We study the problem of determining production quantities in each period of an infinite horizon for a single item produced in a capacity-limited facility. The demand for the product is random, and it is independent and identically distributed from period to period. The demand is observed at the beginning of a time period, but it need not be filled until the end of the period.
Unfilled demand is backordered.
The inventory system is easily understood and managed if we know the distribution of the shortfall V . We find the exact form of the distribution of V when the demand has an exponential tail, or when it has a mixed exponential distribution. For other demand distributions we develop new and improved approximations for the distribution of V , and perform extensive computational tests. Our best approximation works extremely well as long as the coefficient of variation of the demand is less than two. For practical applications this is by far the most interesting case.