On the upper bound of α-lifetime for large sensor networks

In this article, we explore the fundamental limits of sensor network lifetime that all algorithms can possibly achieve. Specifically, under the assumptions that nodes are deployed as a Poisson point process with density λ in a square region with side length ℓ and each sensor can cover a unit-area disk, we first derive the necessary and sufficient condition of the node density in order to maintain complete <i>k</i>-coverage with probability approaching 1. With this result, we obtain that if λ = log ℓ² + (<i>k</i> + 2)loglog ℓ² + <i>c</i>(ℓ), <i>c</i>(ℓ) → −∞, as ℓ → + ∞, the sensor network lifetime (for maintaining complete coverage) is upper bounded by <i>kT</i> with probability approaching 1 as ℓ → + ∞, where <i>T</i> is the lifetime of a single sensor. Second, we derive, given a fixed node density in a finite (but reasonably large) region, the upper bounds of lifetime when only α-portion of the region is covered at any time. We also carry out several sets of experiments to validate the derived theoretical results. Numerical results indicate that the derived upper bounds apply not only to networks of large sizes and homogeneous nodal distributions but also to small size networks with clustering nodal distributions.