In this section, we derive and optimize two key metrics, i.e., the spectral efficiency (SE) and energy efficiency (EE) based on the probability generating functional (PGFL) of the Poisson cluster process in a
K-tier HetNet. Both metrics depend on the SINR in [
3]. As in [
10], the Laplace transform is used to characterize the aggregate interference (the denominator) and the total signal power (the numerator plus the denominator). For comparison, the same denotations have also been employed as in [
10].
For the proposed PCP model, the number of daughter nodes in each cluster is assumed to be fixed, and the parent points are not included. All cooperative BSs are assumed to transmit the same signal to the “typical user" at the origin. The total signal power (the signal plus the interference) received at this user is given by
$$ P= {\sum\limits_{{x} \in {\Phi}} {{p_{k}}{h_{k}}\left\| {{x}} \right\|}^{- {\alpha_{k}}}} = \sum\limits_{k = 1}^{K} {{{\sum\limits_{{x_{k}} \in {\phi_{k}}} {{p_{k}}{h_{k}}\left\| {{x_{k}}} \right\|} }^{- {\alpha_{k}}}}}, $$
(4)
where ϕk denotes the set of BS at the kth tier.
From the above formula, the Laplace transform of total signal power
P is
$$\begin{array}{*{20}l} {{\mathcal{L}}_{P}}(s) &= {E_{P}}[\text{exp}(- sP)]\\ &= \prod \limits_{k = 1}^{K} {E_{{h_{k}},{x_{k}}}}\left[\text{exp}\left(- s{\sum\limits_{{x_{k}} \in {\phi_{k}}} {{p_{k}}{h_{k}}\left\| {{x_{k}}} \right\|}^{- {\alpha_{k}}}}\right)\right]\\ &= \prod \limits_{k = 1}^{K} E\left[\prod \limits_{{x_{k}} \in {\phi_{k}}} {{\mathcal{L}}_{h}}(s{p_{k}}{\left\| {{x_{k}}} \right\|^{- {\alpha_{k}}}})\right]\\ &\mathop = \limits^{(a)} \prod \limits_{k = 1}^{K} E\left[\prod \limits_{{x_{k}} \in {\phi_{k}}} \frac{1}{{1 + s{p_{k}}{{\left\| {{x_{k}}} \right\|}^{- {\alpha_{k}}}}}}\right], \end{array} $$
(5)
where \({{\mathcal {L}}_{h}}(s{p_{k}}{\left \| {{x_{k}}} \right \|^{- {\alpha _{k}}}})\) denotes the Laplace transform of received signal power at xk, and (a) is due to the fact that hk is exponentially distributed with mean one.
On the other hand, the above expression is directly related to the PGFL of PCP as given in [
22]
$$\begin{array}{*{20}l} \mathop G\limits^ \sim (\nu) &= E\left[\prod\limits_{x \in \phi} {v(x)}\right] = \text{exp}\left\{{- {\lambda_{p}}\int_{{\mathbb{R}^{2}}} {\left[1 - \beta {{\left(y \right)}^{c}}\right]dy}} \right\}, \end{array} $$
(6)
wher
$$\beta \left(y \right) = \int_{{\mathbb{R}^{2}}} {v\left({x - y} \right)} f\left(x \right)dx,$$
and
c denotes the number of daughter nodes in each cluster.
The lower bounded PGFL and the upper bounded conditional PGFL of the PCP are given by [
19]. In order to evaluate accurately the spectral efficiency, we derive below the precise expression by algebraic operations.
$$\begin{array}{*{20}l} {{\mathcal{L}}_{P}}(s) = \prod\limits_{k = 1}^{K} {\text{exp}\left\{{ - {\lambda_{p,k}}2\pi \int_{0}^{\infty} {[1 - B_{k}{{(r)}^{{c_{k}}}}]} rdr} \right\}}, \end{array} $$
(7)
where
$$B_{k}\left({r} \right) =\frac{1}{{1 + s{p_{k}}{{\left\| r \right\|}^{- \alpha_{k} }}}}. $$
$$\begin{array}{*{20}l} &\int_{{\mathbb{R}^{2}}} {\left[1 - {\beta_{k}}{{\left(y \right)}^{{c_{k}}}}\right]dy} \\ &= \int_{{\mathbb{R}^{2}}} {\left({1 - {{\left({\int_{{\mathbb{R}^{2}}} {\frac{{f\left(x \right)}}{{1 + s{p_{k}}{{\left\| {x - y} \right\|}^{- \alpha_{k} }}}}} dx} \right)}^{{c_{k}}}}} \right)dy} \\ &\mathop = \limits^{(a)} \int_{{\mathbb{R}^{2}}} {\left({1 - {{\left({\int_{{\mathbb{R}^{2}}} {\frac{{f\left(x \right)}}{{1 + s{p_{k}}{{\left\| {r} \right\|}^{- \alpha_{k} }}}}} dx} \right)}^{{c_{k}}}}} \right)dr}\\ &= \int_{{\mathbb{R}^{2}}} {\left({1 - {{\left({\frac{1}{{1 + s{p_{k}}{{\left\| r \right\|}^{- \alpha_{k} }}}}} \right)}^{{c_{k}}}}} \right)dr}\\ &\mathop = \limits^{(b)}2\pi \int_{0}^{\infty} {\left[ {1 - {{ {B_{k}\left({r} \right)} }^{{c_{k}}}}} \right]} rdr, \end{array} $$
(8)
where (a) uses the change of variables ∥r∥=∥x−y∥, and (b) follows from the polar representation.
Then by substituting (
8) into (
6) and then (
5), we obtain (
7) in Lemma 1. Computing the integral in (
6) numerically is a very arduous task, but (
7) in Lemma 1 is now much more computationally efficient.
4.2 Spectral efficiency
From the information theory, we can achieve Shannon bound
ln(1+SINR) for an instantaneous SINR. Thus, the spectral efficiency in units of nats/s/Hz is calculated as
$$\begin{array}{*{20}l} {\tau}& = {E_{\text{SINR}}}\left[{\ln (1 + \text{SINR})} \right] \\ &= {E_{{h_{k}},{x_{k}}}}\left[ {\ln \left({1 + \frac{{\sum\limits_{x_{k} \in {C_{C}}} {{p_{k}}{h_{k}}{{\left\| {{x_{k}}} \right\|}^{- {\alpha_{k}}}}}}}{{{{\sum\limits_{{x_{k}} \in {C_{I}}} {{p_{k}}{h_{k}}\left\| {{x_{k}}} \right\|} }^{- {\alpha_{k}}}} + {\sigma^{2}}}}} \right)} \right]\\ &= {E_{P,I}}\left[{\ln \left({\frac{{P + {\sigma^{2}}}}{{I + {\sigma^{2}}}}} \right)} \right], \end{array} $$
(9)
where P and I denote the total signal power and aggregate interference power, respectively.
$$\begin{array}{*{20}l} {\tau} = \int_{0}^{\infty} \left[ {\prod\limits_{k = 1}^{K} {\text{exp}\left\{ { {\lambda_{p,k}}2\pi \int_{{R_{k}}}^{\infty} {[B_{k}{{(y)}^{{c_{k}}}}-1]} ydy} \right\}} -} \right. \\ \left. {\prod\limits_{k = 1}^{K} {\text{exp}\left\{ { {\lambda_{p,k}}2\pi \int_{0}^{\infty} {[B_{k}{{(y)}^{{c_{k}}}}-1]} ydy} \right\}}} \right]{\frac{{{e^{{-s\sigma^{2}}}}}}{s}} ds. \end{array} $$
(10)
Although it is not a closed form, the above integral can be analysed numerically. As \(B_{k}{{(y)}^{{c_{k}}}}-1<0\), the spectral efficiency is an increasing function of the cooperative radii Rk. As such, we can design a larger cooperative region of BS in HetNets to meet higher data rates (subject to the overhead incurred).
In order to compare the performances (i.e., SE and EE) under PPP and PCP, we adopt the same setup for a typical user as in a PPP model. Assuming fading coefficient
h∼exp(1), the spectral efficiency under PPP [
10] with the density
λppp,k=
ckλp,k=
λk can be express as
$$\begin{array}{*{20}l} {{\tau}_{ppp}}=\int_{0}^{\infty} {\frac{{{e^{-s{\sigma^{2}}}}}}{s}\left\{{\text{exp} \left[{- \sum\limits_{k = 1}^{K} {\pi {\lambda_{k}}B\left({{R_{k}},s{p_{k}}} \right)}}\right] -}\right.}\\ \qquad\left. {\text{exp} \left[ { - \sum\limits_{k = 1}^{K} {\pi {\lambda_{k}}B\left({0,s{p_{k}}} \right)}} \right]} \right\}ds, \end{array} $$
(11)
where
$$\begin{array}{*{20}l} &B\left({{R_{k}},s{p_{k}}}\right) = \int_{R_{k}^{2}}^{\infty} {\frac{{s{p_{k}}}}{{{u^{\frac{{{\alpha_{k}}}}{2}}} + s{p_{k}}}}}du,\\ &B\left({0,s{p_{k}}}\right) ={\left({s{p_{k}}}\right)^{\frac{2}{{{\alpha_{k}}}}}}\Gamma \left(1 + \frac{2}{{{\alpha_{k}}}}\right)\Gamma\left(1 - \frac{2}{{{\alpha_{k}}}}\right), \end{array} $$
and Γ(·) denotes the gamma function.
Due to the complexity of the spectral efficiency expression in (
10), we now derive a closed form by ignoring the noise (interference is the dominated issue in a dense HetNet).
$$\begin{array}{*{20}l} \tau \mathop \approx \limits^{\left(a \right)} q - \ln f, \end{array} $$
(12)
where
$$ {\begin{aligned} q &= \int_{0}^{\infty} {\left[{e^{- s}} - \prod\limits_{k = 1}^{K} {\text{exp}\left(2\pi {\lambda_{p,k}}\int_{0}^{\infty} [{B_{k}}{{\left(y \right)}^{c_{k}}} - 1]ydy\right)}\right]}\frac{ds}{s},\\ f &= \sum\limits_{k = 1}^{K} {\pi {c_{k}}{\lambda_{p,k}}{p_{k}}\frac{2}{{{\alpha_{k}} - 2}}} {R_{k}^{2 - {\alpha_{k}}}}. \end{aligned}} $$
Note that
q is not related to the cooperative radii, and it is a constant for a given deployment density and transmit power. As shown in Appendix 2,
q−
ln(
f) is in fact the lower bound of
τ in (
10). When
Rk becomes larger, however, the gap between (
10) and
q−
ln(
f) rapidly tapers off, hence the close approximation of (
12) for
τ. Equation (
12) is a closed formula, which makes the following optimization problem easier to solve.
4.3 Minimizing energy consumption via optimizing cooperative radii
For the proposed model, the BSs inside the cooperative clusters can communicate with the typical user located at the origin, and the average power consumption when serving such a user is given by [
10]
$$\begin{array}{*{20}l} {P_{u}} = \sum\limits_{k = 1}^{K} [\pi {R_{k}^{2}}c_{k}\lambda_{p,k}({P_{bh,k}} + {P_{sp,k}} + {a_{k}} \cdot {p_{k}}){+}\\ {\frac{c_{k}\lambda_{p,k}}{\lambda_{u}}} {P_{0,k}}], \end{array} $$
(13)
where
Pbh,k is the backhaul power dissipation when serving one user (i.e., the typical user) in the
kth tier,
Psp,k denotes the corresponding signal processing power consumption, and
ak denotes the slope of the
kth tier power consumption with respect to transmit power
pk. The second term of
Pu in (
13) denotes the average static power for serving one user, and it is independent of the load of BSs.
In the following, we will provide a solution for optimizing the average power consumption of the HetNet for the typical user in (
13). In reality, it is very important to determine the network parameters which can minimize the network power consumption. Since the average energy consumption
Pu per user and the average consumed energy
Pav are related with
Pav=
λu×
Pu, minimizing
Pu is critical to designing an energy-saving network.
For a given cooperative radius
R1 of the first tier, we want to find out the optimal cooperative radius
R2 of the second tier in a two-tier HetNet, which minimizes the energy consumption while ensuring a certain rate to the typical user. Note that the energy consumption in (
13) increases with the cooperative radii, which suggests that we should determine the minimum cooperative radii while guaranteeing the minimum user rate. Intuitively, the user rate should increase with cooperative areas, and the rate in (
10) is indeed an increasing function of the cooperative radii. Assuming that the first-tier cooperative radius
R1 is known, we can determine the optimal value of the second-tier cooperative radius
R2 through dichotomy. The problem will be transformed into a simple problem with a single variable (with the constraint from (
10)):
$$\begin{array}{*{20}l} &\min \begin{array}{*{20}{c}} \end{array}{R_{2}}\\ &s.t.\begin{array}{*{20}{c}} \end{array}\int_{0}^{\infty} \left[{\prod\limits_{k = 1}^{2} {\text{exp}\left\{{{\lambda_{p,k}}2\pi \int_{{R_{k}}}^{\infty} {[B_{k}{{(y)}^{{c_{k}}}}-1]} ydy} \right\}} -} \right.\\ &\left. {\prod\limits_{k = 1}^{2} {\text{exp}\left\{ { {\lambda_{p,k}}2\pi \int_{0}^{\infty} {[B_{k}{{(y)}^{{c_{k}}}}\,-\,1]} ydy} \right\}}} \right]{\frac{{{e^{{-s\sigma^{2}}}}}}{s}}ds \ge {{\tau}_{0 }} \end{array} $$
(14)
The above problem formulation can be solved by the well-known dichotomy algorithm
1.
Now, we want to form the general problem to determine the optimal values for
R1,
R2,···,
RK in a
K-tier network, which minimize the power consumption in (
13) while satisfying the minimum spectral efficiency requirement. Note that the second term of
Pu in (
13) is independent of
R1,
R2,···,
RK, and can be ignored. Hence, by applying (
12) and (
13), the optimization problem can be formulated as follows:
$$\begin{array}{*{20}l} &\mathop{\min}\limits_{\{{R_{1}},{R_{2}}, \cdot \cdot \cdot,{R_{K}}\}} {P_{u1}} = \sum\limits_{k = 1}^{K} {\pi R_{k}^{2}{\lambda_{p,k}}{c_{k}}({P_{bh,k}} + {P_{sp,k}} + {a_{k}}{p_{k}})} \\ &\quad s.t.\begin{array}{*{20}{c}} {}&{q - \ln f = {\tau_{0}}} \end{array} \end{array} $$
(15)
Clearly, both the constraint condition and the optimization objective function are of closed form. The problem can then be solved easily by a linear programming method as follows.
$$\begin{array}{*{20}l} {R_{o,k}} = {\left({\frac{{{p_{k}}{P_{a,i}}}}{{{p_{i}}{P_{a,k}}}}R_{o,i}^{{\alpha_{i}}}} \right)^{\frac{1}{{{\alpha_{k}}}}}} \end{array} $$
(16)
$$\begin{array}{*{20}l} f = \sum\limits_{k = 1}^{K} {\pi {c_{k}}{\lambda_{p,k}}{p_{k}}\frac{2}{{{\alpha_{k}} - 2}}} R_{o,k}^{2 - {\alpha_{k}}} = \exp \left({q - {\tau_{0}}} \right) \end{array} $$
(17)
where Ro,k denotes the optimal value of the kth tier cooperative radius, and Pa,i=Pbh,i+Psp,i+aipi denotes the total power dissipation when serving a typical user in the ith tier.
Combining (
16) and (
17), the optimal value of each tier can be determined by solving the resultant system of nonlinear equations for a given data rate
τ0, which optimizes the PCP networks’ energy consumption.
Once the power
Pu has been minimized, the network energy efficiency
ηee, defined as the ratio of the average spectral efficiency to the average power consumption, can be calculated as
$$\begin{array}{*{20}l} {\eta_{ee}} = \frac{{{\tau}}}{{{P_{u}}}}. \end{array} $$
(18)