On application of a semi-Markov random walk process in logistics

In this work, we will investigate a semi-Markov random walk process for warehouse management in logistics. We consider generating function for a boundary functional for semi-Markov random walk process. By the generating function we can find moments the distribution of warehouse level over time in the context of a semi-Markov random walk process. The random variable is introduced. This random variable representing, the number of steps required to reach a positive level. Here the length of each step follows a gamma distribution. The generating function of the distribution of the random variable is expressed as an integral equation. The purpose of the paper is to reduce the fractional order integral equation to a fractional order differential equation. Finally, the paper aims to derive an explicit form of the generating function.