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We consider a version of the stochastic inventory control problem for aspectrally positive L\'evy demand process, in which the inventory can only bereplenished at independent exponential times. We show the optimality of aperiodic barrier replenishment policy that restocks any shortage below acertain threshold at each replenishment opportunity. The optimal policies andvalue functions are concisely written in terms of the scale functions.Numerical results are also provided.

Optimal periodic replenishment policies for spectrally positive L\'evy demand processes