Multi-server batch-service systems

In this paper we analyse a multi-server batch-service queueing model. Customers arrive one by one according to a Poisson process. They are served in batches under the following threshold policy: when a server becomes available a new batch of waiting customers is taken into service as soon as their number reaches a threshold a. The maximum allowable batch size is equal to b. Two classes of batch service time distributions are considered: Coxian-2 and Erlang-r distributions. In both cases the queueing model can be described by a Markov process. For this process it is shown that the equilibrium probabilities for states with all servers busy can be expressed as a finite sum of geometric terms. This form is used to derive a closed form expression for the waiting time distribution.