Finite dams with inputs forming a Markov chain

1970 ◽  
Vol 7 (2) ◽  
pp. 291-303 ◽  
Author(s):  
M.S. Ali Khan
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Generating Functions ◽  
State Transition ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Time Dependent ◽  
Probability Generating Functions ◽  
State Transition Probability ◽  
Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

Finite dams with inputs forming a Markov chain

1970 ◽  
Vol 7 (02) ◽  
pp. 291-303 ◽  
Author(s):  
M.S. Ali Khan
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Generating Functions ◽  
State Transition ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Time Dependent ◽  
Probability Generating Functions ◽  
State Transition Probability ◽  
Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

scholarly journals Grey theory–based BP-NN co-training for dense sequence long-term tendency prediction

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yuling Hong ◽  
Yingjie Yang ◽  
Qishan Zhang
Keyword(s):  
State Transition ◽  
Transition Probability ◽  
Training Model ◽  
Transition Probability Matrix ◽  
Content Type ◽  
Fine Grained ◽  
State Transition Probability ◽  
Dense Sequence ◽  
Popularity Prediction

PurposeThe purpose of this paper is to solve the problems existing in topic popularity prediction in online social networks and advance a fine-grained and long-term prediction model for lack of sufficient data.Design/methodology/approachBased on GM(1,1) and neural networks, a co-training model for topic tendency prediction is proposed in this paper. The interpolation based on GM(1,1) is employed to generate fine-grained prediction values of topic popularity time series and two neural network models are considered to achieve convergence by transmitting training parameters via their loss functions.FindingsThe experiment results indicate that the integrated model can effectively predict dense sequence with higher performance than other algorithms, such as NN and RBF_LSSVM. Furthermore, the Markov chain state transition probability matrix model is used to improve the prediction results.Practical implicationsFine-grained and long-term topic popularity prediction, further improvement could be made by predicting any interpolation in the time interval of popularity data points.Originality/valueThe paper succeeds in constructing a co-training model with GM(1,1) and neural networks. Markov chain state transition probability matrix is deployed for further improvement of popularity tendency prediction.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

A Comparison of Alternative Approaches for the Synthetic Generation of a Wind Speed Time Series

1991 ◽  
Vol 113 (4) ◽  
pp. 280-289 ◽  
Author(s):  
F. C. Kaminsky ◽  
R. H. Kirchhoff ◽  
C. Y. Syu ◽  
J. F. Manwell
Keyword(s):  
Time Series ◽  
Markov Chain ◽  
Wind Speed ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Statistical Properties ◽  
Embedded Markov Chain ◽  
One Step ◽  
Alternative Approaches

In this paper, alternative approaches for synthetically generating a wind speed time series are discussed. These approaches include: (1) the use of independent values from a specific probability distribution; (2) the use of an algorithm based on the statistical behavior of a one-step Markov chain; (3) the use of an algorithm based on the behavior of a transition probability matrix that describes the next wind speed value statistically as a function of the current wind speed value and the previous wind speed value; (4) the use of Box-Jenkins models; (5) the use of the Shinozuka algorithm; and (6) the use of an embedded Markov chain. The ability of each approach to capture the statistical properties of the desired wind speed time series is discussed. In this context the statistical properties of interest are the probability distribution of the wind speed values, the autocorrelation function of the wind speed values, and the spectral density of the wind speed values.

scholarly journals On the Convergence Rate of Bonus-Malus Systems

1992 ◽  
Vol 22 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Convergence Rate ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Limit Probability ◽  
Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

Stochastic Time-Dependent Deterioration Models for Estimating Residual Service Life of Offshore Jacket-Type Platforms

2021 ◽  
Author(s):  
Ishwarya Srikanth ◽  
M. Arockiasamy
Keyword(s):  
Markov Chain ◽  
Service Life ◽  
Transition Probability ◽  
Steel Corrosion ◽  
Transition Probability Matrix ◽  
Time Dependent ◽  
Residual Service Life ◽  
Jacket Platform ◽  
Offshore Jacket Platform ◽  
Residual Service

Abstract This paper presents deterioration models for maintenance planning of offshore jacket platform based on two methods: i) stochastic Markov-chain based model and ii) stochastic-mechanistic deterioration models based on steel corrosion rates. Markov-chain models require the estimation of transition probability matrix (TPM), which is typically derived from the inspection data. The global structural health condition of the jacket is computed based on the condition of individual elements and their criticality in terms of failure consequence. The criticality factors are established based on nonlinear static redundancy analyses. This method can model deterioration when routine inspection records of jacket members are available. When there is scarcity of inspection records, stochastic-mechanistic deterioration modeling approach can be used. Monte-Carlo simulations with established corrosion wastage models are utilized to estimate the time-dependent deterioration of jacket legs, horizontal and diagonal bracings in splash and immersion zones. This method is proposed when there is scarcity of inspection records. The deterioration models are further utilized to predict the timing for Maintenance, Repair and Rehabilitation (MRR) actions, and estimate the residual service life of the jacket platform. This study demonstrates the application of the proposed deterioration modeling approaches with a case study of a typical 4-legged offshore jacket platform.

Availability models of series mechanical systems considering failure and maintenance dependencies

2018 ◽  
Vol 42 (3) ◽  
pp. 222-232
Author(s):  
Peng Gao ◽  
Liyang Xie
Keyword(s):  
Optimization Design ◽  
Path Dependence ◽  
State Transition ◽  
Transition Probability ◽  
Mechanical Systems ◽  
Transition Probability Matrix ◽  
Strength Degradation ◽  
Statistical Characteristics ◽  
Established Method ◽  
State Transition Probability

Availability models of series mechanical systems based on system working mechanisms are developed by integrating the statistical characteristics of stress, strength, and maintenance parameters. Failure, maintenance, and strength degradation path dependence are taken into consideration in the proposed models without empirical parameters. Moreover, the problem of inconsistency between the failure rate and the availability in time is pointed out, and a method to solve this problem is proposed. Monte Carlo simulations are carried out to verify the proposed models. In addition, numerical examples are given to illustrate the established method. The results show that failure, maintenance, and strength degradation path dependence and time scale inconsistence in the state transition probability matrix have significant influences on the availability of mechanical systems. The proposed models provide an analytical basis for quantitative availability estimation, optimization design, and maintenance strategy decision-making of mechanical systems.

Predict the Volume of Passenger Transport of Railway Based on Grey Markov Chain Model

2014 ◽  
Vol 1030-1032 ◽  
pp. 2069-2072 ◽  
Author(s):  
Ying Li
Keyword(s):  
Markov Chain ◽  
Prediction Accuracy ◽  
State Transition ◽  
Markov Chain Model ◽  
Transition Probability ◽  
Chain Model ◽  
Grey Prediction ◽  
Passenger Transport ◽  
Passenger Flow ◽  
State Transition Probability

Combined with grey model and the characteristics of the Markov chain, based on the grey prediction model, calculating the state transition probability, grey Markov chain model is established. The results show that the grey Markov chain model has higher prediction accuracy than GM (1, 1) model, can offer references for passenger flow organization.

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