The performance measure analysis on the states classification in Markov Chain

Agboola, Sunday. O.; Ayinde, Semiu A.

doi:10.4314/dujopas.v7i4b.4

Cited by 6 publications

(2 citation statements)

References 5 publications

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“…Agboola and Ayoade (2021) investigated a block structure that arises frequently when modeling queueing systems to provide some insight into the solutions of stationary distribution of Markov chain. Agboola and Ayinde (2021) analysed the concept of the classification of groups of states, between states that are recurrent, transient to provide some insight into the performance measure analysis such as the mean first passage time, the mean recurrence time of state as well as recurrence iterative matrix. Agboola (2021) computed the solutions and algorithms for lower -upper triangular matrix approach and the Grassmann -Taksar -Heyman to obtained the stationary distribution vectors in Markov chain.…”

Section: Introductionmentioning

confidence: 99%

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

Agboola¹,

Ayinde²,

Ibikunle³

et al. 2023

dujopas

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The Physical or Mathematical behaviour of this model may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this study, the stationary distribution of Markov chains was solved using iterative methods that begin with an initial estimate of the solution vector and then modified it in a way that brings it closer and closer to the real solution with each step or iteration. These methods also involved matrix operations like multiplication with one or more vectors, which preserves the transition matrices while speeding up the process. We computed the solutions using Jacobi iterative method and Gauss-Seidel iterative method in order to shed more light on the solutions of stationary distribution in Markov chain. This was done with the aid of several already-existing laws, theorems, and formulas of Markov chain and the application of normalization principle and matrix operations such as lower, upper, and diagonal matrices. The stationary distribution vector’s 𝜋𝑖,𝑖=1,2,…,4 are obtained for the illustrative example one as 𝜋(3) = (0.078125,0.109375,0.21875,0.59375) as well as the four eigenvalues of the matrix as 𝜆1=1.0, 𝜆2=−0.7718, 𝜆3,4=−0.1141±0.5576𝑖 using Jacobi iterative technique, and for illustrative example two using Gauss-Siedel method as 𝝅(𝟑) = (0.090909, 0.181818, 0.363636, 0.363636). The research shown that Gauss Siedel method converged faster than Jacobi method.

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Section: Introductionmentioning

confidence: 99%

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

Agboola¹,

Ayinde²,

Ibikunle³

et al. 2023

dujopas

View full text Add to dashboard Cite

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“…The Jacobi iterative method is a fundamental technique that is widely applied in various fields (Chalermwut et al, 2023). Islam (2023) applied Jacobi to solve linear systems using theory, data, and high-performance computing, while Agboola et al (2023) solved the stationary distribution of a markov chain using the Jacobi method. Meanwhile, Zhen et al (2023) found a new way of solving fuzzy linear systems using the Jacobi method, and Huang and Jia (2023) did an approximate orthogonal tensor diagonalization of Jacobi convergence.…”

Section: Introductionmentioning

confidence: 99%

A Third Refinement of Jacobi Method for Solutions to System of Linear Equations

Audu,

Essien,

Zhiri

et al. 2023

FJS

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Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.

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Application of Power Numerical Method for the Stationary Distribution of Markov Chain

Agboola¹,

Adebiyi

2023

FJS

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The evolution of this model is represented by transitions from one state to the next. Also, the physical or mathematical behavior of this system can also be illustrated by identifying all of the possible states and explaining how it transitions between them. The iterative solution approaches for the stationary distribution of Markov chains, which begin with an initial estimate of the solution vector and it becomes closer and closer to the true solution with each iteration are investigated. Our goal is to compute solutions of stationary distribution of Markov chain by utilizing the power iterative method which leaves the transition matrices unchanged and saves time by considering the discretization effect, and the convergency. Matrices operations such as multiplication with one or more vectors, lower, diagonal and upper concepts of matrix, with the help of several existing Markov chain laws, theorems, formulas, and the normalization principle are applied. For the illustrative examples, the stationary distribution vectors and table of convergence are obtained.

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The performance measure analysis on the states classification in Markov Chain

Cited by 6 publications

References 5 publications

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

A Third Refinement of Jacobi Method for Solutions to System of Linear Equations

Application of Power Numerical Method for the Stationary Distribution of Markov Chain

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