Moment-based approximate formulas for stationary characteristics of random walk process with a barrier

This research proposes a moment-based approach to obtain approximate expressions for the expected value and variance of the ergodic distribution of a semi-Markovian random walk process (𝑋(𝑡)) with gamma distributed interference of chance. In the literature, similar problems have generally been examined using asymptotic approaches. However, this study employs moment-based approximations proposed by Kambo instead of asymptotic methods. The accuracy of the moment-based approximation formulas obtained in this work is evaluated through comparisons with Feller's asymptotic expressions using illustrative examples. In particular, it is shown that the approximation formulas based on Kambo's method yield more precise results for relatively small parameter values in the case of a renewal function generated by the Erlang distribution. In this context, the study first derives approximation formulas for the moments of the boundary functional 𝑆𝑁(𝑧) of the process 𝑋(𝑡) and then uses these results to obtain approximations for the expected value and variance of its ergodic distribution. Furthermore, Monte Carlo simulation studies performed on two different distributions (Gaussian and Uniform) demonstrate the practical applicability of the proposed method.