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Mean Interference in Hard-Core Wireless Networks
Martin Haenggi
TL;DR
The paper asks whether Poisson models accurately approximate interference in CSMA networks represented by Matérn hard-core processes. It derives and compares mean interference at a typical process node for types I and II against an equi-dense PPP. Type I exhibits exponentially increasing excess interference, whereas type II remains within 1 dB of the Poisson case.
Problem
The paper examines the accuracy of Poisson approximations for mean interference in CSMA networks modeled by Matérn hard-core processes.
Method
The paper evaluates mean interference at a typical hard-core-process node and compares it with mean interference in an equi-dense Poisson point process using the excess interference ratio.
Results
Type I excess interference increases exponentially in λp and δ for power path loss laws, while type II excess interference never exceeds 1 dB irrespective of the path loss law.
Takeaways & Limitations
The two popular CSMA point-process models behave differently: the Poisson approximation becomes loose for type I but remains a tight lower bound for type II.
Abstract
from arXiv · showhide
Matérn hard core processes of types I and II are the point processes of choice to model concurrent transmitters in CSMA networks. We determine the mean interference observed at a node of the process and compare it with the mean interference in a Poisson point process of the same density. It turns out that despite the similarity of the two models, they behave rather differently. For type I, the excess interference (relative to the Poisson case) increases exponentially in the hard-core distance, while for type II, the gap never exceeds 1 dB.
A. Motivation
The paper evaluates whether Poisson approximations accurately represent interference in CSMA networks modeled by Matérn hard-core processes. It focuses on mean interference at a typical process node and compares it with an equi-dense PPP.
- A. Motivation: CSMA is harder to analyze than ALOHA because concurrent transmitters must be separated by a minimum distance δ, eliminating independence across disjoint areas.Matérn hard-core processes of types I and II model the transmitter set in CSMA.
- A. Motivation: The study evaluates mean interference at a typical hard-core-process node to verify the accuracy of the Poisson approximation.The comparison uses a Poisson process with the same density.
- A. Motivation: Only the type II process produces interference comparable to that of a PPP.This is the paper's stated comparison between the two Matérn models and the Poisson case.
- A. Motivation: Prior work studied mean interference at arbitrary plane locations or used simulations to obtain empirical distributions.The present analysis instead concerns interference measured at a node of the point process.
B. Preliminaries
The paper formulates mean interference at a typical process node using stationary point-process tools and defines the excess interference ratio relative to an equi-dense Poisson baseline.
- B. Preliminaries: The analysis begins by deriving a general expression for mean interference in networks whose nodes form a stationary point process.
- B. Preliminaries: The path loss function must be integrable over R2; otherwise interference is infinite almost surely for any stationary process.
- B. Preliminaries: Mean interference is evaluated at a process node under the reduced Palm distribution, excluding that node's own signal power.By stationarity, the node can be placed at the origin without loss of generality.
- B. Preliminaries: For isotropic processes and radially symmetric path loss, the mean-interference expression is represented conveniently in polar coordinates using the K-function.The radial representation uses K′(r)dr = 2πλK(r)dr.
- B. Preliminaries: The excess interference ratio compares mean interference at a typical hard-core-process point with mean interference in a Poisson process of intensity λ outside the exclusion distance δ.The Poisson reference uses λ(r) = λ1[δ,∞)(r).
II. MEAN INTERFERENCE IN HARD-CORE PROCESSES
Hard-core separation makes the K-function vanish below δ, and the paper compares the resulting interference bounds with a Poisson approximation. The approximation is tight for type II but increasingly loose for type I as δ grows.
- II. MEAN INTERFERENCE IN HARD-CORE PROCESSES: Hard-core processes guarantee a minimum pairwise distance δ, so K(r) = 0 for r < δ.
- II. MEAN INTERFERENCE IN HARD-CORE PROCESSES: The Poisson approximation K′(r) = 2πr1[δ,∞)(r) provides a tight lower bound for type II processes.
- II. MEAN INTERFERENCE IN HARD-CORE PROCESSES: For type I processes, the same approximation becomes increasingly loose as the hard-core distance δ increases.
A. Mat´ern process of type I
For Matérn type I, the hard-core structure creates increasingly many nearby nodes relative to an equi-dense PPP, making the Poisson approximation increasingly loose as the hard-core distance grows. The resulting excess interference grows exponentially with the parent density and hard-core distance.
- The type I process retains parent-PPP points separated by at least δ, with resulting intensity λ = λp exp(−λpπδ2).
- The normalized number of nodes within distance 2δ of a typical point grows exponentially in δ2 and almost exponentially in λp, whereas for a PPP K(2δ) ∝δ2.
- The type I mean-interference bounds are obtained by splitting interference at 2δ and bounding the union-area function over δ ≤ r ≤ 2δ.
- Interference from nodes beyond 2δ is the same as in the equi-dense PPP, so the difference is concentrated in the nearer-node contribution.
- The type I excess interference grows exponentially with λp and δ for power path loss laws; at δ = 2, the reported EIR is about 30 dB, or 31.5 dB from the approximation.
B. Mat´ern process of type II
For Matérn type II processes, the Poisson approximation provides a tight lower bound, and the excess interference ratio remains below 1 dB across path-loss functions and parameters.
- The excess interference ratio for Matérn type II never exceeds 1 dB, regardless of the path-loss function and other parameters.
- For power path-loss laws, the upper bound can be sharpened beyond the general type II bound.
- The pair-retention behavior equals the Poisson case outside distance 2δ, while the relevant ratio increases with λp and δ for δ ≤ r < 2δ.
- Bounding the pair-correlation quantity for all r ≥ δ and substituting it into the mean-interference expression yields the power-law result.
- For α = 3, the excess interference is approximately 0.5 dB.
III. CONCLUSION
The two Matérn hard-core models exhibit markedly different interference behavior relative to a Poisson point process. Type I excess interference grows exponentially under power path loss, whereas type II remains within 1 dB regardless of the path loss law.
- The two Matérn hard-core models behave markedly differently in excess interference relative to a Poisson point process.
- Excess interference for type I increases exponentially in parent process density λp and hard-core distance δ under power path loss laws.
- Type II excess interference never exceeds 1 dB, irrespective of the path loss law.