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Comparing batch delays and customer delays

For a large class of queueing systems in which customers arrive in batches, Halfin (1983) showed that the delay distribution of the last customer in a batch to enter service coincides with the delay distribution of an arbitrary customer when the batch-size distribution is geometric. Halfin's result can be applied to study the performance of complicated communication systems in which messages are divided into packets for transmission. Then packets are customers and the delay of a message is the delay of the last customer in a batch to enter service. If the assumptions are satisfied and if packet delays are easier to analyze, then packet delays can be used to calculate message delays. In this paper, we show that these two delay distributions are stochastically ordered when the batch-size distribution is NBUE or NWUE (new better or worse than used in expectation). The delays of arbitrary customers tend to be less (more) when the batch-size distribution is NBUE (NWUE). In addition to the bounds provided by the stochastic ordering, we also suggest an approximation for the relation between the two expected delays based on known results for the MB/G/1 queue having a batch-Poisson arrival process.