Tandem Queues with Dependent Service Times in Light Traffic

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Abstract
Light traffic results about delay in queue are obtained for r single-channel queues in tandem with Poisson arrivals where the r service times of the same customer at different stations have an arbitrary joint distribution. By light traffic, we mean asymptotic behavior as server utilization approaches zero. A detailed analysis of delay at the second station is presented for r = 2. An expression for expected delay is obtained for r = 3. Methods developed for these purposes may be used for arbitrary r > 3. For arbitrary r, an expression for expected delay is obtained in the special case where the r service times of the same customer are equal. For exponential service, a simple closed form expression is obtained. For r = 2, it is shown that when service times are positively quadrant dependent, expected delay is greater than when service times are independent. In some cases, the ratio between expected delays in the dependent and independent cases is very large, particularly when r is large. This agrees with and generalizes published results in several papers and conflicts with results in one paper. An explanation for this conflict is given.

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