Analysis of MAP/PH(1),PH(2),PH(3)/2 Queueing System with Three Fluctuating Modes of Service, Standby Server, Working Breakdown, Impatient Behaviour of Customers, Vacations under Bernoulli Schedule and Phase Type Repairs

Abstract In this model, we consider the customers arrive according to Markovian Arrival Process (MAP) and their three fluctuating modes of service and repair process of both servers are based on phase-type (PH) distributions. While the main server-1 is offering any one of the modes of service who may affect by breakdown immediately go for phase-type repair process at that moment the standby server will interrupt the main server-1 and carry over the service process up to the phase-type repair completion of main server-1. Likewise, the main server-2 also caused by breakdown while offering any one of the modes of service to customers but the main server-2 has a capacity to provide service at the slower rate for the interrupted customer only before move to the phase-type repair process for rejuvenation due to the concept of working breakdown. At every service completion epoch, the main server-1 has an option to move forward to vacation or else remain in the system to offer service to next customer who is next in the queue. If both the server’s are unavailable in the system, the waiting customers in the queueing line may renege from the system because of impatient behaviour. Using matrix analytic method, we examine the steady-state probability vector, busy period and waiting time distribution. Furthermore, we illustrated some of the numerical and graphical representations of the proposed model.