On M[x]G1G21 queue with optional re-service

We study a single server queue with batch arrivals and two types of heterogeneous service with different general (arbitrary) service time distributions. The server provides either type of service to customers, one by one, on a first come, first served basis. Just before a service starts, a customer has the option to choose either type of service after completion of which the customer may leave the system or may opt for re-service of the service taken by him. We obtain steady-state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue and the system, the average number of customers and the average waiting time in the queue as well as the system. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.