287 research outputs found
Laws of Little in an open queueing network
The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented
The ODE Method for Asymptotic Statistics in Stochastic Approximation and Reinforcement Learning
The paper concerns convergence and asymptotic statistics for stochastic
approximation driven by Markovian noise: in which each
, is a Markov chain on a general state space
X with stationary distribution , and . In
addition to standard Lipschitz bounds on , and conditions on the vanishing
step-size sequence , it is assumed that the associated ODE is
globally asymptotically stable with stationary point denoted , where
with . Moreover, the
ODE@ defined with respect to the vector field, is asymptotically stable. The main contributions are
summarized as follows:
(i) The sequence is convergent if is geometrically ergodic,
and subject to compatible bounds on .
The remaining results are established under a stronger assumption on the
Markov chain: A slightly weaker version of the Donsker-Varadhan Lyapunov drift
condition known as (DV3).
(ii) A Lyapunov function is constructed for the joint process
that implies convergence of in .
(iii) A functional CLT is established, as well as the usual one-dimensional
CLT for the normalized error .
Moment bounds combined with the CLT imply convergence of the normalized
covariance, where
is the asymptotic covariance appearing in the CLT.
(iv) An example is provided where the Markov chain is geometrically
ergodic but it does not satisfy (DV3). While the algorithm is convergent, the
second moment is unbounded
Applications of robust optimization to queueing and inventory systems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 105-111).This thesis investigates the application of robust optimization in the performance analysis of queueing and inventory systems. In the first part of the thesis, we propose a new approach for performance analysis of queueing systems based on robust optimization. We first derive explicit upper bounds on performance for tandem single class, multiclass single server, and single class multi-server queueing systems by solving appropriate robust optimization problems. We then show that these bounds derived by solving deterministic optimization problems translate to upper bounds on the expected steady-state performance for a variety of widely used performance measures such as waiting times and queue lengths. Additionally, these explicit bounds agree qualitatively with known results. In the second part of the thesis, we propose methods to compute (s,S) policies in supply chain networks using robust and stochastic optimization and compare their performance. Our algorithms handle general uncertainty sets, arbitrary network topologies, and flexible cost functions including the presence of fixed costs. The algorithms exhibit empirically practical running times. We contrast the performance of robust and stochastic (s,S) policies in a numerical study, and we find that the robust policy is comparable to the average performance of the stochastic policy, but has a considerably lower standard deviation across a variety of networks and realized demand distributions. Additionally, we identify regimes when the robust policy exhibits particular strengths even in average performance and tail behavior as compared with the stochastic policy.by Alexander Anatolyevich Rikun.Ph.D
The application of non-linear dynamics to teletraffic modelling.
PhDAbstract not availableEngineering and Physical Science Research Council (EPSRC) and NORTE
Diffusion Asymptotics for Sequential Experiments
We propose a new diffusion-asymptotic analysis for sequentially randomized
experiments, including those that arise in solving multi-armed bandit problems.
In an experiment with time steps, we let the mean reward gaps between
actions scale to the order so as to preserve the difficulty of the
learning task as grows. In this regime, we show that the behavior of a
class of sequentially randomized Markov experiments converges to a diffusion
limit, given as the solution of a stochastic differential equation. The
diffusion limit thus enables us to derive refined, instance-specific
characterization of the stochastic dynamics of adaptive experiments. As an
application of this framework, we use the diffusion limit to obtain several new
insights on the regret and belief evolution of Thompson sampling. We show that
a version of Thompson sampling with an asymptotically uninformative prior
variance achieves nearly-optimal instance-specific regret scaling when the
reward gaps are relatively large. We also demonstrate that, in this regime, the
posterior beliefs underlying Thompson sampling are highly unstable over time
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