Non-Homogeneous Binomial Point Processes (NBPP)

• NBPP is a stochastic model that defines a fixed number of points distributed over a spatial region with a non-uniform probability density function.
• It enhances satellite channel analysis by incorporating independent directional marks to model phase and velocity effects accurately.
• Validated against realistic orbital simulations, the NBPP framework offers analytical tractability and improved channel metric predictions over traditional models.

A non-homogeneous binomial point process (NBPP) is a stochastic model for configurations comprising a fixed number of points (N), each independently and identically distributed over a spatial region according to a prescribed, non-uniform probability density function. The NBPP provides a rigorous framework for scenarios where the cardinality of the process is fixed but the spatial distribution exhibits inhomogeneity. The NBPP has found applications in modeling satellite mega-constellations, where a fixed number of satellites are distributed over an inclined orbital shell and additional structure (such as orbital direction or velocity) is naturally encoded via marking. The NBPP is distinct from Poisson point processes (PPPs) in that the number of points is deterministic, and from negative binomial processes (NBPs), which are typically defined in a Bayesian nonparametric setting for count-valued completely random measures.

1. Mathematical Definition and Statistical Structure

Let $\mathcal{R} \subset \mathbb{S}^2$ denote a subset of the two-dimensional sphere (e.g., a spherical shell constrained by orbital inclination). The NBPP consists of $N$ points, with each point $\mathbf{x}_i = (\theta_i, \phi_i)$ independently drawn from a density $f_{\Theta,\Phi}(\theta, \phi)$ supported on $\mathcal{R}$. The process intensity is $$\lambda(\theta, \phi) = N f_{\Theta, \Phi}(\theta, \phi)$$, such that the total intensity $\Lambda = N$. For a Borel subset $B\subseteq\mathcal{R}$, the point count $K_B$ has the binomial distribution: $$\mathbb{P}\{K_B = k\} = \binom{N}{k} \bigl( p_B \bigr)^k \bigl(1 - p_B \bigr)^{N-k}, \qquad p_B = \int_B f_{\Theta, \Phi}(\theta, \phi) d\theta d\phi$$ As a concrete instance, consider satellites occupying the region

$N$0

where $N$1 is the orbital inclination. The longitude $N$2, while the latitude's pdf $N$3 captures the inhomogeneity arising from the projection of inclined orbits: $N$4 This parametric structure fully determines the non-homogeneous point density.

2. Marked NBPP and Directional Augmentation

An extension employed in satellite systems is the marking of each point with an independent random variable, for example, representing the ascending ($N$5) or descending ($N$6) phase of an orbit. The marks $N$7 are i.i.d. with $N$8. The local velocity-direction angle in the tangent plane at each point depends both on the mark and the geographic latitude: $N$9 This augmentation produces a marked NBPP, with marks uncorrelated with spatial locations but imparting critical physical semantics, such as directionality or velocity, necessary for stochastic modeling of propagation phenomena (McBain et al., 27 Jul 2025).

3. Distributional Results for Satellite Channel Characterization

With the marked NBPP model, channel statistics relevant to wireless propagation (e.g., between a stationary Earth user and moving satellites) can be obtained. For a user–satellite geometry characterized by central angle $\mathbf{x}_i = (\theta_i, \phi_i)$0,

• The path gain is $\mathbf{x}_i = (\theta_i, \phi_i)$1, monotone in $\mathbf{x}_i = (\theta_i, \phi_i)$2, with invertibility via $\mathbf{x}_i = (\theta_i, \phi_i)$3.
• Propagation delay is $\mathbf{x}_i = (\theta_i, \phi_i)$4, allowing direct transformation between angle and delay domains.
• Doppler shift is determined by the projected velocity vector, incorporating both position and orbital phase via the mark $\mathbf{x}_i = (\theta_i, \phi_i)$5 and $\mathbf{x}_i = (\theta_i, \phi_i)$6.

The probability laws for these statistics follow directly from the NBPP: the conditional cumulative distribution function for any derived variable $\mathbf{x}_i = (\theta_i, \phi_i)$7 is obtained by integrating $\mathbf{x}_i = (\theta_i, \phi_i)$8 over the region defined by the preimage $\mathbf{x}_i = (\theta_i, \phi_i)$9. Theorems 4, 6, and 7 in (McBain et al., 27 Jul 2025) provide explicit change-of-variables and formulas for power gain, delay, Doppler, and their joint distributions.

4. Scattering Function and Channel-Wide Parameters

The single-satellite spreading function in the delay–Doppler domain is

$f_{\Theta,\Phi}(\theta, \phi)$0

Global second-order statistics are summarized by the (WSSUS) scattering function

$f_{\Theta,\Phi}(\theta, \phi)$1

Empirical moments with respect to this scattering function yield channel metrics:

• Path loss: $f_{\Theta,\Phi}(\theta, \phi)$2
• Mean delay: $f_{\Theta,\Phi}(\theta, \phi)$3
• RMS delay spread: $f_{\Theta,\Phi}(\theta, \phi)$4
• RMS Doppler spread: $f_{\Theta,\Phi}(\theta, \phi)$5 Channel-wide integrals can be computed analytically or via Monte Carlo over NBPP draws (McBain et al., 27 Jul 2025).

5. Model Validation and Practical Considerations

NBPP models were validated against realistic orbit simulations (e.g., Starlink using SGP4 perturbation and TLE data). In empirical comparisons, the NBPP accurately matched the marginal and joint statistics of channel gain, delay, and Doppler, especially at higher user latitudes where longitudinal satellite density is increased and “bucketing” from discrete orbit structures is less pronounced. The NBPP smooths out discrete effects by treating longitude as continuously uniform, which closely matches distributions observed in perturbed constellations. Key simplifications in the NBPP model include the omission of small-scale fading, snapshot-to-snapshot independence, and uniform random selection of the serving satellite. Each can be systematically relaxed or extended within the geometric NBPP framework (McBain et al., 27 Jul 2025).

6. Position of NBPP within Point Process Theory

The NBPP is distinct from the negative binomial process (NBP) and its non-homogeneous variants found in Bayesian nonparametrics (Broderick et al., 2011). The NBPP, as used in satellite constellation modeling, always entails a fixed total number of points and base density, whereas the NBP constructs a completely random measure with negative binomial–distributed multiplicities over a base measure that is typically random and potentially infinite-dimensional. The NBP appears in latent feature modeling and is typically conjugate to beta-process (or three-parameter beta-process) priors within hierarchical generative models. While both NBPP and NBP introduce non-homogeneity through a base measure/density, the former is a finite, multinomial-type process in geometric problems, and the latter a general CRM for count-valued traits in statistical learning contexts.

Property	NBPP	NBP
Number of points	Fixed (N)	Random, possibly infinite
Density/base measure	Prescribed (possibly non-uniform) function	Random CRM, e.g., beta process
Application domain	Geometric, e.g., satellite constellations	Latent feature count modeling, clustering
Marking	Arbitrary, e.g., binary direction	Feature/trial multiplicity

This suggests that in wireless systems with deterministic entity counts and structured spatial distributions, the NBPP offers a natural modeling paradigm, whereas the NBP (and beta–negative binomial process) is tailored to feature allocation and trait-count inference in statistical learning.

7. Extensions and Limitations

In practice, further NBPP generalizations can address deviations from idealized satellite distributions, such as “bucketing” effects from discrete orbits, local non-independence arising from coordinated constellation management, and additional physical channel effects (atmospheric, inter-satellite). Relaxing the i.i.d. or binomial assumptions produces more sophisticated point process models but at the cost of analytical tractability. For the satellite channel modeling application, the NBPP provides closed-form predictions for gain, delay, and Doppler statistics in megaconstellation environments with sufficient realism to match orbital simulation data. A plausible implication is that future geometric channel models for LEO satellite networks will employ NBPPs as the fundamental spatial prior, with additional layers to accommodate physical and network-induced couplings (McBain et al., 27 Jul 2025).