ON A FRACTIONAL INTEGRAL EQUATION DESCRIBING THE SEMI-MARKOV RANDOM WALK
PROCESS WITHIN A BAND
Abstract
One of the important problems in probability theory is finding the
distribution of the time of the sojourn of a system (a process) within a
specified band. With this purpose, in this paper we consider the
semi-Markov random processes with negative drift and positive jumps. An
integral equation for the Laplace transform of conditional distribution
of the time of the system sojourn within a given band is obtained. In
this paper residence time of the system is given by the gamma
distribution with parameters and resulting in a fractional order
integral equation. In the class of gamma distributions, the resulting
general integral equation of convolution type is reduced to a fractional
order differential equation with constant coefficients. And also, in the
presented paper we obtain exact solutions of the fractional differential
equation in the form of infinite series.