Duration of stay inside an interval by the poisson process with a negative exponential component

Abstract

Several two-boundary problems for the Poisson process with an exponential component are solved in the present article. The integral transforms of the joint distribution of the epoch of the first exit from the interval and the value of the overshoot through boundaries at the epoch of the exit are obtained. The Laplace transform of the total duration time of the process’s stay inside the interval is found.