Browsing by Subject "Markov Chain"
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Item Open Access Analyzing large sparse Markov chains of Kronecker products(IEEE, 2009) Dayar, TuğrulKronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. In the Kronecker based approach, the generator matrix underlying the MC is represented using Kronecker products [6] of smaller matrices and is never explicitly generated. The implementation of transient and steady-state solvers rests on this compact Kronecker representation, thanks to the existence of an efficient vector-Kronecker product multiplication algorithm known as the shuffle algorithm [6]. The transient distribution can be computed through uniformization using vector-Kronecker product multiplications. The steady-state distribution also needs to be computed using vector-Kronecker product multiplications, since direct methods based on complete factorizations, such as Gaussian elimination, normally introduce new nonzeros which cannot be accommodated. The two papers [2], [10] provide good overviews of iterative solution techniques for the analysis of MCs based on Kronecker products. Issues related to reachability analysis, vector-Kronecker product multiplication, hierarchical state space generation in Kronecker based matrix representations for large Markov models are surveyed in [5]. Throughout our discussion, we make the assumption that the MC at hand does not have unreachable states, meaning it is irreducible. And we take an algebraic view [7] to discuss recent results related to the analysis of MCs based on Kronecker products independently from modeling formalisms. We provide background material on the Kronecker representation of the generator matrix underlying a CTMC, show that it has a rich structure which is nested and recursive, and introduce a small CTMC whose generator matrix is expressed as a sum of Kronecker products; this CTMC is used as a running example throughout the discussion. We also consider preprocessing of the Kronecker representation so as to expedite numerical analysis. We discuss permuting the nonzero structure of the underlying CTMC symmetrically by reordering, changing the orders of the nested blocks by grouping, and reducing the size of the state space by lumping. The steady-state analysis of CTMCs based on Kronecker products is discussed for block iterative methods, multilevel methods, and preconditioned projection methods, respectively. The results can be extended to DTMCs based on Kronecker products with minor modifications. Areas that need further research are mentioned as they are discussed. Our contribution to this area over the years corresponds to work along iterative methods based on splittings and their block versions [11], associated preconditioners to be used with projection methods [4], near complete decomposability [8], a method based on iterative disaggregation for a class of lumpable MCs [9], a class of multilevel methods [3], and a recent method based on decomposition for weakly interacting subsystems [1]. © 2009 IEEE.Item Open Access Gibbs measures and phase transitions in one-dimensional models(Bilkent University, 2000) Mallak, SaedIn this thesis we study the problem of limit Gibbs measures in one-dimensional models. VVe investigate uniqueness conditions for the limit Gibbs measures for one-dimensional models. VVe construct a one-dimensional model disproving a uniqueness conjecture formulated before for one-dimensional models. It turns out that this conjecture is correct under some natural regularity conditions. VVe also apply the uniqueness theorem to some one-dimensional models.Item Open Access Joint economic design of EWMA control charts for mean and variance(Elsevier, 2008-01-01) Serel, D. A.; Moskowitz, H.Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi's quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance. (C) 2006 Elsevier B.V. All rights reserved.Item Open Access Maximizing profit per unit time in cointegration based pairs trading(Bilkent University, 2014) Tutal, DuyguItem Open Access State-dependent control of a single stage hybrid system with poisson arrivals(2011) Gokbayrak, K.We consider a single-stage hybrid manufacturing system where jobs arrive according to a Poisson process. These jobs undergo a deterministic process which is controllable. We define a stochastic hybrid optimal control problem and decompose it hierarchically to a lower-level and a higher-level problem. The lower-level problem is a deterministic optimal control problem solved by means of calculus of variations. We concentrate on the stochastic discrete-event control problem at the higher level, where the objective is to determine the service times of jobs. Employing a cost structure composed of process costs that are decreasing and strictly convex in service times, and system-time costs that are linear in system times, we show that receding horizon controllers are state-dependent controllers, where state is defined as the system size. In order to improve upon receding horizon controllers, we search for better state-dependent control policies and present two methods to obtain them. These stochastic-approximation-type methods utilize gradient estimators based on Infinitesimal Perturbation Analysis or Imbedded Markov Chain techniques. A numerical example demonstrates the performance improvements due to the proposed methods. © 2011 Springer Science+Business Media, LLC.