In this study, a semi-Markovian random walk with a discrete interference of chance (X t ) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X t are obtained when the random variable 1 , which describes a discrete interference of chance, has a gamma distribution with parameters > 1 > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X t , as → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X t are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

In this study, we constructed a stochastic process (X.t// that expresses a semi-Markovian inventory model of type (s,S/ and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n th order moments .n D 1; 2; 3; : : :/ of ergodic distribution of the process X.t/, as S s ! 1. Finally, we tested how close the obtained approximation formulas are to the exact expressions.

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