In this paper, we study the so-called k-out-of-n queueing-inventory system with a single repair unit, identical elements that are subject to failure, stock of spare elements, and state-dependent replenishment policy. The finite state space Markov chain model is described, and key stationary performance measures are defined. The key focus of this research is on the non-Markov case, in which the random repair and replenishment times may have infinite means, which may affect the positive recurrence of the states of the model. This case is investigated numerically.