A Semi-Markov Random Walk Process and Its Connection with Fractional Order Differential Equation

In this study, a semi-Markov random walk processes with negative drift, positive jumps and two delaying screen is investigated. The random variable—the numbers of the steps for the first moment of reaching level zero is introduced. We provide a mathematical modeling of the semi-Markov random walk processes with two delaying screens, expressed in general form through an integral equation. In this paper, the residence time of the system is given by the gamma distribution with parameters α>0 and β>0 resulting in the fractional order integral equation. The purpose of this paper is to reduce an integral equation for the generating function of the conditional distribution of the random variable to a fractional order differential equation.