1,070 research outputs found
A Tandem Fluid Network with L\'evy Input in Heavy Traffic
In this paper we study the stationary workload distribution of a fluid tandem
queue in heavy traffic. We consider different types of L\'evy input, covering
compound Poisson, -stable L\'evy motion (with ), and
Brownian motion. In our analysis we separately deal with L\'evy input processes
with increments that have finite and infinite variance. A distinguishing
feature of this paper is that we do not only consider the usual heavy-traffic
regime, in which the load at one of the nodes goes to unity, but also a regime
in which we simultaneously let the load of both servers tend to one, which, as
it turns out, leads to entirely different heavy-traffic asymptotics. Numerical
experiments indicate that under specific conditions the resulting simultaneous
heavy-traffic approximation significantly outperforms the usual heavy-traffic
approximation
Exact asymptotics for fluid queues fed by multiple heavy-tailed on-off flows
We consider a fluid queue fed by multiple On-Off flows with heavy-tailed
(regularly varying) On periods. Under fairly mild assumptions, we prove that
the workload distribution is asymptotically equivalent to that in a reduced
system. The reduced system consists of a ``dominant'' subset of the flows, with
the original service rate subtracted by the mean rate of the other flows. We
describe how a dominant set may be determined from a simple knapsack
formulation. The dominant set consists of a ``minimally critical'' set of
On-Off flows with regularly varying On periods. In case the dominant set
contains just a single On-Off flow, the exact asymptotics for the reduced
system follow from known results. For the case of several
On-Off flows, we exploit a powerful intuitive argument to obtain the exact
asymptotics. Combined with the reduced-load equivalence, the results for the
reduced system provide a characterization of the tail of the workload
distribution for a wide range of traffic scenarios
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
A fluid queue with a finite buffer and subexponential input
We consider a fluid model similar to that of Kella and Whitt [33], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established in [33] are extended to the finite buffer case. It is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N on-off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue
Conditional limit theorems for regulated fractional Brownian motion
We consider a stationary fluid queue with fractional Brownian motion input.
Conditional on the workload at time zero being greater than a large value ,
we provide the limiting distribution for the amount of time that the workload
process spends above level over the busy cycle straddling the origin, as
. Our results can be interpreted as showing that long delays occur
in large clumps of size of order . The conditional limit result
involves a finer scaling of the queueing process than fluid analysis, thereby
departing from previous related literature.Comment: Published in at http://dx.doi.org/10.1214/09-AAP605 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotic analysis of Lévy-driven tandem queues
We analyze tail asymptotics of a two-node tandem queue with spectrally-positive Lévy input. A first focus lies in the tail probabilities of the type P(Q 1>α x,Q 2>(1−α)x), for α∈(0,1) and x large, and Q i denoting the steady-state 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
Asymptotic analysis of Levy-driven tandem queues
We analyze tail asymptotics of a two-node tandem queue with spectrally-positive L\'evy input. A first focus lies on tail probabilities of the type , for and large, and denoting the steady-state workload in the th 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 to the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed L\'evy input. It is also indicated how the results can be extended to tandem queues with more than two nodes
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