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Existence of a linear flows particle tracking model with a stochastic waiting time depending on the Gaussian jump length

This paper discusses the existence of the tracking model to detect a linear flows particle with a stochastic waiting time depending on the Gaussian jump length. This model is useful to measure the impurity quantification such as Radionuclides and Toxic Chemicals (particle) within the interaction medium (fluid) with minimum time and maximum probability. The particle flows linearly toward the origin (either from the left or the right). The flow line contains a nano programmed sensor (or nano UV detector). This sensor starts the tracking process for the particle (target) from the origin (filtration point) with speed equals one. We obtain the expected value of the tracking time until the sensor return to the origin with the target. Some competitive analysis depends on the uncertain values of the traveled distances which are derived to get necessary conditions for the sensor’s optimal distances. The numerical results demonstrate the efficiency of this model.