## Linear Algebra, Markov Chains, and Queueing Models

The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space. Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion of locally interacting Markov chains. This corresponds to the situation when the state space has a Cartesian- product form. See interacting particle system and stochastic cellular automata probabilistic cellular automata.

See for instance Interaction of Markov Processes  or . In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states. For example, let X be a non-Markovian process. Then define a process Y , such that each state of Y represents a time-interval of states of X. Mathematically, this takes the form:. An example of a non-Markovian process with a Markovian representation is an autoregressive time series of order greater than one.

Then the matrix P t satisfies the forward equation, a first-order differential equation. The solution to this equation is given by a matrix exponential. However, direct solutions are complicated to compute for larger matrices. The fact that Q is the generator for a semigroup of matrices.

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The stationary distribution for an irreducible recurrent CTMC is the probability distribution to which the process converges for large values of t. Observe that for the two-state process considered earlier with P t given by. Observe that each row has the same distribution as this does not depend on starting state. The player controls Pac-Man through a maze, eating pac-dots. Meanwhile, he is being hunted by ghosts.

## Linear Algebra, Markov Chains, and Queueing Models | Carl D. Meyer | Springer

For convenience, the maze shall be a small 3x3-grid and the monsters move randomly in horizontal and vertical directions. A secret passageway between states 2 and 8 can be used in both directions. Entries with probability zero are removed in the following transition matrix:. This Markov chain is irreducible, because the ghosts can fly from every state to every state in a finite amount of time. Due to the secret passageway, the Markov chain is also aperiodic, because the monsters can move from any state to any state both in an even and in an uneven number of state transitions.

The hitting time is the time, starting in a given set of states until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition. By Kelly's lemma this process has the same stationary distribution as the forward process. A chain is said to be reversible if the reversed process is the same as the forward process. Kolmogorov's criterion states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.

Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a jump process. Each element of the one-step transition probability matrix of the EMC, S , is denoted by s ij , and represents the conditional probability of transitioning from state i into state j. These conditional probabilities may be found by. S may be periodic, even if Q is not.

Markov Chains, Part 2

Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, medicine, music, game theory and sports. Markovian systems appear extensively in thermodynamics and statistical mechanics , whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description. The paths, in the path integral formulation of quantum mechanics, are Markov chains.

Markov chains are used in lattice QCD simulations. Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property.

## Queueing Theory

For example, imagine a large number n of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially independent of each other, the number of molecules in state A or B at a time is n times the probability a given molecule is in that state.

The classical model of enzyme activity, Michaelis—Menten kinetics , can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains. An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals in silico towards a desired class of compounds such as drugs or natural products. It is not aware of its past that is, it is not aware of what is already bonded to it.

It then transitions to the next state when a fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds. Also, the growth and composition of copolymers may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated e.

Due to steric effects , second-order Markov effects may also play a role in the growth of some polymer chains. Similarly, it has been suggested that the crystallization and growth of some epitaxial superlattice oxide materials can be accurately described by Markov chains. Several theorists have proposed the idea of the Markov chain statistical test MCST , a method of conjoining Markov chains to form a " Markov blanket ", arranging these chains in several recursive layers "wafering" and producing more efficient test sets—samples—as a replacement for exhaustive testing. MCSTs also have uses in temporal state-based networks; Chilukuri et al.

Solar irradiance variability assessments are useful for solar power applications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov chains,     also including modeling the two states of clear and cloudiness as a two-state Markov chain.

Hidden Markov models are the basis for most modern automatic speech recognition systems. Markov chains are used throughout information processing. Claude Shannon 's famous paper A Mathematical Theory of Communication , which in a single step created the field of information theory , opens by introducing the concept of entropy through Markov modeling of the English language. Such idealized models can capture many of the statistical regularities of systems.

Even without describing the full structure of the system perfectly, such signal models can make possible very effective data compression through entropy encoding techniques such as arithmetic coding. They also allow effective state estimation and pattern recognition. Markov chains also play an important role in reinforcement learning. Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks which use the Viterbi algorithm for error correction , speech recognition and bioinformatics such as in rearrangements detection .

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The LZMA lossless data compression algorithm combines Markov chains with Lempel-Ziv compression to achieve very high compression ratios. Markov chains are the basis for the analytical treatment of queues queueing theory. Agner Krarup Erlang initiated the subject in Numerous queueing models use continuous-time Markov chains. The PageRank of a webpage as used by Google is defined by a Markov chain. Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.

Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo MCMC. In recent years this has revolutionized the practicability of Bayesian inference methods, allowing a wide range of posterior distributions to be simulated and their parameters found numerically. Markov chains are used in finance and economics to model a variety of different phenomena, including asset prices and market crashes.

The first financial model to use a Markov chain was from Prasad et al. Hamilton , in which a Markov chain is used to model switches between periods high and low GDP growth or alternatively, economic expansions and recessions. Calvet and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models.

Dynamic macroeconomics heavily uses Markov chains. An example is using Markov chains to exogenously model prices of equity stock in a general equilibrium setting. Credit rating agencies produce annual tables of the transition probabilities for bonds of different credit ratings. Markov chains are generally used in describing path-dependent arguments, where current structural configurations condition future outcomes.

An example is the reformulation of the idea, originally due to Karl Marx 's Das Kapital , tying economic development to the rise of capitalism. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the middle class , the ratio of urban to rural residence, the rate of political mobilization, etc. Markov chains also have many applications in biological modelling, particularly population processes , which are useful in modelling processes that are at least analogous to biological populations.

The Leslie matrix , is one such example used to describe the population dynamics of many species, though some of its entries are not probabilities they may be greater than 1. Another example is the modeling of cell shape in dividing sheets of epithelial cells. Markov chains are also used in simulations of brain function, such as the simulation of the mammalian neocortex.

Markov chains have been used in population genetics in order to describe the change in gene frequencies in small populations affected by genetic drift , for example in the diffusion equation method described by Motoo Kimura. Markov chains can be used to model many games of chance. Cherry-O ", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state on a given square and from there has fixed odds of moving to certain other states squares.

Markov chains are employed in algorithmic music composition , particularly in software such as Csound , Max , and SuperCollider. In a first-order chain, the states of the system become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix see below. An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be MIDI note values, frequency Hz , or any other desirable metric. A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table.

Higher, n th-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of phrasal structure, rather than the 'aimless wandering' produced by a first-order system. Markov chains can be used structurally, as in Xenakis's Analogique A and B. Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory.

In order to overcome this limitation, a new approach has been proposed. Markov chain models have been used in advanced baseball analysis since , although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners.

Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team. Markov processes can also be used to generate superficially real-looking text given a sample document. Markov processes are used in a variety of recreational " parody generator " software see dissociated press , Jeff Harrison, [] Mark V.

Shaney , [] [] and Academias Neutronium. In the bioinformatics field, they can be used to simulate DNA sequences. Markov chains have been used for forecasting in several areas: for example, price trends, [] wind power, [] and solar irradiance. From Wikipedia, the free encyclopedia. This article may be too long to read and navigate comfortably. Please help to improve this section by introducing more precise citations. February Learn how and when to remove this template message. Main article: Phase-type distribution. Michaelis-Menten kinetics.

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7. The enzyme E binds a substrate S and produces a product P. Each reaction is a state transition in a Markov chain. Main article: Queueing theory. Dynamics of Markovian particles Markov chain approximation method Markov chain geostatistics Markov chain mixing time Markov decision process Markov information source Markov random field Quantum Markov chain Semi-Markov process Stochastic cellular automaton Telescoping Markov chain Variable-order Markov model.

Oxford Dictionaries English. Retrieved Basics of Applied Stochastic Processes. Archived from the original on 23 March Rozanov 6 December Markov Random Fields. Applied Probability and Queues. Citations Publications citing this paper. On the stability of the computation of the stationary probabilities of Markov chains using Perron complements Michael Neumann , Jianhong Xu. Catral , Stephen J. On optimal condition numbers for Markov chains Stephen J. A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions Stephen J.

Kirkland , Michael Neumann , Jianhong Xu. References Publications referenced by this paper. Kirkland , Michael Neumann. Kirkland , Michael Neumann , Bryan L. Nonnegative matrices in the mathematical science Andrew Berman , Robert J. NOTE: We are unable to offer combined shipping for multiple items purchased. This is because our items are shipped from different locations. Please contact Customer Services and request "Return Authorisation" before you send your item back to us. Unauthorised returns will not be accepted. Returns must be postmarked within 4 business days of authorisation and must be in resellable condition.

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