Mean Interference in Hard-Core Wireless Networks

Mat'ern 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.

💡 Research Summary

The paper investigates the average interference experienced by a typical transmitter in wireless networks that employ carrier‑sense multiple access (CSMA). To capture the spatial exclusion inherent in CSMA, the authors model the set of concurrent transmitters using Matérn hard‑core point processes of type I and type II, which are obtained by thinning an underlying homogeneous Poisson point process (PPP) of intensity λ₀. Both processes enforce a minimum separation distance rₕ (the hard‑core distance), but they differ in the thinning rule.

Matérn type I removes every point that lies within distance rₕ of any other point. The resulting intensity is λ_I = λ₀ exp(−λ₀πrₕ²). The pair‑correlation function g_I(r) equals 0 for 0 ≤ r < rₕ and quickly approaches 1 for r ≥ rₕ. Consequently, points are completely absent inside the exclusion ball, while beyond that radius the spatial statistics revert to those of a PPP.

Matérn type II assigns an independent uniform mark to each point of the PPP and retains only the point with the smallest mark within each ball of radius rₕ. Its intensity is λ_II = (1 − exp(−λ₀πrₕ²))/(πrₕ²), which is higher than λ_I for the same rₕ. The pair‑correlation function g_II(r) also vanishes for r < rₕ but rises to 1 more gradually than g_I, reflecting a softer spatial regularity.

The authors define the average interference at the origin (under the reduced Palm distribution) as

 Ī = 𝔼⁰