On the Problem of Batch Arrivals and its Application to a Scheduling System

The single-server queuing problem where customers arrive in batches is considered with general types of service time and batch interarrival time distributions. Following an argument similar to that of Lindley, transient and steady-state time distribution functions are obtained for the two processes: (i) a random batch size, and (ii) a periodic batch size. It is shown how to apply the general solution of the process with a random batch size in obtaining particular solutions for the cases with p₂ = 1, q₁ = 1, and (1) M/G/1 or (2) D/E/1. The latter solution has been used in a comparison of two scheduling systems.