A queueing system with n -phases of service and (n-1)(n-1)-types of retrial customers

A queueing system with a single server providing n-phases of service in succession is considered. Every customer receives service in all phases. Arriving customers join a single ordinary queue, waiting to start the service procedure. When a customer completes his service in the i  th phase he decides either to proceed to the next phase or to join the KiKi retrial box (i=1,2,…,n-1)(i=1,2,…,n-1), from where he repeats the demand for the (i+1)(i+1)th phase of service after a random amount of time and independently to the other customers in the system. Every customer can join during his service procedure a number of retrial boxes before departs from the system. When at the moment that a customer, either departs from the system or joins a retrial box and so releases the server, there are no other customers waiting in the ordinary queue, then the server departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system, the mean number of customers in the ordinary queue and in each retrial box separately are obtained, and used to investigate numerically system performance.