Analysis of $GI/M/1/N$ and $GI/Geo/1/N$ queues with balking and vacation interruptions

This paper addresses renewal input continuous  and  discrete time  queues with balking and vacation interruptions. An arriving client may join the system or  balk with some state-dependent probability. Whenever the server finds an empty system, he leaves for a working vacation. During working vacations, if there are clients to be served at a service completion instant, the server interrupts the  working vacation and switches to regular service period. The embedded Markov chain technique has been adopted for evaluating pre-arrival epoch probabilities and  supplementary variable approach is employed to evaluate arbitrary instant probabilities. Few performance characteristics and sojourn time distribution have also been demonstrated. Finally,  numerical investigations have been figured out to depict the impact of the model variables on the performance indices.