15,692 research outputs found
Phi-entropy inequalities for diffusion semigroups
We obtain and study new -entropy inequalities for diffusion semigroups,
with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From
this study we derive the asymptotic behaviour of a large class of linear
Fokker-Plank type equations under simple conditions, widely extending previous
results. Nonlinear diffusion equations are also studied by means of these
inequalities. The criterion of D. Bakry and M. Emery appears as a
main tool in the analysis, in local or integral forms.Comment: 31 page
Universality of Load Balancing Schemes on Diffusion Scale
We consider a system of parallel queues with identical exponential
service rates and a single dispatcher where tasks arrive as a Poisson process.
When a task arrives, the dispatcher always assigns it to an idle server, if
there is any, and to a server with the shortest queue among randomly
selected servers otherwise . This load balancing scheme
subsumes the so-called Join-the-Idle Queue (JIQ) policy and the
celebrated Join-the-Shortest Queue (JSQ) policy as two crucial
special cases. We develop a stochastic coupling construction to obtain the
diffusion limit of the queue process in the Halfin-Whitt heavy-traffic regime,
and establish that it does not depend on the value of , implying that
assigning tasks to idle servers is sufficient for diffusion level optimality
Learning Algorithms for Minimizing Queue Length Regret
We consider a system consisting of a single transmitter/receiver pair and
channels over which they may communicate. Packets randomly arrive to the
transmitter's queue and wait to be successfully sent to the receiver. The
transmitter may attempt a frame transmission on one channel at a time, where
each frame includes a packet if one is in the queue. For each channel, an
attempted transmission is successful with an unknown probability. The
transmitter's objective is to quickly identify the best channel to minimize the
number of packets in the queue over time slots. To analyze system
performance, we introduce queue length regret, which is the expected difference
between the total queue length of a learning policy and a controller that knows
the rates, a priori. One approach to designing a transmission policy would be
to apply algorithms from the literature that solve the closely-related
stochastic multi-armed bandit problem. These policies would focus on maximizing
the number of successful frame transmissions over time. However, we show that
these methods have queue length regret. On the other hand, we
show that there exists a set of queue-length based policies that can obtain
order optimal queue length regret. We use our theoretical analysis to
devise heuristic methods that are shown to perform well in simulation.Comment: 28 Pages, 11 figure
Heavy traffic limit for a processor sharing queue with soft deadlines
This paper considers a GI/GI/1 processor sharing queue in which jobs have
soft deadlines. At each point in time, the collection of residual service times
and deadlines is modeled using a random counting measure on the right
half-plane. The limit of this measure valued process is obtained under
diffusion scaling and heavy traffic conditions and is characterized as a
deterministic function of the limiting queue length process. As special cases,
one obtains diffusion approximations for the lead time profile and the profile
of times in queue. One also obtains a snapshot principle for sojourn times.Comment: Published at http://dx.doi.org/10.1214/105051607000000014 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic
A result of Ward and Glynn (2005) asserts that the sequence of scaled offered
waiting time processes of the queue converges weakly to a
reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the
traffic intensity approaches one. As a consequence, the stationary distribution
of a ROU process, which is a truncated normal, should approximate the scaled
stationary distribution of the offered waiting time in a queue;
however, no such result has been proved. We prove the aforementioned
convergence, and the convergence of the moments, in heavy traffic, thus
resolving a question left open in Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (1961) showing that an exponential
distribution approximates the scaled stationary offered waiting time
distribution in a queue in heavy traffic, our result confirms that
the addition of customer abandonment has a non-trivial effect on the queue
stationary behavior.Comment: 29 page
Queuing with future information
We study an admissions control problem, where a queue with service rate
receives incoming jobs at rate , and the decision maker is
allowed to redirect away jobs up to a rate of , with the objective of
minimizing the time-average queue length. We show that the amount of
information about the future has a significant impact on system performance, in
the heavy-traffic regime. When the future is unknown, the optimal average queue
length diverges at rate , as . In sharp contrast, when all future arrival and service times are revealed
beforehand, the optimal average queue length converges to a finite constant,
, as . We further show that the finite limit of
can be achieved using only a finite lookahead window starting from the current
time frame, whose length scales as , as
. This leads to the conjecture of an interesting duality between
queuing delay and the amount of information about the future.Comment: Published in at http://dx.doi.org/10.1214/13-AAP973 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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