Optimal Radius for Connectivity in Duty-Cycled Wireless Sensor Networks

Abstract

We investigate the condition on transmission radius needed to achieve connectivity in duty-cycled wireless sensor networks (briefly, DC-WSNs). First, we settle a conjecture of Das et al. [2012] and prove that the connectivity condition on random geometric graphs (RGGs), given by Gupta and Kumar [1989], can be used to derive a weakly sufficient condition to achieve connectivity in DC-WSNs. To find a stronger result, we define a new vertex-based random connection model that is of independent interest. Following a proof technique of Penrose [1991], we prove that when the density of the nodes approaches infinity, then a finite component of size greater than 1 exists with probability 0 in this model. We use this result to obtain an optimal condition on node transmission radius that is both necessary and sufficient to achieve connectivity and is hence optimal . The optimality of such a radius is also tested via simulation for two specific duty-cycle schemes, called the contiguous and the random selection duty-cycle schemes. Finally, we design a minimum-radius duty-cycling scheme that achieves connectivity with a transmission radius arbitrarily close to the one required in random geometric graphs. The overhead in this case is that we have to spend some time computing the schedule.