Analysis of a two-phase queueing system with a fixed-size batch policy

We consider a single-server, two-phase queueing system with a fixed-size batch policy. Customers arrive at the system according to a Poisson process and receive batch service in the first-phase followed by individual services in the second-phase. The batch service in the first-phase is applied for a fixed number (k) of customers. If the number of customers waiting for the first-phase service is less than k when the server completes individual services, the system stays idle until the queue length reaches k. We derive the steady state distribution for the system's queue length. We also show that the stochastic decomposition property can be applied to our model. Finally, we illustrate the process of finding the optimal batch size that minimizes the long-run average cost under a linear cost structure.