By putting the p.d.f. and c.d.f. of Weibull distribution and using $\Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj} \le \gamma_{th} } \right)$ leads to:

$$ \begin{aligned} P_{outj} & = \mathop \int \limits_{{z_{j} = 0}}^{\frac{d}{a}} f_{{\left| {h_{j} } \right|^{2} }} \left( {z_{j} } \right)dz_{j} + \mathop \int \limits_{{z = \frac{d}{a}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( {z_{j} } \right)\left[ {1 - e^{{ - \left( {\frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}.\frac{1}{{\lambda_{gj} }}} \right)^{{k_{gj} }} }} } \right]dz_{j} \\ P_{outj} & = 1 - \mathop \int \limits_{{z_{j} = \frac{d}{a}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( {z_{j} } \right)e^{{ - \left( {\frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}.\frac{1}{{\lambda_{gj} }}} \right)^{{k_{gj} }} }} dz_{j} \\ P_{outj} & = 1 - \mathop \int \limits_{{z_{j} = \frac{d}{a}}}^{\infty } \frac{{k_{hj} }}{{\lambda_{hj} }}\left( {\frac{{z_{j} }}{{\lambda_{hj} }}} \right)^{{k_{hj} - 1}} e^{{ - \left( {{\raise0.7ex\hbox{${z_{j} }$} \!\mathord{\left/ {\vphantom {{z_{j} } {\lambda_{hj} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\lambda_{hj} }$}}} \right)^{{k_{hj} }} }} e^{{ - \left( {\frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}.\frac{1}{{\lambda_{gj} }}} \right)^{{k_{gj} }} }} dz_{j} \\ & = 1 - \frac{{k_{hj} }}{{\lambda_{hj}^{{k_{hj} }} }}\mathop \int \limits_{{z = \frac{d}{a}}}^{\infty } z_{j}^{{k_{hj} - 1}} e^{{ - \left\{ {\left( {{\raise0.7ex\hbox{${z_{j} }$} \!\mathord{\left/ {\vphantom {{z_{j} } {\lambda_{hj} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\lambda_{hj} }$}}} \right)^{{k_{hj} }} + \left( {\frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}.\frac{1}{{\lambda_{gj} }}} \right)^{{k_{gj} }} } \right\}}} dz_{j} \\ \end{aligned} $$

(65)

Now using $c{z}_{j}+f=c\left({z}_{j}+\frac{f}{c}\right)=c({z}_{j}+\frac{d}{a{\gamma }_{th}})$ and $p_{j} = \left( {z_{j} /\lambda _{{hj}} } \right)^{{k_{{hj}} }}$ Eq. (66) can be obtained.

$${P}_{outj}=1-{\int }_{{\left(\frac{d}{a{\lambda }_{hj}}\right)}^{{k}_{hj}}}^{\infty}{e}^{-\left\{{p}_{j}+{\left(\frac{({{p}_{j}}^{\frac{1}{{k}_{hj}}}+\frac{d}{a{\lambda }_{hj}{\gamma }_{th}})}{\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}- \frac{d}{a{\lambda }_{hj}}\right)\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}+ \frac{b}{a{\lambda }_{hj}}\right)}.\frac{c}{{a\lambda }_{gj}{\lambda }_{hj}}\right)}^{{k}_{gj}}\right\}}d{p}_{j}$$

(66)

Now, using the Eqs. (17)–(20), (33)–(35) and (57)–(58) the Eqs. (67)–(69) are formulated.

$$\frac{d}{a{\lambda }_{hj}}=\frac{{{L}_{1j}}^{2}{\gamma }_{th}\left((1-{\rho }_{j}){I}_{Rj}+{\sigma }_{Rj}^{2}\right)}{{ P}_{s}.(1-{\rho }_{j}){\lambda }_{hj}}={M}_{j}$$

(67)

$$\frac{c}{{a\lambda }_{gj}{\lambda }_{hj}}=\frac{{{L}_{1j}}^{2}{{L}_{2j}}^{2}{\gamma }_{th}\left({I}_{D}+{\sigma }_{D}^{2}\right)(1-{\alpha }_{j}-{\beta }_{j})}{{\lambda }_{hj}{\lambda }_{gj}({{\eta }_{Tj}\alpha }_{j}+{\eta }_{Pj}{\rho }_{j}{\beta }_{j}){ P}_{s}}={N}_{j}$$

(68)

$$\frac{b}{a{\lambda }_{hj}}=\frac{{{L}_{1j}}^{2}{I}_{Rj}}{{P}_{s }{\lambda }_{hj}}={X}_{j}$$

(69)

Equations (67)–(69) lead to Eq. (70).

$${P}_{outj}=1-{\int }_{{{M}_{j}}^{{k}_{hj}}}^{\infty}{e}^{-\left\{{p}_{j}+{\left(\frac{({{p}_{j}}^{\frac{1}{{k}_{hj}}}+\frac{{M}_{j}}{{\gamma }_{th}})}{\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}-{M}_{j}\right)\left({{p}_{j}}^{\frac{1}{{k}_{hj}}}+ {X}_{j}\right)}.{N}_{j}\right)}^{{k}_{gj}}\right\}}d{p}_{j}$$

(70)

Equation (70) shows the outage probability of ${j}{\text{th}}$ relay in AF protocol.

Equations (54) and (70) are in very similar forms with the same constants. Therefore, the AF and DF in the proposed model can be analysed similarly.

Appendix B

2.1 B.1: Outage probability for hybrid AF–DF protocol

Outage probability for ${j}{\text{th}}$ relay for Hybrid AF–DF is given in Eq. (71).

$$ \begin{aligned} P_{outj} & = \underbrace {{\Pr \left( {\gamma_{SRj} \le \gamma_{th} ,\gamma_{SRj} \ge Th_{j} } \right) + \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj - DF} \le \gamma_{th} ,\gamma_{SRj} \ge Th_{j} } \right) + }}_{{{\text{DF}}\,{\text{component}}}} \\ & \underbrace {{\Pr \left( {\gamma_{SRj} \le \gamma_{th} ,\gamma_{SRj} < Th_{j} } \right) + \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj - AF} \le \gamma_{th} ,\gamma_{SRj} < Th_{j} } \right)}}_{{{\text{AF}}\,{\text{component}}}} \\ \end{aligned} $$

(71)

Let us define

$${\gamma }_{Hyj}=\frac{{Th}_{j}}{{\gamma }_{th}}.$$

(72)

• I. If ${\gamma }_{th}\ge {Th}_{j}$, i.e., ${\gamma }_{Hyj}\le 1$, we get Eqs. (73)–(76).

$$\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th},{\gamma }_{SRj}\ge {Th}_{j}\right)=\mathrm{Pr}\left({Th}_{j}\le {\gamma }_{SRj}\le {\gamma }_{thj}\right).$$

(73)

$$\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj-DF}\le {\gamma }_{th},{\gamma }_{SRj}\ge {Th}_{j}\right)=\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj-DF}\le {\gamma }_{thj}\right).$$

(74)

$$\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th},{\gamma }_{SRj}<{Th}_{j}\right)=\mathrm{Pr}\left({\gamma }_{SRj}<{Th}_{j}\right).$$

(75)

$$\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj-AF}\le {\gamma }_{th},{\gamma }_{SRj}<{Th}_{j}\right)=0.$$

(76)

Putting the values of Eqs. (73)–(76) into Eq. (71), outage probability can be written as Eq. (77).

$$ \begin{aligned} P_{outj} & = \Pr \left( {Th_{j} \le \gamma_{SRj} \le \gamma_{th} } \right) + \Pr \left( {\gamma_{SRj} < Th_{j} } \right)\\ &\quad + \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = \Pr \left( {\gamma_{SRj} \le \gamma_{th} } \right) + \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = P_{outj - DF} \\ \end{aligned} $$

(77)

So, if ${\gamma }_{th}\ge {Th}_{j}$ (${\gamma }_{Hyj}\le 1$), then the outage probability of the Hybrid AF–DF protocol is equal to the outage probability of the DF protocol.

• II. If ${\gamma }_{th}<{Th}_{j}$, i.e., ${\gamma }_{Hyj}>1$,

$$\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th},{\gamma }_{SRj}\ge {Th}_{j}\right)=0.$$

(78)

$$\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj-DF}\le {\gamma }_{th},{\gamma }_{SRj}\ge {Th}_{j}\right)=\mathrm{Pr}\left({\gamma }_{SRj}\ge {Th}_{j} ,{ \gamma }_{RDj-DF}\le {\gamma }_{th}\right).$$

(79)

$$\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th},{\gamma }_{SRj}<{Th}_{j}\right)=\mathrm{Pr}\left({\gamma }_{SRj}\le {\gamma }_{th}\right).$$

(80)

$$\mathrm{Pr}\left({\gamma }_{SRj}>{\gamma }_{th} ,{ \gamma }_{RDj-AF}\le {\gamma }_{th},{\gamma }_{SRj}<{Th}_{j}\right)=\mathrm{Pr}\left({\gamma }_{th }<{\gamma }_{SRj}<{Th}_{j},{ \gamma }_{RDj-AF}\le {\gamma }_{th}\right).$$

(81)

Putting the values of Eqs. (78)–(81) into Eq. (71), outage probability can be written as Eq. (82).

$$ \begin{aligned} P_{outj} & = \Pr \left( {\gamma_{SRj} \le \gamma_{th} } \right) + \Pr \left( {\gamma_{SRj} \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & \quad + \Pr \left( {\gamma_{th } < \gamma_{SRj} < Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right) \\ & = \Pr \left(\gamma_{SRj} \le \gamma_{th} \right) + \Pr \left( {\gamma_{SRj} > \gamma_{th} , \gamma_{RDj - AF} \le \gamma_{th} } \right) \\ & \quad + [\Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right)\\ &\quad - \Pr \left( {\gamma_{SRj} \ge Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right)] \\ & = P_{outj - AF} - [\Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right)\\ &\quad - \Pr \left( {\gamma_{SRj} \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right)] \\ \end{aligned} $$

(82)

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right) \\ & = \Pr \left( {\gamma_{SRj } \ge \gamma_{th} .\frac{{Th_{j} }}{{\gamma_{th} }}, \gamma_{RDj - AF} \le \gamma_{th} } \right) \\ & = \Pr \left(|h_{j} |^{2} \ge \frac{d}{a}.\frac{{Th_{j} }}{{\gamma_{th} }} ,\left| {g_{j} } \right|^{2} \left\{ {a|h_{j} |^{4} + b\left| {h_{j} } \right|^{2} - d|h_{j} |^{2} - e} \right\}\right.\\ &\le \left.\left( {c |h_{j} |^{2} + f} \right)\right) \\ & = \Pr \left(|h_{j} |^{2} \ge \frac{d}{a}. \gamma_{Hyj} ,\left| {g_{j} } \right|^{2} \left\{ {a|h_{j} |^{4} + b\left| {h_{j} } \right|^{2} - d|h_{j} |^{2} - e} \right\}\right.\\ &\le \left.\left( {c |h_{j} |^{2} + f} \right)\right) \\ \end{aligned} $$

(83)

Equation (83) is similar to Eq. (62).

Now, as ${\gamma }_{Hyj}>1$, therefore ${{|h}_{j}|}^{2}\ge \frac{d}{a}. {\gamma }_{Hyj}>\frac{d}{a}$.

$$\mathrm{Pr}\left({\gamma }_{SRj }\ge {Th}_{j},{ \gamma }_{RDj-AF}\le {\gamma }_{th}\right)={\int }_{{z}_{j}={\gamma }_{Hyj}.\frac{d}{a}}^{\infty}{f}_{{\left|{h}_{j}\right|}^{2}}\left({z}_{j}\right)Pr\left({\left|{g}_{j}\right|}^{2}\le \frac{c{z}_{j}+f}{a\left({z}_{j}-\frac{d}{a}\right)\left({z}_{j}+\frac{b}{a}\right)}\right)d{z}_{j}$$

(84)

Equation (84) is similar to Eq. (67).

Putting the parameters of Weibull distribution in Eqs. (84), (85) is obtained.

$$ \begin{aligned} & {\text{Pr}}\left( {\gamma _{{SRj}} \ge Th_{j} ,\gamma _{{RDj - AF}} \le \gamma _{{th}} } \right)\\ &\quad = \mathop {\mathop \smallint \limits^{\infty } }\limits_{{z_{j} = \gamma _{{Hyj}} .\frac{d}{a}}} \frac{{k_{{hj}} }}{{\lambda _{{hj}} }}\left( {\frac{{z_{j} }}{{\lambda _{{hj}} }}} \right)^{{k_{{hj}} - 1}} \\ &\qquad e^{{ - \left( {z_{j} /\lambda _{{hj}} } \right)^{{k_{{hj}} }} }} \left[ {1 - e^{{ - \left( {\frac{{cz_{j} + f}}{{a\left( {z_{j} - \frac{d}{a}} \right)\left( {z_{j} + \frac{b}{a}} \right)}}.\frac{1}{{\lambda _{{gj}} }}} \right)^{{k_{{gj}} }} }} } \right]dz_{j} \\ \end{aligned} $$

(85)

Substituting $p_{j} = \left( {{\raise0.7ex\hbox{${{\text{z}}_{{\text{j}}} }$} \!\mathord{\left/ {\vphantom {{{\text{z}}_{{\text{j}}} } {{{\lambda }}_{{{\text{hj}}}} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{{\lambda }}_{{{\text{hj}}}} }$}}} \right)^{{{\text{k}}_{{{\text{hj}}}} }}$ in Eqs. (85), 86) is obtained.

$$\mathrm{Pr}\left({\gamma }_{SRj }\ge {Th}_{j},{ \gamma }_{RDj-AF}\le {\gamma }_{th}\right)$$

$$ \begin{aligned} & = \mathop \int \limits_{{\left( {\frac{d}{{a\lambda_{hj} }}.\gamma_{Hyj} } \right)^{{k_{hj} }} }}^{\infty } e^{{ - p_{j} }} dp_{j} - \mathop \int \limits_{{\left( {\frac{d}{{a\lambda_{hj} }}.\gamma_{Hyj} } \right)^{{k_{hj} }} }}^{\infty }\\ &\quad e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{d}{{a\lambda_{hj} \gamma_{th} }}} \right)}}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} - \frac{d}{{a\lambda_{hj} }}} \right)\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{b}{{a\lambda_{hj} }}} \right)}}.\frac{c}{{a\lambda_{gj} \lambda_{hj} }}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ & = e^{{ - \left( {\frac{d}{{a\lambda_{hj} }}.\gamma_{Hyj} } \right)^{{k_{hj} }} }} - \mathop \int \limits_{{\left( {\frac{d}{{a\lambda_{hj} }}.\gamma_{Hyj} } \right)^{{k_{hj} }} }}^{\infty } \\ &\quad e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{d}{{a\lambda_{hj} \gamma_{th} }}} \right)}}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} - \frac{d}{{a\lambda_{hj} }}} \right)\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{b}{{a\lambda_{hj} }}} \right)}}.\frac{c}{{a\lambda_{gj} \lambda_{hj} }}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ \end{aligned} $$

(86)

Using (67)–(69), Eq. (87) is obtained.

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right) \\ &\quad = e^{{ - \left( {\gamma_{Hyj} M_{j} } \right)^{{k_{hj} }} }} - \mathop \int \limits_{{(\gamma_{Hyj} M_{j} )^{{k_{hj} }} }}^{\infty }\\ &\qquad e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{{M_{j} }}{{\gamma_{th} }}} \right)}}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} - M_{j} } \right)\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + X_{j} } \right)}}.N_{j} } \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ \end{aligned} $$

(87)

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = \Pr \left( {\gamma_{SRj } \ge \gamma_{th} .\frac{{Th_{j} }}{{\gamma_{th} }}, \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = \Pr \left( {\left| {h_{j} } \right|^{2} > \frac{{\gamma_{th} }}{{P_{j} }}.\frac{{Th_{j} }}{{\gamma_{th} }} ,\left| {g_{j} } \right|^{2} \le \frac{{\gamma_{th} }}{{Q_{j} \left( {\left| {h_{j} } \right|^{2} + U_{j} } \right)}}} \right) \\ & = \mathop \int \limits_{{\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} }}}}^{\infty } f_{{\left| {h_{j} } \right|^{2} }} \left( z \right){\text{Pr}}\left( {\left| {g_{j} } \right|^{2} \le \frac{{\gamma_{th} }}{{Q_{j} \left( {z_{j} + U_{j} } \right)}}} \right)dz \\ \end{aligned} $$

(88)

Equation (88) is similar to Eq. (48).

Putting the p.d.f. and c.d.f. of Weibull distribution Eq. (89) is obtained.

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = \frac{{K_{hj} }}{{\lambda_{hj}^{{k_{hj} }} }}\mathop \int \limits_{{\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} }}}}^{\infty } z_{j}^{{k_{hj} - 1}} e^{{ - \left( {\frac{{z_{j} }}{{\lambda_{hj} }}} \right)^{{k_{hj} }} }} \left[ {1 - e^{{ - \left\{ {\left( {\frac{{\gamma_{th} }}{{Q_{j} \lambda_{gj} \left( {z_{j} + U_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} } \right]dz_{j} \\ \end{aligned} $$

(89)

Substitution of ${p_j} = {\left( {{{\text{z}}_{\text{j}}}/{{{\lambda }}_{{\text{hj}}}}} \right)^{{{\text{k}}_{{\text{hj}}}}}}$ into Eq. (90) can be derived.

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = \mathop \int \limits_{{\left( {\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} \lambda_{hj} }}} \right)^{{k_{hj} }} }}^{\infty } e^{{ - p_{j} }} dp_{j} - \mathop \int \limits_{{\left( {\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} \lambda_{hj} }}} \right)^{{k_{hj} }} }}^{\infty } e^{{ - \left\{ {p_{j} + \left( {\frac{{\gamma_{th} }}{{Q_{j} \lambda_{gj} \left( {p_{j}^{{\frac{1}{{k_{hj} }}}} \lambda_{hj} + U_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ & = e^{{ - \left( {\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} \lambda_{hj} }}} \right)^{{k_{hj} }} }} - \mathop \int \limits_{{\left( {\gamma_{Hyj} .\frac{{\gamma_{th} }}{{P_{j} \lambda_{hj} }}} \right)^{{k_{hj} }} }}^{\infty } e^{{ - \left\{ {p_{j} + \left( {\frac{{\gamma_{th} }}{{Q_{j} \lambda_{gj} \left( {p_{j}^{{\frac{1}{{k_{hj} }}}} \lambda_{hj} + U_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ \end{aligned} $$

(90)

Using the Eqs. (51) -(53), Eq. (91) is formulated.

$$ \begin{aligned} & \Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right) \\ & = e^{{ - \left( {\gamma_{Hyj} .M_{j} } \right)^{{k_{hj} }} }} - \mathop \int \limits_{{(\gamma_{Hyj} M_{j} )^{{k_{hj} }} }}^{\infty } e^{{ - \left\{ {p_{j} + \left( {\frac{{N_{j} }}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + X_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ \end{aligned} $$

(91)

Combining Eqs. (70), (87), (91) we get,

$$ P_{outj} = P_{outj - AF} - [\Pr \left( {\gamma_{SRj } \ge Th_{j} , \gamma_{RDj - AF} \le \gamma_{th} } \right) - \Pr \left( {\gamma_{SRj} \ge Th_{j} , \gamma_{RDj - DF} \le \gamma_{th} } \right)] $$

$$ \begin{aligned} & = \left[ {1 - \mathop {\mathop \smallint \limits^{\infty } }\limits_{{M_{j} ^{{k_{{hj}} }} }} e^{{ - \left\{ {p_{j} + \left( {\frac{{(p_{j} ^{{\frac{1}{{k_{{hj}} }}}} + \frac{{M_{j} }}{{\gamma _{{th}} }})}}{{\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} - M_{j} } \right)\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} + X_{j} } \right)}}.N_{j} } \right)^{{k_{{gj}} }} } \right\}}} dp_{j} } \right] \\ & \quad + \mathop {\mathop \smallint \limits^{\infty } }\limits_{{(\gamma _{{Hyj}} M_{j} )^{{k_{{hj}} }} }} e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} + \frac{{M_{j} }}{{\gamma _{{th}} }}} \right)}}{{\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} - M_{j} } \right)\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} + X_{j} } \right)}}.N_{j} } \right)^{{k_{{gj}} }} } \right\}}} dp_{j} \\ & \quad - \mathop {\mathop \smallint \limits^{\infty } }\limits_{{(\gamma _{{Hyj}} M_{j} )^{{k_{{hj}} }} }} e^{{ - \left\{ {p_{j} + \left( {\frac{{N_{j} }}{{\left( {p_{j} ^{{\frac{1}{{k_{{hj}} }}}} + X_{j} } \right)}}} \right)^{{k_{{gj}} }} } \right\}}} dp_{j} \\ \end{aligned} $$

$$ \begin{aligned} & = 1 - \int\limits_{{M_{j}^{{k_{{hj}} }} }}^{{(\gamma _{{Hyj}} M_{j} )^{{k_{{hj}} }} }} {e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j}^{{\frac{1}{{k_{{hj}} }}}} + \frac{{M_{j} }}{{\gamma _{{th}} }}} \right)}}{{\left( {p_{j}^{{\frac{1}{{k_{{hj}} }}}} - M_{j} } \right)\left( {p_{j}^{{\frac{1}{{k_{{hj}} }}}} + X_{j} } \right)}}.N_{j} } \right)^{{k_{{gj}} }} } \right\}}} } dp_{j} \\ & \quad - \int\limits_{{(\gamma _{{Hyj}} M_{j} )^{{k_{{hj}} }} }}^{\infty } {e^{{ - \left\{ {p_{j} + \left( {\frac{{N_{j} }}{{\left( {p_{j}^{{\frac{1}{{k_{{hj}} }}}} + X_{j} } \right)}}} \right)^{{k_{{gj}} }} } \right\}}} } dp_{j} \\ \end{aligned} $$

(92)

Therefore if ${\gamma }_{Hyj}\le 1$ then,${P}_{outj}={ P}_{outj-DF}$; else

$$ \begin{aligned} P_{outj} & = 1 - \mathop \int \limits_{{M_{j}^{{k_{hj} }} }}^{{(\gamma_{Hyj} M_{j} )^{{k_{hj} }} }} e^{{ - \left\{ {p_{j} + \left( {\frac{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + \frac{{M_{j} }}{{\gamma_{th} }}} \right)}}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} - M_{j} } \right)\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + X_{j} } \right)}}.N_{j} } \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ & \quad - \mathop \int \limits_{{(\gamma_{Hyj} M_{j} )^{{k_{hj} }} }}^{\infty } e^{{ - \left\{ {p_{j} + \left( {\frac{{N_{j} }}{{\left( {p_{j}^{{\frac{1}{{k_{hj} }}}} + X_{j} } \right)}}} \right)^{{k_{gj} }} } \right\}}} dp_{j} \\ \end{aligned} $$

(93)

From Eqs. (76) and (93), we can note that if ${\gamma }_{Hyj}\le 1$, then the system behaves like DF and when ${\gamma }_{Hyj}$ is sufficiently large, then the system approaches AF.