The Behavior of Unbounded Path-loss Models and the Effect of Singularity on Computed Network Interference (2007)

In this paper we address the utility of the unbounded path-loss model <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G<sub>1</sub>(&#x01d4cd;) = &#x01d4cd;<sup>-α</sup></i> in wireless networking research problems. It is known that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G<sub>1</sub>(&#x01d4cd;)</i> is not valid for small values of 𝓍 due to the singularity at 0. We compare <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G<sub>1</sub></i> 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniformly</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed</i> over the network domain. In particular, we show that the interference probability density function becomes <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">heavy-</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tailed</i> under the unbounded path-loss model. However, it decays to zero exponentially fast under the bounded path- loss model. We also prove that a phase transition occurs in the interference behavior at a critical value <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α*</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</i> . For <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α ≤ α*</i> , as the network size grows to infinity, interference converges (either <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">probability</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">or</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distribution</i> ) only if we scale it by an appropriate sequence of constants <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&#x01d4b8;<sub>&#x01d4c3;</sub></i> with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&#x01d4b8;<sub>&#x01d4c3;</sub>→∞</i> as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&#x01d4c3; →∞</i> . On the other hand, it naturally converges <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distribution</i> to a real valued random variable without needing any scaling constants for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α &gt; α*</i> . All of our results are invariant under any finite node density <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ &gt; 0</i> .