Ideal flow of Markov Chain

2018 ◽  
Vol 10 (06) ◽  
pp. 1850073
Author(s):  
Kardi Teknomo
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Scaling Factor ◽  
Matrix Perturbation ◽  
Flow Network ◽  
Ideal Flow ◽  
The Matrix ◽  
Strongly Connected ◽  
Flow Equilibrium ◽  
Matrix Perturbation Analysis

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

scholarly journals A note on the matrix renewal function

1972 ◽  
Vol 13 (4) ◽  
pp. 417-422 ◽  
Author(s):  
A. M. Kshirsagar ◽  
Y. P. Gupta
Keyword(s):  
Generalized Inverse ◽  
Transition Probability ◽  
Renewal Theory ◽  
Transition Probability Matrix ◽  
Markov Renewal Process ◽  
Renewal Function ◽  
Stieltjes Transform ◽  
Imbedded Markov Chain ◽  
The Matrix ◽  
Passage Times

AbstractThe Laplace-Stieltjes Transform m(s) of the matrix renewal function M(t) of a Markov Renewal process is expanded in powers of the argument s, in this paper, by using a generalized inverse of the matrix I–P0, where P0 is the transition probability matrix of the imbedded Markov chain. This helps in obtaining the values of moments of any order of the number of renewals and also of the moments of the first passage times, for large values of t, the time. All the results of renewal theory are hidden under the Laplacian curtain and this expansion helps to lift this curtain at least for large values of t and is thus useful in predicting the number of renewals.

scholarly journals The Diameter and Laplacian Eigenvalues of Directed Graphs

10.37236/1142 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Fan Chung
Keyword(s):  
Random Walk ◽  
Directed Graphs ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Undirected Graphs ◽  
Laplacian Eigenvalues ◽  
Strongly Connected ◽  
Perron Vector

For undirected graphs it has been known for some time that one can bound the diameter using the eigenvalues. In this note we give a similar result for the diameter of strongly connected directed graphs $G$, namely $$ D(G) \leq \bigg \lfloor {2\min_x \log (1/\phi(x))\over \log{2\over 2-\lambda}} \bigg\rfloor +1 $$ where $\lambda$ is the first non-trivial eigenvalue of the Laplacian and $\phi$ is the Perron vector of the transition probability matrix of a random walk on $G$.

A note on cumulative sums of markovian variables

1965 ◽  
Vol 5 (2) ◽  
pp. 285-287 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Finite Number ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
The Matrix ◽  
Initial Probability Distribution ◽  
Regular Markov Chain ◽  
Image Position ◽  
Cumulative Sums

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .

On the matrix occurring in a linear search problem

1991 ◽  
Vol 28 (2) ◽  
pp. 336-346 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Search Problem ◽  
Linear Search ◽  
Natural Position ◽  
The Matrix

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj, ···, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.

Stochastic Response of a Vibro-Impact System via a New Impact-to-Impact Mapping

2021 ◽  
Vol 31 (08) ◽  
pp. 2150139
Author(s):  
Liang Wang ◽  
Bochen Wang ◽  
Jiahui Peng ◽  
Xiaole Yue ◽  
Wei Xu
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Significant Feature ◽  
Stochastic Response ◽  
Impact Surface ◽  
Impact System ◽  
The Matrix ◽  
Step Transition ◽  
The One ◽  
The Impact

In this paper, a new impact-to-impact mapping is constructed to investigate the stochastic response of a nonautonomous vibro-impact system. The significant feature lies in the choice of Poincaré section, which consists of impact surface and codimensional time. Firstly, we construct a new impact-to-impact mapping to calculate the one-step transition probability matrix from a given impact to the next. Then, according to the matrix, we can investigate the stochastic responses of a nonautonomous vibro-impact system at the impact instants. The new impact-to-impact mapping is smooth and it effectively overcomes the nondifferentiability caused by the impact. A linear and a nonlinear nonautonomous vibro-impact systems are analyzed to verify the effectiveness of the strategy. The stochastic P-bifurcations induced by the noise intensity and system parameters are studied at the impact instants. Compared with Monte Carlo simulations, the new impact-to-impact strategy is accurate for nonautonomous vibro-impact systems with arbitrary restitution coefficients.

On the matrix occurring in a linear search problem

1991 ◽  
Vol 28 (02) ◽  
pp. 336-346 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Search Problem ◽  
Linear Search ◽  
Natural Position ◽  
The Matrix

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj , ···, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.

scholarly journals THE DEVIATION MATRIX OF A CONTINUOUS-TIME MARKOV CHAIN

2002 ◽  
Vol 16 (3) ◽  
pp. 351-366 ◽  
Author(s):  
Pauline Coolen-Schrijner ◽  
Erik A. van Doorn
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Deviation Matrix ◽  
The Matrix ◽  
Birth Death

The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(·) and ergodic matrix Π is the matrix D ≡ ∫0∞(P(t) − Π) dt. We give conditions for D to exist and discuss properties and a representation of D. The deviation matrix of a birth–death process is investigated in detail. We also describe a new application of deviation matrices by showing that a measure for the convergence to stationarity of a stochastically increasing Markov chain can be expressed in terms of the elements of the deviation matrix of the chain.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

Further enhancement on H∞ sliding mode control design for singular Markovian jump systems with partly known transition probabilities

2021 ◽  
pp. 107754632198920
Author(s):  
Zeinab Fallah ◽  
Mahdi Baradarannia ◽  
Hamed Kharrati ◽  
Farzad Hashemzadeh
Keyword(s):  
Transition Probabilities ◽  
Sliding Mode ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Linear Matrix ◽  
Sliding Mode Controller ◽  
Sliding Surface ◽  
Markovian Jump ◽  
Markovian Jump System ◽  
Singular Markovian Jump Systems

This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.

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