In this paper, we investigate the semi-Markovian random walk processes with jumps and delaying screen in zero. The Laplace transform on time, Laplace-Stieltjes transform on phase of the conditional distribution of semi-Markovian random walk processes with jumps is found. We get a mathematical modeling of the semi-Markov random walk processes with a delaying screen in zero, given in general form by means of integral equation. In this paper, the residence time of the system is given by the gamma distribution with the parameters and resulting in a fractional order integral equation. The purpose of this paper is to reduce the fractional order integral equation to a fractional order differential equation. Finally, we find the exact solution of fractional order differential equation. KEYWORDS fractional order differential equation, gamma distribution, Laplace transform, semi-Markovian random walk process Math Meth Appl Sci. 2018;41:9301-9311.wileyonlinelibrary.com/journal/mma

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Given a four-dimensional sequence of independent identically located and positive random variables, a process with differential semi-Markov walk delayed by a screen at zero is considered. An explicit form of the Laplace transform in time, the Laplace-Stieltjes transform of the ergodic distribution of a process with differential semi-Markov walk with delaying screen at zero is found.

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