86 research outputs found
Convergence to Equilibrium States for Fluid Models of Many-server Queues with Abandonment
Fluid models have become an important tool for the study of many-server
queues with general service and patience time distributions. The equilibrium
state of a fluid model has been revealed by Whitt (2006) and shown to yield
reasonable approximations to the steady state of the original stochastic
systems. However, it remains an open question whether the solution to a fluid
model converges to the equilibrium state and under what condition. We show in
this paper that the convergence holds under a mild condition. Our method builds
on the framework of measure-valued processes developed in Zhang (2013), which
keeps track of the remaining patience and service times
Separation of timescales in a two-layered network
We investigate a computer network consisting of two layers occurring in, for
example, application servers. The first layer incorporates the arrival of jobs
at a network of multi-server nodes, which we model as a many-server Jackson
network. At the second layer, active servers at these nodes act now as
customers who are served by a common CPU. Our main result shows a separation of
time scales in heavy traffic: the main source of randomness occurs at the
(aggregate) CPU layer; the interactions between different types of nodes at the
other layer is shown to converge to a fixed point at a faster time scale; this
also yields a state-space collapse property. Apart from these fundamental
insights, we also obtain an explicit approximation for the joint law of the
number of jobs in the system, which is provably accurate for heavily loaded
systems and performs numerically well for moderately loaded systems. The
obtained results for the model under consideration can be applied to
thread-pool dimensioning in application servers, while the technique seems
applicable to other layered systems too.Comment: 8 pages, 2 figures, 1 table, ITC 24 (2012
Dual Instrumental Method for Confounded Kernelized Bandits
The contextual bandit problem is a theoretically justified framework with
wide applications in various fields. While the previous study on this problem
usually requires independence between noise and contexts, our work considers a
more sensible setting where the noise becomes a latent confounder that affects
both contexts and rewards. Such a confounded setting is more realistic and
could expand to a broader range of applications. However, the unresolved
confounder will cause a bias in reward function estimation and thus lead to a
large regret. To deal with the challenges brought by the confounder, we apply
the dual instrumental variable regression, which can correctly identify the
true reward function. We prove the convergence rate of this method is
near-optimal in two types of widely used reproducing kernel Hilbert spaces.
Therefore, we can design computationally efficient and regret-optimal
algorithms based on the theoretical guarantees for confounded bandit problems.
The numerical results illustrate the efficacy of our proposed algorithms in the
confounded bandit setting
Provably Efficient Learning in Partially Observable Contextual Bandit
In this paper, we investigate transfer learning in partially observable
contextual bandits, where agents have limited knowledge from other agents and
partial information about hidden confounders. We first convert the problem to
identifying or partially identifying causal effects between actions and rewards
through optimization problems. To solve these optimization problems, we
discretize the original functional constraints of unknown distributions into
linear constraints, and sample compatible causal models via sequentially
solving linear programmings to obtain causal bounds with the consideration of
estimation error. Our sampling algorithms provide desirable convergence results
for suitable sampling distributions. We then show how causal bounds can be
applied to improving classical bandit algorithms and affect the regrets with
respect to the size of action sets and function spaces. Notably, in the task
with function approximation which allows us to handle general context
distributions, our method improves the order dependence on function space size
compared with previous literatures. We formally prove that our causally
enhanced algorithms outperform classical bandit algorithms and achieve orders
of magnitude faster convergence rates. Finally, we perform simulations that
demonstrate the efficiency of our strategy compared to the current
state-of-the-art methods. This research has the potential to enhance the
performance of contextual bandit agents in real-world applications where data
is scarce and costly to obtain.Comment: 47 page
Stochastic Graph Bandit Learning with Side-Observations
In this paper, we investigate the stochastic contextual bandit with general
function space and graph feedback. We propose an algorithm that addresses this
problem by adapting to both the underlying graph structures and reward gaps. To
the best of our knowledge, our algorithm is the first to provide a
gap-dependent upper bound in this stochastic setting, bridging the research gap
left by the work in [35]. In comparison to [31,33,35], our method offers
improved regret upper bounds and does not require knowledge of graphical
quantities. We conduct numerical experiments to demonstrate the computational
efficiency and effectiveness of our approach in terms of regret upper bounds.
These findings highlight the significance of our algorithm in advancing the
field of stochastic contextual bandits with graph feedback, opening up avenues
for practical applications in various domains.Comment: arXiv admin note: text overlap with arXiv:2010.03104 by other author
Staffing under Taylor's Law: A Unifying Framework for Bridging Square-root and Linear Safety Rules
Staffing rules serve as an essential management tool in service industries to
attain target service levels. Traditionally, the square-root safety rule, based
on the Poisson arrival assumption, has been commonly used. However, empirical
findings suggest that arrival processes often exhibit an ``over-dispersion''
phenomenon, in which the variance of the arrival exceeds the mean. In this
paper, we develop a new doubly stochastic Poisson process model to capture a
significant dispersion scaling law, known as Taylor's law, showing that the
variance is a power function of the mean. We further examine how
over-dispersion affects staffing, providing a closed-form staffing formula to
ensure a desired service level. Interestingly, the additional staffing level
beyond the nominal load is a power function of the nominal load, with the power
exponent lying between (the square-root safety rule) and (the linear
safety rule), depending on the degree of over-dispersion. Simulation studies
and a large-scale call center case study indicate that our staffing rule
outperforms classical alternatives.Comment: 55 page
Study on Spatial Distribution of Soil Available Microelement in Qujing Tobacco Farming Area, China
AbstractDescriptive analysis characteristics and spatial variation characteristics of soil available microelements were studied based on SPSS and GIS Soil available microelements spatial distribution maps were created with ordinary Kriging method. The results indicate that, 7 available microelements in tobacco soil obey lognormal distribution, all the available microelements were intermediate variability; Anisotropic structure of available microelements of tobacco soil varies evidently, spatial variability of available B was mainly caused by random factors, and others’ spatial variability were caused by structural factors and random factors; Spatial distribution maps show that, available B was widely deficient in tobacco soil of Qujing farming area, ‘lower level’ and ‘low level’ taken 7.74% and 68.20%, respectively available Zn distribution was moderate, only 1.32% of the area lack of Zn, available Cu, available Fe and available Mn were extremely high in the whole extension, available Mo was deficient in part of the region with 28.38%, water soluble Cl was higher than critical value(30mgkg−1)in the most of Qujing farming area, which taken 38.86%
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