A semi-Markov random walk process is analyzed in this study within the framework of warehouse management for logistics. We analyze the generating function of a boundary functional within this process, which enables the determination of moments of the warehouse level distribution over time. A random variable is introduced to represent the number of steps required to reach a positive level. We derive an integral equation for the generating function of the distribution of this random variable. The jump length follows a gamma distribution, leading to a non-integer order integral equation. In this paper, we focus on converting the non-integer order integral equation into a non-integer order differential equation, with the final goal of obtaining an explicit expression for the generating function.