Analysis of a Multiserver Queueing-Inventory System

We attempt to derive the steady-state distribution of the M/M/c queueing-inventory system with positive service time. First we analyze the case of c=2 servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the (s,Q) policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair (s,Q) and the corresponding expected minimum cost are computed. As in the case of M/M/c retrial queue with c≥3, we conjecture that M/M/c for c≥3, queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutive s to s transitions of the inventory level (i.e., the first return time to s) is computed. We also obtain several system performance measures