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In this study, a stochastic process which represents a single-item inventory control model with (s,S)-type policy is constructed when the demands of a costumer are dependent on the inter-arrival times between consecutive arrivals. Under the assumption that the demands can be expressed as a monotone convex function of the inter-arrival times, it is proved that this process is ergodic, and closed form of the ergodic distribution is given. Moreover, a sharp lower bound for this distribution is obtained.

A sharp bound for the ergodic distribution of an inventory control model under the assumption that demands and inter-arrival times are dependent