Displacing Random Sensors to Avoid Interference

Abstract

Consider sensors on a line. Assume that for a given parameter s > 0 two sensors’ signals interfere with each other during communication if their distance is ≤ s. We are allowed to move the sensors on the line, if needed, so as to avoid interference. We call total movement the sum of displacements that the sensors have to move so that the distance between any two sensors is > s. We study the following sensor displacement problem for avoiding interference. Assume that n sensors are thrown randomly and independently with the Poisson distribution having arrival rate λ = n in the interval [0, + ∞ ). What is the expected minimum total distance that the sensors have to move from their initial position to a new destination so that any two sensors are at a distance more than s apart? In this paper we study tradeoffs between the interference distance s and the expected minimum total movement, denoted by E(s). (Clearly, the higher the value of s the more the resulting displacement E(s).)

For the line. we prove the following results. 1) If $s \leq \frac{1}{nt}$ then E(s) ≤ min { t ²/(t − 1)³ , (n − 1)/2t } , where t > 1. 2) For s ≥ 1/n + Ω(n ^− α) we show that E (s) ∈ Ω(n ^2 − α) , 2 ≥ α ≥ 0, while for | s − 1/n | ∈ Θ( n ^− 3/2 ), we show that $E(s) \in \Theta (\sqrt{n})$. These results show a critical regime for the expected minimum total displacement E(s), for s in the interval [ 1/n − 1/n ^3/2, 1/n + 1/n ^3/2 ].

Similar results concerning the expcted optimal sum of displacements are obtained when the sensors are located on the plane and their coordinates are generated by two independent identical Poisson processes. In the critical regime for sensors on the plane, we show that E(s) ∈ Θ(n ^3/4) provided that s is in the interval [ 1/n ^1/2 − 1/n ^3/4, 1/n ^1/2 + 1/n ^3/4 ].

Research supported in part by NSERC grants.