5,146 research outputs found
Some Asymptotic Results for the Transient Distribution of the Halfin-Whitt Diffusion Process
We consider the Halfin-Whitt diffusion process , which is used, for
example, as an approximation to the -server queue. We use recently
obtained integral representations for the transient density of this
diffusion process, and obtain various asymptotic results for the density. The
asymptotic limit assumes that a drift parameter in the model is large,
and the state variable and the initial condition (with
) are also large. We obtain some alternate representations for
the density, which involve sums and/or contour integrals, and expand these
using a combination of the saddle point method, Laplace method and singularity
analysis. The results give some insight into how steady state is achieved, and
how if the probability mass migrates from to the range
, which is where it concentrates as , in the limit we
consider. We also discuss an alternate approach to the asymptotics, based on
geometrical optics and singular perturbation techniques.Comment: 43 pages and 8 figure
A functional central limit theorem for the M/GI/ queue
In this paper, we present a functional fluid limit theorem and a functional
central limit theorem for a queue with an infinity of servers M/GI/.
The system is represented by a point-measure valued process keeping track of
the remaining processing times of the customers in service. The convergence in
law of a sequence of such processes after rescaling is proved by
compactness-uniqueness methods, and the deterministic fluid limit is the
solution of an integrated equation in the space of
tempered distributions. We then establish the corresponding central limit
theorem, that is, the approximation of the normalized error process by a
-valued diffusion. We apply these results to provide
fluid limits and diffusion approximations for some performance processes.Comment: Published in at http://dx.doi.org/10.1214/08-AAP518 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Influence Maximization: Near-Optimal Time Complexity Meets Practical Efficiency
Given a social network G and a constant k, the influence maximization problem
asks for k nodes in G that (directly and indirectly) influence the largest
number of nodes under a pre-defined diffusion model. This problem finds
important applications in viral marketing, and has been extensively studied in
the literature. Existing algorithms for influence maximization, however, either
trade approximation guarantees for practical efficiency, or vice versa. In
particular, among the algorithms that achieve constant factor approximations
under the prominent independent cascade (IC) model or linear threshold (LT)
model, none can handle a million-node graph without incurring prohibitive
overheads.
This paper presents TIM, an algorithm that aims to bridge the theory and
practice in influence maximization. On the theory side, we show that TIM runs
in O((k+\ell) (n+m) \log n / \epsilon^2) expected time and returns a
(1-1/e-\epsilon)-approximate solution with at least 1 - n^{-\ell} probability.
The time complexity of TIM is near-optimal under the IC model, as it is only a
\log n factor larger than the \Omega(m + n) lower-bound established in previous
work (for fixed k, \ell, and \epsilon). Moreover, TIM supports the triggering
model, which is a general diffusion model that includes both IC and LT as
special cases. On the practice side, TIM incorporates novel heuristics that
significantly improve its empirical efficiency without compromising its
asymptotic performance. We experimentally evaluate TIM with the largest
datasets ever tested in the literature, and show that it outperforms the
state-of-the-art solutions (with approximation guarantees) by up to four orders
of magnitude in terms of running time. In particular, when k = 50, \epsilon =
0.2, and \ell = 1, TIM requires less than one hour on a commodity machine to
process a network with 41.6 million nodes and 1.4 billion edges.Comment: Revised Sections 1, 2.3, and 5 to remove incorrect claims about
reference [3]. Updated experiments accordingly. A shorter version of the
paper will appear in SIGMOD 201
Stein's method for steady-state diffusion approximations
Diffusion approximations have been a popular tool for performance analysis in
queueing theory, with the main reason being tractability and computational
efficiency. This dissertation is concerned with establishing theoretical
guarantees on the performance of steady-state diffusion approximations of
queueing systems. We develop a modular framework based on Stein's method that
allows us to establish error bounds, or convergence rates, for the
approximations. We apply this framework three queueing systems: the Erlang-C,
Erlang-A, and systems.
The former two systems are simpler and allow us to showcase the full
potential of the framework. Namely, we prove that both Wasserstein and
Kolmogorov distances between the stationary distribution of a normalized
customer count process, and that of an appropriately defined diffusion process
decrease at a rate of , where is the offered load. Futhermore,
these error bounds are \emph{universal}, valid in any load condition from
lightly loaded to heavily loaded. For the Erlang-C model, we also show that a
diffusion approximation with state-dependent diffusion coefficient can achieve
a rate of convergence of , which is an order of magnitude faster when
compared to approximations with constant diffusion coefficients.Comment: PhD Thesi
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