scholarly journals Markov Chain Models for the Stochastic Modeling of Pitting Corrosion

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
A. Valor ◽  
F. Caleyo ◽  
L. Alfonso ◽  
J. C. Velázquez ◽  
J. M. Hallen
Keyword(s):  
Markov Process ◽  
Pitting Corrosion ◽  
Probability Function ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Closed Form Solution ◽  
Form Solution ◽  
Extreme Value Statistics ◽  
Pit Depth ◽  
Transition Probability Function

The stochastic nature of pitting corrosion of metallic structures has been widely recognized. It is assumed that this kind of deterioration retains no memory of the past, so only the current state of the damage influences its future development. This characteristic allows pitting corrosion to be categorized as a Markov process. In this paper, two different models of pitting corrosion, developed using Markov chains, are presented. Firstly, a continuous-time, nonhomogeneous linear growth (pure birth) Markov process is used to model external pitting corrosion in underground pipelines. A closed-form solution of the system of Kolmogorov's forward equations is used to describe the transition probability function in a discrete pit depth space. The transition probability function is identified by correlating the stochastic pit depth mean with the empirical deterministic mean. In the second model, the distribution of maximum pit depths in a pitting experiment is successfully modeled after the combination of two stochastic processes: pit initiation and pit growth. Pit generation is modeled as a nonhomogeneous Poisson process, in which induction time is simulated as the realization of a Weibull process. Pit growth is simulated using a nonhomogeneous Markov process. An analytical solution of Kolmogorov's system of equations is also found for the transition probabilities from the first Markov state. Extreme value statistics is employed to find the distribution of maximum pit depths.

Markov Chain Model Helps Predict Pitting Corrosion Depth and Rate in Underground Pipelines

2010 ◽  
Author(s):  
F. Caleyo ◽  
J. C. Vela´zquez ◽  
J. M. Hallen ◽  
A. Valor ◽  
A. Esquivel-Amezcua
Keyword(s):  
Markov Chain ◽  
Pitting Corrosion ◽  
Probability Function ◽  
Markov Chain Model ◽  
Transition Probability ◽  
Real Life ◽  
Chain Model ◽  
Corrosion Depth ◽  
Pit Depth ◽  
Transition Probability Function

A continuous-time, non-homogenous pure birth Markov chain serves to model external pitting corrosion in buried pipelines. The analytical solution of Kolmogorov’s forward equations for this type of Markov process gives the transition probability function in a discrete space of pit depths. The transition probability function can be completely identified by making a correlation between the stochastic pit depth mean and the deterministic mean obtained experimentally. Previously reported Monte Carlo simulations have been used for the prediction of the evolution of the pit depth distribution mean value with time for different soil types. The simulated pit depth distributions are used to develop a stochastic model based on Markov chains to predict the progression of pitting corrosion depth and rate distributions from the observed soil properties and pipeline coating characteristics. The proposed model can also be applied to pitting corrosion data from repeated in-line pipeline inspections. Real-life case studies presented in this work show how pipeline inspection and maintenance planning can be improved through the use of the proposed Markovian model for pitting corrosion.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

scholarly journals The Strassen invariance principle for certain non-stationary Markov–Feller chains

2020 ◽  
Vol 121 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Dawid Czapla ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Gene Expression ◽  
Mathematical Model ◽  
Population Dynamics ◽  
Invariance Principle ◽  
Probability Function ◽  
Stochastic Dynamics ◽  
Transition Probability ◽  
Type Condition ◽  
Iterated Logarithm ◽  
Transition Probability Function

We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (3) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

scholarly journals Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period

1990 ◽  
Vol 27 (3) ◽  
pp. 713-719 ◽  
Author(s):  
Bruno Sericola
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Fault Tolerant ◽  
Computer Systems ◽  
Closed Form Solution ◽  
Form Solution ◽  
Homogeneous Markov Process ◽  
Markovian Processes ◽  
Finite Observation ◽  
Observation Period

Markov process are widely used to model computer systems. De Souza e Silva and Gail [3] calculated numerically the distribution of the cumulative operational time of repairable computer systems modelled by Markovian processes, that is, the distribution of the total time during which the system was in operation over a finite observation period. An extension of their approach is presented here. A closed-form solution is obtained for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period, which is theoretically and numerically interesting. We also give an application of this result to a fault-tolerant system.

Transition Probabilities for Markov Process on a Line Segment Located within a Quarter-Plane

2021 ◽  
pp. 4-24
Author(s):  
A.V. Kalinkin
Keyword(s):  
Markov Process ◽  
Random Process ◽  
Generating Function ◽  
Line Segment ◽  
Transition Probabilities ◽  
Special Functions ◽  
Transition Probability ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Quarter Plane

The paper considers a quadratic birth-death Markov process. The points on a line segment located within a quarter-plane represent the states of the random process. We designate the set of vectors that have integer non-negative coordinates as our quarter plane. The process is defined by infinitesimal characteristics, or transition probability densities. These characteristics are determined by a quadratic function of the coordinates at the segment points with integer coordinates. The boundary points of the segment are absorbing; at these points, the random process stops. We investigated a critical case when process jumps are equally probable at the moment of exiting a point. We derived expressions describing transition probabilities of the Markov process as a spectral series. We used a two-dimensional exponential generating function of transition probabilities and a two-dimensional generating function of transition probabilities. The first and second systems of ordinary differential Kolmogorov equations for Markov process transition probabilities are reduced to second-order mixed type partial differential equations for a double generating function. We solve the resulting system of linear equations using separation of variables. The spectrum obtained is discrete. The eigen-functions are expressed in terms of hypergeometric functions. The particular solution constructed is a Fourier series, whose coefficients are derived by means of expo-nential expansion. We employed sums of functional series known in the theory of special functions to construct the exponential expansion required

A method for the calculation of characteristics for the solution to stochastic differential equations

2017 ◽  
Vol 23 (3) ◽  
Author(s):  
Alexander Egorov ◽  
Victor Malyutin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Probability Function ◽  
Transition Probability ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Transition Probability Function ◽  
Fokker Planck

AbstractIn this work, a new numerical method to calculate the characteristics of the solution to stochastic differential equations is presented. This method is based on the Fokker–Planck equation for the transition probability function and the representation of the transition probability function by means of eigenfunctions of the Fokker–Planck operator. The results of the numerical experiments are presented.

Computational and estimation procedures in multidimensional right-shift processes and some applications

1975 ◽  
Vol 7 (2) ◽  
pp. 349-382 ◽  
Author(s):  
Richard J. Kryscio ◽  
Norman C. Severo
Keyword(s):  
Queueing Theory ◽  
Probability Function ◽  
Complete System ◽  
Transition Probability ◽  
Finite State ◽  
Computational Procedures ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Transition Probability Function ◽  
Finite State Space

A right-shift process is a Markov process with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. A multidimensional right-shift process consists of v ≧ 1 concurrent and dependent right-shift processes. In this paper applications of multidimensional right-shift processes to some well-known examples from epidemic theory, queueing theory and the Beetle probblem due to Lucien LeCam are discussed. A transformation which orders the Kolmogorov forward equations into a triangular array is provided and some computational procedures for solving the resulting system of equations are presented. One of these procedures is concerned with the problem of evaluating a given transition probability function rather than obtaining the solution to the complete system of forward equations. This particular procedure is applied to the problem of estimating the parameters of a multidimensional right-shift process which is observed at only a finite number of fixed timepoints.

scholarly journals On the metrical theory of a non-regular continued fraction expansion

2015 ◽  
Vol 23 (2) ◽  
pp. 147-160
Author(s):  
Dan Lascu ◽  
George Cîrlig
Keyword(s):  
Dynamical System ◽  
Continued Fraction ◽  
Probability Function ◽  
Transition Probability ◽  
Continued Fraction Expansion ◽  
Metrical Theory ◽  
Unilateral Shift ◽  
Symbolic Dynamical System ◽  
Continued Fraction Expansions ◽  
Transition Probability Function

Abstract We introduced a new continued fraction expansions in our previous paper. For these expansions, we show the Brodén-Borel-Lévy type formula. Furthermore, we compute the transition probability function from this and the symbolic dynamical system of the natural number with the unilateral shift.

scholarly journals A Markovian Approach To The Modeling Of Sound Propagation In Urban Streets

2012 ◽  
Author(s):  
Zaiton Haron ◽  
David Oldham
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Diffuse Reflection ◽  
Sound Propagation ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Multiple Reflections ◽  
Urban Streets ◽  
Novel Approach ◽  
Markovian Approach

Kertas kerja ini menguji kaedah novel, iaitu Markov untuk tujuan simulasi pengorakan bunyi di jalan raya. Kaedah ini menganggap deretan bangunan di tepi jalan menyerap dan memantulkan bunyi secara berserak. Proses simulasi menganggap proses pengorakan bunyi sebagai proses Markov jujukan pertama bercirikan matrix kebarangkalian perpindahan pancaran bunyi di antara permukaan–permukaan. Keputusan simulasi menggunakan kaedah Markov dibandingkan dengan keputusan diperolehi dari model kommersial RAYNOISE dengan menggunakan pilihan pantulan berserak. Hasil keputusan menunjukkan paras tekanan bunyi di jalan raya yang diramal oleh kaedah Markov mempunyai kesepadanan yang baik dengan ramalan diperolehi dari model RAYNOISE. Ini menunjukkan kaedah Markov mempunyai potensi untuk meramal pantulan berganda bagi keadaan sempadan berserak. Kesan agihan serapan permukaan bangunan juga dikaji, dan dengan skop dan anggapan kajian didapati jalan raya yang mempunyai deretan bangunan berpermukaan menyerap bunyi berupaya menghasilkan pengurangan bunyi kurang dari 1 dB. Kata kunci: Pantulan berserak; proses Markov; kebarangkalian perpindahan; pengorakan bunyi; kawalan bunyi bising This paper examined the capability of the novel approach called Markov in the simulation of sound propagation in streets. The approach assumes that the facades lining the streets absorb and reflect sound diffusely. The simulation process treated the sound propagation process as first order Markov process characterised by a matrix of transition probabilities relating to sound radiation between surfaces. The results of simulation using Markov model were compared with the results obtained from a commercial model, RAYNOISE using the diffuse reflection option. The results showed that sound pressure level in a street predicted by the Markov model was in good agreement with predictions obtained using RAYNOISE model. This suggest that the Markov model has the potential to predict multiple reflections for diffuse boundary conditions. The effects of distribution absorption of building facades were also investigated and within the scope and assumptions in this study; it is shown streets with absorbent building facade result in sound reductions typically less than 1 dB. Key words: Diffuse reflection; Markov process; transition probability; sound propagation; noise control

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