A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.