A Queue with Batch Service

Customers, arriving at random, wait together in a buffer for a server to visit them. Wehn the serve appears it transfers the customers to its own storage and serves them as a batch. Meanwhile the buffer collects new customers. The service returns to the buffer after finishing each batch. If it finds the buffer empty, the service leaves for a random departure time and returns again; it continues to leave until it finds the buffer occupied. In a computer application the customers are jobs, entered at a remote terminal,and the computer uses its departures to do lower-priority jobs that originate elsewhere. To answer questions about buffer storage requirements, this paper examines the probability distribution of batch sizes. An explicit solution is obtained when service times and departure times have exponential distributions. In more general cases a functional equation supplies moments and bounds.