108,103 research outputs found
Lagrangian Duality based Algorithms in Online Energy-Efficient Scheduling
We study online scheduling problems in the general energy model of speed scaling with power down. The latter is a combination of the two extensively studied energy models, speed scaling and power down, toward a more realistic one. Due to the limits of the current techniques, only few results have been known in the general energy model in contrast to the large literature of the previous ones.
In the paper, we consider a Lagrangian duality based approach to design and analyze algorithms in the general energy model. We show the applicability of the approach to problems which are unlikely to admit a convex relaxation. Specifically, we consider the problem of minimizing energy with a single machine in which jobs arrive online and have to be processed before their deadlines. We present an alpha^alpha-competitive algorithm (whose the analysis is tight up to a constant factor) where the energy power function is of typical form z^alpha + g for constants alpha > 2 and g non-negative. Besides, we also consider the problem of minimizing the weighted flow-time plus energy. We give an O(alpha/ln(alpha))-competitive algorithm; that matches (up to a constant factor) to the currently best known algorithm for this problem in the restricted model of speed scaling
SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors
We consider the classical problem of minimizing the total weighted flow-time
for unrelated machines in the online \emph{non-clairvoyant} setting. In this
problem, a set of jobs arrive over time to be scheduled on a set of
machines. Each job has processing length , weight , and is
processed at a rate of when scheduled on machine . The online
scheduler knows the values of and upon arrival of the job,
but is not aware of the quantity . We present the {\em first} online
algorithm that is {\em scalable} ((1+\eps)-speed
-competitive for any constant \eps > 0) for the
total weighted flow-time objective. No non-trivial results were known for this
setting, except for the most basic case of identical machines. Our result
resolves a major open problem in online scheduling theory. Moreover, we also
show that no job needs more than a logarithmic number of migrations. We further
extend our result and give a scalable algorithm for the objective of minimizing
total weighted flow-time plus energy cost for the case of unrelated machines
and obtain a scalable algorithm. The key algorithmic idea is to let jobs
migrate selfishly until they converge to an equilibrium. Towards this end, we
define a game where each job's utility which is closely tied to the
instantaneous increase in the objective the job is responsible for, and each
machine declares a policy that assigns priorities to jobs based on when they
migrate to it, and the execution speeds. This has a spirit similar to
coordination mechanisms that attempt to achieve near optimum welfare in the
presence of selfish agents (jobs). To the best our knowledge, this is the first
work that demonstrates the usefulness of ideas from coordination mechanisms and
Nash equilibria for designing and analyzing online algorithms
Scaling laws for convection and jet speeds in the giant planets
Three-dimensional studies of convection in deep spherical shells have been
used to test the hypothesis that the strong jet streams on Jupiter, Saturn,
Uranus, and Neptune result from convection throughout the molecular envelopes.
Due to computational limitations, these simulations must adopt viscosities and
heat fluxes many orders of magnitude larger than the planetary values. Several
numerical investigations have identified trends for how the mean jet speed
varies with heat flux and viscosity, but no previous theories have been
advanced to explain these trends. Here, we show using simple arguments that if
convective release of potential energy pumps the jets and viscosity damps them,
the mean jet speeds split into two regimes. When the convection is weakly
nonlinear, the equilibrated jet speeds should scale approximately with F/nu,
where F is the convective heat flux and nu is the viscosity. When the
convection is strongly nonlinear, the jet speeds are faster and should scale
approximately as (F/nu)^{1/2}. We demonstrate how this regime shift can
naturally result from a shift in the behavior of the jet-pumping efficiency
with heat flux and viscosity. Moreover, the simulations hint at a third regime
where, at sufficiently small viscosities, the jet speed becomes independent of
the viscosity. We show based on mixing-length estimates that if such a regime
exists, mean jet speeds should scale as heat flux to the 1/4 power. Our
scalings provide a good match to the mean jet speeds obtained in previous
Boussinesq and anelastic, three-dimensional simulations of convection within
giant planets over a broad range of parameters. When extrapolated to the real
heat fluxes, these scalings suggest that the mass-weighted jet speeds in the
molecular envelopes of the giant planets are much weaker--by an order of
magnitude or more--than the speeds measured at cloud level.Comment: 23 pages, 10 figures, in press at Icaru
An `Analytic Dynamical Magnetosphere' formalism for X-ray and optical emission from slowly rotating magnetic massive stars
Slowly rotating magnetic massive stars develop "dynamical magnetospheres"
(DM's), characterized by trapping of stellar wind outflow in closed magnetic
loops, shock heating from collision of the upflow from opposite loop
footpoints, and subsequent gravitational infall of radiatively cooled material.
In 2D and 3D magnetohydrodynamic (MHD) simulations the interplay among these
three components is spatially complex and temporally variable, making it
difficult to derive observational signatures and discern their overall scaling
trends.Within a simplified, steady-state analysis based on overall conservation
principles, we present here an "analytic dynamical magnetosphere" (ADM) model
that provides explicit formulae for density, temperature and flow speed in each
of these three components -- wind outflow, hot post-shock gas, and cooled
inflow -- as a function of colatitude and radius within the closed (presumed
dipole) field lines of the magnetosphere. We compare these scalings with
time-averaged results from MHD simulations, and provide initial examples of
application of this ADM model for deriving two key observational diagnostics,
namely hydrogen H-alpha emission line profiles from the cooled infall, and
X-ray emission from the hot post-shock gas. We conclude with a discussion of
key issues and advantages in applying this ADM formalism toward derivation of a
broader set of observational diagnostics and scaling trends for massive stars
with such dynamical magnetospheres.Comment: 15 pages, 11 figures, accepted for MNRA
Instabilities in the Envelopes and Winds of Very Massive Stars
The high luminosity of Very Massive Stars (VMS) means that radiative forces
play an important, dynamical role both in the structure and stability of their
stellar envelope, and in driving strong stellar-wind mass loss. Focusing on the
interplay of radiative flux and opacity, with emphasis on key distinctions
between continuum vs. line opacity, this chapter reviews instabilities in the
envelopes and winds of VMS. Specifically, we discuss how: 1) the iron opacity
bump can induce an extensive inflation of the stellar envelope; 2) the density
dependence of mean opacity leads to strange mode instabilities in the outer
envelope; 3) desaturation of line-opacity by acceleration of near-surface
layers initiates and sustains a line-driven stellar wind outflow; 4) an
associated line-deshadowing instability leads to extensive small-scale
structure in the outer regions of such line-driven winds; 5) a star with
super-Eddington luminosity can develop extensive atmospheric structure from
photon bubble instabilities, or from stagnation of flow that exceeds the
"photon tiring" limit; 6) the associated porosity leads to a reduction in
opacity that can regulate the extreme mass loss of such continuum-driven winds.
Two overall themes are the potential links of such instabilities to Luminous
Blue Variable (LBV) stars, and the potential role of radiation forces in
establishing the upper mass limit of VMS.Comment: 44 pages, 13 figures. Chapter to appear in the book "Very Massive
Stars in the Local Universe", Springer, J.S. Vink, e
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