Approximate anlysis of an unreliable M/M/c retrial queue with phase merging algorithm

In this paper, we investigate an approximate analysis of unreliable M/M/c retrial queue with cgeq 3 in which all servers are subject to breakdowns and repairs. Arriving customers that are unable to access a server due to congestion or failure can choose to enter a retrial orbit for an exponentially distributed amount of time and persistently attempt to gain access to a server, or abandon their request and depart the system. Once a customer is admitted to a service station, they remain there for a random duration until service is complete and then depart the system. However, if the server fails during service, i.e., an active breakdown, the customer may choose to abandon the system or proceed directly to the retrial orbit while the server begins repair immediately. In the unreliable model, there are no exact solutions when the number of servers exceeds one. Therefore, we seek to approximate the steady-state joint distribution of the number of customers in orbit and the status of the $c$ servers for the case of Markovian arrival and service times. Our approach to deriving the approximate steady-state probabilities employs a phase-merging algorithm.