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A functional normal approximation for stochastic internal demand process is described to address problems in solving multi-echelon inventory systems with a large-scale external demand and an internal demand that is a function of the system's structure, the control policy and the external demand. The approximation, which is applicable to multi-product demand processes, is applied to modified single-product single-location whose demand function is modeled as a Wiener process with drift.

An inventory model with impulse control and a resetting rule and in which the dynamic system needs to be maintained around a specific standard level and within a particular tolerance region is presented. In situations involving spherical tolerance regions and a dynamic system characterized by an n-dimensional vector, average cost in a special two-product inventory system was derived by combining the expected process cost and an integral with respect to the von Mises distribution.