Mathematical Analysis of Queueing-Inventory Model with Compliment and Multiple Working Vacations

In this paper, we provide independent continuous review of stock for two commodities, namely commodity 1(C1) and 2 (C2). The type 1(T1) customer demands C1 and Type 2(T2) customer demands C2 but C2 is also given to T1 customer as a compliment (i.e. C2 is also given as a compliment to C1). The arrival pattern of customers is from a finite source of population. The ordering policies of both commodities are independent and also, each customer demands service with positive time. A single finite retrial orbit is allowed for any customer, when the demanding item is stocked out or the server is busy. The pre-emptive priority service policy is assigned to T1 customer over T2 customer. The server may go for the working vacation as C1 becomes zero, in which T2 customer who is taking his service is allowed to orbit whenever T1 customer demands the service with both commodities and when the orbit is not full. The joint limiting distributions of four random variables are studied under steady state. Long run expected total cost rate and measures of system performance are derived.