A geometric process model for M/PH(M/PH)/1/K queue with new service machine procurement lead time

In this article, we consider a geometric process model for M/PH(M/PH)/1/K queue with new service machine procurement lead time. A maintenance policy (N???1,?N) based on the number of failures of the service machine is introduced into the system. Assuming that a failed service machine after repair will not be ‘as good as new’, and the spare service machine for replacement is only available by an order. More specifically, we suppose that the procurement lead time for delivering the spare service machine follows a phase-type (PH) distribution. Under such assumptions, we apply the matrix-analytic method to develop the steady state probabilities of the system, and then we obtain some system performance measures. Finally, employing an important Lemma, the explicit expression of the long-run average cost rate for the service machine is derived, and the direct search method is also implemented to determine the optimal value of N for minimising the average cost rate.