An M/G/1 Retrial Queue with Non-Persistent Customers, a Second Optional Service and Different Vacation Policies

In this paper, we study an M/G/1 queue with two phases of heterogeneous service. A first essential service is provided to all arriving customers. Upon completion of this service, a customer can either opt for a second phase of service or can leave the system. A customer, who finds the server busy, either leaves the system with probability (1-α) or joins an orbit with probability α. From the orbit the customer makes repeated attempts to obtain service. We assume the inter retrial times are exponential random variables. We also assume that upon completion of a service, the server either remains in the system with probability β0 or leaves the system for an ith type of vacation with probability βi(1 ≤ i ≤ M) where ∑Mi=0 βi = 1. We obtain the probability generating functions of the system size distribution as well as the orbit size distribution in the steady state. We obtain a stochastic decomposition of the system size distribution and an expression for the additional increase in the congestion due to the presence of retrials, in the steady state. We discuss some particular cases.