# Matrix geometric approach for random walks: stability condition and equilibrium distribution

S. Kapodistria (Corresponding author), Z.B. Palmowski

2 Citaties (Scopus)

### Uittreksel

In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix $\mathbf{R}$ appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix $\mathbf{R}$.
Taal Engels 572-597 Stochastic Models 33 4 10.1080/15326349.2017.1359096 Gepubliceerd - 31 aug 2017

### Vingerafdruk

Equilibrium Distribution
Geometric Approach
Stability Condition
Random walk
Infinite Matrices
Simple Random Walk
Product Form
Spectral Properties
Lyapunov Function
Eigenvector
Nearest Neighbor
Boundary Value Problem
Eigenvalue
Calculate
Lyapunov functions
Series
Eigenvalues and eigenfunctions
Alternatives
Boundary value problems
Class

• math.PR

### Citeer dit

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In: Stochastic Models, Vol. 33, Nr. 4, 31.08.2017, blz. 572-597.

TY - JOUR

T1 - Matrix geometric approach for random walks

T2 - Stochastic Models

AU - Kapodistria,S.

AU - Palmowski,Z.B.

PY - 2017/8/31

Y1 - 2017/8/31

N2 - In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix $\mathbf{R}$ appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix $\mathbf{R}$.

AB - In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix $\mathbf{R}$ appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix $\mathbf{R}$.

KW - math.PR

U2 - 10.1080/15326349.2017.1359096

DO - 10.1080/15326349.2017.1359096

M3 - Article

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JO - Stochastic Models

JF - Stochastic Models

SN - 1532-6349

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