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.