The proposed investigation is an inventory system tied to a single-server M/M/1/N facility that allows impatient customers according to the ordering policy (r, Q). The order is triggered when the stock level drops to a positive integer 'r < Q' for a fixed number 'Q' of units. Here: 1) queueing inventory system (QIS) with lost customers; 2) QIS with back-order are investigated. For each type, we study the long-run moments of the X variable representing the observed queue length, and the Y variable representing observed inventory level. We show that the combined sequence Z = (X, Y) constitutes a level-dependent quasi-birth and death process (LDQBD) in the state space E = {(n, j): n = 0, 1, ..., N, and j = 0, 1, ..., (Q + r)}. We obtain the general probability distribution of the Z process by matrix analysis. Numerical illustrations are provided to find the economic order quantity

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