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Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.
The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.