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Opportunistic TPSR cooperative spectrum sharing protocol with secondary user selection for 5G wireless network

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Abstract

In this paper, we consider a cognitive radio system, where multiple secondary users coexist with one primary user. To improve the transmission efficiency of the system, we propose a cooperative spectrum sharing protocol with secondary user selection by using decode-and-forward and two-path successive relaying techniques. Furthermore, we derive the close-form of outage probability of the primary system. The upper bound of outage probability of the secondary system is also derived. In addition, the theoretical results are verified via numerical simulations. Numerical results show that the primary outage performance is much better than conventional selective relaying schemes. In the low secondary Signal-to-Noise Ratio region, the secondary outage performance is also better than conventional selective relaying schemes.

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Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (NSFC) under Grants no. 61801171, no. 61701172, no. 61771185 and no. 61772175, in part by Key scientific research projects of the University of Henan Province (No.16A510005, No.17A520005 and No. 18A510009).

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Authors and Affiliations

  1. Information Engineering College, Henan University of Science and Technology, Luoyang, 471023, China

    Ping Xie, Jianghui Liu, Gaoyuan Zhang, Ling Xing & Honghai Wu

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  1. Ping Xie
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  2. Jianghui Liu
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  3. Gaoyuan Zhang
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  4. Ling Xing
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  5. Honghai Wu
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Correspondence to Ping Xie.

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This article is part of the Topical Collection: Special Issue on Future Networking Applications Plethora for Smart Cities

Guest Editors: Mohamed Elhoseny, Xiaohui Yuan, and Saru Kumari

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Appendices

APPENDIX 1

From Eqs. (2) and (3), the probability of the two-path successive relaying being activated is given by

$$ {\displaystyle \begin{array}{l}\Pr \left\{{R}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^1\ge {R}_{\mathrm{P}},{R}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^2\ge {R}_{\mathrm{P}},{R}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^1\ge {R}_{\mathrm{P}},{R}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^2\ge {R}_{\mathrm{P}}\right\}\\ {}=\Pr \Big\{\frac{\left({2}^{R_P}-1\right){\sigma}^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^{-\alpha }}\le {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}\right|}^2\le \frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}\right|}^2}{\left({2}^{R_P}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^{-\alpha }}-\frac{\sigma^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^{-\alpha }},\\ {}\kern0.75em \frac{\left({2}^{R_P}-1\right){\sigma}^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^{-\alpha }}\le {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}\right|}^2\le \frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}\right|}^2}{\left({2}^{R_P}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^{-\alpha }}-\frac{\sigma^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^{-\alpha }}\Big\}\\ {}=\frac{\Omega_{1i}{\Omega}_{1j}\exp \left(-b1/{\Omega}_{1i}-b2/{\Omega}_{1j}-{x}_0\left(1/{\Omega}_{3 ij}-a1/{\Omega}_{1i}-a2/{\Omega}_{1j}\right)\right)}{\Omega_{1i}{\Omega}_{1j}+a1{\Omega}_{3 ij}{\Omega}_{1j}+a2{\Omega}_{1i}{\Omega}_{3 ij}}\\ {}+\exp \left(-\frac{c1}{\varOmega_{1i}}-\frac{c2}{\varOmega_{1j}}-\frac{x_0}{\varOmega_{3 ij}}\right)-\exp \left(-\frac{b2}{\varOmega_{1j}}-\frac{c1}{\varOmega_{1i}}-\frac{x_0}{\varOmega_{3 ij}}-\frac{a2{x}_0}{\varOmega_{1j}}\right)\\ {}\times \frac{\varOmega_{1j}}{\varOmega_{1j}+a2{\varOmega}_{3 ij}}-\frac{\varOmega_{1i}}{\varOmega_{1i}+a1{\varOmega}_{3 ij}}\exp \left(-\frac{b1}{\varOmega_{1i}}-\frac{c2}{\varOmega_{1j}}-\frac{x_0}{\varOmega_{3 ij}}-\frac{a1{x}_0}{\varOmega_{1i}}\right)\end{array}} $$
(23)

where x0 = max {(c1 − b1)/a1, (c2 − b2)/a2},\( b1=-{\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_i}^{-\alpha } \), \( a1={P}_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}^{-\alpha }/\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_i}^{-\alpha } \), \( c1=\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_i}^{-\alpha } \), \( a2={P}_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_i,{\mathrm{S}\mathrm{T}}_j}^{-\alpha }/\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_j}^{-\alpha } \), \( c2=\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_j}^{-\alpha } \), \( b2=-{\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_j}^{-\alpha } \).

APPENDIX 2

From Eqs. (7) and (8), we have

  • \( {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2\le \frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2-\frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2-\frac{\sigma^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }} \),

  • \( {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2\ge \frac{\left({2}^{R_{\mathrm{P}}}-1\right){P}_S{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2+\frac{\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha }} \),

  • \( {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2\le \frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2-\frac{P_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2-\frac{\sigma^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }} \), and

  • \( {\left|{h}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2\ge \frac{\left({2}^{R_{\mathrm{P}}}-1\right){P}_S{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }}{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2+\frac{\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2}{P_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }} \).

From (Eq. 21), the upper bound probability thatSTScan meet the Interference constraints is given by

$$ {\displaystyle \begin{array}{l}\Pr \left\{{P}_{\mathrm{S}}{g}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{ub}\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}},{P}_{\mathrm{S}}{g}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{ub}\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right\}\\ {}=\Pr \Big\{{P}_{\mathrm{S}}{\left(\left\lfloor \frac{K-1}{2}\right\rfloor {d}_3\right)}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^1,{P}_{\mathrm{S}}{\left(\left\lfloor \frac{K-1}{2}\right\rfloor {d}_3\right)}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}\right|}^2\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^2,\\ {}{P}_{\mathrm{S}}{\left(\left\lfloor \frac{K-1}{2}\right\rfloor {d}_3\right)}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^1,{P}_{\mathrm{S}}{\left(\left\lfloor \frac{K-1}{2}\right\rfloor {d}_3\right)}^{-\alpha }{\left|{h}_{{\mathrm{S}\mathrm{T}}_{\mathrm{S}},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}\right|}^2\le {I}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^2\Big\}\\ {}={\lambda}_1+{\lambda}_2+{\lambda}_3+{\lambda}_4.\end{array}} $$
(24)

The calculation result of the upper bound probability consists of the following four components.

$$ {\displaystyle \begin{array}{l}{\lambda}_1=\frac{\Omega_{1\left(\mathrm{P}1\right)}{\Omega}_{1\left(\mathrm{P}2\right)}\exp \left(-c1/{\Omega}_{1\left(\mathrm{P}1\right)}-c2/{\Omega}_{1\left(\mathrm{P}2\right)}\right)}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\left({\varOmega}_{1(P1)}+b1{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)\left({\varOmega}_{1(P2)}+b2{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}\Big[-\exp \left(\left(\frac{b1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)\right)\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)}\\ {}\kern1em +\frac{\exp \left(-x0\left(a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a1/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\right)\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a1/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}-\exp \left(\left(\frac{b2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)\right)\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)}\\ {}\kern1em +\frac{\exp \left(-\frac{x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}-\frac{a1\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)/x0}-\frac{a2\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)/x0}\right)}{\frac{a1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{a2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}+\frac{a1\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}}\\ {}\kern1em \times \exp \left(\left(\frac{b1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}+\left(\frac{b2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)-\frac{a1\cdot x0}{\Omega_{1\left(\mathrm{P}1\right)}}-\frac{a2\cdot x0}{\Omega_{1\left(\mathrm{P}2\right)}}\Big],\end{array}} $$
$$ {\displaystyle \begin{array}{l}{\lambda}_2=\frac{\Omega_{1\left(\mathrm{P}1\right)}{\Omega}_{1\left(\mathrm{P}2\right)}\exp \left(-t1/{\Omega}_{1\left(\mathrm{P}1\right)}-t2/{\Omega}_{1\left(\mathrm{P}2\right)}\right)}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\left({\Omega}_{1\left(\mathrm{P}1\right)}+s1{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)\left({\Omega}_{1\left(\mathrm{P}2\right)}+s2{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}\Big[-\exp \left(\left(\frac{s2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)\right)\right)}{1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)}-\exp \left(\frac{-x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)\\ {}\kern1em \times \exp \left(\left(\frac{s1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}\right)\frac{\exp \left(-x0\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)\right)}{1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)}\\ {}\kern1em +{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\exp \left(\frac{-x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)+\exp \left(\left(\frac{s1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}+\left(\frac{s2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(\frac{1}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}+\frac{a1\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}\right)\right)}{\frac{1}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}+\frac{a1\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}}\Big],\end{array}} $$
$$ {\displaystyle \begin{array}{l}{\lambda}_3=\frac{-{\Omega}_{1\left(\mathrm{P}1\right)}{\Omega}_{1\left(\mathrm{P}2\right)}\exp \left(-c1/{\Omega}_{1\left(\mathrm{P}1\right)}-t2/{\Omega}_{1\left(\mathrm{P}2\right)}\right)}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\left({\Omega}_{1\left(\mathrm{P}1\right)}+b1{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)\left({\Omega}_{1\left(\mathrm{P}2\right)}+s2{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}\Big[\frac{\exp \left(-x0a1/{\Omega}_{1\left(\mathrm{P}1\right)}\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\\ {}\kern1em \times \exp \left(\frac{-x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)+\exp \left(\left(\frac{b1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}+\left(\frac{s2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(\frac{a1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{a1\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}\right)\right)}{\frac{a1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}+\frac{a1\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}}\\ {}\kern1em \times \exp \left(\frac{-x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)-\frac{\exp \left(-x0\left(a1/{\Omega}_{1\left(\mathrm{P}1\right)}+\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)\right)\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)}\\ {}\kern1em \times \exp \left(\left(\frac{b1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}-\frac{x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)-\exp \left(\left(\frac{s2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(a1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)\right)\right)}{a1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)}\Big],\end{array}} $$
$$ {\displaystyle \begin{array}{l}{\lambda}_4=\frac{-{\Omega}_{1\left(\mathrm{P}1\right)}{\Omega}_{1\left(\mathrm{P}2\right)}\exp \left(-t1/{\Omega}_{1\left(\mathrm{P}1\right)}-c2/{\Omega}_{1\left(\mathrm{P}2\right)}\right)}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\left({\Omega}_{1\left(\mathrm{P}1\right)}+s1{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)\left({\Omega}_{1\left(\mathrm{P}2\right)}+b2{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}\Big[-\exp \left(\frac{-x0a2}{\varOmega_{1(P2)}}+\frac{-x0}{\varOmega_{3(P1)(P2)}}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\frac{a1\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\left(\frac{s1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}\right)}{a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)a1/\left(s1-b1\right)}\\ {}\kern1em +\frac{\exp \left(-x0\left(\frac{a2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{a1\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}\right)\right)}{\frac{a2}{\varOmega_{1(P2)}}+\frac{1}{\varOmega_{3(P1)(P2)}}+\frac{a1\left(s1/{\Omega}_{1\left(\mathrm{P}1\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}1\right)}\right)}{\left(s1-b1\right)}+\frac{a2\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)}{\left(s2-b2\right)}}\\ {}\kern1em \times \exp \left(\left(\frac{s1}{\Omega_{1\left(\mathrm{P}1\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}1\right)}}\right)\frac{t1-c1}{s1-b1}+\left(\frac{b2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}-\frac{x0}{\Omega_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}\right)\\ {}\kern1em +\frac{\exp \left(-x0\left(a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}\right)\right)}{a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}}-\exp \left(\left(\frac{b2}{\Omega_{1\left(\mathrm{P}2\right)}}+\frac{1}{\Omega_{3\mathrm{S}\left(\mathrm{P}2\right)}}\right)\frac{t2-c2}{s2-b2}\right)\\ {}\kern1em \times \frac{\exp \left(-x0\left(a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)\right)\right)}{a2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\left(\mathrm{P}1\right)\left(\mathrm{P}2\right)}+\left(b2/{\Omega}_{1\left(\mathrm{P}2\right)}+1/{\Omega}_{3\mathrm{S}\left(\mathrm{P}2\right)}\right)a2/\left(s2-b2\right)}\Big].\end{array}}. $$

Where x0 = max {(t1 − c1)/a1, t2 − c2/a2},\( a2={P}_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }/\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha } \),\( t1=\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_{\mathrm{P}1}}^{-\alpha } \),\( t2=\left({2}^{R_{\mathrm{P}}}-1\right){\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_{\mathrm{P}2}}^{-\alpha } \), \( a1={P}_{\mathrm{S}}{d}_{{\mathrm{S}\mathrm{T}}_{\mathrm{P}1},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha }/\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha } \), \( s1=\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{S}}{\left(\left\lfloor \left(K-1\right)/2\right\rfloor {d}_3\right)}^{-\alpha }/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha } \),\( s2=\left({2}^{R_{\mathrm{P}}}-1\right){P}_{\mathrm{S}}{\left(\left\lfloor \left(K-1\right)/2\right\rfloor {d}_3\right)}^{-\alpha }/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha } \),\( b1=-{P}_{\mathrm{S}}{\left(\left\lfloor \left(K-1\right)/2\right\rfloor {d}_3\right)}^{-\alpha }/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}1}}^{-\alpha } \),\( c1=-{\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_{\mathrm{P}1}}^{-\alpha } \),\( b2=-{P}_{\mathrm{S}}{\left(\left\lfloor \left(K-1\right)/2\right\rfloor {d}_3\right)}^{-\alpha }/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{S}\mathrm{T}}_{\mathrm{P}2}}^{-\alpha } \),\( c2=-{\sigma}^2/{P}_{\mathrm{P}}{d}_{\mathrm{P}\mathrm{T},{\mathrm{ST}}_{\mathrm{P}2}}^{-\alpha } \).

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Xie, P., Liu, J., Zhang, G. et al. Opportunistic TPSR cooperative spectrum sharing protocol with secondary user selection for 5G wireless network. Peer-to-Peer Netw. Appl. 13, 1272–1283 (2020). https://doi.org/10.1007/s12083-019-00843-y

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