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

In this paper, we develop a general framework to analyze polling systems with either the autonomous‐server or the time‐limited service discipline. According to the autonomous‐server discipline, the server continues servicing a queue for a certain period of time. According to the time‐limited service discipline, the server continues servicing a queue for a certain period of time or until the queue becomes empty, whichever occurs first. 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 systems. 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‐lengths 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.