Why 95% One-Coverage Is Not 95% Two-Coverage

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Why 95% One-Coverage Is Not 95% Two-Coverage

A 10,000 square metre random deployment with a 10 m sensing radius needs about 96 nodes for 95% one-coverage but about 147 nodes for 99% — a jump the chapter frames as a diminishing return, not simply four percent more nodes. The same Poisson model behind that count also governs k-coverage, where a point’s mean sensor count must clear the required redundancy level, not merely clear zero. This audit asks the question that 96-versus-147 jump invites: at the 95% one-coverage density, what fraction of points actually enjoy two-sensor (k=2) redundancy rather than just one?

Companion to the chapter WSN Coverage Worked Examples — every number here comes from that chapter.

Ada: The depth layer warns that k-coverage is “not a small label added after the count.” I can make that precise with the chapter’s own Poisson model. At the 95% one-coverage density (A = 10,000 m^2, Rs = 10 m), the mean number of sensors covering a point is mu = lambda x pi x Rs^2, which by construction equals -ln(1 - 0.95) = -ln(0.05) = 2.99573, near the chapter’s ~3.0.

Three sensors per point on average sounds like comfortable redundancy. But the count at a point is Poisson with that mean, so I have to ask for the probability of fewer than two:

  • P(0 sensors) = e^(-mu) = e^(-2.99573) = 0.05000 – exactly the 5% the design left uncovered.
  • P(exactly 1) = mu x e^(-mu) = 2.99573 x 0.05000 = 0.149787.
  • P(fewer than 2) = 0.05000 + 0.149787 = 0.199787, about 20.0%.

So the identical node count that delivers 95% one-coverage leaves roughly 20% of points with only a single sensor – only 80% actually enjoy k=2. The design meaning is the section’s exact caution: redundancy is a claim about P(fewer than k), not P(zero), so a k=2 requirement must lift the mean well above two rather than borrow the one-coverage count, and the review must say whether the redundancy is for resilience, localization, or just reassurance.

Every number above is taken from the chapter’s own material and re-derived step by step.