Bibliografías temáticas / Markoc chain

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Literatura académica sobre el tema "Markoc chain"

Autor: Grafiati
Publicado: 25 de mayo de 2024
Última modificación: 19 de julio de 2025

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Artículos de revistas sobre el tema "Markoc chain"

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Guyo, X., and C. Hardouin†. "Markow chain markov field dynamics:models and statistics." Statistics 35, no. 4 (2001): 593–627. http://dx.doi.org/10.1080/02331880108802756.

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Valenzuela, Mississippi. "Markov chains and applications." Selecciones Matemáticas 9, no. 01 (2022): 53–78. http://dx.doi.org/10.17268/sel.mat.2022.01.05.

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This work has three important purposes: first it is the study of Markov Chains, the second is to show that Markov chains have different applications and finally it is to model a process of this behaves. Throughout this work we will describe what a Markov chain is, what these processes are for and how these chains are classified. We will describe a Markov Chain, that is, analyze what are the primary elements that make up a Markov chain, among others.
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Barker, Richard J., and Matthew R. Schofield. "Putting Markov Chains Back into Markov Chain Monte Carlo." Journal of Applied Mathematics and Decision Sciences 2007 (October 30, 2007): 1–13. http://dx.doi.org/10.1155/2007/98086.

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Markov chain theory plays an important role in statistical inference both in the formulation of models for data and in the construction of efficient algorithms for inference. The use of Markov chains in modeling data has a long history, however the use of Markov chain theory in developing algorithms for statistical inference has only become popular recently. Using mark-recapture models as an illustration, we show how Markov chains can be used for developing demographic models and also in developing efficient algorithms for inference. We anticipate that a major area of future research involving
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Marcus, Brian, and Selim Tuncel. "The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains." Ergodic Theory and Dynamical Systems 11, no. 1 (1991): 129–80. http://dx.doi.org/10.1017/s0143385700006052.

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AbstractWe study Markov chains via invariants constructed from periodic orbits. Canonical extensions, based on these invariants, are used to establish a constraint on the degree of finite-to-one block homomorphisms from one Markov chain to another. We construct a polytope from the normalized weights of periodic orbits. Using this polytope, we find canonically-defined induced Markov chains inside the original Markov chain. Each of the invariants associated with these Markov chains gives rise to a scaffold of invariants for the original Markov chain. This is used to obtain counterexamples to the
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Xiang, Xuyan, Xiao Zhang, and Xiaoyun Mo. "Statistical Identification of Markov Chain on Trees." Mathematical Problems in Engineering 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/2036248.

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The theoretical study of continuous-time homogeneous Markov chains is usually based on a natural assumption of a known transition rate matrix (TRM). However, the TRM of a Markov chain in realistic systems might be unknown and might even need to be identified by partially observable data. Thus, an issue on how to identify the TRM of the underlying Markov chain by partially observable information is derived from the great significance in applications. That is what we call the statistical identification of Markov chain. The Markov chain inversion approach has been derived for basic Markov chains
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APOSTOLOV, S. S., Z. A. MAYZELIS, O. V. USATENKO, and V. A. YAMPOL'SKII. "HIGH-ORDER CORRELATION FUNCTIONS OF BINARY MULTI-STEP MARKOV CHAINS." International Journal of Modern Physics B 22, no. 22 (2008): 3841–53. http://dx.doi.org/10.1142/s0217979208048589.

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Two approaches to studying the correlation functions of the binary Markov sequences are considered. The first of them is based on the study of probability of occurring different "words" in the sequence. The other one uses recurrence relations for correlation functions. These methods are applied for two important particular classes of the Markov chains. These classes include the Markov chains with permutative conditional probability functions and the additive Markov chains with the small memory functions. The exciting property of the self-similarity (discovered in Phys. Rev. Lett.90, 110601 (20
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Masuyama, Hiroyuki. "Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions." Advances in Applied Probability 47, no. 1 (2015): 83–105. http://dx.doi.org/10.1239/aap/1427814582.

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In this paper we study the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases, where an original Markov chain itself is not necessarily block monotone but is blockwise dominated by a block-monotone Markov chain. Finally, we discuss the application of our results to GI/G/1-type Markov chains.
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Masuyama, Hiroyuki. "Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions." Advances in Applied Probability 47, no. 01 (2015): 83–105. http://dx.doi.org/10.1017/s0001867800007710.

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In this paper we study the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases, where an original Markov chain itself is not necessarily block monotone but is blockwise dominated by a block-monotone Markov chain. Finally, we discuss the application of our results to GI/G/1-type Markov chains.
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Ledoux, James. "A geometric invariant in weak lumpability of finite Markov chains." Journal of Applied Probability 34, no. 4 (1997): 847–58. http://dx.doi.org/10.2307/3215001.

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We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.
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Ledoux, James. "A geometric invariant in weak lumpability of finite Markov chains." Journal of Applied Probability 34, no. 04 (1997): 847–58. http://dx.doi.org/10.1017/s0021900200101561.

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We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.
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Tesis sobre el tema "Markoc chain"

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Chen, Yi ting. "Random generation of executions of concurrent systems." Electronic Thesis or Diss., Sorbonne université, 2022. https://accesdistant.sorbonne-universite.fr/login?url=https://theses-intra.sorbonne-universite.fr/2022SORUS071.pdf.

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La concurrence joue un rôle important dans les systèmes et la programmation modernes. Il révèle le phénomène selon lequel plusieurs calculs s'exécutent simultanément. Ces exécutions entrelacées entraînent le "problème d'explosion d'états". Dans cette thèse, nous visons à construire un cadre probabiliste sur les exécutions de systèmes concurrents à des fins de génération aléatoire. La mesure uniforme des exécutions s'inspire des monoïdes de traces définis sur des traces infinies. La théorie des traces a une solide base combinatoire autour du polynôme de Möbius. L'irréductibilité des monoïdes de
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Yildirak, Sahap Kasirga. "The Identificaton Of A Bivariate Markov Chain Market Model." Phd thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/1257898/index.pdf.

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This work is an extension of the classical Cox-Ross-Rubinstein discrete time market model in which only one risky asset is considered. We introduce another risky asset into the model. Moreover, the random structure of the asset price sequence is generated by bivariate finite state Markov chain. Then, the interest rate varies over time as it is the function of generating sequences. We discuss how the model can be adapted to the real data. Finally, we illustrate sample implementations to give a better idea about the use of the model.
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Lindahl, John, and Douglas Persson. "Data-driven test case design of automatic test cases using Markov chains and a Markov chain Monte Carlo method." Thesis, Malmö universitet, Fakulteten för teknik och samhälle (TS), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mau:diva-43498.

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Large and complex software that is frequently changed leads to testing challenges. It is well established that the later a fault is detected in software development, the more it costs to fix. This thesis aims to research and develop a method of generating relevant and non-redundant test cases for a regression test suite, to catch bugs as early in the development process as possible. The research was executed at Axis Communications AB with their products and systems in mind. The approach utilizes user data to dynamically generate a Markov chain model and with a Markov chain Monte Carlo method,
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Bakra, Eleni. "Aspects of population Markov chain Monte Carlo and reversible jump Markov chain Monte Carlo." Thesis, University of Glasgow, 2009. http://theses.gla.ac.uk/1247/.

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Holenstein, Roman. "Particle Markov chain Monte Carlo." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/7319.

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Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods have emerged as the two main tools to sample from high-dimensional probability distributions. Although asymptotic convergence of MCMC algorithms is ensured under weak assumptions, the performance of these latters is unreliable when the proposal distributions used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. In this thesis we propose a new Monte Carlo framework in which we build efficient high-dimensional proposal distributions using SMC methods. This allows us to
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Byrd, Jonathan Michael Robert. "Parallel Markov Chain Monte Carlo." Thesis, University of Warwick, 2010. http://wrap.warwick.ac.uk/3634/.

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The increasing availability of multi-core and multi-processor architectures provides new opportunities for improving the performance of many computer simulations. Markov Chain Monte Carlo (MCMC) simulations are widely used for approximate counting problems, Bayesian inference and as a means for estimating very highdimensional integrals. As such MCMC has found a wide variety of applications in fields including computational biology and physics,financial econometrics, machine learning and image processing. This thesis presents a number of new method for reducing the runtime of Markov Chain Monte
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Frühwirth-Schnatter, Sylvia, Stefan Pittner, Andrea Weber, and Rudolf Winter-Ebmer. "Analysing plant closure effects using time-varying mixture-of-experts Markov chain clustering." Institute of Mathematical Statistics, 2018. http://dx.doi.org/10.1214/17-AOAS1132.

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In this paper we study data on discrete labor market transitions from Austria. In particular, we follow the careers of workers who experience a job displacement due to plant closure and observe - over a period of 40 quarters - whether these workers manage to return to a steady career path. To analyse these discrete-valued panel data, we apply a new method of Bayesian Markov chain clustering analysis based on inhomogeneous first order Markov transition processes with time-varying transition matrices. In addition, a mixtureof- experts approach allows us to model the probability of belong
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Michalaros, Anastasios. "Engagement of Individual Performance in the Application of Markov Chains Models in Hellenic Navys Chain of Command." Thesis, Monterey, California. Naval Postgraduate School, 2012. http://hdl.handle.net/10945/6835.

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The recent financial-crisis that Greece (Hellas) suffers has restricted and reduced the budgets of many organizations. Among those, the Hellenic ministry of defense has begun examining ways to reduce costs while maintaining operational readiness. Retirement legislation is the first area the Hellenic ministry of defense is examining. Variables such as years of service required to receive a pension, years of service by pay grade, and the skills officers should possess for promotion were examined and recorded in ordinances (directives) issued by the president of the Hellenic Republic. However, th
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Planting, Ralf. "The use of the DWV3 classification system in manufacturing companies for evaluating a market-specific supply chain strategy - A case study at Atlas Copco Industrial Technique." Thesis, KTH, Industriell ekonomi och organisation (Inst.), 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-103925.

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The research topic of this study is market-specific supply chain strategy, and the research problem is defined as, how manufacturing companies can use the DWV3 classification system to evaluate the opportunity for a market-specific supply chain strategy. What has been written about the DWV3 classification system is somewhat general in its nature and the practitioner is left without detailed instructions on how to proceed with the analytical analysis. Key elements of the DWV3 classification system that is not explicitly described in the literature is (1) how to measure each of the classificatio
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Martin, Russell Andrew. "Paths, sampling, and markov chain decomposition." Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/29383.

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Libros sobre el tema "Markoc chain"

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Gagniuc, Paul A. Markov Chains. John Wiley & Sons, Inc., 2017. http://dx.doi.org/10.1002/9781119387596.

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Brémaud, Pierre. Markov Chains. Springer New York, 1999. http://dx.doi.org/10.1007/978-1-4757-3124-8.

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Sericola, Bruno. Markov Chains. John Wiley & Sons, Inc., 2013. http://dx.doi.org/10.1002/9781118731543.

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Douc, Randal, Eric Moulines, Pierre Priouret, and Philippe Soulier. Markov Chains. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-97704-1.

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Graham, Carl. Markov Chains. John Wiley & Sons, Ltd, 2014. http://dx.doi.org/10.1002/9781118881866.

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Brémaud, Pierre. Markov Chains. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-45982-6.

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Ching, Wai-Ki, Ximin Huang, Michael K. Ng, and Tak-Kuen Siu. Markov Chains. Springer US, 2013. http://dx.doi.org/10.1007/978-1-4614-6312-2.

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Hermanns, Holger. Interactive Markov Chains. Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/3-540-45804-2.

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Privault, Nicolas. Understanding Markov Chains. Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-0659-4.

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Hartfiel, Darald J. Markov Set-Chains. Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0094586.

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Capítulos de libros sobre el tema "Markoc chain"

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Camacho Olmedo, M. T., and J. F. Mas. "Markov Chain." In Geomatic Approaches for Modeling Land Change Scenarios. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-60801-3_25.

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Fürnkranz, Johannes, Philip K. Chan, Susan Craw, et al. "Markov Chain." In Encyclopedia of Machine Learning. Springer US, 2011. http://dx.doi.org/10.1007/978-0-387-30164-8_510.

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Hu, Fuyan. "Markov Chain." In Encyclopedia of Systems Biology. Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_443.

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Weik, Martin H. "Markov chain." In Computer Science and Communications Dictionary. Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_11086.

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Bandyopadhyay, Susmita. "Markov Chain." In Decision Support System. CRC Press, 2023. http://dx.doi.org/10.1201/9781003307655-8.

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Zhang, Hao. "Markov Chain." In Models and Methods for Management Science. Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-1614-4_11.

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Rodrigues, Eliane Regina, and Jorge Alberto Achcar. "Markov Chain Models." In Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-4645-3_2.

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He, Xindong. "Markov Chain Analysis." In Geographic Data Analysis Using R. Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-97-4022-2_9.

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Møller, Jan Kloppenborg, Marcel Schweiker, Rune Korsholm Andersen, et al. "Markov chain models." In Statistical Modelling of Occupant Behaviour. Chapman and Hall/CRC, 2023. http://dx.doi.org/10.1201/9781003340812-9.

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Stewart, William J. "Markov Chain Equations." In Encyclopedia of Operations Research and Management Science. Springer US, 2013. http://dx.doi.org/10.1007/978-1-4419-1153-7_578.

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Actas de conferencias sobre el tema "Markoc chain"

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Brenna, Andrea, Luciano Lazzari, and Marco Ormellese. "Probabilistic Model Based on Markov Chain for the Assessment of Localized Corrosion of Stainless Steels." In CORROSION 2014. NACE International, 2014. https://doi.org/10.5006/c2014-4091.

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Abstract Pitting, crevice and stress corrosion cracking are the most damaging corrosion forms of stainless steels in industrial applications. Generally, pitting and crevice susceptibility depends on a variety of factors related to the metal (chemical composition, differences in the metallurgical structure, inclusions), the environment (chloride content, pH, temperature, differential aeration) and the geometry of the system. Due to their unpredictable occurrence, localized corrosion events cannot be explained without using a proper statistical method. In this work a probabilistic approach based
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Aminof, Benjamin, Linus Cooper, Sasha Rubin, Moshe Y. Vardi, and Florian Zuleger. "Probabilistic Synthesis and Verification for LTL on Finite Traces." In 21st International Conference on Principles of Knowledge Representation and Reasoning {KR-2023}. International Joint Conferences on Artificial Intelligence Organization, 2024. http://dx.doi.org/10.24963/kr.2024/3.

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We study synthesis and verification of probabilistic models and specifications over finite traces. Probabilistic models are formalized in this work as Markov Chains and Markov Decisions Processes. Motivated by the recent attention given to, and importance of, finite-trace specifications in AI, we use linear-temporal logic on finite traces as a specification formalism for properties of traces with finite but unbounded time horizons. Since there is no bound on the time horizon, our Markov chains generate infinite traces, and we consider two possible semantics: “existential (resp. universal) pref
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Luo, Jian-qiang, and Yan-ping Zhao. "Research on the Supply Chain Product Market Forecasting Based on Markov Chain." In 2010 International Conference on E-Product E-Service and E-Entertainment (ICEEE 2010). IEEE, 2010. http://dx.doi.org/10.1109/iceee.2010.5660723.

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Morris, Lloyd, Juan Luis Arias, Alonso Toro, Andrés Martínez, and Homero Murzi. "Information management for the projection of productive capacities articulated to export scenarios." In Human Interaction and Emerging Technologies (IHIET-AI 2022) Artificial Intelligence and Future Applications. AHFE International, 2022. http://dx.doi.org/10.54941/ahfe100924.

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This article proposes the management of information through the study and analysis of Colombian export scenarios that serve as a reference to articulate strategic nodes, for which an epistemological approach is proposed from a perspective of triangulation of techniques that allows a comprehensive approach in relation to the study of the external market of Colombia. This process is achieved through a comparative and complementary analysis of three techniques: Markov chains, forecasting techniques and mathematical functions. The behavior of sales is transformed into estimates of future behavior
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Zhou, Qing-xin. "The Application of Markov Chain in China's Market Economy." In 2013 International Conference on Advances in Social Science, Humanities, and Management. Atlantis Press, 2013. http://dx.doi.org/10.2991/asshm-13.2013.20.

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Cogill, Randy, and Erik Vargo. "The Poisson equation for reversible Markov chains: Analysis and application to Markov chain samplers." In 2012 IEEE 51st Annual Conference on Decision and Control (CDC). IEEE, 2012. http://dx.doi.org/10.1109/cdc.2012.6425978.

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Arkov, V. Yu, G. G. Kulikov, and T. V. Breikin. "Application of Markov Chains to Identification of Turbine Engine Dynamic Models." In ASME Turbo Expo 2002: Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30038.

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The paper addresses the problem of dynamic modelling of gas turbines for condition monitoring purposes. Identification of dynamic models is performed using a novel Markov chain technique. This includes identifiability analysis and model estimation. When identifying the model, experimental data should be sufficiently informative for identification. So far, identifiability analysis is weak formed and workable solutions are still to be developed. A possible technique is proposed based on non-parametric models in the form of controllable Markov chains. The second step in systems identification is
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Talvitie, Topi, Teppo Niinimäki, and Mikko Koivisto. "The Mixing of Markov Chains on Linear Extensions in Practice." In Twenty-Sixth International Joint Conference on Artificial Intelligence. International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/74.

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We investigate almost uniform sampling from the set of linear extensions of a given partial order. The most efficient schemes stem from Markov chains whose mixing time bounds are polynomial, yet impractically large. We show that, on instances one encounters in practice, the actual mixing times can be much smaller than the worst-case bounds, and particularly so for a novel Markov chain we put forward. We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders. Our empirical results suggest tha
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Kozma, Tímea, Katalin Faragóné Lepp, and Rita Szabó-Geletóczki. "Relationship Factors Between Supply Chain Actors." In 7th FEB International Scientific Conference. University of Maribor, University Press, 2023. http://dx.doi.org/10.18690/um.epf.3.2023.26.

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The competition between companies is now in fact competition between value chains and supply chains. Nowadays, companies that participate in an efficiently managed supply chain based on partnerships are leading the increasingly competitive market. The companies must have recognising that the development of close strategic relationships and network cooperation between chain members is an essential part of market competition. The problem is that few company realise this yet. In the framework of the Budapest Business School Supply Chain and Value Chain Research Group, we are trying to explore the
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Zhang, Yu, and Mitchell Bucklew. "Max Markov Chain." In Thirty-Second International Joint Conference on Artificial Intelligence {IJCAI-23}. International Joint Conferences on Artificial Intelligence Organization, 2023. http://dx.doi.org/10.24963/ijcai.2023/639.

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In this paper, we introduce Max Markov Chain (MMC), a novel model for sequential data with sparse correlations among the state variables. It may also be viewed as a special class of approximate models for High-order Markov Chains (HMCs). MMC is desirable for domains where the sparse correlations are long-term and vary in their temporal stretches. Although generally intractable, parameter optimization for MMC can be solved analytically. However, based on this result, we derive an approximate solution that is highly efficient empirically. When compared with HMC and approximate HMC models, MMC co
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Informes sobre el tema "Markoc chain"

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Zhao, Yao. Supply Chain Analytics and Stock Market Intelligence. Instats Inc., 2025. https://doi.org/10.61700/8cx0l3vsfg5ea1342.

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This seminar delves into the intersection of supply chain management and stock market intelligence, highlighting how data analytics can reveal the influence of supply chains on financial and stock market performance. Participants will acquire advanced techniques, data and tools to analyze these dynamics, enhancing their research capabilities and understanding the strategic implications for businesses and investors.
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Gelfand, Alan E., and Sujit K. Sahu. On Markov Chain Monte Carlo Acceleration. Defense Technical Information Center, 1994. http://dx.doi.org/10.21236/ada279393.

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Krebs, William B. Markov Chain Simulations of Binary Matrices. Defense Technical Information Center, 1992. http://dx.doi.org/10.21236/ada249265.

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Banomyong, Ruth. Supply Chain Dynamics in Asia. Inter-American Development Bank, 2009. http://dx.doi.org/10.18235/0011303.

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Supply chain management in Asia is a relatively novel topic but a key challenge for all Asian based manufacturers and traders when trying to integrate into the "global market". The purpose of the paper is to describe key supply chain issues faced in Asia. These issues are related to supply chain security that forces Asian firms to comply with numerous requirements as well as the importance of a properly managed supply chain in enhancing firms' competitiveness. The critical role played by Asian based logistics providers in facilitating supply chain integration is explored. Logistics providers m
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Kocherlakota, Narayana. Sluggish Inflation Expectations: A Markov Chain Analysis. National Bureau of Economic Research, 2016. http://dx.doi.org/10.3386/w22009.

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Calvin, James M. Markov Chain Moment Formulas for Regenerative Simulation. Defense Technical Information Center, 1989. http://dx.doi.org/10.21236/ada210684.

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Ketterer, Juan Antonio, and Agustina Calatayud. Integrated Value Chain Risk Management. Inter-American Development Bank, 2016. http://dx.doi.org/10.18235/0010631.

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Resumen
A widespread view in the private sector is that the lack of access to finance significantly limits the entry into and the performance of value chains. Access to finance is expensive, scarce, and short term in countries in Latin America and the Caribbean, and it hampers firms' investment and the financial management required to gain entry and remain as participants in a value chain. The lack of access to finance is a consequence of a series of market failures that form the basis for public policy intervention. The region's development banks and specialized agencies have thus designed programs t
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Wereley, Norman M., and Bruce K. Walker. Approximate Evaluation of Semi-Markov Chain Reliability Models,. Defense Technical Information Center, 1988. http://dx.doi.org/10.21236/ada194669.

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Safta, Cosmin, Mohammad Khalil, and Habib N. Najm. Transitional Markov Chain Monte Carlo Sampler in UQTk. Office of Scientific and Technical Information (OSTI), 2020. http://dx.doi.org/10.2172/1606084.

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Dabrowski, Christopher, and Fern Hunt. Markov chain analysis for large-scale grid systems. National Institute of Standards and Technology, 2009. http://dx.doi.org/10.6028/nist.ir.7566.

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