Analysis of MAP / PH /1 queue with differentiated vacation, vacation interruption under N-policy, optional service, breakdown, repair, setup and discouragement of customers

This study looks at a single server queueing system with differentiated vacations, vacation interruption under N-policy, optional services, setup, discouragement, breakdown, and repair. For customers' arrival, the Markovian arrival process (MAP) is applied, while phase type distributions (PH) are used for service and repair times. The server offers essential services to customers who are logged into the system on a first-come, first-served basis. The customer has the option to request another service (an additional service), if necessary, from the server after the essential service is complete. Here, we consider two types of vacation such as single and multiple vacations with distinct durations. Customers' arrivals may be discouraged due to impatience during both vacation periods. The matrix analytical technique is used to determine the steady-state probability vector for this model. The stability condition, busy period analysis, cost analysis and system effectiveness are all found in this model. The outcome of numerical values and graphical representations are represented in the view of two dimensional and three dimensional graphs.