Min–Max User-Pair Association Criterion and Outage Performance of K-Tier Relay-Based Heterogeneous Networks

Abstract

This paper focuses on the user-pair association for multi-tier relay-based dual-hop heterogeneous networks (HetNets) and proposes a novel min–max user-pair association (MM-UPA) criterion by integrating the bias factors, which is an evolution of conventional single-hop user association based on receive signal strength (RSS). The core idea of the proposed MM-UPA is that, in each tier the nearest relay to a typical user-pair is characterized by the maximum of the RSS reciprocals ${1 / {P_{SR}^{j} }}$ and ${1/ {P_{RD}^{j} }}$ by the source and destination from relay. Then, a typical user-pair is associated to the relay based on minimizing these maximums of each tier. This proposed MM-UPA criterion is fit especially for the relay-based HetNets due to effectively exploiting the source-relay and relay-destination links simultaneously. With the MM-UPA criterion, by using stochastic geometry and homogeneous Poisson point processes, we present the analytical expression of the probability that a typical user-pair is associated with a relay of the kth tier as well as the corresponding statistical description of the distances from the source and destination of a typical user-pair to its associated relay. Finally, the average outage probability of a typical user-pair relay communication link is derived. The presented simulations and numerical results validate the derivations.

Appendices

Appendix 1: Proof of (15)

With (14), the product term in (14) is written as

$$\begin{aligned} & \mathop \prod \limits_{j = 1,j \ne k}^{K} 1 - \left( {1 - \exp \left( { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} \left( x \right)^{{ - \frac{2}{{\alpha_{j} }}}} } \right)} \right)^{2} = \mathop \prod \limits_{j = 1,j \ne k}^{K} 2\exp \left( { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} \left( x \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\left( {1 - y_{j} } \right) \\ & \quad = 2^{K - 1} \exp \left( { - \sum\limits_{j = 1,j \ne k}^{K} {} \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} \left( x \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\mathop \prod \limits_{j = 1,j \ne k}^{K} \left( {1 - y_{j} } \right) \\ \end{aligned}$$

(29)

where $y_{j}$ is defined as

$$y_{j} = \frac{1}{2}\exp \left( { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} \left( x \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)$$

(30)

Furthermore, the product term of the right hand in (29) can be described in a more tractable form with the help of the identity product given by [27]

$$\mathop \prod \limits_{j = 1,j \ne k}^{K} \left( {1 - y_{j} } \right) = \sum\limits_{l = 0}^{K} {\frac{{( - 1)^{l} }}{l!}} \underbrace {{\sum\limits_{{n_{1} = 1}}^{K} { \ldots \sum\limits_{{n_{l} = 1}}^{K} {} } }}_{{n_{1} \ne n_{2} \ne \cdots \ne n_{l} \ne k}}\prod\limits_{t = 1}^{l} {y_{{n_{t} }} }$$

(31)

From (30) and (31), we can describe (29) by

$$2^{K - 1} \exp \left( { - \sum\limits_{j = 1,j \ne k}^{K} {} \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} \left( x \right)^{{\frac{2}{{\alpha_{j} }}}} } \right)\sum\limits_{l = 0}^{K} {\frac{{( - 1)^{l} }}{l!}} \underbrace {{\sum\limits_{{n_{1} = 1}}^{K} { \ldots \sum\limits_{{n_{l} = 1}}^{K} {} } }}_{{n_{1} \ne n_{2} \ne \cdots \ne n_{l} \ne k}}\prod\limits_{t = 1}^{l} {\left( {\frac{1}{2}\exp \left( { - \pi \lambda_{{n_{t} }} \left( {\widehat{P}_{R}^{{n_{t} }} \widehat{\beta }_{{n_{t} }} } \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} \left( x \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} } \right)} \right)}$$

(32)

The product term in (32) is given by

$$\prod\limits_{t = 1}^{l} {\left( {\frac{1}{2}\exp \left( { - \pi \lambda_{{n_{t} }} \left( {\widehat{P}_{R}^{{n_{t} }} \widehat{\beta }_{{n_{t} }} } \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} \left( x \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} } \right)} \right)} = 2^{ - l} \exp \left( { - \sum\limits_{t = 1}^{l} {} \pi \lambda_{{n_{t} }} \left( {\widehat{P}_{R}^{{n_{t} }} \widehat{\beta }_{{n_{t} }} } \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} \left( x \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} } \right)$$

(33)

Combining (33) and (32) leads to (15).

Appendix 2: Proof of (19)

For the joint probability $\Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,n = k} \right\}$, using the definition $R_{SD}^{k} = \hbox{max} \left( {\left( {R_{SR}^{k} } \right)_{{}}^{{\alpha_{k} }} ,\left( {R_{RD}^{k} } \right)_{{}}^{{\alpha_{k} }} } \right)$ in (6) and the idea of the proposed MM-UPA criterion we have

$$\begin{aligned} \Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,n = k} \right\} & = \Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,P_{k} (R_{SD}^{k} (R_{SR}^{k} ,R_{RD}^{k} )) < \mathop {\hbox{min} }\limits_{{j = \left\{ {1,2, \ldots ,K} \right\}\backslash k}} \left( {P_{j} } \right){\kern 1pt} {\kern 1pt} } \right\} \\ & \mathop = \limits^{(a)} \int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} \Pr \left\{ {P_{j} > P_{k} (R_{SD}^{k} (u,v))} \right\}f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv} } \\ & = \int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} \Pr \left\{ {\hbox{max} \left( {\left( {R_{SR}^{j} } \right)_{{}}^{{\alpha_{j} }} ,\left( {R_{RD}^{j} } \right)_{{}}^{{\alpha_{j} }} } \right) > \widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} \left( {r_{0} } \right)^{{\alpha_{j} - \alpha_{k} }} {\kern 1pt} (R_{SD}^{k} (u,v))} \right\}f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv} } \\ & { = }\int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} \left[ {1 - \Pr \left\{ {\hbox{max} \left( {\left( {R_{SR}^{j} } \right)_{{}}^{{\alpha_{j} }} ,\left( {R_{RD}^{j} } \right)_{{}}^{{\alpha_{j} }} } \right) < \widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} \left( {r_{0} } \right)^{{\alpha_{j} - \alpha_{k} }} {\kern 1pt} (R_{SD}^{k} (u,v))} \right\}} \right]f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv} } \\ & & { = }\int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} \left[ {1 - \Pr \left\{ {\left( {R_{SR}^{j} } \right)_{{}}^{{\alpha_{j} }} < \widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} \left( {r_{0} } \right)^{{\alpha_{j} - \alpha_{k} }} {\kern 1pt} (R_{SD}^{k} (u,v))} \right\} \times \Pr \left\{ {\left( {R_{RD}^{j} } \right)_{{}}^{{\alpha_{j} }} < \widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} \left( {r_{0} } \right)^{{\alpha_{j} - \alpha_{k} }} {\kern 1pt} (R_{SD}^{k} (u,v))} \right\}} \right]f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv} } \\ \end{aligned}$$

(34)

where (a) follows from the independence assumption, $f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)$ is the joint PDF of the RVs $R_{SR}^{k}$ and $R_{RD}^{k}$. Then, using the law of statistics [25] and the independence between $R_{SR}^{j}$ and $R_{RD}^{j}$, the Eq. (34) is further written as

$$\begin{aligned} \Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,n = k} \right\} & { = }\int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} 1 - \left( {1 - \Pr \left\{ {R_{SR}^{j} > \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{1}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{1 - \frac{1}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} \left( u \right)_{{}}^{{\frac{1}{{\widehat{\alpha }_{j} }}}} } \right\} \times \Pr \left\{ {R_{SR}^{j} > \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{1}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{1 - \frac{1}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} {\kern 1pt} \left( v \right)_{{}}^{{\frac{1}{{\widehat{\alpha }_{j} }}}} } \right\}} \right)} } \\ & \quad \times \left( {1 - \Pr \left\{ {R_{RD}^{j} > \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{1}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{1 - \frac{1}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} \left( u \right)_{{}}^{{\frac{1}{{\widehat{\alpha }_{j} }}}} } \right\} \times \Pr \left\{ {R_{RD}^{j} > \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{1}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{1 - \frac{1}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} {\kern 1pt} \left( v \right)_{{}}^{{\frac{1}{{\widehat{\alpha }_{j} }}}} } \right\}} \right)f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv \\ \end{aligned}$$

(35)

With the consideration that the CDFs of $R_{SR}^{j}$ and $R_{RD}^{j}$ are $F_{{R_{RD}^{j} (R_{SR}^{j} )}} \left( x \right) = 1 - \exp \left( { - \pi \lambda_{j} x^{2} } \right)$, we have the joint probability

$$\begin{aligned} \Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,n = k} \right\} & = \int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} 1 - \left( {1 - \exp \left\{ { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{2}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} \left( {\left( u \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} + \left( v \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} } \right)} \right\}} \right)^{2} } } f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv \\ & = \int\limits_{0}^{x} {\int\limits_{0}^{y} {\mathop \prod \limits_{j = 1,j \ne k}^{K} \exp \left\{ { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{2}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} \left( {\left( u \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} + \left( v \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} } \right)} \right\} \times \left( {1 - \frac{1}{2}\exp \left\{ { - \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} {\kern 1pt} } \right)^{{\frac{2}{{\alpha_{j} }}}} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{j} }}}} {\kern 1pt} \left( {\left( u \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} + \left( v \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{j} }}}} } \right)} \right\}} \right)} } f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv \\ & = \sum\limits_{l = 1}^{K} {\frac{{( - 1)^{l} }}{l!}} \underbrace {{\sum\limits_{{n_{1} = 1}}^{K} { \ldots \sum\limits_{{n_{l} = 1}}^{K} {} } }}_{{n_{1} \ne n_{2} \ne \cdots \ne n_{l} \ne k}}2^{K - 1 - l} \int\limits_{0}^{x} {\int\limits_{0}^{y} {\exp \left( { - (C_{l} + D_{K} )} \right)} } f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right)dudv \\ \end{aligned}$$

(36)

Where the similar argument as in (32) is used, and $A_{lh}$ and $B_{lh}$ are defined as

$$C_{l} = \sum\limits_{t = 1}^{l} {} \pi \lambda_{{n_{t} }} \left( {\widehat{P}_{R}^{{n_{t} }} \widehat{\beta }_{{n_{t} }} } \right)^{{\frac{2}{{\alpha_{{n_{t} }} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} {\kern 1pt} \left( {\left( u \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} + \left( v \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{{n_{t} }} }}}} } \right),\quad D_{k} = \sum\limits_{j = 1,j \ne k}^{K} {} \pi \lambda_{j} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} {\kern 1pt} \left( {r_{0} } \right)^{{2 - \frac{2}{{\widehat{\alpha }_{{_{j} }} }}}} \left( {\left( u \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{{_{j} }} }}}} + \left( v \right)_{{}}^{{\frac{2}{{\widehat{\alpha }_{{_{j} }} }}}} } \right)$$

(37)

With the independence between $R_{SR}^{k}$ and $R_{RD}^{k}$, using Lemma 1 we have the joint CDF of $R_{SR}^{k}$ and $R_{RD}^{k}$

$$F_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right){\text{ = Pr}}\left\{ {R_{SR}^{k} < u,R_{RD}^{k} < v} \right\} = \left( {1 - \exp \left( { - \pi \lambda_{k} u^{2} } \right)} \right) \times \left( {1 - \exp \left( { - \pi \lambda_{k} v^{2} } \right)} \right)$$

(38)

As a result, the joint PDF of $R_{SR}^{k}$ and $R_{RD}^{k}$ is given by

$$f_{{R_{SR}^{k} ,R_{RD}^{k} }} \left( {u,v} \right) = \left( {2\pi \lambda_{k} } \right)^{2} (uv)\exp \left( { - \pi \lambda_{k} \left( {u^{2} + v^{2} } \right)} \right)$$

(39)

Finally, by substituting (37) and (39) into (36) and taking the derivative of $\Pr \left\{ {R_{SR}^{k} \le x,R_{RD}^{k} \le y,n = k} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt}$ with respect to $x$ and $y$, we have Theorem 2.

Appendix 3: Proof of (26)

Here, we only present the proof for the case $X_{SR}^{k} \ge Y_{RD}^{k}$. The results for $X_{SR}^{k} < Y_{RD}^{k}$ are straightforward. With the definition $SINR_{SR}^{k}$ in (21), by defining the total interference term $I_{SR}^{k} = \sum\nolimits_{j = 1}^{K} {\sum\nolimits_{{i \in \varPhi_{j} \backslash k}} {P_{S}^{j} h_{{SR_{i} }}^{j} \left( {Y_{{SR_{i} }}^{j} } \right)^{{ - \alpha_{j} }} } }$ we have

$$\begin{aligned} \Pr \left\{ {SINR_{SR}^{k} > \tau } \right\} & { = }\Pr \left\{ {\frac{{P_{S}^{k} h_{SR}^{k} \left( {X_{SR}^{k} } \right)^{{ - \alpha_{k} }} }}{{I_{SR}^{k} + {W \mathord{\left/ {\vphantom {W {L_{0} }}} \right. \kern-0pt} {L_{0} }}}} > \tau } \right\}{\kern 1pt} {\kern 1pt} \mathop = \limits^{\left( a \right)} \exp \left( { - \left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau {W \mathord{\left/ {\vphantom {W {L_{0} }}} \right. \kern-0pt} {L_{0} }}} \right)E_{{I_{SR}^{k} }} \left[ {\exp \left( { - \left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau I_{SR}^{k} } \right)} \right] \\ & {\kern 1pt} \mathop = \limits^{\left( b \right)} { \exp }\left( { - \left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau {W \mathord{\left/ {\vphantom {W {L_{0} }}} \right. \kern-0pt} {L_{0} }}} \right)\mathop \prod \limits_{j = 1}^{K} \mathcal{L}_{{I_{SRj}^{k} }} \left( {\left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right) \\ \end{aligned}$$

(40)

where (a) follows from the Rayleigh fading assumption $h_{SR}^{k} \sim \exp \left( 1 \right)$, (b) follows from the definition of Laplace transform, and $\mathcal{L}_{{I_{SRj}^{k} }} \left( \cdot \right)$ is the Laplace transform of the RV $I_{SRj}^{k} = \sum\limits_{{i \in \varPhi_{j} \backslash R}} {P_{S}^{j} h_{{SR_{i} }}^{j} \left( {Y_{{SR_{i} }}^{j} } \right)}^{{ - \alpha_{j} }}$. With the definition $I_{SRj}^{k}$, we have the Laplace transform

$$\mathcal{L}_{{I_{SRj}^{k} }} \left( {\left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = E_{{\varPhi_{j} }} \left[ {\mathop \prod \limits_{{i \in \varPhi_{j} \backslash k}} \left( {1 + \frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \left( {Y_{{SR_{i} }}^{j} } \right)^{{ - \alpha_{j} }} \tau } \right)^{ - 1} } \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop { = }\limits^{\left( b \right)} \exp \left( { - 2\pi \lambda_{j} \int\limits_{{Z_{1} }}^{\infty } {\left( {\frac{1}{{1 + \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{ - 1} y^{{\alpha_{j} }} }}} \right)} ydy} \right)$$

(41)

where (b) follows from the mapping theorem. The integral limits of (41) are achieved by the following fact. When the typical user-pair is associated with the preferable relay in the kth tier, the conditions $X_{SD}^{k} = \hbox{max} \left( {\left( {X_{SR}^{k} } \right)_{{}}^{{\alpha_{k} }} ,\left( {Y_{RD}^{k} } \right)_{{}}^{{\alpha_{k} }} } \right)$ and $\frac{1}{{P_{R}^{k} L_{0} \beta_{k} r_{0}^{{\alpha_{k} }} }}X_{SD}^{k} < \frac{1}{{P_{R}^{j} L_{0} \beta_{j} r_{0}^{{\alpha_{j} }} }}\hbox{max} \left( {\left( {R_{SR}^{j} } \right)_{{}}^{{\alpha_{j} }} ,\left( {R_{RD}^{k} } \right)_{{}}^{{\alpha_{j} }} } \right)$ should be always satisfied. Since we consider the case $X_{SR}^{k} \ge Y_{RD}^{k}$, it is achieved that $\frac{1}{{P_{R}^{k} \beta_{k} r_{0}^{{\alpha_{k} }} }}\left( {X_{SR}^{k} } \right)_{{}}^{{\alpha_{k} }} < \frac{1}{{P_{R}^{j} \beta_{j} r_{0}^{{\alpha_{j} }} }}\left( y \right)_{{}}^{{\alpha_{j} }}$ and the lower bound of $y$ can be given by $Z_{1} = \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{1}{{\alpha_{j} }}}} r_{0}^{{1 - \frac{{\alpha_{k} }}{{\alpha_{j} }}}} \left( {X_{SR}^{k} } \right)^{{\frac{{\alpha_{k} }}{{\alpha_{j} }}}}$. Using the variable transformation $u = \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{ - \frac{2}{{\alpha_{j} }}}} y^{2}$ leads to (41) given by

$$\mathcal{L}_{{I_{Rj}^{k} }} \left( {\left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \int\limits_{{Z_{2} }}^{\infty } {\frac{1}{{1 + u^{{{{\alpha_{j} } \mathord{\left/ {\vphantom {{\alpha_{j} } 2}} \right. \kern-0pt} 2}}} }}du} } \right)$$

(42)

where $Z_{2} = \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} r_{0}^{{2 - \frac{{2\alpha_{k} }}{{\alpha_{j} }}}} \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\tau } \right)^{{ - \frac{2}{{\alpha_{j} }}}}$. The Eq. (42) can be further written as

$$\mathcal{L}_{{I_{Rj}^{k} }} \left( {\left( {P_{S}^{k} } \right)^{ - 1} \left( {R_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \frac{2}{{\alpha_{j} }}\int_{{Z_{3} }}^{\infty } {\frac{{t^{{\frac{2}{{\alpha_{j} }} - 1}} }}{1 + t}dt} } \right)$$

(43)

where $Z_{3} = Z_{2}^{{\frac{{\alpha_{j} }}{2}}} = \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)r_{0}^{{\alpha_{j} - \alpha_{k} }} \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\tau } \right)^{ - 1}$, by using (3. 194.2) in [26] we have

$$\mathcal{L}_{{I_{Rj}^{k} }} \left( {\left( {P_{S}^{k} } \right)^{ - 1} \left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{S}^{j} }}{{P_{S}^{k} }}\left( {X_{SR}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \times \frac{2}{{\alpha_{j} - 2}}Z{_{3}^{{\frac{2}{{\alpha_{j} }} - 1}}}{_{2}} F_{1} \left( {1,1 - \frac{2}{{\alpha_{j} }};2 - \frac{2}{{\alpha_{j} }}; - \frac{1}{{Z_{3} }}} \right)} \right)$$

(44)

Similarly, for the probability $\Pr \left\{ {SINR_{RD}^{k} > \tau } \right\}$ we have

$$\Pr \left\{ {SINR_{RD}^{k} > \tau } \right\}{ = }\Pr \left\{ {\frac{{P_{R}^{k} h_{RD}^{k} \left( {Y_{RD}^{k} } \right)^{{ - \alpha_{k} }} }}{{\sum\nolimits_{j = 1}^{K} {\sum\nolimits_{{i \in \varPhi_{j} \backslash k}} {P_{R}^{j} h_{{R_{i} D}}^{j} \left( {Y_{{R_{i} D}}^{j} } \right)^{{ - \alpha_{j} }} } } + {W \mathord{\left/ {\vphantom {W {L_{0} }}} \right. \kern-0pt} {L_{0} }}}} > \tau } \right\}{\kern 1pt} {\kern 1pt} = { \exp }\left( { - \left( {P_{R}^{k} } \right)^{ - 1} \left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau {W \mathord{\left/ {\vphantom {W {L_{0} }}} \right. \kern-0pt} {L_{0} }}} \right)\mathop \prod \nolimits_{j = 1}^{K} \mathcal{L}_{{I_{RDj}^{k} }} \left( {\left( {P_{R}^{k} } \right)^{ - 1} \left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)$$

(45)

where $I_{RDj}^{k} = \sum\nolimits_{{i \in \varPhi_{j} \backslash k}} {P_{R}^{j} h_{{R_{i} D}}^{j} \left( {Y_{{R_{i} D}}^{j} } \right)^{{ - \alpha_{j} }} }$. The Laplace transformation $\mathcal{L}_{{I_{RDj}^{k} }} \left( {\left( {P_{R}^{k} } \right)^{ - 1} \left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)$ is

$$\mathcal{L}_{{I_{RDj}^{k} }} \left( {\left( {P_{R}^{k} } \right)^{ - 1} \left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{R}^{j} }}{{P_{R}^{k} }}\left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \int\limits_{{Z_{4} }}^{\infty } {\frac{dv}{{1 + v^{{\alpha_{j} /2}} }}} } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{R}^{j} }}{{P_{R}^{k} }}\left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \frac{2}{{\alpha_{j} }}\int\limits_{{Z_{5} }}^{\infty } {\frac{{t^{{\frac{2}{{\alpha_{j} }} - 1}} dt}}{1 + t}} } \right)$$

(46)

Obviously, $Z_{4} = \left( {\frac{{P_{R}^{j} }}{{P_{R}^{k} }}\left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{ - \frac{2}{{\alpha_{j} }}}} \left( {\widehat{P}_{R}^{j} \widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} r_{0}^{{2 - 2\frac{{\alpha_{k} }}{{\alpha_{j} }}}} \left( {X_{SR}^{k} } \right)^{{\frac{{2\alpha_{k} }}{{\alpha_{j} }}}} = \left( {\frac{{X_{SR}^{k} }}{{Y_{RD}^{k} }}} \right)^{{\frac{{2\alpha_{k} }}{{\alpha_{j} }}}} \left( \tau \right)^{{ - \frac{2}{{\alpha_{j} }}}} \left( {\widehat{\beta }_{j} } \right)^{{\frac{2}{{\alpha_{j} }}}} r_{0}^{{2 - 2\frac{{\alpha_{k} }}{{\alpha_{j} }}}}$, and $Z_{5} = \left( {\frac{{X_{SR}^{k} }}{{Y_{RD}^{k} }}} \right)^{{\alpha_{k} }} \left( \tau \right)^{ - 1} \left( {\widehat{\beta }_{j} } \right)r_{0}^{{\alpha_{j} - \alpha_{k} }}$.

$$\mathcal{L}_{{I_{Rj}^{k} }} \left( {\left( {P_{R}^{k} } \right)^{ - 1} \left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right) = \exp \left( { - \pi \lambda_{j} \left( {\frac{{P_{R}^{j} }}{{P_{R}^{k} }}\left( {Y_{RD}^{k} } \right)^{{\alpha_{k} }} \tau } \right)^{{\frac{2}{{\alpha_{j} }}}} \times \frac{2}{{\alpha_{j} - 2}}\left( {Z_{5}^{{}} } \right){_{{}}^{{\frac{2}{{\alpha_{j} }} - 1}}}{_{2}} F_{1} \left( {1,1 - \frac{2}{{\alpha_{j} }};2 - \frac{2}{{\alpha_{j} }}; - \frac{1}{{Z_{5}^{{}} }}} \right)} \right)$$

(47)

Finally, with the substitution of (47), (45), (44), (40) into (25), we have (27). With the similar argument line, the result (28) can be achieved for the case $X_{SR}^{k} < Y_{RD}^{k}$.