Service by a Queue and a Cart

Items arrive randomly at a production facility that functions as a single-server queueing system. The items might represent parts, raw material, etc. and the server might be a factory worker or a machine in a production line. Following service, items are placed in a buffer where they are accumulated before delivery to a customer or some downstream activity in a production line. In practice, the buffer might be called a hopper; it may take the form of a cart or a pallet moved by a forklift. For simplicity the discussion here keeps with the cart terminology.

The cart is delivered at times to be determined; during its absence the queue will in general grow by new arrivals. An item's average time in system, from arrival to delivery, is to be made small. The system must compromise between infrequent deliveries to avoid long delays in the queue and frequent deliveries to avoid long waits in the cart. This problem has a simple relation with standard batch-sizing problems in production scheduling.

The cart delivery (batch-sizing) strategy considered here depends on two integers M and N. Delivery begins when N are in the cart or when the queue is empty and at least M are in the cart (M ≤ N). Items are assumed to arrive by a Poisson process, and their service times have a general distribution. Generating functions are derived which determine probability distributions for the numbers k in the cart, q in the queue, and k + q in the system. Numerical results are given for special cases M = N, N = ∞, and M = 0.