On periodic phenomena and stationary distribution of queueing system M/G/1 with group arrivals and batch service

In this paper several models of queueing system M/G/1 with group arrivals and batch service are considered, and the following fundamental questions are considered: (1) what is the structure of the phase space of the imbedded Markov chain? (2) what are the necessary and sufficient conditions causing the imbedded Markov chain to be reducible or irreducible, and periodic or aperiodic? (3) what are the necessary and sufficient conditions of the existence of stationary distribution? The generating function of stationary distribution is obtained.