29,759 research outputs found
Asymptotic optimality of maximum pressure policies in stochastic processing networks
We consider a class of stochastic processing networks. Assume that the
networks satisfy a complete resource pooling condition. We prove that each
maximum pressure policy asymptotically minimizes the workload process in a
stochastic processing network in heavy traffic. We also show that, under each
quadratic holding cost structure, there is a maximum pressure policy that
asymptotically minimizes the holding cost. A key to the optimality proofs is to
prove a state space collapse result and a heavy traffic limit theorem for the
network processes under a maximum pressure policy. We extend a framework of
Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams
[Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing
network setting to the stochastic processing network setting to prove the state
space collapse result and the heavy traffic limit theorem. The extension can be
adapted to other studies of stochastic processing networks.Comment: Published in at http://dx.doi.org/10.1214/08-AAP522 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Many-server queues with customer abandonment: numerical analysis of their diffusion models
We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for many-server queues, sometimes
for queues with as few as twenty servers
Positive recurrence of reflecting Brownian motion in three dimensions
Consider a semimartingale reflecting Brownian motion (SRBM) whose state
space is the -dimensional nonnegative orthant. The data for such a process
are a drift vector , a nonsingular covariance matrix
, and a reflection matrix that specifies the boundary
behavior of . We say that is positive recurrent, or stable, if the
expected time to hit an arbitrary open neighborhood of the origin is finite for
every starting state. In dimension , necessary and sufficient conditions
for stability are known, but fundamentally new phenomena arise in higher
dimensions. Building on prior work by El Kharroubi, Ben Tahar and Yaacoubi
[Stochastics Stochastics Rep. 68 (2000) 229--253, Math. Methods Oper. Res. 56
(2002) 243--258], we provide necessary and sufficient conditions for stability
of SRBMs in three dimensions; to verify or refute these conditions is a simple
computational task. As a byproduct, we find that the fluid-based criterion of
Dupuis and Williams [Ann. Probab. 22 (1994) 680--702] is not only sufficient
but also necessary for stability of SRBMs in three dimensions. That is, an SRBM
in three dimensions is positive recurrent if and only if every path of the
associated fluid model is attracted to the origin. The problem of recurrence
classification for SRBMs in four and higher dimensions remains open.Comment: Published in at http://dx.doi.org/10.1214/09-AAP631 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spectrum and Duration of Delayed MeV-GeV Emission of Gamma-Ray Bursts in Cosmic Background Radiation Fields
We generally analyze prompt high-energy emission above a few hundreds of GeV
due to synchrotron self-Compton scattering in internal shocks. However, such
photons cannot be detected because they may collide with cosmic infrared
background photons, leading to electron/positron pair production.
Inverse-Compton scattering of the resulting electron/positron pairs off cosmic
microwave background photons will produce delayed MeV-GeV emission, which may
be much stronger than a typical high-energy afterglow in the external shock
model. We expand on the Cheng & Cheng model by deriving the emission spectrum
and duration in the standard fireball shock model. A typical duration of the
emission is ~ 10^3 seconds, and the time-integrated scattered photon spectrum
is nu^{-(p+6)/4}, where p is the index of the electron energy distribution
behind internal shocks. This is slightly harder than the synchrotron photon
spectrum, nu^{-(p+2)/2}. The lower energy property of the scattered photon
spectrum is dependent on the spectral energy distribution of the cosmic
infrared background radiation. Therefore, future observations on such delayed
MeV-GeV emission and the higher-energy spectral cutoff by the Gamma-Ray Large
Area Space Telescope (GLAST) would provide a probe of the cosmic infrared
background radiation.Comment: 5 pages, accepted for publication in Ap
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