Stochastic Geometry Analysis for Distributed RISs-Assisted mmWave Communications

Millimeter wave (mmWave) technology has become the core technology for the next generation of mobile communication networks due to its abundant spectrum resources and low spectrum interference[1, 2, 3]. The high-frequency nature of mmWave massive multiple-input multiple-output technology enables high spatial resolution and directivity, facilitating concentrated power transmission of communication signals. However, it renders the signals highly susceptible to atmospheric absorption and rain attenuation, resulting in significant penetration loss and path loss[4, 5]. To address these challenges, one feasible approach is to deploy distributed reconfigurable intelligent surfaces (RISs), each consisting of numerous low-cost passive reflective elements[6, 7, 8]. Using an intelligent controller, each unit or element can manipulate amplitudes and phase shifts to effectively change incident electromagnetic waves, thereby reconfiguring the wireless propagation environment. Moreover, the distributed RISs provide additional transmission links, effectively guaranteeing the quality of service for users[9]. Therefore, we are interested in the performance gain that distributed RISs can bring to communication systems.

Nevertheless, analyzing the communication performance within a distributed RISs-assisted system remains a challenge. Recently, stochastic geometry has proven to be a successful mathematical framework for simulating the operation of diverse types of wireless networks[10, 11, 12]. In [13], stochastic geometry was utilized to stochastically model the locations of receivers and transmitters using Poisson point processes (PPPs). By considering typical receivers positioned at the origin, the analysis of average network performance was simplified, provided performance metric bounds, and approached realistic scenarios. In [14], the authors leveraged the line-of-sight (LoS) probability function proposed in [13] to model the locations of LoS and non-LoS base stations (BSs) as two independent inhomogeneous PPPs, and further analyzed the dense network case by applying a simplified sphere model. In [15], the authors proposed an analytical framework that incorporates realistic channel models and antenna element radiation patterns. In [16], the authors disregarded the small-scale channel fading model, deployed RISs as coatings on a portion of the blockages, and utilized stochastic geometry to analyze the impact of coating blockages with RISs on the cellular network coverage probability. The authors in [17] used stochastic geometry to analyze the performance of a realistic two-step user association-based mmWave network, consisting of multiple users, transmitters, and one-hop reflection from a large intelligent surface.

Building on the discussion above, this paper investigates the performance analysis in distributed RISs-assisted mmWave communication systems using stochastic geometry, and numerical results validate the gains from the deployment of distributed RISs. In more detail, we start with establishing the scene model and the system model using stochastic geometry. Next, we define the association criterion and decompose the RIS distribution model into several inhomogeneous PPPs using the LoS probability. Afterward, we analyze the two cases of direct association between the users with the serving BS and association through RIS reflection, and derive the association probabilities, the distance distributions, and the conditional coverage probabilities. Using the total probability theorem and Campbell’s theorem, we combine the two cases to derive the ergodic coverage probability. Moreover, we derive closed-form expressions for the sum rate by transformations between the expectation and distribution functions.

Simulation results verify the gains in the ergodic coverage probability and the sum rate brought by the distributed RISs. Specifically, distributed RISs significantly improve the ergodic coverage probability by 45.4% and the sum rate by over 1.5 times when the blockage density is $\lambda_{b}=1.59e$-$3/$m². These results indicate how many distributed RISs should be deployed to reach the performance limit, which provide a valuable guideline for the practical deployment of RISs in cellular networks.

We consider a downlink distributed RISs-assisted mmWave communication system as illustrated in Fig. 1, including the single BS serving multiple users. It is assumed that a BS can associate with several users, but only serves one user in a time slot, whether through a direct link or a RIS reflected link, providing the maximum channel gain. Considering the irregular shapes of real cells, we introduce the concept of the cell virtual radius, which is the radius of an equivalent circle of average cell size, denoted as $r_{v}$.

Disregarding the height information, the cell is modeled as a planar area with a virtual radius of $r_{v}=R$. The BS is positioned at the center of this area, equipped with $N_{BS}$ antennas. Assuming users remain static within a short time interval, their positions are modeled by a PPP with a density of $\lambda_{u}(\xi)$, denoted as $\Phi_{u}=\{u_{i}\}\in\mathbb{R}^{2}$, where $u_{i}$ is the position coordinate of the $i$-th user, and $\Phi_{u}$ is the set that comprises all user coordinate points. The parameter $\xi$ denotes the distance between the BS and users, and each user is equipped with $N_{u}$ antennas. The positions of RISs are denoted as the PPP $\Phi_{R}=\{R_{i}\}\in\mathbb{R}^{2}$, with a density of $\lambda_{R}$. Here, $R_{i}$ is the position of the $i$-th RIS, $\Phi_{R}$ comprises all RIS coordinate points, and each RIS consists of $N_{R}$ elements. The horizontal and vertical spaces between these elements/antennas are $\lambda/2$, where $\lambda$ represents the signal wavelength. In addition, it is assumed that there is always a LoS link between the BS and the RISs, and that the BS can select different steering vectors as its transmit beams toward the RISs.

Blockages, characterized by their spatial randomness and irregular shapes, especially for buildings in urban areas, are typically modeled as random object processes (ROPs) in the theory of random shapes[18]. However, analyzing ROPs can be complex, especially when considering correlations between objects, as well as their shape, position, and orientation. To make it analyzable, we approximate blockages using the line Boolean model[13], i.e., a line segment models each blockage.

As for the signal model, it is assumed that the data transmission links between the BS and the users have already been beam-aligned using existing beamforming techniques like zero forcing[19], minimum mean square error[20], single value decomposition[21], etc. However, interfering signals may arrive through non-beam-aligned links.

Thus, the signal-to-interference-plus-noise ratio (SINR) of the direct links and the RIS reflection links can be defined as follows. For a user located at a distance $\xi$ from the BS, the large-scale fading channel gain of the direct link is denoted as $g_{d}=\frac{10^{\alpha}}{\xi^{\beta}}$. The received signal at the user is denoted as $y_{d}$ with power $|y_{d}|^{2}=\frac{10^{\alpha}\cdot P_{0}\cdot h_{d}}{\xi^{\beta}}$, where $P_{0}$ is the transmit power of the BS to each user and $h_{d}\sim\mathbf{Exp}(N_{BS}N_{u})$ is the small-scale fading channel gain of the BS-user link. It is assumed that the interference from other users has been eliminated using existing techniques. Therefore, the SINR of the direct link is expressed as

where $\sigma^{2}$ is the noise power.

For the case where the BS reflects the signal to users through the $l$-th RIS $R_{l}$, we define the channel as the $l$-th reflected LoS link. The large-scale fading channel gain and received signal power of the $l$-th reflected link are expressed as $g_{I_{l}}=\frac{10^{2\alpha}}{(s_{l}r_{l})^{\beta}}$ and $|y_{I_{l}}|^{2}=\frac{10^{2\alpha}P_{0}h_{s_{l}}h_{r_{l}}}{(s_{l}r_{l})^{\beta}}$, respectively. Here, $s_{l}$ and $h_{s_{l}}\sim\mathbf{Exp}(N_{BS}N_{R})$ are the distance and the small-scale fading channel gain of the BS-$R_{l}$ link, $r_{l}$ and $h_{r_{l}}\sim\mathbf{Exp}(N_{R}N_{u})$ are the distance and the small-scale fading channel gain of the $R_{l}$-user link, respectively. The SINR of the BS-$R_{l}$-user link can be expressed as

In this section, a theoretical performance analysis is presented leveraging the inherent void probability and contact distribution[22] of the PPP, the location-dependent thinning[23], and the LoS probability.

In the following, we start with deriving the probability that RISs can assist the communication for any user within the set $\Phi_{u}$. It is assumed that RISs can both reflect and transmit to achieve 360-degree coverage. It is further assumed that RISs only assist communication in scenarios where the direct link is non-line-of-sight (NLoS).

Next, we define the association criterion.

Proof: First, we calculate the number of RISs capable of providing reflective LoS links, which requires a cell-wide integration over the density of LoS RISs ${\lambda}^{L}_{R}$. Since ${\lambda}^{L}_{R}$ depends on the RIS-user distance $r$, which requires transforming the area into an expression on $r$. To facilitate this transformation, we introduce the angle $\psi=2\pi-\vartheta$, and the shaded area in Fig. 2 can be approximated as $\frac{r^{2}\mathrm{d}\psi}{2}$. Utilizing the law of cosines, we can obtain $\cos(2\pi-\psi)=\cos\vartheta=\frac{\xi^{2}+r^{2}-R^{2}}{2\xi r}$. Further, we derive that $r=\sqrt{R^{2}-\xi^{2}\sin^{2}\psi}-\xi\cos\psi$. Hence, the number of RISs that can provide reflective LoS links is

Thus, the probability is further expressed as

Proof: As shown in Fig. 2, the angle between the BS-RIS and BS-user links is defined as $\theta$, because the location-dependent thinning is only related to the distance but not to the angle, it can be assumed that $\cos\theta\sim\mathcal{U}[-1,1]$, then we have $r_{l}=\sqrt{s_{l}^{2}+\xi_{l}^{2}-2s_{l}\xi_{l}\cos\theta}$. In Eq. (10), the lower bound of equal sign $(a)$ is established because $\min\limits_{l}\{s_{l}\cdot r_{l}\}>x$. This implies that for $\forall l\in\{1,...R\}$, there is no $s_{l}\cdot r_{l}<x$. Omitting the subscript $l$, $s\cdot r<x$ is equivalent to $\cos\theta>\frac{s^{4}+s^{2}\xi^{2}-x^{2}}{2s^{3}\xi}$. The upper bound is derived from the fact that $r^{2}=s^{2}+\xi^{2}-2s\xi\cos\theta>0$. Thus, within the region satisfying $\frac{s^{4}+s^{2}\xi^{2}-x^{2}}{2s^{3}\xi}<\cos\theta<\frac{s^{2}+\xi^{2}}{2s\xi}$ and in a radius $R$, there exists no RIS capable of assisting communication. Consequently, the number of points in ${\Phi}^{L}_{R}$ is 0.

Proof: Since the multiplicative path loss of the RIS reflection link is several orders of magnitude larger than that of the direct link, the association criterion degenerates. When a direct LoS link between the user and the serving BS exists, there is a direct association. On the other hand, when the direct link is NLoS and a LoS reflection link exists, the association criterion is an association through RIS reflection.

Then, the conditional coverage probability of the direct link is derived as

Since the criterion prioritizes the link with the largest received signal power, we obtain the associated RIS as

The SINR of the associated link is expressed as $\gamma_{I}=\frac{10^{2\alpha}P_{0}h_{s_{l^{*}}}h_{r_{l^{*}}}}{(s_{l^{*}}r_{l^{*}})^{\beta}\sigma^{2}}$. Thus, the conditional coverage probability of the reflected LoS link is

Therefore, the conditional coverage probability of users at a distance $\xi$ from the BS is

The ergodic coverage probability of the cell is

where $(b)$ is based on Campbell’s theorem[24].

Proof: Obviously, the random variable $W\log_{2}(1+\gamma)$ is positive. According to Proposition 2, the ergodic rate of users at a distance $\xi$ from the BS is

Since the distance $\xi$ between each user and the BS is also a random variable, the sum rate of the cell can be expressed as

In this section, we provide numerical results for the coverage probability and the sum rate in distributed RISs-assisted mmWave wireless communication systems. We assume that the mmWave cellular network operates at a frequency of $f_{c}=3$ GHz. The parameters in the large-scale fading channel gain model are set to $\alpha=3.8$ and $\beta=2.2$. The average length of the blockage line Boolean model is $E[L]=15$ m. The BS antenna array consists of $N_{BS}=64$ antennas. We set the virtual radius as $R=100$ m and the density of users as $\lambda_{u}=3.18e$-$3/$m². Each user is equipped with $N_{u}=4$ antennas and is allocated a bandwidth of $W=200$ MHz.

Fig. 3 plots the ergodic coverage probability for different densities of RISs with a blocking density of $\lambda_{b}=1.59e$-$3/$m². The solid line represents the ergodic coverage probability in Theorem 1, while the corresponding dashed line is obtained by Monte Carlo simulation. The simulation results can overlap the theorem well, which proves the accuracy of our derived formulation. Moreover, we observe that deploying more RISs increases the coverage probability, but the rate of improvement decreases as the density of RISs increases, eventually reaching saturation. Specifically, deploying RISs with a density of $\lambda_{R}=1.59e$-$4/$m² (around the number of $5$) can improve the coverage probability by 27.3%, while deploying RISs with a density of $\lambda_{R}=9.55e$-$4/$m² (around $30$) improves the coverage probability by 45.4%.

Fig. 4 plots the effect of the density of deployed RISs on the sum rate of all users in the cell for varying blockage densities. The results of Theorem 2, denoted by the solid line, and the Monte Carlo scenario simulation, denoted by the dashed line, are closely matched. As the blockages become denser, the sum rate decreases, yet the improvement in the sum rate is more significant with the same density of RISs deployed. For example, when RISs with density $\lambda_{R}=1.59e$-$3/$m² (around $50$) are deployed, the sum rate increases by $70.5$ bps/Hz (1.15 times) when $\lambda_{b}=3.18e$-$4/$m², and by $241.8$ bps/Hz (1.91 times) when the blockage density $\lambda_{b}=1.59e$-$3/$m². However, it is worth noting that the curve tends to flatten out as the density of RISs increases, which means that the sum rate does not increase indefinitely as the density of RISs increases, but rather reaches an upper limit.

In this paper, we leveraged stochastic geometry to investigate the ergodic coverage probability and the sum rate in distributed RISs-assisted wireless communication systems. Firstly, the system model was established stochastically, including the distribution models of blockages, RISs, and users. Subsequently, the association criterion was defined, and the inhomogeneous PPPs for RISs were derived based on the LoS probability. Then, the association probabilities, the distance distributions, and the conditional coverage probabilities were obtained for the two cases of direct association and reflective association by RISs. Finally, we combined the two cases and obtained the closed-form expressions for the ergodic coverage probability and the sum rate.

The stochastic geometry analysis and simulation results can provide insights into how many RISs should be distributed to achieve a given performance at different blockage densities, which essentially facilitates the optimal design and cost control for practical RIS deployment.