Why 95% One-Coverage Is Not 95% Two-Coverage
Why 95% One-Coverage Is Not 95% Two-Coverage
Ada re-derives this chapter’s own numbers step by step, at full precision
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.
See the relationship before changing it
The figure reads from left to right. The blue card is mean sensors covering a point. The middle card applies this page's rule. The green card is two-or-more coverage. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only mean sensors covering a point, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 3 sensors.
- 2
Name the relationship. P(K >= 2) = (1 - e^-lambda x (1 + lambda)) x 100
- 3
Substitute with units. (1 - e^-3 x (1 + 3)) x 100 = 80.1%
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change mean sensors covering a point
Try Predict the direction of P(K >= 2) = (1 - e^-lambda x (1 + lambda)) x 100. Test another mean sensors covering a point, then compare two-or-more coverage.
Observe One-coverage can look strong while two-sensor redundancy still leaves many points exposed. Reset mean sensors covering a point to 3 and compare two-or-more coverage.
Explain One-coverage can look strong while two-sensor redundancy still leaves many points exposed.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
At A = 10000 m², Rs = 10 m, and 95% one-coverage, use mu = -ln(0.05) = 2.99573 to compute P(0) and P(1).
Uncovered probability is 5.0% and exactly-one coverage is about 14.98%, leaving only about 80.0% of points with at least two sensors.
Poisson k-coverage requires subtracting every probability below k; clearing P(0) alone proves one-coverage but says nothing about the sizeable P(1) population.
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 the5%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, about20.0%.
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.
Technical boundaries: This homogeneous Poisson disk model excludes site borders, obstacles, correlated placement, irregular sensing range, terrain, node failure, localisation error, anisotropy, and connectivity between covering sensors.