Probability of interruption: response time beats sensor count

Timely detection#

A detection only matters if a response can arrive before the adversary finishes. Sandia's physical protection methodology, set out in Garcia's textbook, formalizes this as the probability of interruption $P_I$ along an adversary path. The path is a sequence of elements — approach, fence, yard, door, target — each with a detection probability $p_i$ and a delay $t_i$. If detection happens at element $i$, the response must arrive within the remaining delay $T_i={\sum }_{j\ge i}{t}_j$.

where ${\mu }_R,{\sigma }_R$ describe the response force time and $\Phi$ is the standard normal CDF. This is the structure of the EASI method; we assume detection at the start of each element, which is the conservative choice for barriers.

An illustrative remote substation#

Total path delay is 690 s (about 12 minutes). The fence-centric design has a vibration sensor on the fence and cameras on the yard. The approach-detection design adds wide-area radar and a perched EO/IR sentry covering the 300 m of open ground outside the fence. A third variant also raises yard detection with a sentry overhead. Response time standard deviation is 30% of its mean. All values are illustrative (probability_of_interruption.py).

Results#

Three things stand out.

Response time dominates. For the fence-centric site, $P_I$ falls from 0.90 with a 5-minute response to 0.37 at 10 minutes and 0.10 at 15. Remote substations commonly see response times of 20–30 minutes, where $P_I$ is below 0.05 whatever the sensors.

Earlier detection buys a fixed amount of time. Detecting on the approach adds the approach delay (150 s here) to the response window. At a 10-minute response that lifts $P_I$ from 0.37 to 0.62 — the largest single improvement available without changing the response. At 20 minutes it barely matters.

More detection late in the path adds almost nothing. Raising yard detection from 0.3 to 0.6 moved $P_I$ by less than 0.01 at every response time, because by then most adversaries have already been detected and the remaining question is time.

Sensitivity: response-time variance and added delay#

Response times are not just slow; they are variable, and variance hurts. Holding the fence-centric layout fixed and varying the standard deviation of response time (recomputed at build time with the same formula):

When the mean response is comfortably inside the path delay (5 minutes against 11.5), variance costs a little. When the mean is close to the delay (8–10 minutes), a predictable response is worth a great deal: at a 10-minute mean, going from ±50% to ±10% variability changes $P_I$ from 0.40 to 0.30. Pre-positioned mobile officers, rather than a guard who might be anywhere on a large site, are mostly a variance reduction.

Now add delay instead. Upgrading the control-house door from 180 s to 420 s of delay (a hardened door set, which is a facilities project rather than a security-technology one):

Four extra minutes of delay after the first likely detection point is worth more than any sensor in this model at 10–15 minute response times.

What actually moves the number#

Rearranging the formula makes the levers explicit: raise early $p_i$, add delay $t_i$ after the first likely detection point, or cut ${\mu }_R$. For remote infrastructure the third is often impossible, which leaves two options. Add delay — hardened control-house doors and equipment enclosures — or change the response model. Remote deterrence from a perched sentry (light and voice within seconds of detection) is not counted as interruption in EASI terms, but in practice it ends most opportunistic intrusions; and continuous tracking means responders arrive with a live picture rather than to an empty yard. We discuss the deterrence evidence separately because it does not fit a timely-detection model.

Limitations#

Single path, single adversary, normal approximations for delays and response time, detection at element start, no adversary adaptation to sensor placement, illustrative numbers. Full vulnerability assessments use multi-path analysis and site-specific performance data; this model is a teaching tool to show the shape of the trade-offs.

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.