Optimal service rates of a queueing inventory system with finite waiting hall, arbitrary service times and positive lead times

In this article, we consider a finite capacity queueing-inventory system in which the service rate is subject to control. We assume that the customers arrive according to a Poisson process. The customer who arrives to the service station, when the waiting hall is full, is considered to be lost. The inventory attached with the service facility is replenished according to a ( s , Q ) policy and the lead times are exponentially distributed. We calculate the optimal service rates to be employed at each service completion epoch so that the long-run expected cost rate is minimized for fixed maximum inventory level, reorder point and capacity of the waiting hall. This problem is modelled as a semi-Markov decision problem. The stationary optimal policy is obtained using linear programming algorithm and the results are illustrated numerically.