Time-limited polling systems with batch arrivals and phase-type service times

In this paper, we will develop a general framework to analyze polling systems with either the autonomous-server or the time-limited service discipline. We consider Poisson batch arrivals and phase-type service times. It is known that these disciplines do not satisfy the well-known branching property in polling system. Therefore, hardly any exact results exist in the literature. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains