An Examination on Queuing System of General Service Distribution with an Establishment Time and Second Discretionary Administration

We consider a non Markovian Queuing framework with two phases of heterogeneous service gave by a solitary server with various (discretionary) general time circulations. Each clients experiences two phases of service. When the primary mandatory phase service gets over, the client has the alternative of picking the second phase of administration which is discretionary. After service fruition of either first stage or second phase, if there are no clients in the framework, the server may take a Bernoulli vacation. The vacation time are thought to be general. Moreover, we expect a set up time criteria before the initiation of the administration. The time dependent probability producing capacities have been gotten as far as their Laplace changes and the relating enduring state comes about have been acquired expressly. Additionally the mean number of clients in the queue and the framework are likewise inferred. Some specific cases are talked about.