Robust Patrol in a Dispersed Environment

Abstract

We consider a patrol problem in which a patroller moves among several geographically dispersed locations to detect attackers that arrive over time. Once the patroller arrives at a location, they can spend any amount of time searching for attackers at that location---and detect an attacker there with some location-dependent instantaneous detection rate---before moving to a different location. The objective of the patroller is to minimize the expected time an attacker stays undetected at a location, regardless of where and when the attack occurs. In the special case where travel times are negligible, we elucidate an optimal cyclic policy in which the patroller allocates a fixed fraction of their effort to each location continuously. The patrol problem becomes significantly more challenging when travel times cannot be ignored. We introduce two types of cyclic patrol patterns based on common patrol practice for perimeter patrol and border patrol, respectively, and derive formulae for the expected time to detect an attack in both cases. We also provide an algorithm for finding the best search time parameters in both of these cases. We give several examples where these cycle types perform well and numerically demonstrate that the optimal patrol policy depends highly on the structure and parameters of each patrol problem.