Effective User-Pair Association Schemes for Relay-Based Multi-tier Heterogeneous Networks with Physical Layer Security

Abstract

This paper focuses on relay-assisted multi-tier heterogeneous networks (HetNets) in term of feasible user-pair association (UPA) criterions and physical layer secrecy analysis. In the interesting relay-assisted multi-tier HetNets, the randomly located mobile user-pair communicates with the help of the relay of its associated tier. We model the locations of all network elements as independent Poisson point process. For such relay-assisted HetNets, similar to the nearest relay defined for user association in traditional single-hop HetNets, we define the so-called best relay in each tier for a typical mobile user-pair by using the equivalent end-to-end biased received power (BRP). Then based on the defined best-relay, we first propose the max–min user-pair association (MM-UPA) criterion. Due to the fact that the MM-UPA criterion is dominated by the bottleneck link’s BRP and does not exploit the joint effect of both the source-relay and relay-destination links, we present the maximum harmonic mean user-pair association (MHM-UPA) criterion, again. For the two UPA criterions, by using feasible mathematical analysis, we derive the corresponding UPA probabilities. Finally, as an implement of the two proposed UPA criterions, by using stochastic geometry, we perform the secrecy performance analysis of the considered relay-assisted multi-tier HetNets. The presented numerical analysis first validates our derivations through the comparison analysis with traditional single-hop user association criterion. At the same time, we also present the comparison analysis between the two proposed MM-UPA and MHM-UPA criterions. It is found that when the transmission power $P_{R(S)}^{k}$ is small, the MHM-UPA scheme outperforms the MM-UPA one in term of UPA probability. On the contrary, the two schemes achieve approximately the same UPA probability. For the total secrecy probability, we find that when transmit power is small, the MHM-UPA achieves the higher secrecy probabilities. Moreover, the achieved gain by MHM-UPA is increasing with the decrease of transmission power. Contrarily, when the transmission power is large, although the MM-UPA outperforms the MHM-UPA, the achieved gain by MM-UPA cover MHM-UPA is small.

Appendices

Appendix A

Proof of Lemma 1

With (7), the CDF of $P_{MM-SD}^{j}$ is calculated as follows.

$$\begin{aligned} F_{{P_{MM-SD}^{j} }} (x) & = \Pr \left\{ {P_{SD}^{j} \le x} \right\} = \Pr \left\{ {\hbox{min} \left( {\frac{{\left( {P_{S}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} }}{{R_{SR}^{j} }},\frac{{\left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} }}{{R_{RD}^{j} }}} \right) \le x} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop = \limits^{(a)} 1 - \Pr \left\{ {\hbox{min} \left( {\frac{{\left( {P_{S}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} }}{{R_{SR}^{j} }},\frac{{\left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} }}{{R_{RD}^{j} }}} \right) > x} \right\} \\ & \mathop = \limits^{(b)} 1 - \Pr \left\{ {R_{SR}^{j} < \left( {P_{S}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} x^{ - 1} } \right\}\left\{ {R_{SD}^{j} < \left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} x^{ - 1} } \right\}\mathop = \limits^{(c)} 1 - \left( {1 - \exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} x^{ - 2} } \right)} \right)\left( {1 - \exp \left( { - \pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} x^{ - 2} } \right)} \right) \\ \end{aligned}$$

(43)

In (43), (a) follows from the order statistics, (b) from the independent assumption. In addition, (c) follows from the fact that in homogeneous PPP, the number of points follows Poisson distribution and the distance between a generic user and its association relay is Rayleigh, i.e., $f_{{R_{SR}^{j} }} (x) = 2\pi \lambda_{j} \exp \left( { - \pi \lambda_{j} x^{2} } \right)$ and $f_{{R_{RD}^{j} }} (x) = 2\pi \lambda_{j} \exp \left( { - \pi \lambda_{j} x^{2} } \right)$. Note that, in [28] the two results were also adopted. The PDF $f_{{P_{MM-SD}^{j} }} (x)$ is achieved by taking the derivative of $F_{{P_{MM-SD}^{j} }} (x)$.□

Appendix B

Proof of Lemma 2

To achieve the CDF of $P_{MHM-SD}^{j}$, by using the definition (9) we have

$$F_{{P_{MHM-SD}^{j} }} \left( z \right) = \Pr \left\{ {\frac{{X_{j} Y_{j} }}{{X_{j} + Y_{j} }} \le z} \right\} = \Pr \left\{ {X_{j} \le \frac{{zY_{j} }}{{Y_{j} - z}}} \right\}$$

(44)

Using the result in [43] leads to

$$F_{{P_{MHM-SD}^{j} }} \left( z \right) = I_{1} + I_{2}$$

(45)

where we define

$$I_{1} = \int_{0}^{z} {\Pr \left\{ {X_{j} > \frac{zy}{y - z}} \right\}} f_{{Y_{j} }} \left( y \right)dy;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_{2} = \int_{z}^{\infty } {\Pr \left\{ {X_{j} \le \frac{zy}{y - z}} \right\}} f_{{Y_{j} }} \left( y \right)dy$$

(46)

It is easy to see that $I_{1} = F_{{Y_{j} }} \left( z \right)$, since ${ \Pr }\left\{ {X_{j} > \frac{{zY_{j} }}{{Y_{j} - z}}} \right\} = 1$ as $0 \le Y_{j} \le z$. Moreover, with the definition $Y_{j} = \left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} /R_{RD}^{j}$, we have

$$F_{{{\text{Y}}_{j} }} \left( y \right) = { \Pr }\left\{ {\frac{{\left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} }}{{R_{RD}^{j} }} < y} \right\} = { \Pr }\left\{ {R_{RD}^{j} > \left( {P_{R}^{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} y^{ - 1} } \right\} = \exp \left( { - \pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} y^{ - 2} } \right)$$

(47)

The PDF of $Y_{j}$ is

$$F_{{{\text{Y}}_{j} }} \left( y \right) = 2\pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} y^{ - 3} \exp \left( { - \pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} y^{ - 2} } \right)$$

(48)

We have

$$I_{1} = \exp \left( { - \pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} z^{ - 2} } \right)$$

(49)

Therefore, we have (14).

With the similar argument, we have that the CDF of $X_{j}$ is $F_{{X_{j} }} \left( x \right) = \exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} x^{ - 2} } \right)$. This leads to the term $I_{2}$ in (46) written as

$$\begin{aligned} I_{2} & = \int_{z}^{\infty } {\exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \left( {\frac{yz}{y - z}} \right)^{ - 2} } \right)} \times \frac{{2\pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{y^{3} }}\exp \left( { - \pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} y^{ - 2} } \right)dy \\ & = 2\pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} z^{ - 2} } \right)\int_{z}^{\infty } {\frac{1}{{y^{3} }}} \exp \left( { - \pi \lambda_{j} \left( {\frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{y^{2} }} - \frac{{2\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{yz}} \right)} \right)dy{\kern 1pt} \\ \end{aligned}$$

(50)

With the variable substitution $t = 1/y$, (50) can be further written as

$$\begin{aligned} I_{ 2} & = 2\pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} z^{ - 2} } \right)\int_{0}^{{\frac{1}{z}}} t \exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\left( {t^{2} - \frac{{2\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} t}}{{\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)z}}} \right)} \right)dt \\ & {\kern 1pt} = 2\pi \lambda_{j} \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \exp \left( { - \pi \lambda_{j} \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} z^{ - 2} } \right)\exp \left( {\pi \lambda_{j} \frac{{\left( {P_{S}^{j} } \right)^{{\frac{4}{{\alpha_{j} }}}} }}{{z^{2} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right) \times \left( {I_{21} + I_{22} } \right) \\ \end{aligned}$$

(51)

where $I_{21}$ and $I_{22}$ are defined respectively by

$$I_{21} = \int_{0}^{{\frac{1}{z}}} {\left( {t - \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right)} \times \exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\left( {t - \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right)^{2} } \right)dt$$

(52)

$$I_{22} = \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}\int_{0}^{{\frac{1}{z}}} {\exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\left( {t - \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right)^{2} } \right)} dt$$

(53)

The term I₂₁ in (52) is calculated as

$$I_{ 2 1} = \frac{1}{2} \times \frac{1}{{\pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}} \times \left[ {\exp \left( { - \pi \lambda_{j} z^{ - 2} \frac{{\left( {P_{S}^{j} } \right)^{{\frac{4}{{\alpha_{j} }}}} }}{{\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right)} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \left. {\exp \left( { - \pi \lambda_{j} z^{ - 2} \frac{{\left( {P_{R}^{j} } \right)^{{\frac{4}{{\alpha_{j} }}}} }}{{\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}} \right)} \right]$$

(54)

Substituting (54) into (51), we have (15) and (16).

At the same time, in (53) by using the variable substitution $x = t - \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} /z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)$ the term I₂₂ can be calculated as

$$\begin{aligned} I_{22} & = \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}\int_{{ - \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}}}}^{{\frac{1}{z}\frac{{\left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}}} {\exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)x^{2} } \right)} dx \\ & = \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)}} \times \left[ {\int_{0}^{{\frac{1}{z}\frac{{\left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}}} {\exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)x^{2} } \right)} dx} \right.{\kern 1pt} {\kern 1pt} + \left. {\int_{0}^{{\frac{1}{z}\frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}}} {\exp \left( { - \pi \lambda_{j} \left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)x^{2} } \right)} dx} \right] \\ & = \frac{{\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{z\left( {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)^{{\frac{2}{{\alpha_{j} }}}} }} \times \left[ {\frac{\sqrt \pi }{2}\Upphi \left( {\frac{1}{z}\frac{{\sqrt {\pi \lambda_{j} } \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{\sqrt {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } }}} \right) + \left. {\frac{\sqrt \pi }{2}\Upphi \left( {\frac{1}{z}\frac{{\sqrt {\pi \lambda_{j} } \left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} }}{{\sqrt {\left( {P_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} + \left( {P_{R}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} } }}} \right)} \right] \times \frac{1}{{\sqrt {\pi \lambda_{j} } }}} \right. \\ \end{aligned}$$

(55)

where $\Upphi \left( \cdot \right) = erf\left( \cdot \right)$ is the error function defined by (8.250.1) in [46]. Substituting (55) into (51), we have (17) and (18). The PDF of $P_{MHM-SD}^{j}$ can be achieved by taking the derivative of $F_{i} (z)$, $i = 1,2,3,4,5$.□

Appendix C

Proof of Theorem 3

Obviously, to achieve $P_{Sec}^{k}$, $P_{Sec-S}^{k}$ and $P_{Sec-R}^{k}$ in (38) are required. We first consider the term $P_{Sec-S}^{k}$, which is written as

$$\begin{aligned} P_{Sec-S}^{k} & = \Pr \left\{ {\mathop {\hbox{max} }\limits_{{z \in \Upphi_{E} }} \left( {SINR_{SE}^{k} \left( z \right)} \right) \le T_{Sec} } \right\} \\ & {\kern 1pt} = \Pr \left\{ {\bigcap\limits_{{z \in \Upphi_{E} }} {SINR_{SE}^{k} \left( z \right) < T_{Sec} } } \right\}{\kern 1pt} = E_{{\Upphi_{E} }} \left[ {\prod\limits_{{z \in \Upphi_{E} }} {\Pr \left\{ {SINR_{SZ}^{k} \left( z \right) < T_{Sec} \left| z \right.} \right\}} } \right] \\ \end{aligned}$$

(56)

Then, by defining $I_{S} = \sum_{j = 1}^{K} {\sum_{{i \in \Upphi_{j} \backslash S_{k} }} {P_{S}^{j} h_{Siz}^{j} \left( {Y_{Siz}^{j} } \right)^{{ - \alpha_{j} }} } }$ and substituting (35) into (56), we have

$$P_{Sec-S}^{k} = E_{{\Upphi_{E} }} \left[ {\prod\limits_{{z \in \Upphi_{E} }} {\left[ {1 - \exp \left( { - \left( {P_{S}^{k} \left( {Y_{SE}^{k} } \right)^{{ - \alpha_{k} }} } \right)^{ - 1} T_{Sec} \frac{w}{{L_{0} }}} \right)\mathcal{L}_{{I_{S} }} \left( {T_{Sec} \left( {\left( {P_{S}^{k} \left( {Y_{SE}^{k} } \right)^{{ - \alpha_{k} }} } \right)^{ - 1} } \right)} \right)} \right]} } \right]$$

(57)

where $\mathcal{L}_{{I_{S} }} \left( \cdot \right)$ is the Laplace transformation of $I_{S}$. From A.3 in [48] Eq. (57) can be written as

$$P_{Sec-S}^{k} = \exp \left( {\left( { - 2\pi \lambda_{E} } \right)\int_{0}^{\infty } {\exp \left( { - \left( {P_{S}^{k} r^{{ - \alpha_{k} }} } \right)^{ - 1} T_{Sec} \frac{w}{{L_{0} }}} \right)} \mathcal{L}_{{I_{S} }} \left( {T_{Sec} \left( {\left( {P_{S}^{k} r^{{ - \alpha_{k} }} } \right)^{ - 1} } \right)} \right)rdr} \right)$$

(58)

With the definition of $I_{S} = \sum_{j = 1}^{k} {\sum_{{i \in \Upphi_{j} \backslash S_{k} }} {P_{S}^{j} h_{Siz}^{j} \left( {Y_{Siz}^{j} } \right)^{{ - \alpha_{j} }} } }$, we have the Laplace transform $\mathcal{L}_{{I_{S} }} \left( s \right)$ given by

$$\begin{aligned} \mathcal{L}_{{I_{S} }} \left( s \right) & = E_{{\Upphi_{j} ,h}} \left[ {\exp \left( { - s\sum_{j = 1}^{K} {\sum_{{i \in \Upphi_{j} \backslash S_{k} }} {P_{S}^{j} h_{Siz}^{j} \left( {Y_{Siz}^{j} } \right)^{{ - \alpha_{j} }} } } } \right)} \right] = \prod\limits_{j = 1}^{K} {E_{{\Upphi_{j} ,h}} } \left[ {\exp \left( { - s\sum_{{i \in \Upphi_{j} \backslash S_{k} }}^{k} {P_{S}^{j} h_{Siz}^{j} \left( {Y_{Siz}^{j} } \right)^{{ - \alpha_{j} }} } } \right)} \right] \\ & {\kern 1pt} \mathop = \limits^{(a)} \prod\limits_{j = 1}^{K} {\exp \left( { - 2\pi \lambda_{j} \int_{0}^{\infty } {\left( {1 - \frac{1}{{1 + sP_{S}^{j} r^{{ - \alpha_{j} }} }}} \right)rdr} } \right)} = \prod\limits_{j = 1}^{K} {\exp \left( { - 2\pi \lambda_{j} \int_{0}^{\infty } {\frac{rdr}{{1 + \left( {sP_{S}^{j} } \right)^{ - 1} r^{{\alpha_{j} }} }}} } \right)} \\ & = \prod\limits_{j = 1}^{K} {\exp \left( { - \pi \lambda_{j} \left( {sP_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \frac{2}{{\alpha_{j} }}\int_{0}^{\infty } {\frac{{x^{{\frac{2}{{\alpha_{j} }} - 1}} dx}}{1 + x}} } \right)} \mathop = \limits^{(b)} \prod\limits_{j = 1}^{K} {\exp \left( { - \pi^{2} \lambda_{j} \left( {sP_{S}^{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} \times {\text{cosec}}\left( {\frac{2}{{\alpha_{j} }}} \right)} \right)} \\ \end{aligned}$$

(59)

where (a) follows from the mapping theorem [48] and (b) follows from the (3.194.4) in [46]. Then, by substituting (59) into (58) we have (41).

$$P_{Sec-S}^{k} = \exp \left( { - 2\pi \lambda_{E} \int_{0}^{\infty } {\exp \left( { - \left( {P_{S}^{k} r^{{ - \alpha_{k} }} } \right)^{ - 1} T_{Sec} \frac{w}{{L_{0} }} - \sum_{j = 1}^{K} {\pi^{2} \lambda_{j} \left( {\left( {T_{Sec} r^{{ - \alpha_{k} }} } \right)^{ - 1} \frac{{P_{S}^{j} }}{{P_{S}^{k} }}} \right)^{{\frac{2}{{\alpha_{j} }}}} \cos {\text{ec}}\left( {\frac{2}{{\alpha_{j} }}} \right)} } \right)} } \right)$$

(60)

With the consideration of symmetry between $SINR_{RZ}^{k} \left( z \right)$ and $SINR_{SZ}^{k} \left( z \right)$, it is easy to achieve $P_{Sec-R}^{k}$ by replacing $P_{S}^{k}$ and $P_{S}^{j}$ with $P_{R}^{k}$ and $P_{R}^{j}$, respectively. We have (42).□