Tandem Queueing Inventory System With Breakdown Server and Two Types of Service

The two service stations have been connected in series in our tandem queueing inventory model, replicating the real-life environment that provides customer service by providing seperated service without congestion. Customers arrive at Station 1, which has infinite servers (self-service), according to the Markovian arrival process, where they get an information about the inventory. After completion of service in Station 1, the customers can choose whether to move to the buffer in front of Station 2, where a single server gives service for buying inventory, or can exit the system with the predetermined probability. Once the customer who demands an inventory enters the buffer, they are instantly attached to an inventory from the unattached inventory of maximum capacity S, and the inventory replenishment follows
$ (s, Q) $
ordering policy. In Station 2, the server provides two services that the customers can choose with a predefined probability. The server in Station 2 may break down at any time during service. When the server experiences the empty buffer, the server automatically moves to vacation until it finds the customer in the system. Exponentially distributed service, lead, breakdown, and repair times have been considered. The joint probability distribution of the number of customers in Station 1, the available inventory level, and the number of customers in the Station 2 buffer has been found. Numerical calculations have been done to show the sensitive analysis of all parameters.