A bulk service GI/M/1 queue with service rates depending on service batch size

We consider a bulk service GI/M/1 queue with service rates depending on service batch size. If there are n customers waiting at the completion of service, min(n, K) customers enter service. We show that the queue size and the service batch size at points of arrivals form an embedded Markov chain and the steady-state probabilities of this Markov chain have the matrix geometric form. We describe the rate matrix R of the matrix geometric solution procedure in a readily computable form. We obtain explicit analytic expressions for the steady-state queue length distribution at points of arrivals. Further we obtain the Laplace-Stieltjes transform and the moments of the stationary waiting time distribution of an arbitrary customer.