A Study on the Finiteness of a Tracking Method with Reduction in the Collision Time between a D-Dimensional Random Walk Particle and One of Multiple Nano-Sensors

In order to detect radionuclides and dangerous substances in a fractured medium with the least amount of time and maximum probability, this paper provides a revolutionary tracking model. It relies on a coordinated tracking strategy using set of programmed nano-sensors. The radionuclide particle has a linear flow from d-directions to spread in a single direction with Random Walk process. We assume that this particle has been spread over a short period of time along phantomic cylindrical path inside the fluid. Two nano-sensors will track or search along each cylinder’s route, starting from any location in the fragmented medium (space). A vector of random variables with a known probability distribution makes up the particle’s initial position. We add the discounted effort-reward track as a reduction parameter in the distance function. The expected value of the first meeting time between one of the nano-sensors and a Random walk target will be reduced as a result. We extract various supplementary results regarding the optimal tracking strategy in addition to presenting the analytical solution that demonstrates the existence of a finite tracking technique.