Connectivity of Random 1-Dimensional Networks

An important problem in wireless sensor networks is to find the minimal number of randomly deployed sensors making a network connected with a given probability. In practice sensors are often deployed one by one along a trajectory of a vehicle, so it is natural to assume that arbitrary probability density functions of distances between successive sensors in a segment are given. The paper computes the probability of connectivity and coverage of 1-dimensional networks and gives estimates for a minimal number of sensors for important distributions.

💡 Research Summary

The paper tackles the fundamental problem of determining how many randomly deployed sensors are needed to achieve a desired probability of connectivity in a one‑dimensional wireless sensor network. Unlike most prior work that assumes a fixed spacing or a simple uniform model, the authors consider the realistic scenario where a vehicle drops sensors one after another along a trajectory, and the distances between successive sensors follow arbitrary probability density functions (PDFs).

Model and Definitions
A line segment of length (L) is considered. (n) sensors are placed sequentially; the inter‑sensor distances (X_1, X_2, …, X_{n-1}) are independent random variables with PDFs (f_i(x)). The network is said to be connected if the sum of the first (n-1) distances does not exceed (L). This condition leads to a connectivity probability expressed as a multiple convolution:
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