On the semi-Markovian random walk with two reflecting barriers

Abstract

In this paper, the semi-Markovian random walk with two reflecting barriers is constructed mathematically and non-stationary distribution functions of it are expressed by means of the probability characteristics of renewal process {T _n } and random walk {Y _n } without barriers. In particular, when the time between two jump instants has exponential or Erlang distribution, explicit formulae are obtained for non-stationary distribution functions of the process. Moreover, explicit expressions are given for expected value, variance and moment generating function of the first reflection moment, an important boundary functional, of the process from lower reflecting barrier.

Acknowledgments

We wish to express our thanks to Professor A.V. Skorohod for the formulation of the common problems in connection with the semi-Markovian random walk with two screens, and his support and some valuable advice.