<p>This paper investigates a discrete-time <i>Geo/Geo/1</i> queueing-inventory system with (<i>s,S</i>) inventory policy. Assume that demand occurs to the system according to the Bernoulli process and every demand needs single item for service. The service time is assumed to be geometric distribution. An external vendor replenishes the items and the replenishment of items requires positive random time which follows geometric distribution. There are a limited number of places available for demands to wait. We obtain the joint probability distributions for the number of demands and the inventory position at the outside observer’s and random epochs. We compute the waiting time distribution for an arrival demand who are able to join the queue. Various performance measures are obtained to establish a suitable cost function for the system. Moreover, numerical experiments are used to highlight the minimum value of the cost function with respect to the optimal parameter values. We validate the accuracy of discrete-time results through a simulation study. It is demonstrated that the results from this paper tend toward their continuous-time counterpart in the limiting case.</p>