The Markov model for base-stock control of an inventory system with Poisson demand, non-crossing lead times and lost sales

We study base-stock control of a continuous review single-item inventory system with Poisson demand and lost sales. The item is supplied by an exogenous and sequential system with stochastic lead times (LTs) specified by their mean and standard deviation (SD). Define R as the square of mean/SD and define r as the smallest integer which is at least R. When R < r, we introduce a novel tractable LT distribution called Extended Erlangian. When R = r then Extended Erlangian is the same as pure Erlangian and the simple Markov model presented by Johansen (2005) makes it easy to compute and minimize the long-run average cost per unit time. We present an algorithm to compute the steady state distribution of a new Markov model for base-stock control with Extended Erlangian LTs. For fixed base-stock S, this algorithm can be applied to compute the average cost and it is straightforward (but burdensome for large r) to compute the optimal S and the minimum average cost. We also suggest a reasonable S computed in closed form from simple models with pure Erlangian LTs. The reasonable S is easy to compute and it performs well. Our numerical study illustrates that the average cost goes up as SD goes up, which is in sharp contrast to independence of SD when the LTs are i.i.d.