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

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

2011-11-21

arxiv.org/abs/1111.4724v2

Computer Science

Networking and Internet Architecture

Scientific paper

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 $\alpha\in(0,2]$.

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