In this study, an extended model of type (s,S) is considered and a semi-Markovian random walk with Gaussian distribution of summands, which mathematically describes this model, is constructed. Moreover, under some weak assumptions the ergodicity of the process is discussed. In addition, characteristic function of the ergodic distribution of the process is expressed by means of appropriating a boundary functional S N . Using this relation, the exact formulas for the first four moments of ergodic distribution are obtained and the asymptotic expansions are derived for up to three terms, as beta= S - s ->infinity. Moreover, approximations are offered for the coefficients of these asymptotic expansions, which are sufficiently satisfied of the needs of applications. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter beta.