Spatio-Temporal Network Dynamics Framework for Energy-Efficient Ultra-Dense Cellular Networks

The relentless surge in traffic demands has compelled network operators to seek innovative solutions to maximize their performance in terms of users’ data rates and energy expenditures [1, 2]. In parallel to that, $5$G systems are expected to be ultra-dense in nature, i.e. $\lambda_{b}\gg\lambda_{u}$ [3, 4, 5], rendering network optimization highly complex [1, 6]. Resource management techniques such as power control and scheduling in ultra-dense networks (UDNs) become significantly more challenging than in traditional sparse deployments due to spatio-temporal traffic demand fluctuations and channel uncertainties. Current state-of-the-art approaches are merely based on heuristics or system-level simulations lacking fundamental insights.

Recently, mean-field games (MFGs) received significant attention in cellular networks based on the fact that players make their local decisions (based on their own states) while abstracting other players’ strategies using a mean-field (MF) measure obtained after solving a set of coupled partial differential equations (PDEs), known as Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) [7, 8, 9]. Nonetheless, solving a set of coupled PDEs is still cumbersome, so none of the current MFG works incorporate spatial aspects of practical networks such as the randomness of BS locations.

From a stochastic geometry (SG) perspective, while there is a significant body of literature in terms of performance modeling and analysis, most preceding works overlook the temporal network dynamics. Such missing gaps motivate this work which aims at capturing the spatio-temporal nature of wireless networks. To the best of our knowledge, this is the first work combining MFG and SG frameworks within the scope of UDNs.

Our contributions are as follows. Firstly, MF Interference convergence is rigorously proved (Theorem 1), i.e. interference normalized by BS density is finite in UDNs. The result specifies necessary conditions: (i) BS densification (scaling law), (ii) user density, (iii) fading channel, and (iv) LOS dependency, captured by a so-called reception ball size. Secondly, an optimal power control policy that maximizes EE is obtained (Proposition 2) via a closed-form expression of energy-efficiency (EE) (Proposition 1) by leveraging the MF measure and MF interference convergence.

Consider a downlink cellular network where base stations (BSs) are uniformly distributed over a two-dimensional infinite Euclidean plane with density $\lambda_{b}$, leading to a homogeneous Poisson point process (PPP). Similarly, users are uniformly distributed with density $\lambda_{u}$, independent of BS locations. Each user associates with the nearest BS. This results in a set of geographical coverages of BSs, i.e. Voronoi tessellation [10]. A BS having no serving user within its coverage becomes dormant, not transmitting any signals so as to save energy as well as to mitigate interference; otherwise, active.

A single active BS directionally transmits a data signal to the associated user by using $N$ number of antennas. Its beam pattern experienced at a receiving user follows a sectorized uniform linear array antenna model [11] where the main lobe gain $G_{N}$ is $N$ with beam width $\theta_{N}=2\pi/\sqrt{N}$, neglecting side lobes and assuming beam centers point at the associated users.

Transmitted signals from BSs experience path-loss attenuation. The attenuation from the $i$-th BS coordinates $z_{i}$ to the $j$-th user coordinates $y_{j}$ $l_{ij}{(\lambda_{b})}=\min\left(1,\|z_{i}-y_{j}\|^{-\alpha}\right)$ where $\alpha>2$ denotes the path-loss exponent. Its bounded unity value comes from the fact that path-loss attenuation cannot amplify the transmitted signal. In addition to path-loss attenuation, the transmitted signals at time $t\geq 0$ experience spatially i.i.d. but temporally correlated fading with the fading coefficient vector $g_{ij}(t)=\{g_{x_{ij}}(t),g_{y_{ij}}(t)\}$. This fading gain is constant within a block, and evolves from block to block while satisfying the Markov property as in a Gauss-Markov correlated block fading model [12]. Consider the block length is infinitesimal, leading to the following fading evolution law [7]:

where $\mu:=\{\mu_{x},\mu_{y}\}$ for non-negative constants $\mu_{x},\mu_{y}$, $0\leq\eta<\infty$, and $\mathbb{W}_{i}(t)$’s are mutually independent Wiener processes. Adjusting $\mu$ and $\eta$ respectively determines the amounts of temporal correlation and (slow/fast) fading variance (see [7] for further details). Combining the effects of path-loss attenuation and fading yields channel gain $h_{ij}$ from the $i$-th BS to the $j$-th user, given as $|h_{ij}(t,\lambda_{b})|^{2}=l_{ij}{(\lambda_{b})}\times|g_{ij}(t)|^{2}$. In the following, we abuse notation by using the sole subscript $i$ for indicating the associated BS $i$ to user $i$ links.

The $i$-th BS transmission power $0<P_{i}\left\{t,h_{i}(t,\lambda_{b})\right\}\leq P_{\max}$, which is determined only by its own channel state in $h_{i}(t,\lambda_{b})$. Note that different BSs having identical channel state transmit the same power, i.e. homogeneous admissible controls for BS transmission powers.

User $i$ only receives signals from BSs located within a reception ball $b(y_{i},R)$ centered at $y_{i}$ with radius $R>0$. Motivated by the line-of-sight (LOS) ball model [13], adjusting $R$ enables emulating not only sub 6 GHz cellular networks but also LOS dependent networks where guaranteeing LOS is necessary for communication. For LOS dependent networks such as above 6 GHz millimeter-wave cellular networks, $R$ corresponds to the average LOS distance that can be calculated at a given geographical site [13, 5]. For sub 6 GHz networks, $R$ corresponds to the average distance that provides he received signal power larger than noise floor. When $R\rightarrow\infty$, the reception ball model becomes identical to a traditional PPP network model [14].

Interactions of BSs through interference is a major source of the bottlenecks for optimizing wireless networks. MFG allows us to circumvent this problem by changing the entire interactions into a single interaction via MF interference. To achieve this, MF interference should asymptotically converge to zero, verified as in the following.

Let $\Phi_{R}(\lambda)$ denote either (i) point coordinates following homogeneous PPP with density $\lambda$ within $b(o,R)$ or (ii) the number of such points within $b(o,R)$. Consider a randomly selected user, i.e. typical user. Now that the user coordinates are translation-invariant, i.e. stationary, it is always possible to make this typical user located at the origin $o$, i.e. Slyvnyak’s theorem [10]. At the typical user, the following statements hold according to the nearest association and user-void dormant BSs under the reception ball model in Section II.

• •

  If at least a single BS exists within $b(o,R)$ with probability $p_{R}$, the associated BS is the nearest out of the entire BS coordinates $\Phi_{R}(\lambda_{b})$ (recall the associated BS is always active).

• •

  An arbitrary interfering BS $i$ belongs to the set of active BS coordinates $\Phi_{R}(p_{a}\lambda_{b})$ where $p_{a}$ denotes a BS’s active probability.

Utilizing the void probability of a PPP [10] yields $p_{R}=1-\exp\left(-\pi\lambda_{b}R^{2}\right)$. Assuming $p_{a}$’s are homogeneous over BSs provides $p_{a}\approx 1-\left[1+\lambda_{u}/(3.5\lambda_{b})\right]^{-3.5}$ [15]. For sufficiently large $\lambda_{b}/\lambda_{u}$, the accuracy of this approximation is in [16], and asymptotic convergence is verified in [5].

Let the subscript $0$ denote the associated BS. At a typical user, the received power from the associated BS and aggregate interfering BSs with a single antenna are respectively given as follows:

where $\Phi^{o}_{R}(p_{a}\lambda_{b}):=\Phi_{R}(p_{a}\lambda_{b})\backslash\{z_{0}\}$ via Slyvnyak’s theorem.

Signal-to-interference-plus-noise ratio ($\mathsf{SINR}$) with $N$ number of transmit antennas is then:

where $\sigma^{2}$ noise power and the last step follows from $S_{\lambda_{b}}(t)\sim{\lambda_{b}}^{\frac{\alpha}{2}}S_{1}(t)$ via mapping theorem [17]. Note that $\hat{I}_{p_{a}\lambda_{b}}(t)$ is the interference normalized by BS density and the number of antennas.

The goal of this section is to validate the effectiveness of the proposed approach. To this end, we maximize the energy efficiency of a UDN in a tractable manner by virtue of MF interference. For simplicity, this section only considers channel states, incorporating the impact of energy states is deferred to future work. In the following, we may abuse notations by dropping $i$, $t$ and/or $\lambda$.

Consider at first an instantaneous energy efficiency $\mathsf{EE}(t)$ defined as the spatially averaged downlink capacity divided by power consumption at time $t$, i.e. $\mathsf{EE}(t):=\mathsf{E}_{y_{i}}[\log\left(1+\mathsf{SINR}_{i}(t)\right)]/(P_{i}(t)+P_{c})$ where $P_{c}$ denotes the fixed circuit power consumption at BS $i$. In a UDN where MF interference approximation holds, $\mathsf{EE}(t)$ can be represented in closed-form as the following proposition shows.

where $c_{0}:=P(t)|g(t)|^{2}/\left(\frac{\sigma^{2}}{N\sqrt{{\lambda_{b}}^{\alpha}}}+\hat{I}\right)$ and the last step follows from the void probability of the nearest BS [10].

Notice (27) is identical to the CCDF of a Gumbel distributed random variable with parameters $\beta=\left[2(P(t)+P_{c})/\alpha\right]^{-1}$ and $\mu=\beta\log\left(\pi{c_{0}}^{2/\alpha}\right)$. Taking an integration over $v>0$ at (27) thereby leads to the Gumbel mean given as $\mu+\beta\gamma$. Taking an expectation over the associated fading channel provides the spatially averaged EE by using Slyvnyak’s theorem [10] $\blacksquare$

It is worth mentioning that replacing $P(t)+P_{c}$ by unity in Proposition 1 yields the instantaneous spatially averaged downlink rate. Respectively for noise-limited or interference-limited regime, this downlink rate is a logarithmic function of BS density or BS/user density ratio (recall the definition of $\hat{I}$), which is interestingly in line with our preceding result without incorporating MF interference [5].

Calculating $\mathsf{EE}(t)$ is viable by using $g(t)$’s evolutional law given as (1). Given $t>0$, the following verifies $|g(t)|$ follows Rician fading.

Utilizing Proposition 1 with Lemma 1 allows us to consider the EE maximization at $t>0$.

The solution of this problem is given in the following proposition.

This section validates the impact of the proposed power control policy under MF interference (Proposition 2).

Fig. 2 shows the proposed power control (solid blue) provides $1.58$ times higher EE compared to a $P_{\max}/2$ fixed power baseline (dotted red), which corresponds to achieving $98$% of the EE maximum via a full search algorithm (dashed black). For small $t$, transmission power of the proposed scheme fluctuates due to the recursive relationship between $P(t)$ and $\hat{P}(t)$. This quickly converges to the EE optimal power within a couple of iterations for different values of $P(0)$ and $\hat{P}(0)$. The curve variances for the full search and fixed power algorithms solely come from temporally correlated fading, vanishing as time elapses.

Fig. 3 illustrates the maximized EE by the proposed power control increases along with $N$. In addition, the EE grows as $\lambda_{b}$ increases. This numerical evaluation is based on stationary fading, i.e. $\lim_{t\rightarrow\infty}|g(t)|$ following Rice$(|\mu|,\eta)$ according to Lemma 1.

In this work, we have characterized the EE of a UDN taking into account the spatio-temporal dynamics arising from BS locations, channel information, and interference patterns. Closed-form expressions of the MF measure and the utility function have been derived by invoking MF interference convergence as a function of BS/user density ratio, fading channel, and reception ball size, i.e. LOS dependency of networks. Remarkably our result allows operators to optimize their energy-efficient networks with a large number of BSs in a tractable manner under MF constraints. On the basis of the proposed framework, incorporating user mobility is our work in progress [19]. Considering other spatio-temporal SG settings and/or spatial MFG could also be an interesting topic.

This work was supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (No.B0126-16-1017, Spectrum Sensing and Future Radio Communication Platforms).

The following proves Theorem 1 for $N=1$ by sequentially verifying the aforementioned three conditions for MF interference (replacing $\lambda_{b}$ in this proof with $N^{\frac{1}{\alpha}}\lambda_{b}$ straightforwardly leads to the desired result for $N>1$).

Firstly, BS coordinates $z_{i}$’s as well as fading vector $g_{i}$’s are mutually independent, and so are their $h_{i}$’s. Secondly, the BSs’ common rule of user associations and homogeneous admissible power control along with the identical fading distribution lead to identically distributed $h_{i}$’s, ensuring exchangeability.

1. 2-1)

   MF measure existence for $R<\infty$:

   For $\mathsf{E}[|g_{0}|^{2}]<\infty$ and $\lambda_{u}\rightarrow\infty$, consider ${\lambda_{b}}/\lambda_{u}\rightarrow\infty$.

   	$\displaystyle\hskip-20.0pt\underset{\begin{subarray}{c}{\lambda_{b}}/\lambda_{u}\rightarrow\infty\\ \lambda_{u}\rightarrow\infty\end{subarray}}{\lim}\hat{I}_{p_{a}\lambda_{b}}(t)$
   		$\displaystyle\hskip-70.0pt\overset{(a)}{\sim}\hskip-5.0pt\underset{\begin{subarray}{c}{\lambda_{b}}/\lambda_{u}\rightarrow\infty\\ \lambda_{u}\rightarrow\infty\end{subarray}}{\lim}\frac{1}{{\lambda_{b}}^{\frac{\alpha}{2}}}\left[\sum_{i=1}^{|\Phi^{o}_{R}\left({\lambda_{u}}\right)|}\hskip-10.0ptP_{i}\left\{t,h_{i}(t,\lambda_{u})\right\}|h_{i0}(t,\lambda_{u})|^{2}-S_{\lambda_{u}}(t)\right]$		(32)
   		$\displaystyle\hskip-70.0pt=\hskip-5.0pt\underset{\begin{subarray}{c}{\lambda_{b}}/\lambda_{u}\rightarrow\infty\\ \lambda_{u}\rightarrow\infty\end{subarray}}{\lim}\hskip-5.0pt\frac{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}{{\lambda_{b}}^{\frac{\alpha}{2}}}\frac{1}{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}\sum_{i=1}^{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}\hskip-10.0ptP_{i}\left\{t,h_{i}(t,\lambda_{u})\right\}|h_{i0}(t,\lambda_{u})|^{2}$		(33)

   where $(a)$ follows from $\lim_{\lambda_{b}/\lambda_{u}\rightarrow\infty}p_{a}=\lambda_{u}/\lambda_{b}$ by Taylor expansion, in conjunction with the mapping theorem [17]. The last step comes from $S_{\lambda_{u}}(t)\leq P_{\max}|g_{0}|^{2}<\infty$ $a.s$. The last inequality is derived by applying Markov’s inequality to $|g_{0}|^{2}$ as follows.

   $\displaystyle\lim_{v\rightarrow\infty}\Pr\left(|g_{0}|^{2}\geq v\right)\leq\lim_{v\rightarrow\infty}\mathsf{E}[|g_{0}|^{2}]/v=0$

   (34)

   Define $M_{t}$ as

   $\displaystyle M_{t}$

   $\displaystyle:=\frac{1}{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}\sum_{i=1}^{|\Phi^{o}_{R}\left({\lambda_{u}}\right)|}\delta_{h_{i}(t,\lambda_{u})}.$

   (35)

   Note that $|\Phi^{o}_{R}\left({\lambda_{u}}\right)|$ is the number of active BSs within $b(o,R)$, following Poisson distribution with mean ${\lambda_{u}}\pi R^{2}$. It is therefore straightforward $|\Phi^{o}_{R}\left({\lambda_{u}}\right)|\rightarrow\infty$ $a.s.$ as $\lambda_{u}\rightarrow\infty$. In such a condition, the empirical measure $M_{t}$ converges to Borel probability measure $m_{t}$, i.e. $m_{t}=\lim_{\lambda_{u}\rightarrow\infty}M_{t}$ $a.s.$ according to Theorem 11.4.1 in [21]. This proves the existence of $m_{t}$.

2. 2-2)

   MF measure existence for $R\rightarrow\infty$:

   The proof directly follows from (33) while relaxing the necessary condition by replacing $\lambda_{u}\rightarrow\infty$ with $\lambda_{u}>0$. Under this condition, $|\Phi_{R}^{o}(\lambda_{u})|\rightarrow\infty$ as $R\rightarrow\infty$. This leads to $\lim_{R\rightarrow\infty}M_{t}=m_{t}$ $a.s.$

1. 3-1)

   Finite MF interference for $R<\infty$:

   Consider $\lambda_{b}/{\lambda_{u}}^{\frac{4}{\alpha}}\rightarrow\infty$ for $\lambda_{u}\rightarrow\infty$ and $\mathsf{E}[|g_{0}|^{2}]<\infty$. Applying $m_{t}$, the terms after $|\Phi_{R}^{o}\left({\lambda_{u}}\right)|/\left({\lambda_{b}}/\lambda_{u}\right)^{\frac{\alpha}{2}}$ in (33) becomes

   	$\displaystyle\frac{1}{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}\sum_{i=1}^{|\Phi_{R}^{o}\left({\lambda_{u}}\right)|}P_{i}\left\{t,X_{i}(t,\lambda_{u})\right\}|h_{i0}(t,\lambda_{u})|^{2}$
   		$\displaystyle\hskip-200.0pt\overset{(b)}{=}\mathsf{E}_{\chi}\left[P\{t,x\}\right]\mathsf{E}_{g}\left[|g(t)|^{2}\right]\mathsf{E}_{\Phi_{R}^{o}\left({\lambda_{u}}\right)}\left[l_{i0}(\lambda_{u})\right]$		(36)

   where $\mathsf{E}_{\chi}\left[P\{t,x\}\right]=\int_{x\in\chi}P\{t,x\}m_{t}(dx)$. The step $(b)$ comes from the mutual independence of $g_{i0}(t)$’s and $\ell_{i0}(\lambda_{u})$’s.

   The first term $\mathsf{E}_{\chi}\left[P\{t,x\}\right]$ in (36) converges since $\mathsf{E}_{\chi}\left[P\{t,x\}\right]\leq P_{\max}<\infty$ when considering the maximum power transmission while neglecting a generic BS’s remaining energy state. The second term $\mathsf{E}_{g}\left[|g(t)|^{2}\right]$ converges by definition. By using Campbell’s theorem [17], the last term is

   $\displaystyle\mathsf{E}_{\Phi_{R}^{o}\left({\lambda_{u}}\right)}\left[l_{i0}(\lambda_{u})\right]$

   $\displaystyle={\lambda_{u}}\pi\left(1+\frac{1-R^{2-\alpha}}{\alpha-2}\right).$

   (37)

   Applying this to (36) combined with $\lim_{\lambda_{u}\rightarrow\infty}|\Phi_{R}^{o}(\lambda_{u})|=\mathsf{E}\left[|\Phi_{R}^{o}(\lambda_{u})|\right]=\lambda_{u}\pi R^{2}$ makes (33) become

   $\displaystyle\underbrace{\frac{(\lambda_{u}\pi R)^{2}}{{\lambda_{b}}^{\frac{\alpha}{2}}}}_{(c)}\left(1+\frac{1-R^{2-\alpha}}{\alpha-2}\right)\mathsf{E}_{\chi}\left[P\{t,x\}\right]\mathsf{E}_{g}\left[|g(t)|^{2}\right].$

   (38)

   The term $(c)$ converges on $0$ since $\lambda_{b}/{\lambda_{u}}^{\frac{4}{\alpha}}\rightarrow\infty$, verifying $\hat{I}_{p_{a}\lambda_{b}}$ is $a.s.$ finite.

2. 3-2)

   Finite MF interference for $R\rightarrow\infty$:

   Consider $\lambda_{b}/(\lambda_{u}R)^{\frac{4}{\alpha}}\rightarrow\infty$ for $\lambda_{u}>0$ and $\mathsf{E}[|g_{0}|^{2}]<\infty$. Under these conditions, (38) holds and the term $(c)$ converges on $0$, yielding the desired result.

Combining the results of subsections 1, 2, and 3 completes the proof of Theorem 1. $\blacksquare$