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Beyond the Hexagonal Idealisation

Real base station deployments bear little resemblance to a perfect hexagonal grid. Terrain, buildings, zoning regulations, and economic factors produce irregular BS placements. In 2011, Andrews, Baccelli, and Ganti showed that modelling BS locations as a homogeneous Poisson point process (PPP) leads to remarkably tractable — and surprisingly accurate — expressions for coverage and rate. The key insight is that the randomness in both signal and interference, when both come from a PPP, creates cancellations that yield closed-form results. This stochastic geometry approach has become the dominant analytical framework for modern cellular network analysis.

Definition:
Homogeneous Poisson Point Process

A homogeneous Poisson point process (PPP) $\Phi$ on $\mathbb{R}^2$ with intensity $\lambda > 0$ (points per unit area) satisfies two properties:

Poisson counts: For any bounded region $\mathcal{B}$ with area $|\mathcal{B}|$ , the number of points $N(\mathcal{B}) = |\Phi \cap \mathcal{B}|$ is Poisson distributed: $N(\mathcal{B}) \sim \text{Poisson}(\lambda |\mathcal{B}|)$ .
Independent scattering: For disjoint regions $\mathcal{B}_1, \mathcal{B}_2$ , the counts $N(\mathcal{B}_1)$ and $N(\mathcal{B}_2)$ are independent.

In cellular analysis, $\Phi$ models the BS locations and $\lambda$ is the BS density (e.g., $\lambda = 5$ BS/km $^2$ ). The typical user is placed at the origin, and the serving BS is the nearest point of $\Phi$ .

The PPP is analytically powerful because of Slivnyak's theorem: the distribution of $\Phi$ as seen from a typical point of $\Phi$ (or from an independently placed user) is the same as $\Phi$ itself (plus the conditioning point). This enables analysis from the perspective of a "typical user" without loss of generality.

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Definition:
SINR Coverage Probability

The coverage probability $p_c(\tau)$ is the probability that a typical user achieves SINR above a threshold $\tau$ :

$p_c(\tau) = \Pr[\text{SINR} > \tau] = \Pr\!\left[\frac{P h_0 r_0^{-\alpha}} {\sum_{x_i \in \Phi \setminus \{x_0\}} P h_i \|x_i\|^{-\alpha} + \sigma^2} > \tau\right]$

where $r_0$ is the distance to the serving (nearest) BS, $h_0, h_i \sim \text{Exp}(1)$ are i.i.d. Rayleigh fading gains, $\alpha > 2$ is the path-loss exponent, and $\sigma^2$ is the noise power.

The ergodic rate of a typical user is:

$\bar{R} = \mathbb{E}\!\left[\log_2(1 + \text{SINR})\right] = \int_0^{\infty} \frac{p_c(2^t - 1)}{\ln 2}\,dt$

This integral representation connects coverage probability to average throughput.

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Theorem: Coverage Probability under PPP (Andrews--Baccelli--Ganti)

Let base stations form a homogeneous PPP $\Phi$ with intensity $\lambda$ on $\mathbb{R}^2$ . Under Rayleigh fading ( $h_i \sim \text{Exp}(1)$ ), path-loss $\ell(r) = r^{-\alpha}$ with $\alpha > 2$ , and in the interference-limited regime ( $\sigma^2 = 0$ ), the coverage probability of a typical user with nearest-BS association is:

$p_c(\tau) = \frac{1}{1 + \tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^{\infty}\frac{1}{1+u^{\alpha/2}}\,du}$

For the special case $\alpha = 4$ :

$p_c(\tau) = \frac{1}{1 + \sqrt{\tau}\,\arctan(\sqrt{\tau})} \approx \frac{1}{1 + \frac{\pi}{2}\sqrt{\tau}}$

Remarkably, $p_c(\tau)$ is independent of the BS density $\lambda$ . The coverage probability depends only on $\tau$ and $\alpha$ .

The independence from $\lambda$ is the most striking result: adding more base stations increases both the desired signal (shorter distance to the nearest BS) and the aggregate interference (more interferers) in exactly the same proportion. Under the PPP model with Rayleigh fading, these two effects cancel perfectly, making coverage a function of the propagation environment ( $\alpha$ ) and the SINR requirement ( $\tau$ ) alone. This explains why densification improves capacity (more BSs per area) but not coverage (the SINR distribution is invariant).

Proof

Conditional Laplace functional

Conditioning on the serving distance $r_0 = r$ , the SINR is:

$\text{SINR} = \frac{h_0 r^{-\alpha}}{I_r} \quad\text{where}\quad I_r = \sum_{x_i \in \Phi \setminus B(0,r)} h_i \|x_i\|^{-\alpha}$

Since $h_0 \sim \text{Exp}(1)$ :

$\Pr[\text{SINR} > \tau \mid r_0 = r] = \Pr[h_0 > \tau r^{\alpha} I_r] = \mathbb{E}[e^{-\tau r^{\alpha} I_r}] = \mathcal{L}_{I_r}(\tau r^{\alpha})$

where $\mathcal{L}_{I_r}(s)$ is the Laplace transform of $I_r$ .

Laplace transform of PPP interference

By the probability generating functional (PGFL) of the PPP:

$\mathcal{L}_{I_r}(s) = \exp\!\left( -2\pi\lambda \int_r^{\infty} \frac{s v^{-\alpha}}{1 + s v^{-\alpha}}\,v\,dv\right)$

Substituting $s = \tau r^{\alpha}$ and changing variables $u = (v/r)^2$ :

$\mathcal{L}_{I_r}(\tau r^{\alpha}) = \exp\!\left(-\pi\lambda r^2 \tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^{\infty} \frac{1}{1 + u^{\alpha/2}}\,du\right)$

Define $\rho(\tau, \alpha) = \tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^{\infty}\frac{du}{1+u^{\alpha/2}}$ .

Averaging over the serving distance

The PDF of $r_0$ under nearest-BS association in a PPP is:

$f_{r_0}(r) = 2\pi\lambda r \exp(-\pi\lambda r^2)$

Therefore:

$p_c(\tau) = \int_0^{\infty} e^{-\pi\lambda r^2 \rho(\tau,\alpha)}\cdot 2\pi\lambda r\,e^{-\pi\lambda r^2}\,dr$

$= \int_0^{\infty} 2\pi\lambda r\, \exp(-\pi\lambda r^2(1 + \rho))\,dr = \frac{1}{1 + \rho(\tau, \alpha)}$

The dependence on $\lambda$ cancels completely. For $\alpha = 4$ : $\rho(\tau, 4) = \sqrt{\tau}\arctan(\sqrt{\tau})$ . $\blacksquare$

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PPP Coverage Probability: Density Invariance

Side-by-side comparison of sparse (

\lambda = 5

) and dense (

\lambda = 20

) PPP realisations. Despite the 4

\times

difference in BS density, both yield the same coverage probability — demonstrating the remarkable density invariance of the PPP model.

Doubling BS density increases both signal and interference equally, leaving the SINR distribution unchanged.

PPP Coverage Probability

Explore the coverage probability of a typical user in a PPP-distributed cellular network. Adjust the BS density $\lambda$ to verify that coverage probability is indeed invariant to density (in the interference-limited regime). Change the path-loss exponent $\alpha$ to see how propagation conditions affect coverage. Sweep the SINR threshold to trace the full coverage probability curve $p_c(\tau)$ .

Parameters

\lambda

(BS density, per km

^2

)5

Path-loss exponent

\alpha

4

SINR threshold

\tau

(dB)0

Example: Coverage Probability Computation

A cellular network has BS locations modelled as a PPP with $\alpha = 4$ . Compute:

(a) The coverage probability at $\tau = 0$ dB (SINR $> 1$ ). (b) The coverage probability at $\tau = 10$ dB (SINR $> 10$ ). (c) The fraction of the cell area with rate at least 1 bit/s/Hz (i.e., $\log_2(1 + \tau) \geq 1$ ).

Solution

Coverage at 0 dB

(a) For $\alpha = 4$ and $\tau = 1$ :

$\rho(1, 4) = \sqrt{1}\,\arctan(\sqrt{1}) = 1 \cdot \pi/4 = 0.785$

$p_c(1) = \frac{1}{1 + 0.785} = \frac{1}{1.785} = 0.560$

About 56% of users achieve SINR above 0 dB.

Coverage at 10 dB

(b) For $\tau = 10$ :

$\rho(10, 4) = \sqrt{10}\,\arctan(\sqrt{10}) = 3.162 \times 1.264 = 3.997$

$p_c(10) = \frac{1}{1 + 3.997} = \frac{1}{4.997} = 0.200$

Only 20% of users achieve SINR above 10 dB.

Rate threshold

(c) Rate $\geq 1$ bit/s/Hz requires $\log_2(1+\tau) \geq 1$ , so $\tau \geq 1$ (0 dB).

From part (a): $p_c(1) = 0.560$ .

Thus 56% of users achieve at least 1 bit/s/Hz. Note this is independent of BS density $\lambda$ . $\blacksquare$

Quick Check

In the PPP model with Rayleigh fading and $\alpha > 2$ (interference-limited regime), what happens to the coverage probability when the BS density $\lambda$ is doubled?

It doubles, since the serving BS is closer

It stays the same — coverage probability is independent of $\lambda$

It decreases because there are more interferers

It depends on the specific value of $\lambda$

Correction:

It stays the same — coverage probability is independent of

\lambda

Under the PPP model, both the desired signal and the aggregate interference scale identically with $\lambda$ , making $p_c(\tau)$ depend only on $\tau$ and $\alpha$ . This is the key insight of the Andrews--Baccelli--Ganti (2011) framework.

Common Mistake: PPP Coverage Invariance Breaks with Noise or Correlated Fading

Mistake:

Concluding that "densification never improves coverage" based on the PPP result $p_c(\tau)$ being independent of $\lambda$ .

Correction:

The density-invariance of $p_c(\tau)$ holds only in the interference-limited regime ( $\sigma^2 = 0$ ) with Rayleigh fading and single-slope path loss. In practice:

With thermal noise ( $\sigma^2 > 0$ ), densification does improve coverage because the noise term decays with $\lambda$ .
With dual-slope path loss (LOS/NLOS transition), densification can degrade per-link SINR in the ultra-dense regime.
With correlated shadow fading, the PPP result is no longer exact.

The invariance result is a useful benchmark but not a universal law.

Historical Note: The Stochastic Geometry Revolution

2011

The 2011 paper by Andrews, Baccelli, and Ganti, "A Tractable Approach to Coverage and Rate in Cellular Networks," transformed wireless network analysis. By modelling BS locations as a PPP instead of a hexagonal grid, they obtained the first closed-form expression for coverage probability in a general cellular network — and discovered the striking invariance to BS density. The paper has been cited over 8,000 times and spawned an entirely new analytical paradigm. Prior to this work, system-level performance required Monte Carlo simulations with 1,000+ BS drops; after it, many quantities could be computed in closed form. The PPP model was initially criticised as unrealistic (BS locations have some regularity), but subsequent studies showed it provides lower bounds on coverage that are surprisingly tight (within 5--10% of regular deployments).

Poisson Point Process (PPP)

A spatial point process where the number of points in any bounded region follows a Poisson distribution with mean proportional to the region's area, and counts in disjoint regions are independent. In cellular analysis, a homogeneous PPP with intensity $\lambda$ models random BS locations and yields tractable closed-form coverage expressions.

Coverage Probability

The probability $p_c(\tau) = \Pr[\text{SINR} > \tau]$ that a typical user achieves SINR above a threshold $\tau$ . Under a PPP model with Rayleigh fading and path-loss exponent $\alpha > 2$ , the coverage probability is independent of BS density and depends only on $\tau$ and $\alpha$ .

Why This Matters: Point Process Theory in the FSP Book

The Poisson point process used in this section is a fundamental object in probability theory. The FSP (Foundations of Stochastic Processes) book develops the full mathematical framework:

Poisson process properties: thinning, superposition, marking, and the Campbell-Mecke theorem
Beyond the PPP: Matérn hard-core processes (repulsive), determinantal point processes, and Ginibre point processes
Palm calculus: rigorous definition of the "typical point" perspective used in Slivnyak's theorem
Convergence and scaling limits: how point processes behave as intensity $\lambda \to \infty$

Readers pursuing research in stochastic geometry for wireless networks will benefit from the rigorous probabilistic foundations.

Stochastic Geometry

A mathematical framework that models the spatial distribution of network nodes (base stations, users) using random point processes, enabling probabilistic analysis of coverage, rate, and interference in wireless networks without assuming a deterministic geometry.

Stochastic Geometry and PPP