A Two Server Queueing Inventory Model With Two Types of Customers and a Dedicated Server

We consider a two server queueing inventory model with two types of customers. Type-I customers can form an infinite queue, while type-II customers join a finite buffer of size N. There are two servers. Server-1 is a dedicated server, giving service only to the type-I customers. Server-2 provides service to both type-I and type-II customers. Service of each customer requires an inventory item. Type-I customers are served one by one, by both servers on a FIFO basis. Type-II customers are served in batches of varying sizes by server-2. A random clock is set for type-II customers, which starts ticking with the arrival of first type-II customer to an empty buffer. When the number of type-II customers becomes N or when the random clock realizes, server-2 starts the service of that batch of type-II customers. In such a situation, if server-2 is giving service to a type-I customer, its service is frozen and the service of that batch of type-II begins. The arrival of both types of customers are according to a stationary Marked Markovian Arrival Process. The service times of both type-I and type-II customers follow phase type distribution. The inventory replenishment follows the (s,S) policy with positive leadtime. If the required inventory is not there at the time of service of a batch of type-II customers, local purchase is done.