scholarly journals Heavy-tailed asymptotics of stationary probability vectors of Markov chains of gi/g/1 type

2005 ◽  
Vol 37 (02) ◽  
pp. 482-509 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Transition Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Necessary And Sufficient Condition ◽  
Hopf Equation ◽  
Novel Approach ◽  
Matrix Sequence ◽  
Heavy Tailed ◽  
Necessary And Sufficient ◽  
Stationary Probability Vector

In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.

scholarly journals Heavy-tailed asymptotics of stationary probability vectors of Markov chains of gi/g/1 type

2005 ◽  
Vol 37 (2) ◽  
pp. 482-509 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Transition Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Necessary And Sufficient Condition ◽  
Hopf Equation ◽  
Novel Approach ◽  
Matrix Sequence ◽  
Heavy Tailed ◽  
Necessary And Sufficient ◽  
Stationary Probability Vector

In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.

scholarly journals The Extrapolation-Accelerated Multilevel Aggregation Method in PageRank Computation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bing-Yuan Pu ◽  
Ting-Zhu Huang ◽  
Chun Wen ◽  
Yi-Qin Lin
Keyword(s):  
Stochastic Matrix ◽  
Numerical Results ◽  
Extrapolation Method ◽  
Stationary Probability ◽  
Probability Vector ◽  
Aggregation Method ◽  
Vector Extrapolation ◽  
Stationary Probability Vector

An accelerated multilevel aggregation method is presented for calculating the stationary probability vector of an irreducible stochastic matrix in PageRank computation, where the vector extrapolation method is its accelerator. We show how to periodically combine the extrapolation method together with the multilevel aggregation method on the finest level for speeding up the PageRank computation. Detailed numerical results are given to illustrate the behavior of this method, and comparisons with the typical methods are also made.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (04) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by theR-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. TheRG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals Constant slope, entropy, and horseshoes for a map on a tame graph

2019 ◽  
Vol 40 (11) ◽  
pp. 2970-2994
Author(s):  
ADAM BARTOŠ ◽  
JOZEF BOBOK ◽  
PAVEL PYRIH ◽  
SAMUEL ROTH ◽  
BENJAMIN VEJNAR
Keyword(s):  
Topological Entropy ◽  
Transition Matrix ◽  
Sufficient Condition ◽  
Branch Points ◽  
Necessary And Sufficient Condition ◽  
Peano Continuum ◽  
Monotone Maps ◽  
Markov Map ◽  
Necessary And Sufficient ◽  
Constant Slope

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

Processes ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 2146
Author(s):  
V. Vinitha ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
K. Jeganathan ◽  
Gyanendra Prasad Joshi ◽  
...  
Keyword(s):  
Steady State ◽  
Lead Time ◽  
Arrival Process ◽  
Inventory Level ◽  
Stationary Probability ◽  
Probability Vector ◽  
Steady State Analysis ◽  
Proposed Model ◽  
Stationary Probability Vector ◽  
Bernoulli Schedule

This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

Synchronization Analysis of Boolean Network

2013 ◽  
Vol 432 ◽  
pp. 528-532
Author(s):  
Cheng Chen ◽  
Wei Zhu
Keyword(s):  
Regulatory Network ◽  
Transition Matrix ◽  
Boolean Network ◽  
Necessary And Sufficient Condition ◽  
Complete Synchronization ◽  
Boolean Control Network ◽  
Large Gene ◽  
Gene Regulatory ◽  
Necessary And Sufficient ◽  
Product Of Matrices

Boolean network and its synchronization have been gradually used to the global behavior analysis of large gene regulatory network. Network synchronization depends mainly on the dynamics of each node and the topology of the network. In this paper, using the semi-tensor product of matrices, a necessary and sufficient condition based on transition matrix for Boolean network complete synchronization is presented. The synchronization of Boolean control network is also discussed. Two examples are given to illustrate the theoretical result.

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