Performance Analysis of Decode-and-Forward Scheme with Relay Ordering for Secondary Spectrum Access

Abstract

The communication efficiency of primary networks in cognitive radio depends on wireless environments, such as obstacles (e.g. buildings), distances between transmitter and receiver, and limited transmit power. A cooperative model between primary and secondary networks has the potential to overcome these problems. In this paper, we propose and analyze the performance of a decode-and-forward scheme with relay ordering for secondary spectrum access. In this scheme, a primary transmitter communicates with a primary receiver with the help of two secondary transmitters. Each secondary transmitter relays primary signals from the primary transmitter to primary receiver, and follows an optimal order to ensure the best communication capacity of the primary network and to find opportunities to transmit its own signals. The performance of primary and secondary networks is evaluated by theoretical analysis in terms of outage probability. Monte Carlo simulations are presented to verify the theoretical analysis and to compare the performance of the proposed protocol with that of a direct transmission protocol and a decode-and-forward protocol with a relay selection scheme.

Appendices

Appendix A: Solving Formula (23)

We use symbols as following: $\mu _1 = \frac{\theta _p}{\alpha _1 - \theta _p \times (1- \alpha _1)}$, $\mu _2 = \frac{\theta _p}{2\alpha _1 - \theta _p \times (1- \alpha _1)}$, $\mu _3 = \theta _p-\frac{\alpha _1}{(1- \alpha _1)}$, $\mu _4 = \frac{\theta _p}{\alpha _2 - \theta _p \times (1- \alpha _2)}$, $\mu _5 = \frac{\theta _p}{2\alpha _2 - \theta _p \times (1- \alpha _2)}$, $\mu _6 = \theta _p-\frac{\alpha _2}{(1- \alpha _2)}$.

It is easy to find that

$$\begin{aligned} {P_{12}} = {P_{13}} = {P_{14}} = {P_{21}} = {P_{22}} = {P_{24}} = {P_{31}} = {P_{33}} = {P_{41}} = {P_{42}} = 0 \end{aligned}$$

(39)

1. Derivation $P_{11}$

$$\begin{aligned} {P_{11}}&= \Pr (\max (I_1^1,I_2^1) < {R_p},{\gamma _{11}} < {\theta _p},\gamma {}_{12} < {\theta _p})\nonumber \\&= \Pr ({\gamma _{11}} < {\theta _p})\times \Pr (\gamma {}_{12} < {\theta _p})\, = (1 - {e^{ - {\lambda _{11}}{\theta _p}}})\times (1 - {e^{ - {\lambda _{12}}{\theta _p}}}) \end{aligned}$$

(40)

2. Derivation $P_{23}$

$$\begin{aligned} {P_{23}}&= \Pr \left\{ {\max (I_1^2,I_2^3) < {R_p},\,\,\gamma {}_{12} + \frac{{{\alpha _1} \times {\gamma _5}}}{{(1 - {\alpha _1}) \times {\gamma _5} + 1}} < {\theta _p},{\gamma _{12}} < {\theta _p},{\gamma _{11}} > {\theta _p}} \right\} \nonumber \\&= \underbrace{\Pr \left( \frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}} < {\theta _p}\right) }_{{P_{23a}}} \times \underbrace{\Pr ({\gamma _{11}} > {\theta _p})}_{{P_{23b}}}\nonumber \\&\times \underbrace{\Pr \left( {\gamma _{12}} < {\theta _p},\gamma {}_{12} + \frac{{{\alpha _1} \times {\gamma _5}}}{{(1 - {\alpha _1}) \times {\gamma _5} + 1}} < {\theta _p}\right) }_{{P_{23c}}}\, \end{aligned}$$

(41)

• In the case where ${\alpha _1} = 1$ (${\textit{ST}}_1$ does not send signals), $P_{23}$ is given as:

  $$\begin{aligned} {P_{23}} = \Pr ({\gamma _{21}} < {\theta _p}) \times \Pr ({\gamma _{11}} > {\theta _p}) \times \Pr ({\gamma _{12}} < {\theta _p},\gamma {}_{12} + {\gamma _5} < {\theta _p}) \end{aligned}$$

  (42)

  where

  $$\begin{aligned} \Pr ({\gamma _{12}} \!<\! {\theta _p},\gamma {}_{12} \!+\! {\gamma _5} \!<\! {\theta _p})&= \int \limits _{x = 0}^{{\theta _p}} {\int \limits _{z =0}^{{\theta _p} \!-\! x} {{f_{{\gamma _5}}}(z) \!\times \! {f_{{\gamma _{12}}}}(x)dzdx}}\nonumber \\&= 1 \!-\! {e^{ \!-\! {\lambda _{12}}{\theta _p}}} \!+\! \frac{{{\lambda _{12}}}}{{{\lambda _5} \!-\! {\lambda _{12}}}} \!\times \! ({e^{ \!-\! {\lambda _5}{\theta _p}}} \!-\! {e^{ \!-\! {\lambda _{12}}{\theta _p}}}) \end{aligned}$$

  (43)

  Thus, in this case:

  $$\begin{aligned} {P_{23}} = \left( {1 - {e^{ - {\lambda _{21}}{\theta _p}}}} \right) \times {e^{ - {\lambda _{11}}{\theta _p}}} \times \left\{ {1 - {e^{ - {\lambda _{12}}{\theta _p}}} + \frac{{{\lambda _{12}}}}{{{\lambda _5} - {\lambda _{12}}}} \times ({e^{ - {\lambda _5}{\theta _p}}} - {e^{ - {\lambda _{12}}{\theta _p}}})} \right\} \nonumber \\ \end{aligned}$$

  (44)

• When ${\alpha _1} < 1$:

  $$\begin{aligned} {P_{23a}} = \left\{ \begin{array}{l@{\quad }l} 1- e^{-\lambda _{21} \mu _1}, &{} \theta _p < \alpha _1/(1-\alpha _1)\\ 1, &{} \theta _p \ge \alpha _1/(1-\alpha _1)\\ \end{array} \right. \end{aligned}$$

  (45)

Following [11], Eqs. (25) and (29)], we have:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} P_{23c}^{{\textit{LB}}} \!=\! \left( 1 \!-\! e^{\!-\! \lambda _{12}\frac{\theta _p}{2}}\right) (1\!-\!e^{\!-\!\lambda _5 \mu _2}) \!\le \! {P_{23c}} \!\le \! P_{23c}^{{\textit{UB}}} \!=\! (1\!-\!e^{\!-\! \lambda _{12} \theta _p}) (1\!-\!e^{\!-\!\lambda _5 \mu _1}), &{} \theta _p \!<\! \frac{\alpha _1}{1\!-\!\alpha _1}\\ P_{23c}^{{\textit{LB}}} \!=\! (1\!-\!e^{-\lambda _{12} \mu _3}) \!\le \! \mathrm{P}_{23c} \!\le \! P_{23c}^{{\textit{UB}}} \!=\! (1 \!-\!e^{\!-\!\lambda _{12} \theta _p}), &{} \theta _p \ge \frac{\alpha _1}{1\!-\!\alpha _1} \end{array}\right. \end{aligned}$$

(46)

Therefore, from (41), (45) and (46), we have:

$$\begin{aligned} P_{23}^{{\textit{LB}}} \le {P_{23}} \le P_{23}^{UP} \end{aligned}$$

(47)

where

$$\begin{aligned} P_{23}^{{\textit{LB}}}&\!=\!&\left\{ \begin{array}{l@{\quad }l} (1\!-\!e^{\!-\!\lambda _{21} \mu _1}) \!\times \! e^{\!-\!\lambda _{11} \theta _p} \!\times \! \left( 1\!-\!e^{\!-\!\lambda _{12}\frac{\theta _p}{2}}\right) \times (1\!-\!e^{\!-\!\lambda _5 \mu _2}),&{} \theta _p < \alpha _1/1\!-\!\alpha _1\\ e^{-\lambda _{11} \theta _p} \times (1\!-\!e^{-\lambda _{12} \mu _3}), &{}\theta _p \!\ge \! \alpha _1/1\!-\!\alpha _1\\ \end{array}\right. \end{aligned}$$

(48)

$$\begin{aligned} \quad P_{23}^{{\textit{UB}}}&\!=\!&\left\{ \begin{array}{l@{\quad }l} ({1\!-\!e^{-\lambda _{21} \mu _1}}) \!\times \! e^{-\lambda _{11} \theta _p} \!\times \! (1\!-\!e^{-\lambda _{12} \theta _p}) \!\times \! (1\!-\!e^{-\lambda _5 \mu _1}),&{}\theta _p < \alpha _1/1\!-\!\alpha _1\\ e^{-\lambda _{11} \theta _p} \times (1\!-\!e^{-\lambda _{12} \theta _p}), &{}\theta _p \!\ge \! \alpha _1/1-\alpha _1\\ \end{array}\right. \end{aligned}$$

(49)

3. Derivation $P_{32}$ Similar to $P_{23}$, we receive the results as:

$$\begin{aligned} {P_{32}}&= \underbrace{\Pr \left( \frac{{{\alpha _2}{\gamma _{22}}}}{{(1 \!-\! {\alpha _2}){\gamma _{22}} \!+\! 1}} \!<\! {\theta _p}\right) }_{{P_{32a}}}\!\times \! \underbrace{\Pr ({\gamma _{12}} \!>\! {\theta _p})}_{{P_{32b}}}\!\times \! \underbrace{\Pr \left( {\gamma _{11}} < {\theta _p},\gamma {}_{11} \!+\! \frac{{{\alpha _2}{\gamma _5}}}{{(1 \!-\! {\alpha _2}){\gamma _5} \!+\! 1}} \!<\! {\theta _p}\right) }_{{P_{32c}}}\nonumber \\&= \left\{ \begin{array}{l@{\quad }l} (1-e^{-\lambda _{22} \theta _p}) \times e^{-\lambda _{12} \theta _p} \times \left\{ 1-e^{-\lambda _{11} \theta _p} + \frac{\lambda _{11}}{{\lambda _5}-{\lambda _{11}}} (e^{-\lambda _5 \theta _p} - {e^{-\lambda _{11} \theta _p}})\right\} ,&{}\alpha _2 = 1\\ P_{32}^{{\textit{LB}}} \le {P_{32}} \le P_{32}^{{\textit{UB}}}, &{} \alpha _2 < 1 \end{array}\right. \nonumber \\ \end{aligned}$$

(50)

The lower and upper bounds of $P_{32c}$ is given as:

$$\begin{aligned} \left\{ \begin{array}{ll} P_{32c}^{{\textit{LB}}} \!=\! \left( 1\!-\!e^{\frac{\!-\!\lambda _{11}\theta _p}{2}}\right) (1\!-\!e^{\!-\!\lambda _5 \mu _5}) \!\le \! P_{32c} \!\le \! P_{32c}^{{\textit{UB}}} \!=\! (1\!-\! e^{-\lambda _{11} \theta _p}) (1\!-\!e^{\!-\!\lambda _5 \mu _4}), &{}\theta _p < \frac{\alpha _2}{1\!-\!\alpha _2}\\ P_{32c}^{{\textit{LB}}} \!=\! (1\!-\!e^{-\lambda _{11} \mu _6}) \!\le \! \mathrm{P}_{32c} \!\le \! P_{32c}^{{\textit{UB}}} = (1\!-\!e^{-\lambda _{11} \theta _p}),&{} \theta _p \!\ge \! \frac{\alpha _2}{1\!-\!\alpha _2} \end{array} \right. \nonumber \\ \end{aligned}$$

(51)

Hence, from (50) and (51), the lower and upper bounds of $P_{32}$ as:

$$\begin{aligned} P_{32}^{{\textit{LB}}}&= \left\{ \begin{array}{ll} (1-e^{-\lambda _{22} \mu _4}) \times e^{-\lambda _{12} \theta _p} \times \left( 1-e^{-\lambda _{11}\frac{\theta _p}{2}}\right) \times (1-e^{-\lambda _5 \mu _5}), &{} \theta _p < \alpha _2/(1-\alpha _2)\\ e^{-\lambda _{12} \theta _p} \times (1-e^{-\lambda _{11}\mu _6}), &{}\theta _p \ge \alpha _2/(1-\alpha _2) \end{array}\right. \nonumber \\\end{aligned}$$

(52)

$$\begin{aligned} P_{32}^{{\textit{UB}}}&= \left\{ \begin{array}{ll} (1-e^{-\lambda _{22} \mu _4}) \times e^{-\lambda _{12} \theta _p} \times (1-e^{-\lambda _{11}\theta _p}) \times (1-e^{-\lambda _5 \mu _4}), &{} \theta _p < \alpha _2/(1-\alpha _2)\\ e^{-\lambda _{12} \theta _p} \times (1-e^{-\lambda _{11}\theta _p}), &{}\theta _p \ge \alpha _2/(1-\alpha _2) \end{array}\right. \nonumber \\ \end{aligned}$$

(53)

4. Derivation $P_{34}$

$$\begin{aligned} P_{34}&= \underbrace{\Pr \left( \frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}} + \frac{{{\alpha _2} \times {\gamma _{22}}}}{{(1 - {\alpha _2}) \times {\gamma _{22}} + 1}} < {\theta _p}\right) }_{{P_{34a}}}\nonumber \\&\times \underbrace{\Pr ({\gamma _{12}} > {\theta _p})}_{{P_{34b}}} \times \underbrace{\Pr ({\gamma _{11}} < {\theta _p},\gamma {}_{11} + \frac{{{\alpha _2} \times {\gamma _5}}}{{(1 - {\alpha _2}) \times {\gamma _5} + 1}} > {\theta _p})}_{{P_{34c}}} \end{aligned}$$

(54)

• In the special case when ${\alpha _1} = 1 \mathrm { and }\,{\alpha _2} = 1$, by following (43) similarly:

  $$\begin{aligned} {P_{34a}}&= 1 - {e^{ - {\lambda _{21}}{\theta _p}}} + \frac{{{\lambda _{21}}}}{{{\lambda _{22}} - {\lambda _{21}}}} \times ({e^{ - {\lambda _{22}}{\theta _p}}} - {e^{ - {\lambda _{21}}{\theta _p}}})\end{aligned}$$

  (55)
  $$\begin{aligned} {P_{34c}}&\!=\!&\Pr ({\gamma _{11}} \!<\! {\theta _p})\! -\! \Pr ({\gamma _{11}} \!<\! {\theta _p},\gamma {}_{11} \!+\! {\gamma _r} < {\theta _p}) \!=\! \frac{{{\lambda _{11}}}}{{{\lambda _{11}} \!-\! {\lambda _5}}} \!\times \! ({e^{ - {\lambda _5}{\theta _p}}} - {e^{ - {\lambda _{11}}{\theta _p}}})\nonumber \\ \end{aligned}$$

  (56)

From (54), (55) and (56), we have the result:

$$\begin{aligned} P_{34} = \left\{ {1 - {e^{ - {\lambda _{21}}{\theta _p}}} + \frac{{{\lambda _{21}} \times ({e^{ - {\lambda _{22}}{\theta _p}}} - {e^{ - {\lambda _{21}}{\theta _p}}})}}{{{\lambda _{22}} - {\lambda _{21}}}}} \right\} \times \frac{{{e^{ - {\lambda _{12}}{\theta _p}}} \times {\lambda _{11}} \times ({e^{ - {\lambda _5}{\theta _p}}} - {e^{ - {\lambda _{11}}{\theta _p}}})}}{{{\lambda _{11}} - {\lambda _5}}}\nonumber \\ \end{aligned}$$

(57)

• When only ${\alpha _2} = 1$($0 \le {\alpha _1} < 1$):

$P_{34a}$ is solved in the same way with $P_{23c}$ in (46), the lower and upper bounds of $P_{34a}$ are obtained as:

$$\begin{aligned} \left\{ \begin{array}{ll} P_{34a}^{{\textit{LB}}} \!=\! \left( 1\!-\!e^{\frac{\!-\!\lambda _{22}{\theta _p}}{2}}\right) (1\!-\!e^{\!-\!\lambda _{21} \mu _2}) \le {P_{34a}} \le P_{34a}^{{\textit{UB}}} = (1\!-\!e^{\!-\!\lambda _{22} \theta _p}) (1\!-\!e^{-\lambda _{21} \mu _1}), &{} {\theta _p} < \frac{\alpha _1}{1 \!-\! {\alpha _1}}\\ P_{34a}^{{\textit{LB}}} \!=\! (1\!-\!e^{-\lambda _{22} \mu _3}) \!\le \! \mathrm{P}_{34a} \!\le \! P_{34a}^{{\textit{UB}}} \!=\! (1\!-\!e^{-\lambda _{22} \theta _p}),&{} \theta _p \!\ge \! \frac{\alpha _1}{1\!-\!{\alpha _1}} \end{array} \right. \end{aligned}$$

(58)

Then, from (54), (56) and (58), the lower and upper bounds of $P_{34}$ are given as:

$$\begin{aligned} P_{34}^{{\textit{LB}}}&= \left\{ \begin{array}{ll} \frac{\lambda _{11}}{\lambda _{11}-\lambda _5}\left( 1-e^{\frac{-\lambda _{22} \theta _p}{2}}\right) (1-e^{-\lambda _{21} \mu _2}) e^{-\lambda _{12} \theta _p}(e^{-\lambda _5 \theta _p}- e^{-\lambda _{11} \theta _p}),&{} \theta _p < \frac{\alpha _1}{1-\alpha _1}\\ \frac{\lambda _{11}}{\lambda _{11}-\lambda _5}(1-e^{-\lambda _{22} \mu _3}) e^{-\lambda _{12} \theta _p}(e^{-\lambda _5 \theta _p}-e^{-\lambda _{11} \theta _p}),&{} \theta _p \ge \frac{\alpha _1}{1-\alpha _1} \end{array} \right. \nonumber \\ \end{aligned}$$

(59)

$$\begin{aligned} P_{34}^{{\textit{UB}}}&= \left\{ \begin{array}{ll} \frac{\lambda _{11}}{\lambda _{11}-\lambda _5} (1-e^{-\lambda _{22} \theta _p}) (1-e^{-\lambda _{21} \mu _1}) e^{-\lambda _{12} \theta _p}(e^{-\lambda _5 \theta _p}- e^{-\lambda _{11} \theta _p}),&{} \theta _p < \frac{\alpha _1}{1-\alpha _1}\\ \frac{\lambda _{11}}{\lambda _{11}-\lambda _5}(1-e^{-\lambda _{22} \theta _p}) e^{-\lambda _{12} \theta _p}(e^{-\lambda _5 \theta _p}-e^{-\lambda _{11} \theta _p}),&{} \theta _p \ge \frac{\alpha _1}{1-\alpha _1} \end{array} \right. \nonumber \\ \end{aligned}$$

(60)

• When ${\alpha _1} < 1 \mathrm { and }\,\alpha _2 < 1$, we use the upper and lower bounds for $P_{34}$, yielding

  $$\begin{aligned} \max \left\{ \begin{array}{l} \frac{\alpha _1 \gamma _{21}}{(1-{\alpha _1}) \gamma _{21}+1},\\ \frac{\alpha _2 \gamma _{22}}{(1 - {\alpha _2}) \gamma _{22}+1} \end{array} \right\}&\le \left\{ \frac{\alpha _1 \gamma _{21}}{(1-\alpha _1) \gamma _{21}+1} +\frac{\alpha _2 \gamma _{22}}{(1- \alpha _2) \gamma _{22}+1}\right\} \nonumber \\&\le 2\max \left\{ \begin{array}{l} \frac{\alpha _1 \gamma _{21}}{(1-\alpha _1) \gamma _{21}+1},\\ \frac{\alpha _2 \gamma _{22}}{(1-{\alpha _2}) \gamma _{22}+1} \end{array} \right\} \end{aligned}$$

  (61)

Then, lower and upper bounds of $P_{34a}$ are obtained as:

$$\begin{aligned} P_{34a}^{{\textit{LB}}}&= \Pr \left[ 2\max \left( \frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}},\frac{{{\alpha _2} \times {\gamma _{22}}}}{{(1 - {\alpha _2}) \times {\gamma _{22}} + 1}}\right) \, < {\theta _p}\right] \nonumber \\&= \Pr \left( 2\frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}} < {\theta _p}\right) \times \Pr \left( 2\frac{{{\alpha _2} \times {\gamma _{22}}}}{{(1 - {\alpha _2}) \times {\gamma _{22}} + 1}} < {\theta _p}\right) \end{aligned}$$

(62)

$$\begin{aligned} P_{34a}^{{\textit{UB}}}&= \Pr \left[ \max \left( \frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}},\frac{{{\alpha _2} \times {\gamma _{22}}}}{{(1 - {\alpha _2}) \times {\gamma _{22}} + 1}}\right) < {\theta _p}\right] \nonumber \\&= \Pr \left( \frac{{{\alpha _1} \times {\gamma _{21}}}}{{(1 - {\alpha _1}) \times {\gamma _{21}} + 1}} < {\theta _p}\right) \times \Pr \left( \frac{{{\alpha _2} \times {\gamma _{22}}}}{{(1 - {\alpha _2}) \times {\gamma _{22}} + 1}} < {\theta _p}\right) \end{aligned}$$

(63)

Similar to (45), we have:

When ${\alpha _1} > {\alpha _2}$:

$$\begin{aligned} P_{34a}^{{\textit{LB}}}&= \left\{ \begin{array}{l@{\quad }l} (1-e^{-\lambda _{21}\mu _2})\times (1-e^{-\lambda _{22}\mu _5}),&{} \theta _p < 2\alpha _2/(1-\alpha _2)\\ (1-e^{-\lambda _{21}\mu _2}),&{}2\alpha _2/(1-\alpha _2)\le \theta _p<2\alpha _1/(1-\alpha _1)\\ 1,&{}\theta _p \ge 2\alpha _1/(1-\alpha _1)\\ \end{array}\right. \nonumber \\ P_{34a}^{{\textit{UB}}}&= \left\{ \begin{array}{l@{\quad }l} (1-e^{-\lambda _{21} \mu _1})\times (1-e^{-\lambda _{22} \mu _4}), &{}\theta _p < \alpha _2/(1-\alpha _2)\\ (1-e^{-\lambda _{21} \mu _1}),&{} \alpha _2/(1-\alpha _2)\le \theta _p < \alpha _1/(1-\alpha _1)\\ 1,&{}\theta _p \ge \alpha _1/(1-\alpha _1)\\ \end{array}\right. \end{aligned}$$

(64)

When ${\alpha _1} \le {\alpha _2}$:

$$\begin{aligned} P_{34a}^{{\textit{LB}}}&= \left\{ \begin{array}{l@{\quad }l} (1-e^{-\lambda _{21}\mu _2})\times (1-e^{-\lambda _{22}\mu _5}),&{} \theta _p < 2\alpha _1/(1-\alpha _1)\\ (1-e^{-\lambda _{22}\mu _5}),&{}2\alpha _1/(1-\alpha _1)\le \theta _p<2\alpha _2/(1-\alpha _2)\\ 1,&{}\theta _p \ge 2\alpha _2/(1-\alpha _2)\\ \end{array}\right. \nonumber \\ P_{34a}^{{\textit{UB}}}&= \left\{ \begin{array}{l@{\quad }l} (1-e^{-\lambda _{21} \mu _1})\times (1-e^{-\lambda _{22} \mu _4}), &{}\theta _p < \alpha _1/(1-\alpha _1)\\ (1-e^{-\lambda _{22} \mu _4}),&{} \alpha _1/(1-\alpha _1)\le \theta _p < \alpha _2/(1-\alpha _2)\\ 1,&{}\theta _p \ge \alpha _2/(1-\alpha _2)\\ \end{array}\right. \end{aligned}$$

(65)

In addition:

$$\begin{aligned} {P_{34c}}&= \Pr ({\gamma _{11}} < {\theta _p}) - \Pr ({\gamma _{11}} < {\theta _p},\gamma {}_{11} + \frac{{{\alpha _2} \times {\gamma _5}}}{{(1 - {\alpha _2}) \times {\gamma _5} + 1}} < {\theta _p})\nonumber \\&= \left( {1 - {e^{ - {\lambda _{11}}{\theta _p}}}} \right) - {P_{32c}} \end{aligned}$$

(66)

Using the results of (51) with the same condition, the upper and lower bounds of $P_{34c}$ are given as:

$$\begin{aligned} P_{34c}^{{\textit{LB}}} = \left( {1 - {e^{ - {\lambda _{11}}{\theta _p}}}} \right) - P_{32c}^{{\textit{UB}}} \le {P_{34c}} \le P_{34c}^{{\textit{UB}}} = \left( {1 - {e^{ - {\lambda _{11}}{\theta _p}}}} \right) - P_{32c}^{{\textit{LB}}} \end{aligned}$$

(67)

From (54), (64), (65) and (67), we have the bounds:

$$\begin{aligned} P_{34}^{{\textit{LB}}} = {e^{ - {\lambda _{12}}{\theta _p}}} \times P_{34a}^{{\textit{LB}}} \times P_{34c}^{{\textit{LB}}} \le {P_{34}} \le P_{34}^{{\textit{UB}}} = {e^{ - {\lambda _{12}}{\theta _p}}} \times P_{34a}^{{\textit{UB}}} \times P_{34c}^{{\textit{UB}}} \end{aligned}$$

(68)

5. Derivation $P_{43}$: Similar to $P_{34}$, we also receive the results as:

$$\begin{aligned} {P_{43}}&= \underbrace{\Pr \left( \frac{{{\alpha _1}{\gamma _{21}}}}{{(1 - {\alpha _1}){\gamma _{21}} + 1}} + \frac{{{\alpha _2}{\gamma _{22}}}}{{(1 - {\alpha _2}){\gamma _{22}} + 1}} < {\theta _p}\right) }_{{P_{43a}}}\nonumber \\&\quad \quad \times \underbrace{\Pr ({\gamma _{11}} > {\theta _p})}_{{P_{43b}}}\underbrace{\Pr ({\gamma _{12}} < {\theta _p},\gamma {}_{12} + \frac{{{\alpha _1}{\gamma _5}}}{{(1 - {\alpha _1}){\gamma _5} + 1}} > {\theta _p})}_{{P_{43c}}}\nonumber \\&= \left\{ \begin{array}{l@{\quad }l} \frac{\lambda _{12}}{\lambda _{12}-\lambda _5}\left[ 1-e^{-\lambda _{21} \theta _p}+\frac{\lambda _{21}}{\lambda _{22}-\lambda _{21}}(e^{-\lambda _{22} \theta _p}-e^{-\lambda _{21}\theta _p})\right] \\ \quad e^{-\lambda _{11} \theta _p}(e^{-\lambda _5 \theta _p}-e^{-\lambda _{12}\theta _p}), &{}\alpha _1 = \alpha _2 = 1\\ P_{43}^{{\textit{LB}}}\le {P_{43}}\le P_{43}^{{\textit{UB}}},&{} others \end{array}\right. \end{aligned}$$

(69)

• When ${\alpha _1} = 1,\,0 \le {\alpha _2} < 1$:

The same way as $P_{34}$, the lower and upper bounds of $P_{43}$ are given as:

$$\begin{aligned} P_{43}^{{\textit{LB}}}&= \left\{ \begin{array}{l@{\quad }l} \frac{\lambda _{12}}{\lambda _{12}\!-\!\lambda _5}\left( 1\!-\!e^{\frac{-\lambda _{21} \theta _p}{2}}\right) (1\!-\!e^{-\lambda _{22} \mu _2}) e^{-\lambda _{11} \theta _p}(e^{-\lambda _5 \theta _p} \!-\! e^{-\lambda _{12} \theta _p}),&{} \theta _p < \frac{\alpha _2}{1\!-\!\alpha _2}\\ \frac{\lambda _{12}}{\lambda _{12}\!-\!\lambda _5}(1\!-\!e^{-\lambda _{21} \mu _3})e^{- \lambda _{11} \theta _p}(e^{-\lambda _5 \theta _p}\!-\!e^{-\lambda _{12} \theta _p}),&{} \theta _p\!\ge \! \frac{\alpha _2}{1\!-\!\alpha _2} \end{array} \right. \qquad \end{aligned}$$

(70)

$$\begin{aligned} P_{43}^{{\textit{UB}}}&= \left\{ \begin{array}{l@{\quad }l} \frac{\lambda _{12}}{\lambda _{12}-\lambda _5}(1-e^{-\lambda _{21} \theta _p})(1-e^{-\lambda _{22} \mu _1}) e^{-\lambda _{11} \theta _p}(e^{-\lambda _5 \theta _p} - e^{-\lambda _{12} \theta _p}),&{} \theta _p < \frac{\alpha _2}{1-\alpha _2}\\ \frac{\lambda _{12}}{\lambda _{12}-\lambda _5}(1-e^{-\lambda _{21} \theta _p})e^{- \lambda _{11} \theta _p}(e^{-\lambda _5 \theta _p}-e^{-\lambda _{11} \theta _p}),&{} \theta _p\ge \frac{\alpha _2}{1-\alpha _2} \end{array} \right. \qquad \end{aligned}$$

(71)

• When ${\alpha _1} < 1$ and ${\alpha _2} < 1$: $P_{43a}$ is also $P_{34a}$; $P_{43c}$ is calculated similar to $P_{34c}$, we have the result:

  $$\begin{aligned} P_{43c}^{{\textit{LB}}} = \left( {1 - {e^{ - {\lambda _{12}}{\theta _p}}}} \right) - P_{23c}^{{\textit{UB}}} \le {P_{43c}} \le P_{43c}^{{\textit{UB}}} = \left( {1 - {e^{ - {\lambda _{12}}{\theta _p}}}} \right) - P_{23c}^{{\textit{LB}}} \end{aligned}$$

  (72)

The lower and upper bounds of $P_{43}$ are given as:

$$\begin{aligned} P_{43}^{{\textit{LB}}} = {e^{ - {\lambda _{11}}{\theta _p}}} \times P_{34a}^{{\textit{LB}}} \times P_{43c}^{{\textit{LB}}} \le {P_{43}} \le P_{43}^{{\textit{UB}}} = {e^{ - {\lambda _{11}}{\theta _p}}} \times P_{34a}^{{\textit{UB}}} \times P_{43c}^{{\textit{UB}}} \end{aligned}$$

(73)

6. Derivation $P_{44}$

$$\begin{aligned} {P_{44}}&= \Pr \left( \frac{{{\alpha _1}{\gamma _{21}}}}{{(1 - {\alpha _1}){\gamma _{21}} + 1}} + \frac{{{\alpha _2}{\gamma _{22}}}}{{(1 - {\alpha _2}){\gamma _{22}} + 1}} < {\theta _p}\right) \nonumber \\&\times \Pr \left( {\gamma _{11}} > {\theta _p},\gamma {}_{11} + \frac{{{\alpha _2}{\gamma _5}}}{{(1 - {\alpha _2}){\gamma _5} + 1}} > {\theta _p}\right) \nonumber \\&\times \Pr \left( {\gamma _{12}} > {\theta _p},\gamma {}_{12} + \frac{{{\alpha _1}{\gamma _5}}}{{(1 - {\alpha _1}){\gamma _5} + 1}} > {\theta _p}\right) \nonumber \\&= \Pr \left( \frac{{{\alpha _1}{\gamma _{21}}}}{{(1 - {\alpha _1}){\gamma _{21}} + 1}} + \frac{{{\alpha _2}{\gamma _{22}}}}{{(1 - {\alpha _2}){\gamma _{22}} + 1}} < {\theta _p}\right) \times \Pr ({\gamma _{11}} > {\theta _p})\times \Pr ({\gamma _{12}} > {\theta _p})\nonumber \\&= \left\{ \begin{array}{l@{\quad }l} \left[ 1-e^{-\lambda _{21} \theta _p}+\frac{\lambda _{21}}{\lambda _{22}-\lambda _{21}} (e^{-\lambda _{22} \theta _p}-e^{-\lambda _{21} \theta _p})\right] e^{-\lambda _{11} \theta _p} e^{-\lambda _{12} \theta _p}, &{} \alpha _1 = \alpha _2= 1\\ P_{44}^{{\textit{LB}}} = e^{-\lambda _{11} \theta _p} e^{-\lambda _{12} \theta _p} P_{34a}^{{\textit{LB}}} \le {P_{44}} \le P_{44}^{{\textit{UB}}} = e^{-\lambda _{11} \theta _p} e^{-\lambda _{12} \theta _p} P_{34a}^{{\textit{UB}}}, &{} \mathrm {others} \end{array}\right. \nonumber \\ \end{aligned}$$

(74)

where $P_{43a}^{{\textit{LB}}},\,P_{43a}^{{\textit{UB}}}$ is calculated from (64) and (65)

Appendix B: Solving Formula (36)

Formula (36) is manipulated as:

$$\begin{aligned}&P_{{\textit{DFRSSA}}}^{{\textit{out}}}\nonumber \\&\quad = \Pr \left\{ {\min [I({\textit{PT}},{{\textit{ST}}_1}),I({{\textit{ST}}_1},{\textit{PR}})] < {R_p},\min [I({\textit{PT}},{{\textit{ST}}_2}),I({{\textit{ST}}_2},{\textit{PR}})] < {R_p}} \right\} \nonumber \\&\quad = \Pr \left\{ {\min [I({\textit{PT}},{{\textit{ST}}_1}),I({{\textit{ST}}_1},{\textit{PR}})] \!<\! {R_p}} \right\} \!\times \! \Pr \left\{ {\min [I({\textit{PT}},{{\textit{ST}}_2}),I({{\textit{ST}}_2},{\textit{PR}})] \!<\! {R_p}} \right\} \nonumber \\&\quad = \underbrace{\left\{ {1 - \Pr \left\{ {\min [I({\textit{PT}},{{\textit{ST}}_1}),I({{\textit{ST}}_1},{\textit{PR}})] > {R_p}} \right\} } \right\} }_{P_{{\textit{DFRSSAa}}}^{{\textit{out}}}}\nonumber \\&\qquad \times \underbrace{\left\{ {1 - \Pr \left\{ {\min [I({\textit{PT}},{{\textit{ST}}_2}),I({{\textit{ST}}_2},{\textit{PR}})] > {R_p}} \right\} } \right\} }_{P_{{\textit{DFRSSAb}}}^{{\textit{out}}}} \end{aligned}$$

(75)

From (14) and (15), $P_{{\textit{DFRSSAa}}}^{{\textit{out}}}$ is calculated as:

$$\begin{aligned}&P_{{\textit{DFRSSAa}}}^{{\textit{out}}}\nonumber \\&\quad = 1 - \Pr \left[ {I({\textit{PT}},{{\textit{ST}}_1}) > {R_p}} \right] \times \Pr \left[ {I({{\textit{ST}}_1},{\textit{PR}}) > {R_p}} \right] \nonumber \\&\quad = 1 - \Pr \left[ {{{\log }_2}\left[ {1 + {\textit{SNR}}({\textit{PT}},{{\textit{ST}}_1})} \right] > {R_p}} \right] \times \Pr \left[ {{{\log }_2}\left[ {1 + {\textit{SINR}}({{\textit{ST}}_1},{\textit{PR}})} \right] > {R_p}} \right] \nonumber \\&\quad = 1 - \Pr ((3/2) \times \gamma _{11} > \theta _p) \times \Pr \left[ \frac{(3/2) \times \alpha _1 \times \gamma _{21}}{(3/2) \times (1-\alpha _1) \times \gamma _{21}+1} > \theta _p\right] \nonumber \\&\quad =\left\{ \begin{array}{l@{\quad }l} 1-e^{-2(\lambda _{11} \theta _p + \lambda _{21} \mu _1)/3},&{} \theta _p < \alpha _1/(1-\alpha _1)\\ 1,&{}\theta _p \ge \alpha _1/(1-\alpha _1) \end{array}\right. \end{aligned}$$

(76)

Similar as (76), we obtain as

$$\begin{aligned} P_{{\textit{DFRSSAb}}}^{{\textit{out}}} = \left\{ \begin{array}{l@{\quad }l} 1-e^{-2(\lambda _{12} \theta _p + \lambda _{22} \mu _4)/3},&{} \theta _p < \alpha _2/(1-\alpha _2)\\ 1,&{}\theta _p < \alpha _2/(1-\alpha _2) \end{array}\right. \end{aligned}$$

(77)

where ${\mu _1},\,\,{\mu _4}$ are defined in “Appendix A”

Substituting (76) and (77) into (75), we have the exact closed-form expression of the outage probability of DFRSSA protocol as (37) and (38).