Sojourn time analysis of a queueing system with two-phase service and server vacations

We consider a two-phase service queueing system with batch Poisson arrivals and server vacations denoted by M^X/G1−G2/1. The first phase service is an exhaustive or a gated bulk service, and the second phase is given individually to the members of a batch. By a reduction to an M^X/G/1 vacation system and applying the level-crossing method to a workload process with two types of vacations, we obtain the Laplace–Stieltjes transform of the sojourn time distribution in the M^X/G1−G2/1 with single or multiple vacations. The decomposition expression is derived for the Laplace–Stieltjes transform of the sojourn time distribution, and the first two moments of the sojourn time are provided.