Normal approximations for the ergodic distribution of a semi-Markovian inventory model of type (s,S)

This paper develops new closed form normal approximations for the ergodic distribution of a renewal–reward process $X(t)$ describing a semi-Markovian inventory model of type $(s, S)$. When demand sizes follow a Weibull distribution with shape parameter α ≥ 3, the renewal function can be well approximated by that of a normal distribution, as shown by Cui and Xie. We use this observation to derive a new numerical method for computing the ergodic distribution QX(x) of the process $X(t)$. The resulting formulas are easy to evaluate and avoid repeated numerical computation of the Weibull renewal function. Numerical examples show that the approximations are accurate over a wide range of parameter values and that, as the replenishment threshold increases, the ergodic distribution of the inventory level approaches a uniform distribution.