Nuisance alarms, sensor fusion and operator load

Two numbers that fight each other#

Every perimeter intrusion detection system is judged on probability of detection $P_d$ and nuisance alarm rate (NAR) — alarms from wind, animals, debris and weather rather than people. Raising sensitivity raises both. Garcia's treatment of physical protection systems makes the point that an alarm nobody assesses is not a detection; this note puts numbers on how quickly assessment capacity runs out.

Model#

Consider 40 perimeter zones, each with a fence-vibration sensor F and a radar R. Model nuisance alarms in each sensor as independent Poisson processes with rates ${\lambda }_F,{\lambda }_R$ per zone-hour, and real intrusions as detected with probabilities $p_F=0.9,p_R=0.95$. Rates (assumed, illustrative): calm night ${\lambda }_F=0.2,{\lambda }_R=0.1$; windy night ${\lambda }_F=1.5,{\lambda }_R=0.4$.

OR fusion

Alert if either sensor fires:

AND (coincidence) fusion

Alert only if both fire within a window $w$. For two independent Poisson streams with $\lambda w\ll 1$, the rate of chance coincidences is

(each F event has probability $\approx 2{\lambda }_{R}w$ of an R event within ±$w$).

OR + verification

Every OR alert is checked automatically by a sentry's EO/IR classifier before reaching a human. With $P(\text{confirm}\mid \text{real})=0.97$ and $P(\text{pass}\mid \text{nuisance})=0.03$:

Operator capacity

If assessing an alert takes 90 s and sustained utilization above 50% degrades attention, one operator handles $c=3600\times 0.5/90=20$ alerts per hour.

Results#

On a windy night, OR logic produces 76 alerts an hour — 3.8× what one operator can assess. Real systems in this state get their sensitivity turned down or their alarms ignored, which silently converts a 99.5% $P_d$ into something much lower.

AND logic is seductive: alerts fall to 0.4 per hour. But $P_d$ falls to 85.5%: roughly one intruder in seven now gets through, and nothing on the console says so.

OR with automated verification keeps $P_d$ at 96.5% while holding alerts to 2.28 per hour, well inside one operator's capacity even on the bad night.

Coincidence window sensitivity#

The analytic rate grows linearly with $w$. A 10,000-hour simulation of the Poisson streams (last column) agrees with the formula to within sampling noise, which is the check that the approximation holds at these rates. Shrinking the window cuts chance coincidences but risks missing real intrusions whose two signatures are separated in time — a person who cuts the fence, waits, then walks into radar coverage.

Designing to an alarm budget#

Inverting the model gives a design rule. If one operator can assess $c$ alerts per hour, a perimeter of $Z$ zones with per-zone alert rate ${\lambda }_{\text{alert}}$ needs

Design to the windy column, not the calm one. An OR-fused perimeter that one operator handles comfortably on a still night (66 zones) needs about four operators for the same perimeter on a gusty one — and nobody staffs for that, so the system gets detuned. Verification moves the windy-night limit to 350 zones per operator.

The practical sequence: run sensors in OR mode, record raw nuisance rates per zone through a full range of weather during a 30–90 day baseline, then set fusion and verification so the 95th-percentile night stays inside the alarm budget. Zones that cannot be brought inside budget are usually telling you about a sensor placement problem — a fence line under a tree, a radar looking across a road — not a tuning problem.

The failure mode the model hides#

The coincidence formula assumes the two sensors' nuisance alarms are independent. Wind breaks that assumption: gusts shake the fence and move vegetation in front of the radar at the same moment. Correlated nuisance sources make the real AND rate higher than $2{\lambda }_F{\lambda }_{R}w$, sometimes by an order of magnitude on gusty nights. Fusion buys the most when the two sensors respond to physically different phenomena — vibration versus Doppler versus thermal contrast — and the least when one weather event drives both. The same logic argues for a verification stage that looks at the scene rather than at a second sensor's alarm.

Limitations#

All rates are assumed for illustration, chosen within ranges commonly discussed for fence and radar sensors; your site's numbers will differ and should be measured during a baseline period. Assessment time and the 50% utilization ceiling are planning assumptions, not human-factors results. The verification stage's 97%/3% performance is an assumption about a good classifier in good conditions; it degrades in fog and heavy rain.

Frequently asked questions#

References#

1. M. L. Garcia, Design and Evaluation of Physical Protection Systems, 2nd ed., Butterworth-Heinemann / Elsevier, 2007. Timely detection, probability of interruption and the EASI method developed at Sandia National Laboratories.

Reproducibility: model code is published at /research-code/ with a README. Results are model outputs under the assumptions stated above, not field trials.