OBTAINING A FRACTIONAL DIFFERENTIAL EQUATION FOR THE LAPLACE TRANSFORM OF THE BOUNDARY FUNCTIONAL OF A STEPWISE SEMI-MARKOV PROCESS
K. K. Omarova, Elshan A. Ibayev, Aykhan Bandaliyev · 2024
Abstract. In this study a stepwise semi-Markov process with delaying barrier is considered. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. The solutions obtained from the integral equation allow for the interpretation of key characteristics of the stepwise Markov process, including the impact of the delaying barrier on the process dynamics. In this work, the residence time of the system, representing the time spent in a particular state or between transitions, is modelled using generalized exponential distributions with different parameters. The main goal of this work is to transformation of some integral equation for the boundary functional of a stepwise semi-Markov process to fractional order differential equation. Keywords: random variable, semi-Markov random walk process, Laplace transform, RiemannLiouville integral.