We analyze tail asymptotics of a two-node tandem queue with spectrallypositive Lévy input. A first focus lies in the tail probabilities of the type, for α ∈ (0, 1) and x large, and Q i denoting the steadystate workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.

We consider the flow-level performance of a linear network supporting elastic traffic, where the service capacity is shared among the various classes of users according to a weighted alpha-fair policy. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the dynamics of the user population may be described by a Markov process. While valuable stability results have been established for the family of alpha-fair policies, the distribution of the number of active users has remained intractable in all but a few special cases. In order to gain further insight in the flow-level performance in more general scenarios, we develop approximations for the mean number of users based on the assumption that one or two of the nodes experience heavy-traffic conditions.In case of just a single 'bottleneck' node, we exploit the fact that this node approximately behaves as a two-class Discriminatory Processor-Sharing model. In the case that there are two nodes critically loaded, we rely on the observation that the joint workload process at these nodes is asymptotically independent of the fairness coefficient alpha, provided all classes have equal weights. In particular, the distribution of the joint workload process is roughly equal to that for an unweighted Proportional Fair policy, which is exactly known. In both cases, the numbers of users at nonbottleneck nodes can be approximated by that in an M/M/1 queue with reduced service capacity. Extensive numerical experiments indicate that the resulting approximations tend to be reasonably accurate across a wide range of parameters, even at relatively moderate load values. The approximations for the mean number of users also provide useful estimates for the mean transfer delays and user throughputs.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsGeneralized processor sharing: characterization of the admissible region and selection of optimal weights ABSTRACT We consider a two-class Generalized Processor Sharing (GPS) queueing system, in which each class has its specific traffic characteristics and Quality-of-Service (QoS) requirements. Traffic of both classes is assumed to be Gaussian (a versatile family of models that covers both longrange dependent and short-range dependent traffic). In this paper we address the question how to select the GPS weight values. To do so, we first characterize the admissible region of the system for fixed weights. Then we obtain the realizable region by taking the union of the admissible regions over all possible weight values. The results indicate that, under a broad variety of traffic characteristics and QoS requirements, nearly the entire realizable region can be obtained by strict priority scheduling disciplines. In addition, we indicate how the buffer thresholds, QoS requirements and the traffic characteristics of the two classes determine which class should get high priority.2000 Mathematics Subject Classification: 60G15;60K25;68M20;90B18

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We consider a system with two service classes with heterogeneous traffic characteristics and Quality-of-Service requirements. The available bandwidth is shared between the two traffic classes in accordance with the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, provide a popular mechanism for service differentiation among heterogeneous traffic classes. While the performance of GPS for given weights has been thoroughly examined, the problem of selecting weight values that maximize the traffic-carrying capacity, has only received limited attention so far. In the present paper, we address the latter problem for the case of general Gaussian traffic sources. Gaussian models cover a wide variety of both long-range dependent and short-range dependent processes, and are especially suitable at relatively high levels of aggregation. In particular, we determine the realizable region, i.e., the combinations of traffic sources that can be supported for given Quality-of-Service requirements in terms of loss and delay metrics. The results yield the remarkable observation that simple priority scheduling strategies achieve nearly the full realizable region. 1 .