Induced burstiness in generalized processor sharing queues with long-tailed traffic flows

We analyze the queueing behavior of long-tailed traffic flows under the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, play a major role in achieving differentiated quality-of-service in integrated-services networks. We prove that, in certain scenarios, a flow may be strongly affected by the activity of `heavier'-tailed flows, and may inherit their traffic characteristics, causing induced burstiness. This phenomenon contrasts with previous results which show that, under certain conditions, an individual flow with long-tailed traffic characteristics is effectively served at a constant rate. In particular, the flow is then essentially immune from excessive activity of flows with `heavier'-tailed traffic characteristics. The sharp dichotomy in qualitative behavior illustrates the crucial importance of the weight parameters in protecting individual flows

Heavy-Tailed Limits for Medium Size Jobs and Comparison Scheduling

We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k>=0 out of the preceding m>=k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1-\rho)/(k+1) for PS/FBPS, and (1-\rho) for SRPT, respectively, where \rho is the traffic intensity. Hence, the introduced notion of conditional limits allows us to distinguish the asymptotic performance of the studied schedulers by showing that SRPT exhibits considerably better asymptotic behavior for relatively smaller jobs than PS/FBPS. Inspired by the preceding results, we propose an approximation to the SRPT discipline based on a novel adaptive job grouping mechanism that uses relative size comparison of a newly arriving job to the preceding m arrivals. Specifically, if the newly arriving job is smaller than k and larger than m-k of the previous m jobs, it is routed into class k. Then, the classes of smaller jobs are served with higher priorities using the static priority scheduling. The good performance of this mechanism, even for a small number of classes m+1, is demonstrated using the asymptotic queueing analysis under the heavy-tailed job requirements. We also discuss refinements of the comparison grouping mechanism that improve the accuracy of job classification at the expense of a small additional complexity.Comment: 26 pages, 2 figure

GPS queues with heterogeneous traffic classes

We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic classes are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behavior of the light-tailed class for the situation where its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed class served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is in fact asymptotically equivalent to that in the isolated system, multiplied with a certain pre-factor, which accounts for the interaction with the heavy-tailed class. Specifically, the pre-factor represents the probability that the heavy-tailed class is backlogged long enough for the light-tailed class to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Tail asymptotics for processor sharing queues

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues

Bandwidth sharing with heterogeneous service requirements

We consider a system with two heterogeneous traffic classes. The users from both classes randomly generate service requests, one class having light-tailed properties, the other one exhibiting heavy-tailed characteristics. The heterogeneity in service requirements reflects the extreme variability in flow sizes observed in the Internet, with a vast majority of small transfers ('mice') and a limited number of exceptionally large flows ('elephants'). The active traffic flows share the available bandwidth in a Processor-Sharing (PS) fashion. The PS discipline has emerged as a natural paradigm for modeling the flow-level performance of bandwidth-sharing protocols like TCP. The number of simultaneously active traffic flows is limited by a threshold on the maximum system occupancy. We obtain the exact asymptotics of the transfer delays incurred by the users from the light-tailed class. The results show that the threshold mechanism significantly reduces the detrimen

Sojourn time asymptotics in processor sharing queues

This paper addresses the sojourn time asymptotics for a GI/GI/• queue operating under the Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize distribution has a light tail. Whereas upper bounds on the decay rate can be derived under fairly general conditions, the establishment of the corresponding lower bounds requires that the service process satisfies a samplepath large-deviation principle. We show that the class of allowed service processes includes the case where the service rate is modulated by a Markov process. Finally, we extend our results to a similar system operation under the Discriminatory Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations techniques