In this section, we first introduce the performance metrics that are considered in this paper. Then, we proceed to derive the coverage probability for both CUEs and D2D users which are needed to compute these metrics.
Performance metrics
In this paper, two main performance metrics for the network are considered: the average sum rate (ASR) and energy efficiency (EE). The metrics used here are aligned to requirements that are demanded in 5G [1, 46, 47]. We investigate this scenario in order to get an understanding of how such coexistence would perform. Another important fact is that to the best of our knowledge no one has compared the EE and ASR performance of D2D communication in a massive MIMO system. Both of these technologies are known to bring high ASR and are likely to be more energy-efficient. However, there is no work in literature showing the impact of high number of antennas and cellular users along with the density of D2D users in such a setting.
The ASR is obtained from total rates of both D2D users and CUEs as
$$ \text{ASR}= U_{c} \bar{R}_{c} + \pi R^{2} \lambda_{d} \bar{R}_{d}, $$
(9)
where π
R
2
λ
d
is the average number of D2D users in the cell and \(\bar {R}_{t}\) with t∈{c,d} denotes the average rates of the CUEs and D2D users, respectively. \(\bar {R}_{t}\) for both cellular and D2D users is computed as the successful transmission rate by
$$ \bar{R}_{t} = \underset{\beta_{t} \geq 0}{\text{sup}}~B_{w} \log_{2}(1 + \beta_{t}) \mathrm{P}^{t}_{\text{cov}} (\beta_{t}). $$
(10)
This ASR metric is referred to as the transmission capacity [45, 48] which guarantees the highest spatial reuse under a maximum outage constraint. In (10),
$$ \mathrm{P}^{t}_{\text{cov}} (\beta_{t}) = \text{Pr}\left\{\text{SINR}_{t} \geq \beta_{t} \right\} $$
(11)
is the coverage probability when the received SINR is higher than a specified threshold β
t
needed for successful reception. Note that SINR
t
contains random channel fading and random user locations. Finding the supremum guarantees the best constant rate for the D2D users and the CUEs. If we know the coverage probability (\(\mathrm {P}^{t}_{\text {cov}} (\beta _{t})\)), (10) can easily be computed by using line search for each user type independently. Moreover, (10) is easily achievable in practice since the modulation and coding is performed without requiring that every transmitter knows the interference characteristics at its receiver.
Energy efficiency is defined as the benefit-cost ratio between the ASR and the total consumed power:
$$ \text{EE} = \frac{\text{ASR}}{\text{Total power}}. $$
(12)
For the total power consumption, we consider a detailed model described in [7]:
$$ \begin{aligned} \text{Total power}&=\frac{1}{\eta} \left(P_{c} +\lambda_{d} \pi R^{2} P_{d}\right) + C_{0} + T_{c} C_{1} \\ & \quad + \left(U_{c} + 2 \lambda_{d} \pi R^{2}\right)C_{2}, \end{aligned} $$
(13)
where P
c
+λ
d
π
R
2
P
d
is the total transmission power averaged over the number of D2D users, η is the amplifier efficiency (0<η≤1), C
0 is the load independent power consumption at the BS, C
1 is the power consumption per BS antenna, C
2 is the power consumption per user device, and U
c
+2λ
d
π
R
2 is the average number of active users.
In order to calculate the ASR and EE, we need to derive the coverage probability for both cellular and D2D users. The analytic derivation of these expressions is one of the main contributions of this paper.
Coverage probability of D2D users
We first derive the expression for the coverage probability of D2D users.
Proposition 1
The approximate coverage probability for a typical D2D user is given by
$$\begin{array}{*{20}l} \mathrm{P}^{d}_{\text{cov}}(\beta_{d}) &\approx \frac{(\kappa\beta_{d})^{2/\alpha_{c}}}{R^{2}}\left(y^{U_{c} + \frac{2}{\alpha_{c}}-1} (1-y)^{- \frac{2}{\alpha_{c}}} \right.\notag \\ &\;\quad \left. - \! \left(\!U_{c} \,+\, \frac{2}{\alpha_{c}}\,-\,1\right) \mathcal{B}\left(\! y; U_{c} + \frac{2}{\alpha_{c}}-1, 1-\frac{2}{\alpha_{c}}\right) \right) \notag \\ &\;\quad \cdot \exp\left(- \frac{\pi \lambda_{d} R_{0,0}^{2}}{{\text{sinc}}\left(\frac{2}{\alpha_{d}}\right)} \beta_{d}^{2/\alpha_{d}}\right) \exp\left(\frac{-\beta_{d}} {\bar{\gamma}_{d}}\right), \end{array} $$
(14)
where \(\kappa \triangleq \frac {\zeta }{P_{d} A_{d} R_{0,0}^{-\alpha _{d}}}\) with ζ defined in (5), \(y \triangleq \frac {1}{\kappa \beta _{d} R^{-\alpha _{c}} + 1}\), \(\text {sinc}(x) = \frac {\sin (\pi x)}{\pi x}\), \(\bar {\gamma }_{d} = \frac {A_{d} R_{0,0}^{-\alpha _{d}}P_{d}}{N_{0}}\) is the average D2D SNR, and \(\mathcal {B}(x;a,b)\) is the incomplete Beta function.
Proof
The proof is given in Appendix 1: Proof of Proposition 1. □
The coverage probability expression in Proposition 1 allows us to compute the average data rate of a typical D2D user in (10). The approximation in this proposition is due to neglecting the spatial interference correlation resulting from the fact that multiple interfering streams are coming from the same location (more details can be found in Appendix 1: Proof of Proposition 1). We note that (14) is actually a tight approximation and its tightness is evaluated in Section 4. From the expression in (14), we make several observations as listed below.
Remark 1
In the high-SNR regime for the D2D users where \(\bar {\gamma }_{d} \gg \beta _{d}\), the last term in (14) converges to one, i.e., \(\exp \left (- \frac {\beta _{d}} {\bar {\gamma }_{d}}\right) \to 1\), and we have
$$ {\begin{aligned} \mathrm{P}^{d}_{{\text{cov}}}(\beta_{d}) &= \frac{(\kappa\beta_{d})^{2/\alpha_{c}}}{R^{2}}\left(y^{U_{c} + \frac{2}{\alpha_{c}}-1} (1-y)^{- \frac{2}{\alpha_{c}}} - \left(U_{c} + \frac{2}{\alpha_{c}}-1\right) \right. \\ &\;\quad \cdot \left. \mathcal{B}\left(y; U_{c} + \frac{2}{\alpha_{c}}-1, 1-\frac{2}{\alpha_{c}}\right) \right) \exp\left(- \frac{\pi \lambda_{d} R_{0,0}^{2}}{{\text{sinc}}\left(\frac{2}{\alpha_{d}}\right)} \beta_{d}^{2/\alpha_{d}}\right). \end{aligned}} $$
(15)
This can also be referred to as the interference-limited regime.
Remark 2
The coverage probability of a typical D2D user is a decreasing function of the D2D density λ
d
. Because higher λ
d
results in more interference among D2D users. In particular, it can be seen that \(\mathrm {P}^{d}_{ {\text {cov}}}\) in (14) is a function of λ
d
through exp(−C
λ
d
) with \(C\triangleq \frac {\pi R_{0,0}^{2} \beta _{d}^{2/\alpha _{d}}}{{\text {sinc}}\left (\frac {2}{\alpha _{d}}\right)} > 0\). Thus, if λ
d
→∞, \(\mathrm {P}^{d}_{{\text {cov}}} \to 0\). Recall that in our model, the D2D Rx is associated to the D2D Tx which is located at a fixed distance away. However, if we had assumed that the D2D Rx’s association to a D2D Tx is based on, for example, the shortest distance or the maximum SINR, then the \(\mathrm {P}^{d}_{{\text {cov}}}\) would have been unaffected by the D2D density (in the high-interference regime). Similar observation can be found in [
49
,
50
].
Now, considering the number of BS antennas or the number of CUEs as variables, we have the following behavior of the D2D coverage probability.
Remark 3
\(\mathrm {P}^{d}_{{\text {cov}}}\) is not affected by the number of BS antennas T
c
. The BS antennas are used to cancel out the interference among CUEs and they do not have any impact on D2D users’ performance as long as the number of CUEs U
c
is constant and does not vary with the number of BS antennas T
c
. The coverage probability of a typical D2D user \(\mathrm {P}^{d}_{{\text {cov}}}\) is a decreasing function of U
c
. However, increasing the number of CUEs have a small effect on D2D users’ performance. This is due to the fact that the resulting interference from the BS to D2D users does not change significantly by increasing the number of CUEs as the transmit power of the BS is the same irrespective of the number of users and the precoding is independent of the D2D channels. Thus, a change of U
c
will only change the distribution of the interference but not its average.
Next, we comment on how changes in the transmit powers of the BS and D2D Tx as well as the distance between D2D user pairs affect the coverage probability of D2D users.
Remark 4
\(\mathrm {P}^{d}_{{\text {cov}}}\) is a decreasing function of the ratio between the transmit power of the BS and of the D2D users, i.e., \(\frac {P_{c}}{P_{d}}\), which is part of the first term in (14) and corresponds to the interference from the BS. For instance, if we fix P
c
and decrease P
d
, the coverage probability for D2D users decreases as the interference from the BS would be the dominating factor. At the same time, if we decrease P
c
, it would improve the coverage of D2D users.
Remark 5
\(\mathrm {P}^{d}_{{{\text {cov}}}}\) is a decreasing function of the distance between D2D Tx-Rx pairs R
0,0 and the cell radius R. Increasing the cell radius with the same D2D user density reduces the effect of the interference from the BS. Also by decreasing the distance between D2D Tx-Rx pairs, it is evident that a better performance for D2D users can be obtained.
Using Proposition 1, the following corollary provides the optimal D2D user density that maximizes the D2D ASR, i.e., \(\pi R^{2} \lambda _{d} \bar {R}_{d}\), where \(\bar {R}_{d}\) is given in (10). The optimal density is also evident in our numerical results in Section 4.
Corollary 1
For a given SINR threshold β
d
, the optimal density of D2D users \(\lambda _{d}^{*}\) that maximizes the D2D ASR is
$$\begin{array}{*{20}l} \lambda_{d}^{*}(\beta_{d}) = \frac{{\text{sinc}}\left(\frac{2}{\alpha_{d}}\right)}{\pi R_{0,0}^{2}}\beta_{d}^{-2/\alpha_{d}}. \end{array} $$
(16)
Proof
Given the SINR threshold β
d
and using (9)–(10), the D2D ASR is
$$ \pi R^{2} \lambda_{d} B_{w}\log_{2} (1+\beta_{d}) \mathrm{P}^{d}_{\text{cov}}(\beta_{d}), $$
(17)
where \(\mathrm {P}^{d}_{\text {cov}}(\beta _{d})\) is given in (14) and depends on λ
d
through an exponential function. Taking the second derivative of (17) with respect to λ
d
, we observe that for \(\lambda _{d} < 2 \frac {\text {sinc}\left (\frac {2}{\alpha _{d}}\right)}{\pi R_{0,0}^{2}}\beta _{d}^{-2/\alpha _{d}}\), the function is concave. Therefore, setting the first derivative of (17) with respect to λ
d
to zero yields the optimal D2D user density \(\lambda _{d}^{*}(\beta _{d})\) given in (16) that maximizes the D2D ASR. □
Coverage probability of cellular users
Next, we compute the coverage probability for CUEs.
Proposition 2
The coverage probability for a typical cellular user is given by
$$ {\begin{aligned} \mathrm{P}^{c}_{{\text{cov}}}(\beta_{c})\! &=\! \mathbb{E}_{D_{0,\text{BS}}}\!\left[\! e^{-\frac{N_{0}}{A_{d}}s} \sum_{k=0}^{T_{c} - U_{c}} \frac{s^{k}}{k!} \sum_{i=0}^{k} {k \choose i}\! \left(\frac{N_{0}}{A_{d} }\right)^{k-i}\!(-1)^{i} \;\Upsilon(\lambda_{d},s,i)\right], \end{aligned}} $$
(18)
with
$$ {\begin{aligned} \Upsilon(\lambda_{d},s,i) &= \exp\left(-C_{d} \lambda_{d} s^{2/\alpha_{d}} \right) \\ &\;\quad \cdot \sum_{(j_{1},\ldots,j_{i})\in\mathcal{J}}\! i! \prod_{\ell=1}^{i}\frac{1}{j_{\ell}!(\ell!)^{j_{\ell}}}\!\left(\!-C_{d} \lambda_{d} s^{\frac{2}{\alpha_{d}}-\ell}\prod_{q=0}^{\ell-1}\left(\frac{2}{\alpha_{d}}-q\right)\!\right)^{j_{\ell}}, \end{aligned}} $$
(19)
where \(s\triangleq \frac {A_{d}}{\zeta } D_{0, {\text {BS}}}^{\alpha _{c}}\beta _{c}\) with ζ defined in (5), \(C_{d} \triangleq \frac {\pi P_{d}^{2/\alpha _{d}}}{{\text {sinc}}\left (\frac {2}{\alpha _{d}}\right)}\), and
$$\mathcal{J}\triangleq \left\{(j_{1},\ldots,j_{i}): j_{\ell} \in \mathbb{Z}_{\geq 0}, \; \sum_{\ell=1}^{i} \ell j_{\ell} = i \right\}. $$
Proof
The proof is given in Appendix Appendix 2: Proof of Proposition 2. □
This proposition gives an expression for the coverage probability of CUEs in which there is only one random variable left. The expectation in (18) with respect to D
0,BS is intractable to derive analytically but can be computed numerically. The analytical results of Proposition 1 and Proposition 2 have been verified by Monte Carlo simulations in Section 4. A main benefit of the analytic expressions (as compared to pure Monte Carlo simulations with respect to all sources of randomness) is that they can be computed much more efficiently, which basically is a prerequisite for the multi-variable system analysis carried out in Section 4.
Next, we present some observations from the result in Proposition 2 as follows.
Remark 6
In the interference-limited regime where I
d,c
≫N
0, the coverage probability in (18) for a typical cellular user is simplified to
$$\begin{array}{*{20}l} \mathrm{P}^{c}_{{\text{cov}}}(\beta_{c}) &= \mathbb{E}_{D_{0,{\text{BS}}}}\left[\sum_{k=0}^{T_{c} - U_{c}} \frac{(-s)^{k}}{k!}\Upsilon(\lambda_{d},s,k)\right]. \end{array} $$
(20)
The result obtained in Remark 6 has a lower computational complexity compared to the expression in Proposition 2 and at the same time it is a tight approximation for Proposition 2. This can be observed from the denominator of the (7) where the term \(\frac {N_{0}}{A_{d} }\approx 0\).
Remark 7
The coverage probability of a typical CUE \(\mathrm {P}^{c}_{\text {cov}}(\beta _{c})\) is a decreasing function of the D2D user density λ
d
. From Proposition 2, only Υ(λ
d
,s,i) is a function of λ
d
which is composed of an exponential term in λ
d
multiplied by a polynomial term in λ
d
. Thus, if λ
d
→∞, the exponential term which has a negative growth dominates the polynomial term and \(\mathrm {P}^{c}_{{\text {cov}}}(\beta _{c}) \to 0\).
We proceed to analyze the behavior of Proposition 2 by considering a number of special cases. The impact of these special cases is also corroborated in our numerical results in Section 4.
Corollary 2
If T
c
=U
c
, which is a classical MU-MIMO scenario as indicated in [
51
], the coverage probability for a typical cellular user is given by
$$\begin{array}{*{20}l} \mathrm{P}^{c}_{{\text{cov}}}(\beta_{c}) &= \mathbb{E}_{D_{0,{\text{BS}}}}\left[\exp\left(-\frac{N_{0}}{A_{d}}s - C_{d} \lambda_{d} s^{2/\alpha_{d}}\right)\right], \end{array} $$
(21)
where \(s = \frac {A_{d}}{\zeta } D_{0,{\text {BS}}}^{\alpha _{c}} \beta _{c}\) and \(C_{d}=\frac {\pi P_{d}^{2/\alpha _{d}}}{{\text {sinc}}\left (\frac {2}{\alpha _{d}}\right)}\).
Proof
(21) follows directly from (18) by setting T
c
−U
c
=0. □
Corollary 2 applies for any case of MU-MIMO and massive MIMO is a form of MU-MIMO [52,53]. The important distinction is that MU-MIMO has traditionally been considered for the case of equal number of antennas and users, while massive MIMO employs a large number of antennas compared to the number of users [52
,
53]. As a rule-of-thumb, T
c
>50 and T
c
/U
c
>2 are required to exploit the favorable propagation of massive MIMO [52]. Next, we consider the case where massive number of antennas are deployed in the BS.
Corollary 3
If (T
c
−U
c
)→∞, the coverage probability for a typical cellular user tends to one, that is,
$$\begin{array}{*{20}l} {\lim}_{(T_{c} - U_{c}) \to \infty}~\mathrm{P}^{c}_{{\text{cov}}}(\beta_{c}) = 1. \end{array} $$
(22)
Proof
Let m=T
c
−U
c
. Substituting SINR
c
from (7) into (11), we have
$${ \begin{aligned} {\lim}_{m \to \infty}~\mathrm{P}^{c}_{\text{cov}}(\beta_{c}) &= {\lim}_{m \to \infty}~\text{Pr}\left\{ |\mathbf{h}_{0}^{H} \mathbf{v}_{0}|^{2} \geq \frac{A_{d}}{\zeta} D_{0,\text{BS}}^{\alpha_{c}}\left(I_{d,c} + \frac{N_{0}}{A_{d}}\right)\beta_{c} \right\} \\[-3pt] &\overset{(a)}= {\lim}_{m \to \infty}~\mathbb{E}_{D_{0,\text{BS}}, I_{d,c}}\left[e^{-\frac{A_{d}}{\zeta} D_{0,\text{BS}}^{\alpha_{c}}\left(I_{d,c} + \frac{N_{0}}{A_{d}}\right)\beta_{c}} \right.\\[-3pt] &\;\quad\cdot \left.\sum_{k=0}^{m} \frac{1}{k!}\left(\frac{A_{d}}{\zeta} D_{0,\text{BS}}^{\alpha_{c}}\left(I_{d,c} + \frac{N_{0}}{A_{d}}\right)\beta_{c}\right)^{k}\right] \\[-3pt] &\overset{(b)}= {\lim}_{m \to \infty}~\mathbb{E}_{D_{0,\text{BS}}, I_{d,c}}\left[e^{-z} \sum_{k=0}^{m} \frac{z^{k}}{k!}\right] \\[-3pt] &\overset{(c)}= \mathbb{E}_{D_{0,\text{BS}}, I_{d,c}}\left[{\lim}_{m \to \infty} e^{-z} \sum_{k=0}^{m} \frac{z^{k}}{k!}\right] \\[-3pt] &\overset{(d)}= \mathbb{E}_{D_{0,\text{BS}}, I_{d,c}}\left[e^{-z} e^{z} \right] = 1, \end{aligned}} $$
where (a) follows from the CCDF of \(|\mathbf {h}_{0}^{H} \mathbf {v}_{0}|^{2}\) with \(2|\mathbf {h}_{0}^{H} \mathbf {v}_{0}|^{2} \sim {\chi ^{2}_{2}}\) given D
0,BS and I
d,c
(refer to Appendix 5 for more details). Step (b) follows from setting \(z = \frac {A_{d}}{\zeta } D_{0,\text {BS}}^{\alpha _{c}}\left (I_{d,c} + \frac {N_{0}}{A_{d}}\right)\beta _{c}\). Step (c) is obtained from the dominated convergence theorem which allows for an interchange of limit and expectation and step (d) is due to the fact that \(\sum _{k=0}^{\infty } \frac {z^{k}}{k!} = e^{z}\). □
Corollary 3 gives an indication that the desired signal can be amplified by adding more antennas. However, note that even if the power gain becomes much stronger than the D2D interference, it will, in practice, eventually become limited by pilot contamination, hardware distortion, and/or finite modulation sizes.
In the results so far, we have discussed the case where there exist some D2D users as underlay to the cellular network, that is, λ
d
≠0, However, it is interesting to see what can be achieved without D2D users.
Corollary 4
If λ
d
=0, the coverage probability for a typical cellular user is given by
$$ {\begin{aligned} \mathrm{P}^{c}_{{\text{cov}}}\left(\beta_{c}\right) &= \frac{2}{\alpha_{c} R^{2}}\Gamma\left(\frac{2}{\alpha_{c}}\right)\left(\frac{N_{0}}{\zeta} \beta_{c} \right)^{-2/\alpha_{c}} \sum_{k=0}^{T_{c}-U_{c}}{\frac{2}{\alpha_{c}}+k-1 \choose k}, \end{aligned}} $$
(23)
where Γ(·)is the Gamma function and ζ is defined in (5).
Proof
Substituting SINR
c
from (7) into (11) and setting λ
d
=0, we have
$$ {\begin{aligned} \mathrm{P}^{c}_{\text{cov}}(\beta_{c}) &= \text{Pr}\left\{ |\mathbf{h}_{0}^{H} \mathbf{v}_{0}|^{2} \geq \frac{D_{0,\text{BS}}^{\alpha_{c}}}{\zeta} N_{0} \beta_{c} \right\}\\[-3pt] &\overset{(a)}= \mathbb{E}_{z}\left[\sum_{k=0}^{T_{c}-U_{c}}\frac{l^{k}}{k!}z^{k} e^{-lz} \right]\\[-3pt] &\overset{(b)}=\frac{2}{\alpha_{c} R^{2}}\Gamma\left(\frac{2}{\alpha_{c}}\right) \sum_{k=0}^{T_{c}-U_{c}}\frac{(-l)^{k}}{k!} ~\frac{\mathrm{d}^{k}}{{\mathrm{d}l}^{k}} ~l^{-2/\alpha_{c}}, \end{aligned}} $$
(24)
where (a) follows from the CCDF of \(|\mathbf {h}_{0}^{H} \mathbf {v}_{0}|^{2}\) with \(2|\mathbf {h}_{0}^{H} \mathbf {v}_{0}|^{2} \sim {\chi ^{2}_{2}}\) given D
0,BS and setting \(l= \frac {N_{0}}{\zeta } \beta _{c}\) and \(z = D_{0,\text {BS}}^{\alpha _{c}}\) with PDF \(f(z)=\frac {2}{\alpha _{c}R^{2}}z^{\frac {2}{\alpha _{c}}-1}\). Step (b) follows from taking the expectation with respect to z which is similar to the expression in (35) with the Laplace transform \( \mathcal {L}_{z}(l) = \frac {2}{\alpha _{c} R^{2}}\Gamma \left (\frac {2}{\alpha _{c}}\right)l^{-2/\alpha _{c}}\). Simplifying the k-th derivative to \(\frac {\mathrm {d}^{k}}{\mathrm {d}l^{k}} ~l^{-2/\alpha _{c}} = (-1)^{k} l^{-\frac {2}{\alpha _{c}}-k} \prod _{i=0}^{k-1}\left (\frac {2}{\alpha _{c}}+i\right)\) and using the identity \(\frac {1}{k!}\prod _{i=0}^{k-1}\left (\frac {2}{\alpha _{c}}+i\right) = {\frac {2}{\alpha _{c}}+k-1 \choose k}\), (23) follows. □
The closed-form results in Corollary 4 for λ
d
=0 depends only on noise rather than interference and perhaps can result in higher ASR for CUEs. The ASR for λ
d
>0 also depend on noise but its impact is much smaller. However, we note that this result is obtained for a single-cell scenario. Thus, comparing Proposition 2 and Corollary 4 and evaluating the potential performance gain/loss due to introducing D2D communications would make more sense in a multi-cell scenario.
Using the results from Proposition 1 and Proposition 2, we proceed to evaluate the network performance in terms of the ASR and EE from (9) and (12), respectively.