Investigation a boundary functional of a random walk process using fractional order differential equation

In this study, we consider a semi-Markov random walk process with negative drift, positive jumps and delays. We give a mathematical construction of the semi-Markov random walk process with a delaying positive screen. The aim of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of the first passage time of positive level for the semi-Markov random walk process to fractional order differential equation with constant coefficients. The Laplace transform of the conditional distribution of the first passage time for the semi-Markov random walk process with positive delaying screen is obtained