On the Critical Delays of Mobile Networks under Lévy Walks and Lévy Flights

Delay-capacity tradeoffs for mobile networks have been analyzed through a number of research work. However, Lévy mobility known to closely capture human movement patterns has not been adopted in such work. Understanding the delay-capacity tradeoff for a network with Lévy mobility can provide important insights into understanding the performance of real mobile networks governed by human mobility. This paper analytically derives an important point in the delay-capacity tradeoff for Lévy mobility, known as the critical delay. The critical delay is the minimum delay required to achieve greater throughput than what conventional static networks can possibly achieve (i.e., $O(1/\sqrt{n})$ per node in a network with $n$ nodes). The Lévy mobility includes Lévy flight and Lévy walk whose step size distributions parametrized by $\alpha \in (0,2]$ are both heavy-tailed while their times taken for the same step size are different. Our proposed technique involves (i) analyzing the joint spatio-temporal probability density function of a time-varying location of a node for Lévy flight and (ii) characterizing an embedded Markov process in Lévy walk which is a semi-Markov process. The results indicate that in Lévy walk, there is a phase transition such that for $\alpha \in (0,1)$, the critical delay is always $\Theta (n^{1/2})$ and for $\alpha \in [1,2]$ it is $\Theta(n^{\frac{\alpha}{2}})$. In contrast, Lévy flight has the critical delay $\Theta(n^{\frac{\alpha}{2}})$ for $\alpha\in(0,2]$.