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Heavy-traffic extreme-value limits for queues
We consider the maximum waiting time among the first n customers in the GI/G/1 queue. We use strong approximations to prove, under regularity conditions, convergence of the normalized maximum wait to the Gumbel extreme-value distribution when the traffic intensity ρ approaches 1 from below and n approaches infinity at a suitable rate. The normalization depends on the interarrival-time and service-time distributions only through their first two moments, corresponding to the iterated limit in which first ρ approaches 1 and then n approaches infinity. We need n to approach infinity sufficiently fast so that n ( 1 − ρ )2 → ∞. We also need n to approach infinity sufficiently slowly: If the service time has a pth moment for ρ > 2, then it suffices for ( 1 − ρ ) n 1 / p to remain bounded; if the service time has a finite moment generating function, then it suffices to have ( 1 − ρ ) log n → 0. This limit can hold even when the normalized maximum waiting time fails to converge to the Gumbel distribution as n → ∞ for each fixed ρ. Similar limits hold for the queue length process
Lingering Issues in Distributed Scheduling
Recent advances have resulted in queue-based algorithms for medium access
control which operate in a distributed fashion, and yet achieve the optimal
throughput performance of centralized scheduling algorithms. However,
fundamental performance bounds reveal that the "cautious" activation rules
involved in establishing throughput optimality tend to produce extremely large
delays, typically growing exponentially in 1/(1-r), with r the load of the
system, in contrast to the usual linear growth.
Motivated by that issue, we explore to what extent more "aggressive" schemes
can improve the delay performance. Our main finding is that aggressive
activation rules induce a lingering effect, where individual nodes retain
possession of a shared resource for excessive lengths of time even while a
majority of other nodes idle. Using central limit theorem type arguments, we
prove that the idleness induced by the lingering effect may cause the delays to
grow with 1/(1-r) at a quadratic rate. To the best of our knowledge, these are
the first mathematical results illuminating the lingering effect and
quantifying the performance impact.
In addition extensive simulation experiments are conducted to illustrate and
validate the various analytical results
A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering
Routing mechanisms for stochastic networks are often designed to produce
state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the
limiting process to a lower-dimensional subset of its full state space. In a
fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding
mode control that forces the fluid trajectories to a lower-dimensional sliding
manifold. Within deterministic dynamical systems theory, it is well known that
sliding-mode controls can cause the system to chatter back and forth along the
sliding manifold due to delays in activation of the control. For the prelimit
stochastic system, chattering implies fluid-scaled fluctuations that are larger
than typical stochastic fluctuations. In this paper we show that chattering can
occur in the fluid limit of a controlled stochastic network when inappropriate
control parameters are used. The model has two large service pools operating
under the fixed-queue-ratio with activation and release thresholds (FQR-ART)
overload control which we proposed in a recent paper. We now show that, if the
control parameters are not chosen properly, then delays in activating and
releasing the control can cause chattering with large oscillations in the fluid
limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even
when the arrival rates are smaller than the potential total service rate in the
system, a phenomenon referred to as congestion collapse. We show that the fluid
limit can be a bi-stable switching system possessing a unique nontrivial
periodic equilibrium, in addition to a unique stationary point
Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse
We consider a queueing network in which there are constraints on which queues
may be served simultaneously; such networks may be used to model input-queued
switches and wireless networks. The scheduling policy for such a network
specifies which queues to serve at any point in time. We consider a family of
scheduling policies, related to the maximum-weight policy of Tassiulas and
Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936--1948], for single-hop
and multihop networks. We specify a fluid model and show that fluid-scaled
performance processes can be approximated by fluid model solutions. We study
the behavior of fluid model solutions under critical load, and characterize
invariant states as those states which solve a certain network-wide
optimization problem. We use fluid model results to prove multiplicative state
space collapse. A notable feature of our results is that they do not assume
complete resource pooling.Comment: Published in at http://dx.doi.org/10.1214/11-AAP759 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The snowball effect of customer slowdown in critical many-server systems
Customer slowdown describes the phenomenon that a customer's service
requirement increases with experienced delay. In healthcare settings, there is
substantial empirical evidence for slowdown, particularly when a patient's
delay exceeds a certain threshold. For such threshold slowdown situations, we
design and analyze a many-server system that leads to a two-dimensional Markov
process. Analysis of this system leads to insights into the potentially
detrimental effects of slowdown, especially in heavy-traffic conditions. We
quantify the consequences of underprovisioning due to neglecting slowdown,
demonstrate the presence of a subtle bistable system behavior, and discuss in
detail the snowball effect: A delayed customer has an increased service
requirement, causing longer delays for other customers, who in turn due to
slowdown might require longer service times.Comment: 23 pages, 8 figures -- version 3 fixes a typo in an equation. in
Stochastic Models, 201
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
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