Technical note | An Intermittent Sensor versus a Target that Emits Glimpses as a Homogenous Poisson Process

Abstract

This technical note considers a sensor that alternates randomly between working and broken versus a target that reluctantly gives away glimpses as a homogenous Poisson process. Over any interval of time, the sensor has a probability of detecting n glimpses, of detecting the k-th glimpse, and of detecting the k-th glimpse when there are n glimpses in that interval. We devise closed-form approximations to the distributions for those probabilities, prove that the approximations become perfect as the time interval becomes infinitely long (asymptotic distributions, pointwise convergence), and compare the approximations with empirical results obtained from simulations.

Executive Summary

The work in this report was motivated by studies of operations to understand the performance that may be required from future systems; for example, unmanned aerial vehicles hunting for time-sensitive targets and submarines standing off from counter-detection. When collapsed to their essentials, the operations could be modelled in terms of a sensor that alternates between working and broken at random times, and is looking for a target that reluctantly gives away glimpses as a homogenous Poisson process. We refer to this situation as the intermittent sensor homogenous glimpses model.

In studies of such operations, the key measures of performance include the probability of detecting n glimpses, of detecting the k-th glimpse, and of detecting the k-th glimpse when there are n glimpses in the time interval. This report establishes that if we accept the intermittent sensor homogenous glimpses model then the measures of performance have approximations that are easy to calculate, and the approximations are close to reality when the time interval is long. So while the model is evidently an abstraction of real life, it can be sufficiently valid for a first, ‘back of the envelope’ analysis. Moreover the approximations provide insight into how performance will behave overall, something that can be difficult to obtain from stochastic simulation only.

The results can be applied to analysis of operations whenever the intermittent sensor homogenous glimpses model is a valid abstraction of the operation being studied. The analysis proves that the approximations become perfect in the technical sense of ‘pointwise convergence’ and uses simulation to compare the approximations with reality. The report will be of interest to analysts who are considering the intermittent sensor homogenous glimpses model for their work.