Source-linked AI summary
The Ginibre Point Process as a Model for Wireless Networks with Repulsion
Na Deng, Wuyang Zhou, Martin Haenggi
TL;DR
The paper proposes the β-GPP, a thinned and re-scaled GPP, for wireless networks with repulsive nodes. It derives interference and coverage results, finding that β-GPP better captures node distributions than PPP and can accurately model real base-station deployments while balancing accuracy, tractability, and practicability.
Problem
The PPP’s independence assumption is unsuitable for wireless networks with repulsion, motivating a model that better captures node spatial distributions.
Method
The paper proposes the β-GPP, a thinned and re-scaled GPP, and derives mean and variance of interference plus a computable coverage-probability representation.
Results
The β-GPP better captures node spatial distributions than PPP, and fitted β-GPP models have nearly the same coverage probability as given base-station point sets.
Takeaways & Limitations
β-GPP balances accuracy, tractability, and practicability for wireless networks with repulsion and provides an accurate model for real base-station deployments in terms of coverage probability.
Abstract
from arXiv · showhide
The spatial structure of transmitters in wireless networks plays a key role in evaluating the mutual interference and hence the performance. Although the Poisson point process (PPP) has been widely used to model the spatial configuration of wireless networks, it is not suitable for networks with repulsion. The Ginibre point process (GPP) is one of the main examples of determinantal point processes that can be used to model random phenomena where repulsion is observed. Considering the accuracy, tractability and practicability tradeoffs, we introduce and promote the $β$-GPP, an intermediate class between the PPP and the GPP, as a model for wireless networks when the nodes exhibit repulsion. To show that the model leads to analytically tractable results in several cases of interest, we derive the mean and variance of the interference using two different approaches: the Palm measure approach and the reduced second moment approach, and then provide approximations of the interference distribution by three known probability density functions. Besides, to show that the model is relevant for cellular systems, we derive the coverage probability of the typical user and also find that the fitted $β$-GPP can closely model the deployment of actual base stations in terms of the coverage probability and other statistics.
I. INTRODUCTION
The paper addresses the mismatch between tractable PPP models and wireless deployments with repulsive, spatially correlated nodes by proposing the β-GPP. It develops tractable interference and coverage analyses and fits the model to real cellular base-station locations.
- The β-GPP is a thinned and re-scaled GPP that interpolates between the GPP and the PPP while preserving the original intensity.
- The paper derives classical β-GPP statistics, interference mean and variance through Palm and reduced-second-moment approaches, and interference-distribution approximations.
- It also derives a computable cellular coverage representation and fits β-GPPs to real base-station deployments.
- Fitted β-GPPs closely model actual base-station coverage, with fitted β values near 1 in urban regions and between 0.2 and 0.4 in rural regions.
- The β-GPP balances accuracy, analytical tractability, and practicability for repulsive wireless networks.
- PPP tractability comes at the cost of assuming independent node locations, which is questionable for spatially correlated real deployments.
- Repulsive models such as the MHCP and Strauss process better capture real network structure, but their limited analytical tractability restricts wireless-network analysis.
D. Mathematical Preliminaries
This section introduces determinantal point processes and the Ginibre construction, then defines the β-GPP and its scaled version for controlling intensity.
- A determinantal point process is specified through joint intensities represented by determinants of a kernel matrix.
- The determinant formulation implies repulsiveness because joint intensity vanishes when two locations coincide.
- The GPP is a determinantal process on the complex plane with a Gaussian-based kernel representation.
- The GPP is repulsive at all distances because its pair-correlation function is below 1 for every distance.
- The β-GPP independently retains each GPP point with probability β and rescales the retained process by homothety ratio √β to preserve intensity.
- The 1-GPP equals the GPP, while β-GPPs interpolate toward a PPP of intensity 1/π as β approaches 0.
- The scaled β-GPP introduces intensity λ = c/π through a scaling parameter c > 0.
E. Organization
The paper is organized around β-GPP properties and wireless-network applications, supported by distributional and Palm-measure representations and comparisons with hard-core processes.
- Organization: The paper studies β-GPP interference, coverage probability, and fitting to actual network deployments across successive sections.
- Basic properties: For a scaled β-GPP, squared point moduli have the distribution of independently retained gamma variables.
- Basic properties: The Palm version is obtained by adding the origin and deleting a point according to the stated β-dependent construction.
- K and L functions: The K function converges to the PPP form K(r) = πr2 as β approaches 0.
- K and L functions: The GPP is soft-core: repulsion exists, but no minimum interpoint distance is strictly forbidden.
- K and L functions: For equal intensity, the MHCP is less regular than the GPP at large distances, and the GPP becomes more regular than the compared processes once r is approximately at least δ.
2) L function:
The β-GPP’s eL function places it between the GPP and PPP, reflecting an intermediate degree of spatial regularity while retaining soft-core behavior.
- The modified eL function is used to highlight the β-GPP’s soft-core properties.
- eL(r) approaches −1 for every β > 0 as r approaches 0.
- For different β values, the β-GPP eL function lies between the GPP and PPP eL functions.
- Contact distribution: The contact distance is the distance from a location u to the nearest point of the process, and its empty-space function is the corresponding cumulative distribution function.
- Nearest-neighbor distance: The nearest-neighbor distance is the distance from a point to its nearest other point, with Gx(r) giving its distribution function.
- J function: The J function is introduced as a measure of how close a point process is to a reference process.
3) The J function:
The β-GPP interpolates between the PPP and the GPP, with increasing β producing stronger repulsion and regularity. The paper derives tractable interference moments and examines how path loss, intensity, and repulsion affect them.
- The J function: As β → 0, the β-GPP approaches the PPP, while its J function exceeds 1 and increases toward β = 1.The increase in J reflects increasing regularity as repulsion strengthens.
- Interference analysis: The paper derives mean and variance interference using Palm-measure and reduced-second-moment approaches for β-GPP networks.It also considers gamma, inverse Gaussian, and inverse gamma approximations for the interference distribution.
- Path-loss effects: With bounded path loss ℓ(r) = (max{r0, r})−α, larger α attenuates interference faster from nodes with r > r0, reducing mean interference.Nodes closer than r0 have the same bounded path-loss contribution regardless of their exact distance.
- Mean interference: For r0 = 0, increasing α raises interference from nearby nodes but lowers interference from distant nodes.The observed balance depends on repulsion and node intensity; at high intensity, inter-node distances may not sufficiently reduce path-loss effects.
- Mean interference: Increasing β decreases mean interference, whereas increasing intensity c increases it in the examined cases.These effects reflect the competing influence of stronger repulsion and more transmitters.
B. Approach II - The Reduced Second Moment Measure
The reduced second moment measure provides a more general route to interference analysis than the tractable Palm-measure approach. Applied to the GPP, it reproduces the mean-interference result obtained by the first approach.
- Approach II: The reduced second moment measure applies to many spatial point processes, whereas the tractable Palm-measure approach is limited to the GPP.The paper uses the reduced measure to provide an alternative derivation for the GPP.
- Approach II: For the GPP, specializing the reduced-second-moment expression to ℓ(r) = (max{r0, r})−α yields the same mean interference as Theorem 1.The polar representation is convenient because the GPP is motion-invariant.
1) Comparison with the Poisson point process:
Compared with the PPP, the 1-GPP has a different density-dependent mean interference because repulsion changes the spatial arrangement of interferers. The difference grows with intensity and is bounded in the high-density limit.
- Comparison with the PPP: For β → 0, the β-GPP has mean interference proportional to c, as does the PPP.The β-GPP tends to the PPP in this limit.
- Comparison with the PPP: The difference between the 1-GPP and PPP mean interferences increases with c and approaches a maximum as c → ∞.The limiting maximum is associated with the bounded path-loss term r0^−α.
2) Comparison with the Mat´ern hard-core process:
The paper compares GPP interference with hard-core and Poisson models and evaluates parametric fits to simulated interference distributions. Repulsion changes the intensity dependence, while fit quality depends on α and the candidate density.
- Comparison with the MHCP: Figure 6 compares β = 1 GPP mean interference across α and r0 with type-I and type-II MHCPs and the PPP at common intensity c/π.The comparison isolates differences among spatial models while holding intensity equal.
- Comparison with the MHCP: Unlike the PPP, the GPP and MHCP mean interferences are not proportional to intensity c.Their reduced second moment measures are not proportional to intensity, unlike the PPP measure K(b(o, r)) = cr2.
IV. COVERAGE PROBABILITY
The paper derives coverage probability for the typical user in a β-Ginibre wireless network and fits β-GPP models to actual cellular base-station deployments using several spatial and performance metrics.
- Coverage probability: Rayleigh fading uses independent exponential coefficients with mean 1, and the path-loss function is ℓ(r) = r^-α for α > 2.The typical user is placed at the origin using motion-invariance of the GPPs.
- Coverage probability: The coverage probability of the typical user is derived for an SINR threshold θ in a β-Ginibre wireless network.The result applies to cellular networks where users associate with the closest base station.
- Coverage probability: For β = 1, the coverage-probability result retrieves the previously known result in.
- Fitting the β-GPP to actual network deployments: The β-GPP is fitted to actual urban and rural base-station deployments by adjusting β, which controls repulsion, while matching the data-set density.The deployments were obtained from Ofcom data, and the fitted model uses the estimated density as its intensity.
- Fitting the β-GPP to actual network deployments: The fitting uses the L function, J function, and coverage probability; the first two are classical stochastic-geometry statistics, while coverage probability is a key performance metric.The fitting objective minimizes the vertical average squared error between experimental and β-GPP curves.
- Fitting the β-GPP to actual network deployments: The urban deployment matches a β-GPP with β = 0.86, and both urban and rural regions are less regular than the triangular lattice.The rural region’s empirical J function tends toward that of the PPP, while the compared lattice is more regular.
B. Fitting Result for Coverage Probability
The fitted β-GPP closely matches actual cellular deployments in coverage probability, while tuning β captures differences between urban and rural regularity. The section also summarizes tractable interference analysis and distributional approximations.
- B. Fitting Result for Coverage Probability: For smaller α, an analytical interference term accounts for far-away interferers absent from the finite fitting region.
- B. Fitting Result for Coverage Probability: The fitting procedure minimizes vertical average squared error and evaluates coverage using SINR samples from simulated β-GPP point distances.
- B. Fitting Result for Coverage Probability: Urban deployments are fairly regular, with β close to 1, while rural deployments are quite irregular and have smaller β values.
- B. Fitting Result for Coverage Probability: The fitted β-GPP and the point-set coverage-probability curves match extremely well, supporting its use for modeling actual cellular networks.
- B. Fitting Result for Coverage Probability: For bounded path loss, mean and variance of interference are finite when α > 2; for unbounded path loss, they are finite on different α intervals.
- B. Fitting Result for Coverage Probability: Interference distributions are approximated with gamma, inverse Gaussian, and inverse gamma PDFs; inverse gamma fits best for α = 3, while inverse Gaussian fits best for α = 4.
APPENDIX A
The appendix presents expressions for the mean and variance of interference and notes an asymptotic behavior of the mean.
- APPENDIX A: The appendix gives an expression for the mean interference and states that it approaches an affine function as c →∞.
- APPENDIX A: It also presents a separate expression for the variance of the interference.
- APPENDIX A: The notation 1_A denotes the indicator function for set A.