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Exploiting Randomly-located Blockages for Large-Scale Deployment of Intelligent Surfaces

Mustafa A. Kishk, Mohamed-Slim Alouini

arXiv:2001.10766v3cs.ITcs.NI

TL;DR

The paper studies how RIS-equipped blockages can restore indirect LoS links in cellular networks with blocked user-BS paths. Using stochastic geometry and a line Boolean blockage model, it derives coverage-related metrics and shows that RIS deployment can sharply reduce blind spots, with required density depending strongly on blockage density and deployment strategy.

  • Problem

    The paper addresses how large-scale RIS deployment on communication blockages can improve cellular coverage for user-BS pairs without direct LoS links.

  • Method

    It uses stochastic geometry with a line Boolean blockage model to analyze RIS-equipped blockages and derive coverage, association, and path-loss metrics.

  • Results

    6 RISs/km2 suffice at 300 blockage/km2, whereas 490 RISs/km2 are required at 700 blockage/km2 to reach E = 10^-5.

  • Takeaways & Limitations

    RIS deployment improves coverage by providing indirect LoS links for blocked user-BS pairs, while deployment effectiveness depends on blockage density and planning.

  • Takeaways & Limitations

    Random RIS deployment provides a lower performance bound than well-planned deployment, whose deployment criteria remain an open research problem.

Abstract

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One of the promising technologies for the next generation wireless networks is the reconfigurable intelligent surfaces (RISs). This technology provides planar surfaces the capability to manipulate the reflected waves of impinging signals, which leads to a more controllable wireless environment. One potential use case of such technology is providing indirect line-of-sight (LoS) links between mobile users and base stations (BSs) which do not have direct LoS channels. Objects that act as blockages for the communication links, such as buildings or trees, can be equipped with RISs to enhance the coverage probability of the cellular network through providing extra indirect LoS-links. In this paper, we use tools from stochastic geometry to study the effect of large-scale deployment of RISs on the performance of cellular networks. In particular, we model the blockages using the line Boolean model. For this setup, we study how equipping a subset of the blockages with RISs will enhance the performance of the cellular network. We first derive the ratio of the blind-spots to the total area. Next, we derive the probability that a typical mobile user associates with a BS using an RIS. Finally, we derive the probability distribution of the path-loss between the typical user and its associated BS. We draw multiple useful system-level insights from the proposed analysis. For instance, we show that deployment of RISs highly improves the coverage regions of the BSs. Furthermore, we show that to ensure that the ratio of blind-spots to the total area is below 10^5, the required density of RISs increases from just 6 RISs/km2 when the density of the blockages is 300 blockage/km^2 to 490 RISs/km^2 when the density of the blockages is 700 blockage/km^2.

I. INTRODUCTION

RISs are presented as controllable reflective surfaces for addressing blocked wireless links, especially by equipping existing blockages to create indirect LoS paths. The paper positions stochastic-geometry analysis of randomly located RIS-equipped blockages as a gap in prior work.

  • RISs use adjustable phase shifts to steer reflected waves toward intended directions while consuming less energy than comparable relays.
  • Blocked user-BS links can gain an indirect LoS path when an RIS has LoS with both endpoints.
  • Coating buildings or trees with RISs exploits existing communication blockages to increase BS coverage and reduce blind-spot areas.
  • The paper develops a stochastic-geometry framework for large-scale RIS deployment, emphasizing LoS probability and average path-loss improvement.
  • Prior studies considered RIS-enabled systems, blockage effects, or fixed RIS locations, whereas stochastic-geometry modeling of RIS locations remains scarce.

B. Contributions

The paper analyzes randomly RIS-equipped blockages in a line Boolean cellular-network model and derives coverage, blind-spot, indirect-service, deployment-efficiency, and path-loss metrics. Its results show strong benefits from RIS coating, strategic placement, and more meta-surfaces, while random deployment provides a lower performance bound.

  • Large scale deployment of RISs: The model uses a line Boolean blockage model and equips a fraction µ of blockages with RISs to provide indirect LoS links.
  • Performance Analysis: The paper derives LoS probability, blind-spot area, RIS deployment efficiency, and path-loss performance as functions of RIS coverage conditions.
  • System-level insights: Increasing the number of meta-surfaces per RIS significantly reduces the required fraction of RIS-equipped blockages.
  • System-level insights: 2% of blockages need RISs at 300 km^-2, versus 70% at 700 km^-2, to significantly reduce blind-spot area.
  • System-level insights: Strategic RIS placement can reduce required deployment density by more than 70% in high-blockage environments and more than 80% in low-blockage environments.
  • Performance metrics: The analysis derives blind-spot ratio E, indirect-service probability Ai, and threshold-based path-loss probability Pcov.
  • Deployment assumptions: Random RIS deployment yields a lower performance bound than well-planned deployment, whose strategic criteria remain an open research problem.

A. Path-Loss Model

The path-loss model characterizes indirect RIS-assisted links in high-frequency settings where meta-surfaces are larger than the wavelength. Received power scales with the squared meta-surface count and the total two-hop distance.

  • The model assumes high-frequency operation with RIS meta-surfaces considerably larger than the signal wavelength.
  • Indirect-path received power scales as M^2(dU-I + dI-B)^-α, using the user-RIS and RIS-BS distances.
  • The analysis treats M as the number of meta-surfaces in the RIS and α as the path-loss exponent.

B. Number of Meta-Surfaces per RIS

The paper models RISs with either a fixed or random number of meta-surfaces and defines association and coverage using average path-loss. A realization shows that equipping blockages with RISs can sharply reduce blind-spot areas.

  • B. Number of Meta-Surfaces per RIS: The analysis considers fixed M = MF for every RIS or random M with PMF P(M = k) = ρk.The fixed-M case is a special case of the random-M scenario.
  • B. Number of Meta-Surfaces per RIS: For random meta-surface counts, RISs with k meta-surfaces have density λR,k = ρkµλb.Comparisons between fixed and random scenarios use the same average number of meta-surfaces per unit area: MFλR = E[M]λR.
  • B. Number of Meta-Surfaces per RIS: Users associate with the BS providing the lowest average path-loss through either a direct or indirect LoS link.RISs are assumed to support only NLoS links, so they are not used when a direct LoS link exists.
  • B. Number of Meta-Surfaces per RIS: Equipping 5% of blockages with RISs highly reduces blind-spots, while µ = 0.4 makes the blind-spot areas disappear in the shown realization.The realization uses λb = 500 blockages/km2, λR = µλb, 10 BSs/km2, and a 1 km-square area.
  • B. Number of Meta-Surfaces per RIS: Users can be associated through a direct link, an indirect link, or fall in a blind-spot; coverage is defined by path-loss below threshold τ.The RIS association probability estimates the ratio of utilized RISs.

III. PRELIMINARIES ON STOCHASTIC GEOMETRY

This section introduces stochastic-geometry tools used in the paper, including Poisson point processes, void probabilities, thinning, and isotropic inhomogeneous PPPs.

  • III. PRELIMINARIES ON STOCHASTIC GEOMETRY: A homogeneous PPP has Poisson point counts with mean λh|B| in any area B, while an inhomogeneous PPP uses spatial density λi(x).These models represent point-process locations used in the analysis.
  • III. PRELIMINARIES ON STOCHASTIC GEOMETRY: The void probability gives the probability of zero points in an area and supports computation of contact-distance distributions.For a homogeneous PPP in B, the void probability is exp(−λh|B|).
  • III. PRELIMINARIES ON STOCHASTIC GEOMETRY: Independent thinning removes each point independently, while location-dependent thinning removes x with probability 1−g(x).The retention probability depends only on the point location and not on other point locations.
  • III. PRELIMINARIES ON STOCHASTIC GEOMETRY: Location-dependent thinning of a homogeneous PPP with density λ produces an inhomogeneous PPP with density ˜λ(x) = g(x)λ.The proposition is repeatedly used in the paper’s analytical derivations.
  • III. PRELIMINARIES ON STOCHASTIC GEOMETRY: An inhomogeneous PPP with radially symmetric density is isotropic, meaning its distribution is invariant under rotations around the origin.Radially symmetric intensity depends only on ∥x∥.

IV. PERFORMANCE ANALYSIS

The performance analysis derives blind-spot fractions, indirect-path probabilities, visible-BS densities, and shortest visible-path distributions for RIS-assisted cellular networks.

  • IV. PERFORMANCE ANALYSIS: The analysis derives mathematical expressions for the paper’s formally defined performance metrics, assuming the typical user is at the origin.The blockage model uses uniformly distributed orientations with length L and density λb.
  • IV. PERFORMANCE ANALYSIS: Lemma 1 models an RIS’s ability to provide an indirect LoS path through user-RIS LoS, RIS-BS LoS, and a suitable orientation.The RIS distance t creates a trade-off: orientation effects weaken as t increases, while LoS probabilities decrease.
  • IV. PERFORMANCE ANALYSIS: RISs serving a specific area need not be the closest RISs because increasing t reduces orientation sensitivity but also reduces LoS-link probabilities.The insight applies to random, unplanned RIS deployment.
  • IV. PERFORMANCE ANALYSIS: The visible-BS density becomes λBSPv(r), where Pv(r) = PLoS(r) + PNLoS(r)PI(r), combining direct and indirect visibility.RIS deployment adds λBSPNLoS(r)PI(r) to the pre-deployment visible-BS density λBSPLoS(r).
  • IV. PERFORMANCE ANALYSIS: Deploying RISs reduces the fraction of blind-spots by a factor, with the fraction decreasing because PI(r) increases with µ.Theorem 1 provides the blind-spot fraction expression.

B. Indirect LoS-link Length Distribution

The paper derives distributions for shortest indirect paths through RISs and combines them with direct-path distributions to obtain the shortest visible path-length distribution.

  • B. Indirect LoS-link Length Distribution: The analysis studies indirect LoS-link length statistics before deriving association through direct or indirect links.The target is the shortest visible path, which may be direct or indirect.
  • B. Indirect LoS-link Length Distribution: Lemma 4 gives the distributions of Ri,k|r and Ri, representing shortest indirect paths through RISs with k meta-surfaces or any RIS.The conditional path length is for a BS at distance r from the typical user.
  • B. Indirect LoS-link Length Distribution: The derivation uses location-dependent thinning so retained BSs represent those whose shortest indirect RIS path is shorter than x.The resulting BS process is modeled as an inhomogeneous PPP with density λBSFRi,k|r(x).
  • B. Indirect LoS-link Length Distribution: Lemma 5 provides the distributions of Ri,k and Ri by combining conditional indirect-path distributions with the BS process.These distributions support the subsequent shortest visible-path result.
  • B. Indirect LoS-link Length Distribution: Theorem 2 derives the CDF of the shortest path-length between the typical user and a visible BS, whether the path is direct or indirect.The result combines FRd(x) with FRi(x).

C. Association Probability and RIS Deployment Efficiency

A typical user is partitioned among blind-spot, direct-path, and RIS-assisted associations, with the RIS-assisted probability determining associated-user density and deployment efficiency.

  • Users fall into blind-spot, direct-path, or RIS-assisted association states whose probabilities sum to one.
  • Theorem 3 gives the probability that a typical user associates with a BS through an RIS.
  • Given user density λu, RIS-associated users have density Aiλu.
  • When Aiλu ≫ λR, all deployed RISs are used for coverage; when Aiλu ≤ λR, at least λR−Aiλu RISs are unused.
  • RIS deployment efficiency is defined as η = min{1, λuAi…The supplied passage truncates the displayed definition after λuAi.
  • The efficiency expression provides an upper bound on the average number of RISs associated with at least one user.

D. Coverage Analysis

The paper derives coverage-related metrics for RIS-assisted cellular networks and evaluates how blockage density, RIS deployment, and RIS size affect visibility, blind spots, deployment efficiency, and coverage probability.

  • Coverage probability: Theorem 4 gives the probability that the average path-loss between a typical user and its associated BS is below threshold τ.The coverage probability is expressed using the direct-path-loss distribution and the function H(x) from Theorem 3.
  • Visibility probability: Increasing the RIS-equipped blockage fraction µ significantly increases visibility probability at lower blockage densities, but its influence decreases as blockage density rises.This behavior is evaluated through the visibility probability Pv(r) versus user–BS distance r for blockage densities of 300, 500, and 700 km^-2.
  • Blind spots: At 300 blockage/km2, equipping 2% of blockages with RISs reduces the blind-spot ratio E to 10^-5, requiring 6 RISs/km2.At 700 blockage/km2, reaching E = 10^-5 requires µ = 70%, or 490 RISs/km2.
  • Deployment efficiency: Increasing µ lowers deployment efficiency because the fraction of utilized RISs decreases under random, unplanned deployment.The analysis relates this inefficiency to the random selection of equipped blockages and contrasts it with well-planned deployment.
  • Coverage probability: At 700 blockage/km2, reaching Pcov = 0.75 requires µ = 1 for M = 1, µ = 0.1 for M = 2, and µ = 0.05 for M = 3.Increasing M from 1 to 3 also raises the maximum achievable Pcov at µ = 1 from 0.8 to 1.
  • RIS configuration: For the same mean number of meta-surfaces per RIS, uniformly distributed M produces higher Pcov than fixed M.The comparison considers fixed M = MF versus uniformly distributed M with E[M] = MF for MF = 2 and MF = 3.

VI. CONCLUSIONS

The paper develops a stochastic-geometry framework for large-scale RIS deployment on randomly blocked cellular links and derives several performance metrics. Its results provide deployment guidance while identifying the unresolved problem of optimally selecting RIS-equipped blockages.

  • The paper provides a stochastic geometry-based performance evaluation for large-scale RIS deployment in cellular networks with randomly located blockages.
  • The analysis derives indirect path probability, visible-BS density, blind-spot area, RIS-based association probability, and average path-loss distribution.
  • Well-planned RIS deployment can reduce the required deployment density for achieving target performance levels compared with random deployment.
  • The number of meta-surfaces per RIS significantly affects cellular-network performance.
  • The derived system-level insights support pre-deployment design, including guidelines for RIS density under different blockage densities.
  • Selecting the optimal subset of blockages to equip with RISs remains an open research problem for a fixed RIS-equipped fraction.

APPENDIX A

The appendix characterizes when a blockage-mounted RIS can provide an indirect path and derives the associated path-length distribution. The construction combines geometric feasibility conditions with stochastic modeling of capable RIS locations.

  • The analysis computes RIS-to-BS distance and angular conditions from the user-BS geometry using the cosine law.
  • An indirect RIS path requires the user to face the RIS-equipped side and the user and BS to lie on the same side of the RIS.
  • The RIS-to-BS angle must satisfy θ ≥ ψ for the RIS to provide an indirect path.
  • It derives the probability that at least one capable RIS provides an indirect path between a typical user and a BS.
  • For a given user-BS distance, the indirect path length through an RIS is t + dI−B, with its distribution derived for the shortest capable path.

APPENDIX D

The appendix models direct and indirect visible BSs as inhomogeneous point processes and combines their path-length distributions. It then uses these distributions to analyze association and coverage-related quantities.

  • Direct-LoS BSs and NLoS BSs with at least one indirect path are modeled using inhomogeneous PPPs.
  • The shortest path to a visible BS may be either direct or indirect, and the CDF of its length is derived accordingly.
  • Conditioned on the nearest direct-LoS distance, the analysis evaluates whether a BS at distance r provides lower path-loss through an RIS.
  • Averaging over the nearest direct-LoS distance yields the probability that the typical user associates with a BS through an RIS.
  • The coverage analysis applies the conditional coverage probability as a thinning probability for the BS process before deriving the final result.
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