On the Critical Delays of Mobile Networks under L\'{e}vy Walks and L\'{e}vy Flights

Delay-capacity tradeoffs for mobile networks have been analyzed through a number of research work. However, L\'{e}vy mobility known to closely capture human movement patterns has not been adopted in such work. Understanding the delay-capacity tradeoff for a network with L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy mobility includes L\'{e}vy flight and L\'{e}vy 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 L\'{e}vy flight and (ii) characterizing an embedded Markov process in L\'{e}vy walk which is a semi-Markov process. The results indicate that in L\'{e}vy 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, L\'{e}vy flight has the critical delay $\Theta(n^{\frac{\alpha}{2}})$ for $\alpha\in(0,2]$.