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Analysis of Blockage Effects on Urban Cellular Networks
Tianyang Bai, Rahul Vaze, Robert W. Heath
TL;DR
Urban buildings complicate cellular-network analysis because blockage effects are especially important at higher frequencies and are often modeled inadequately. The paper uses random shape theory and PPP-based random rectangles to derive blockage statistics and blockage-aware path loss. Its analyses indicate that buildings can improve network performance by blocking more interference than desired signal, while also reducing connectivity in the impenetrable-blockage case.
Problem
Building penetration losses make urban cellular coverage difficult to predict, especially at higher frequencies, while prior models neglect or oversimplify blockage effects.
Method
The paper models buildings as a Boolean scheme of random rectangles and derives blockage-loss distributions and path-loss formulas for cellular-network analysis.
Results
Analytical results indicate that buildings can improve network performance by blocking more interference than desired signal, although impenetrable blockages reduce network connectivity.
Takeaways & Limitations
Blockage-aware stochastic modeling can capture network-level effects in which urban buildings both obstruct links and provide interference suppression.
Abstract
from arXiv · showhide
Large-scale blockages like buildings affect the performance of urban cellular networks, especially at higher frequencies. Unfortunately, such blockage effects are either neglected or characterized by oversimplified models in the analysis of cellular networks. Leveraging concepts from random shape theory, this paper proposes a mathematical framework to model random blockages and analyze their impact on cellular network performance. Random buildings are modeled as a process of rectangles with random sizes and orientations whose centers form a Poisson point process on the plane. The distribution of the number of blockages in a link is proven to be Poisson random variable with parameter dependent on the length of the link. A path loss model that incorporates the blockage effects is proposed, which matches experimental trends observed in prior work. The model is applied to analyze the performance of cellular networks in urban areas with the presence of buildings, in terms of connectivity, coverage probability, and average rate. Analytic results show while buildings may block the desired signal, they may still have a positive impact on network performance since they can block significantly more interference.
I. INTRODUCTION
Urban buildings make cellular coverage difficult to predict, especially at higher frequencies, while prior models often neglect blockages or represent them with distance-independent log-normal shadowing. The paper develops a random-shape framework for blockage-aware path loss and network analysis, finding that buildings can improve performance by blocking more interference than desired signal.
- Building penetration losses make cellular coverage difficult to predict and become more severe at higher frequencies.
- Distance-independent log-normal shadowing does not capture the intuition that longer links intersect more buildings and experience more shadowing.
- Prior stochastic-geometry studies either neglected building penetration losses or incorporated them into log-normal shadowing.
- The paper models urban buildings as rectangles with random sizes and orientations whose centers form a PPP, extending the model to height and general convex shapes.
- The framework is compatible with PPP cellular-network models and derives blockage-loss distributions for analyzing connectivity, coverage probability, and average rate.
- Analytical results indicate that buildings may improve covered-area performance because they block more interference while also blocking line-of-sight links.
- The analysis considers direct propagation while ignoring reflections and assumes independent blockage experiences on links.
II. SYSTEM MODEL
The system model uses random shape theory to represent urban blockages as a tractable Boolean scheme of independently sampled objects. This formulation supports extensions from rectangles to general convex shapes and provides a basis for cellular-network analysis.
- Random shape theory formalizes random spatial objects, and the paper represents randomly located buildings through a process of random rectangles.
- A random object process samples objects, places their centers using a point process, and independently determines their orientations.
- The Boolean scheme specializes this process by using PPP-distributed centers and independent object attributes and locations.
- Independence assumptions make object attributes such as size, location, and orientation tractable in network-model analysis.
- Minkowski sums extend results from rectangular blockages to general convex objects.
B. Cellular Network Model with Random Buildings
The cellular model combines PPP base stations with a Boolean scheme of random rectangular buildings and an interference-limited propagation model. Link power includes fading, distance-based path loss, and multiplicative blockage losses.
- Base stations form a homogeneous PPP on R2 with density µ and fixed transmission power, while a typical user is placed at the origin.
- Building centers form a homogeneous PPP of density λ, with i.i.d. lengths, widths, uniformly distributed orientations, and independent heights.
- A location is indoor when it lies within at least one modeled blockage.
- The model first analyzes blockages in R2 and then provides extensions incorporating building height.
- Each link uses i.i.d. Rayleigh fading and neglects thermal noise under the interference-limited assumption.
- Received power combines fading and path loss with a multiplicative product of per-building penetration-loss ratios.
- For impenetrable buildings, the blockage-loss ratio is zero for every intersecting building, with the case motivated for large buildings and millimeter-wave networks.
III. QUANTIFICATION OF BLOCKAGE EFFECTS
The section derives blockage-count distributions for random rectangles and extends them to broader blockage shapes. It also obtains line-of-sight and blockage-coverage probabilities from these distributions.
- Blockage-count distribution: J(ℓ, w, θ) is Poisson with mean λℓ,w,θ(Rℓ|sin θ| + Rw|cos θ| + ℓw).The result follows by counting rectangle centers in the region whose rectangles intersect the link.
- Blockage-count distribution: K is Poisson with mean βR + p, where β = 2λ(E[W] + E[L]) and p = λE[L]E[W].Thus, the mean blockage count grows linearly with link length R.
- Line-of-sight and coverage: The line-of-sight probability for a link of length R is P(K = 0) = e^−(βR+p).The model therefore predicts exponential decay of line-of-sight probability with distance, consistent with the cited 3GPP models.
- Line-of-sight and coverage: The probability that a location is inside a blockage is 1−e^−p ≈ p, with approximation error caused by overlapping blockages.When blockages are sparse, overlap is unlikely and the error is negligible.
- General blockage shapes: For compact convex blockage shapes, the crossing count remains Poisson with mean λE[V(OX ⊕D)].The Minkowski-sum formulation extends the rectangle result to general compact and convex objects.
B. Incorporating Building Height
The section incorporates transmitter, receiver, and building heights by distinguishing geometric link intersections from effective blockages. Effective blockage counts remain Poisson after height-dependent thinning.
- Height-aware blockage model: A building crossing OX blocks O′X′ only when its height exceeds the path height h_y at its intersection point.Buildings can intersect the ground projection without blocking the actual propagation path.
- Height-aware blockage model: The effective blockage count ˆK is Poisson with mean E[ˆK] = ηE[K], where E[K] = 2λE[L]R.η is the probability that a building crossing the ground-projected link also blocks the three-dimensional propagation path.
- Height-aware blockage model: Height incorporation introduces a constant scaling factor η to results that ignore blockage height.The later formulas can account for heights by incorporating η appropriately.
C. Quantification of Power Losses by Blockages
The section models aggregate penetration loss from random blockages using Poisson blockage counts and Laplace transforms. Because general closed forms are difficult, it also develops a moment-matched beta approximation.
- Aggregate penetration loss: The aggregate power-loss ratio S is analyzed from the Poisson blockage count K and independent per-blockage loss ratios γ_k.The Laplace transform provides a systematic route to the distribution of S for general γ_k distributions.
- Approximation: A beta distribution approximates the continuous part of f_S(x) when the general closed-form inverse Laplace transform is difficult to obtain.Its parameters are determined by matching moments of S.
- Special cases: In the impenetrable case γ_k = 0, S is Bernoulli with P{S = 1} = e^−(βR+p).This equals the probability that the link has no blockage-induced power loss.
- Special cases: For uniformly distributed γ_k on [0, 1], the density of S is expressed using a Dirac delta function and the first-order modified Bessel function I1.The delta term at x = 1 represents the probability of no blockage-induced power loss.
- Approximation: Given β and p, the second-order statistics of γ_k are sufficient to determine the distribution approximation.The model also represents penetration loss in decibels through [S] = Σ_k=1^K [γ_k].
D. Analysis of Blockage Effects on Link Budget
The section derives path-loss formulas for outdoor-to-outdoor and indoor-to-outdoor links while accounting for blockage penetration losses. On average, blockages introduce an additional exponential decay in the link budget, consistent with field experiments.
- Average received power: Blockage effects introduce an additional exponential decay in the average link budget.This observation matches results verified using field-experiment data.
- Average received power: Theorem 7 gives average received-power formulas for both outdoor-to-outdoor and indoor-to-outdoor links.The formulas distinguish whether the transmitter and receiver are covered by blockages.
- Indoor versus outdoor links: An indoor cellular user receives, on average, signals e−p(1−E[γ_k])E[γ_k] weaker than an outdoor user.The paper connects this difference with the deployment of indoor small cells such as femtocells.
IV. ANALYSIS OF BLOCKAGE EFFECTS ON IMPENETRABLE NETWORKS
The section develops tractable connectivity analysis for cellular networks with impenetrable blockages, using visible regions and an independent-link approximation. It derives results for visible area, visible base stations, nearest visible base stations, and silent area, including how base-station density must scale with blockage density.
- Modeling assumptions: The analysis approximates blockage effects on different links as independent, although aligned links can share correlated blockage counts.The approximation enables simple expressions for connectivity and coverage metrics.
- Visible regions: A location’s visible region contains points reachable by a line-of-sight link, and it is empty when the location lies inside a blockage.Users can connect only to base stations located in their visible region.
- Visible regions: Conditioned on the origin being uncovered, the nearest blockage distance in any direction is exponential with parameter β and has a direction-independent distribution.This directional result is used to compute the average visible area.
- Connectivity metrics: The average number of visible base stations equals µE[V(Q0)] when base stations form an independent homogeneous PPP with density µ.The result is finite, implying finitely many visible base stations almost surely for a typical user.
- Connectivity metrics: The silent-area ratio ξ measures connectivity, increasing as users become less likely to connect; maintaining fixed connectivity requires base-station density to scale superlinearly with blockage density.When p is small, the required base-station density scales with λ^2.
B. Coverage Probability
The paper derives coverage probability for users connecting to the nearest visible base station under independent blockage counts on links. Unlike the no-blockage case, coverage depends on base-station density because longer interference links experience more blockages on average.
- Coverage interpretation: Coverage probability can be interpreted as the fraction of network area where the received SIR exceeds threshold T when the base-station process is stationary.Here T is the SIR threshold for successful decoding.
- Coverage formulation: Coverage probability is derived for a user connecting to the nearest visible base station when each link has an independent blockage count.The expression conditions on the distance to the nearest visible base station and then averages over that distance.
- Coverage formulation: The coverage probability depends on base-station density, blockage density, blockage-size statistics, and the user’s connecting strategy.The analysis specifically uses nearest-visible-base-station association for tractability.
- Impact of blockages: In blockage-free interference-limited networks, signal and interference scale together with density, but blockages apply different distance-dependent attenuation to them.The interference links are expected to contain more blockages because they are longer than the desired-signal link.
C. Average Achievable Rate
The average achievable rate is analyzed using the blockage-aware SIR distribution, with the operating SIR bounded by a maximum threshold. The paper notes a convergence issue for rare single-base-station visible regions and motivates the bound using hardware-limited SINR.
- Rate analysis: The average rate is evaluated from the refined blockage-aware SIR expression.The rate calculation uses the coverage probability as an intermediate quantity.
- Rate analysis: If exactly one base station lies in a user’s visible area, the user experiences no interference and the SIR becomes infinite.Although this event is considered unlikely, it makes the convergence of the unbounded rate expression very slow.
- Practical limitation: RF imperfections impose a maximum achievable SINR even when thermal noise and interference are absent.The limitation is associated with error vector magnitude in the transmitter.
- Rate analysis: The operating SIR is assumed to be upper bounded by Tmax for rate calculation.The bound reflects a maximum achievable SINR under practical system limitations.
V. SIMULATION RESULTS
Simulations compare the Boolean blockage model with lattice and real-building distributions, validate analytical coverage results, and examine connectivity, coverage, and rate. Random blockage sizes and orientations affect SIR distributions, while blockages can reduce connectivity yet improve coverage and achievable rate by suppressing interference more strongly.
- Model comparisons: Equal average blockage counts do not produce equal SIR distributions because the Boolean model randomizes blockage sizes and orientations.The simulations identify size and orientation randomness as a differentiating feature of the proposed model.
- Connectivity: Increasing blockage density or size limits line-of-sight range, shrinks visible area, and increases the silent-area fraction.These effects reduce network connectivity.
- Coverage: Blockages may improve coverage because the longer interference links are more likely to intersect buildings than the desired-signal link.Consequently, more interference power can be blocked than signal power.
- Analytical validation: Analytical and Monte Carlo coverage curves differ only slightly because link-blockage correlations are neglected, while analytical evaluation is much faster.The simulations require more than 3 hours for 10000 samples, compared with less than 1 minute for the analytical expression.
- Density effects: With fixed blockage density, increasing base-station density can degrade coverage, implying a finite density that achieves the best performance.As density tends to infinity, coverage converges to the no-blockage case.
- Average rate: Blockages may increase achievable rate, and average rate is no longer invariant with base-station density when blockage effects are included.The rate comparison is reported with Tmax = 40dB.
VI. CONCLUSIONS
The paper develops a stochastic framework for random urban blockages and derives blockage-aware path-loss models. Its analyses show that buildings can reduce connectivity yet sometimes benefit network performance by blocking more interference than desired signal.
- Framework: The framework models urban blockages as a Boolean scheme of rectangles with random sizes, locations, orientations, and heights.
- Framework: The model derives penetration-loss distributions and path-loss formulas that incorporate blockage effects across different scenarios.
- Framework: The proposed path-loss model captures the distance-dependent nature of blockage effects.
- Network effects: Impenetrable blockages can reduce network connectivity and alter cellular-network behavior substantially.
- Network effects: Blockages may benefit cellular networks because longer paths to interfering base stations may contain more blockages.
- Scope and future work: Future work includes multi-tier networks, infrastructure correlated with blockages, and reflections that contribute to urban coverage.