15,500 research outputs found
Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems
In real-time systems, in addition to the functional correctness recurrent
tasks must fulfill timing constraints to ensure the correct behavior of the
system. Partitioned scheduling is widely used in real-time systems, i.e., the
tasks are statically assigned onto processors while ensuring that all timing
constraints are met. The decision version of the problem, which is to check
whether the deadline constraints of tasks can be satisfied on a given number of
identical processors, has been known -complete in the strong sense.
Several studies on this problem are based on approximations involving resource
augmentation, i.e., speeding up individual processors. This paper studies
another type of resource augmentation by allocating additional processors, a
topic that has not been explored until recently. We provide polynomial-time
algorithms and analysis, in which the approximation factors are dependent upon
the input instances. Specifically, the factors are related to the maximum ratio
of the period to the relative deadline of a task in the given task set. We also
show that these algorithms unfortunately cannot achieve a constant
approximation factor for general cases. Furthermore, we prove that the problem
does not admit any asymptotic polynomial-time approximation scheme (APTAS)
unless when the task set has constrained deadlines, i.e.,
the relative deadline of a task is no more than the period of the task.Comment: Accepted and to appear in ISAAC 2018, Yi-Lan, Taiwa
Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology
We present a model of (modified) gravity on spacetimes with fractal structure
based on packing of spheres, which are (Euclidean) variants of the Packed Swiss
Cheese Cosmology models. As the action functional for gravity we consider the
spectral action of noncommutative geometry, and we compute its expansion on a
space obtained as an Apollonian packing of 3-dimensional spheres inside a
4-dimensional ball. Using information from the zeta function of the Dirac
operator of the spectral triple, we compute the leading terms in the asymptotic
expansion of the spectral action. They consist of a zeta regularization of a
divergent sum which involves the leading terms of the spectral actions of the
individual spheres in the packing. This accounts for the contribution of the
points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There
is an additional term coming from the residue at the additional point in the
real dimension spectrum that corresponds to the packing constant, as well as a
series of fluctuations coming from log-periodic oscillations, created by the
points of the dimension spectrum that are off the real line. These terms detect
the fractality of the residue set of the sphere packing. We show that the
presence of fractality influences the shape of the slow-roll potential for
inflation, obtained from the spectral action. We also discuss the effect of
truncating the fractal structure at a certain scale related to the energy scale
in the spectral action.Comment: 38 pages LaTe
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