Abstract
We establish heavy-traffic stochastic-process limits for the queue-length and overflow stochastic processes in the standard single-server queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related single-server models with finite waiting room, such as the finite dam, satisfy the same heavy-traffic stochastic-process limits. As a consequence, we obtain heavy-traffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavy-traffic limits for the steady-state loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavy-traffic limits for the steady-state blocking probabilities.
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Whitt, W. Heavy-Traffic Limits for Loss Proportions in Single-Server Queues. Queueing Systems 46, 507–536 (2004). https://doi.org/10.1023/B:QUES.0000027997.79716.b4
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DOI: https://doi.org/10.1023/B:QUES.0000027997.79716.b4