We consider two types of queues with workload-dependent arrival rate and service speed. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.First, in the M/G/1 case, we compare the steady-state distribution of the workload (both at arbitrary epochs and at arrival instants) in two models, in which the ratio of arrival rate and service speed is equal. Applying level crossing arguments, we show that the steady-state distributions are proportional. Second, we consider a G/G/1-type queue with workload-dependent interarrival times and service speed. Using a stochastic mean value approach, several well-known relations for the workload at various epochs in the ordinary G/G/1 queue are generalized.

We analyze the asymptotic behavior of long-tailed traffic sources under the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated quality-ofservice in integrated-services networks.Under certain conditions, we prove that in an asymptotic sense an individual source with longtailed traffic characteristics is effectively served at a constant rate, which may be interpreted as the maximum feasible average rate for that source to be stable. Thus, asymptotically, the source is only affected by the traffic characteristics of the other sources through their average rate. In particular, the source is essentially immune from excessive activity of sources with 'heavier'-tailed traffic characteristics. This suggests that GPS-based scheduling algorithms provide an effective mechanism for extracting high multiplexing gains, while protecting individual connections.1991 Mathematics Subject Classification: 60K25 (primary), 68M20, 90B12, 90B22 (secondary).

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