Mean Queue Length of a Four-Server Feedback Queuing Model with Customer Returns to Any Server up to Finite Times

A feedback queueing model that considers four servers—one of which is centrally connected to the other three—is presented in this paper.
Only the first server allows customers to access the system; they can then move on to the second, third, or fourth server. If the service does not satisfy the customer's needs, they may return to the previously visited server a limited number of times. The probabilities of going back to the servers is thought to be different with every visit.
The steady state equations have been used to calculate the mean queue lengths of the system. Using both numerical and graphical methods, the variations in the average mean queue lengths of the system have been identified.