System size and service size distributions of a batch service queue

The authors derive the arbitrary time point system size distribution of M/G^B/1 queue in which late arrivals are not allowed to join the on-going service. The distribution is given by P(z)=Pq(z) S*(λ-λz) where Pq(z) is the probability generating function of the queue size and S*(θ) is the Laplace-Stieltjes transform of the service time distribution function. The authors also derive the distribution of the service size at an arbitrary point of time.