Batch Delays Versus Customer Delays

For a large class of contention schemes with messages transmitted by subdividing them into packets, we show that the delay distribution of a message is the same as that for an individual packet. We conclude this from analyzing queueing systems with batch arrivals, where batch sizes have a geometric distribution and the queue discipline is indifferent to batch sizes and service times. There we prove that the customer (packet) delay distribution is the same as the batch (message) delay distribution, where delay is defined to be the delay of the last served customer in the batch. The proof is based on the discrete-time analog of the Poisson Arrivals See Time Averages (PASTA) theorem. We conclude that, in many cases, we can obtain message delays by calculating or measuring the packet delays, which is usually an easier task.