The behavior of unbounded path-loss models and the effect of singularity on computed network interference

In this paper we address the utility of the unbounded path-loss model G₁(x) = x^-α in wireless networking research problems. It is known that G₁(x) is not valid for small values of x due to the singularity at 0. We compare G₁ to a more realistic bounded path-loss model, showing that the effect of the singularity on the total network interference power is significant and cannot be disregarded when the nodes are uniformly distributed over the network domain. In particular, we show that the interference probability density function becomes heavy-tailed under the unbounded path-loss model. However, it decays to zero exponentially fast under the bounded pathloss model. We also prove that a phase transition occurs in the interference behavior at a critical value α* α. For α ≤ α*, as the network size grows to infinity, interference converges (either in probability or in distribution) only if we scale it by an appropriate sequence of constants c_n with c_n → ∞ as n → ∞. On the other hand, it naturally converges in distribution to a real valued random variable without needing any scaling constants for α > α*. All of our results are invariant under any finite node density λ > 0.