On the choice of the stochastic comparison method for multidimensional Markov chains analysis

2012 ◽  
Author(s):  
Hind Castel-Taleb ◽  
Nihal Pekergin
Keyword(s):  
Markov Chains ◽  
Comparison Method ◽  
Stochastic Comparison

scholarly journals Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohamed Boualem ◽  
Natalia Djellab ◽  
Djamil Aïssani
Keyword(s):  
Markov Chains ◽  
Service Time ◽  
Retrial Queue ◽  
Stochastic Ordering ◽  
Stochastic Comparison ◽  
Single Server ◽  
Stochastic Approximations ◽  
Bernoulli Feedback ◽  
Monotonicity Results

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for anM/G/1retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

Reduction techniques for discrete-time Markov chains on totally ordered state space using stochastic comparisons

2000 ◽  
Vol 37 (03) ◽  
pp. 795-806 ◽  
Author(s):  
Laurent Truffet
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Stochastic Comparison ◽  
Homogeneous Markov Chain ◽  
Monotone Functions ◽  
Stochastic Comparisons ◽  
Reduction Techniques ◽  
Ordered State

We propose in this paper two methods to compute Markovian bounds for monotone functions of a discrete time homogeneous Markov chain evolving in a totally ordered state space. The main interest of such methods is to propose algorithms to simplify analysis of transient characteristics such as the output process of a queue, or sojourn time in a subset of states. Construction of bounds are based on two kinds of results: well-known results on stochastic comparison between Markov chains with the same state space; and the fact that in some cases a function of Markov chain is again a homogeneous Markov chain but with smaller state space. Indeed, computation of bounds uses knowledge on the whole initial model. However, only part of this data is necessary at each step of the algorithms.

Near Complete Decomposability: Bounding the Error by a Stochastic Comparison Method

1997 ◽  
Vol 29 (3) ◽  
pp. 830-855 ◽  
Author(s):  
Laurent Truffet
Keyword(s):  
Random Variable ◽  
Stochastic Approach ◽  
Comparison Method ◽  
Stochastic Comparison ◽  
Short Term ◽  
Steady State Probability ◽  
Markovian Approximations ◽  
Aggregation Technique ◽  
Markovian Systems

An aggregation technique of ‘near complete decomposable' Markovian systems has been proposed by Courtois [3]. It is an approximate method in many cases, except for some queuing networks, so the error between the exact and the approximate solution is an important problem. We know that the error is O(ε), where ε is defined as the maximum coupling between aggregates. Some authors developed techniques to obtain a O(ε k) error with k > 1 error with k > 1, while others developed a technique called ‘bounded aggregation’. All these techniques use linear algebra tools and do not utilize the fact that the steady-state probability vector represents the distribution of a random variable. In this work we propose a stochastic approach and we give a method to obtain stochastic bounds on all possible Markovian approximations of the two main dynamics: short-term and long-term dynamics.

Renewal-type behavior of absorption times in Markov chains

1994 ◽  
Vol 26 (4) ◽  
pp. 988-1005 ◽  
Author(s):  
Bernard Van Cutsem ◽  
Bernard Ycart
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Limit Theorem ◽  
Markov Chains ◽  
Transition Matrix ◽  
Renewal Theory ◽  
Stochastic Comparison ◽  
Absorption Time ◽  
Starting Point ◽  
Lower Triangular

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

scholarly journals Results on Varextropy Measure of Random Variables

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 356
Author(s):  
Nastaran Marzban Vaselabadi ◽  
Saeid Tahmasebi ◽  
Mohammad Reza Kazemi ◽  
Francesco Buono
Keyword(s):  
Hazard Rate ◽  
Random Variables ◽  
Record Values ◽  
Comparison Method ◽  
Dispersion Measure ◽  
Stochastic Comparison ◽  
Alternative Measure ◽  
Proportional Hazard ◽  
Hazard Rate Models ◽  
Measure Of Uncertainty

In 2015, Lad, Sanfilippo and Agrò proposed an alternative measure of uncertainty dual to the entropy known as extropy. This paper provides some results on a dispersion measure of extropy of random variables which is called varextropy and studies several properties of this concept. Especially, the varextropy measure of residual and past lifetimes, order statistics, record values and proportional hazard rate models are discussed. Moreover, the conditional varextropy is considered and some properties of this measure are studied. Finally, a new stochastic comparison method, named varextropy ordering, is introduced and some of its properties are presented.

Reduction techniques for discrete-time Markov chains on totally ordered state space using stochastic comparisons

2000 ◽  
Vol 37 (3) ◽  
pp. 795-806 ◽  
Author(s):  
Laurent Truffet
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Stochastic Comparison ◽  
Homogeneous Markov Chain ◽  
Monotone Functions ◽  
Stochastic Comparisons ◽  
Reduction Techniques ◽  
Ordered State

We propose in this paper two methods to compute Markovian bounds for monotone functions of a discrete time homogeneous Markov chain evolving in a totally ordered state space. The main interest of such methods is to propose algorithms to simplify analysis of transient characteristics such as the output process of a queue, or sojourn time in a subset of states. Construction of bounds are based on two kinds of results: well-known results on stochastic comparison between Markov chains with the same state space; and the fact that in some cases a function of Markov chain is again a homogeneous Markov chain but with smaller state space. Indeed, computation of bounds uses knowledge on the whole initial model. However, only part of this data is necessary at each step of the algorithms.

Near Complete Decomposability: Bounding the Error by a Stochastic Comparison Method

1997 ◽  
Vol 29 (03) ◽  
pp. 830-855 ◽  
Author(s):  
Laurent Truffet
Keyword(s):  
Random Variable ◽  
Stochastic Approach ◽  
Comparison Method ◽  
Stochastic Comparison ◽  
Short Term ◽  
Steady State Probability ◽  
Markovian Approximations ◽  
Aggregation Technique ◽  
Markovian Systems

An aggregation technique of ‘near complete decomposable' Markovian systems has been proposed by Courtois [3]. It is an approximate method in many cases, except for some queuing networks, so the error between the exact and the approximate solution is an important problem. We know that the error is O(ε), where ε is defined as the maximum coupling between aggregates. Some authors developed techniques to obtain a O(ε k ) error with k > 1 error with k > 1, while others developed a technique called ‘bounded aggregation’. All these techniques use linear algebra tools and do not utilize the fact that the steady-state probability vector represents the distribution of a random variable. In this work we propose a stochastic approach and we give a method to obtain stochastic bounds on all possible Markovian approximations of the two main dynamics: short-term and long-term dynamics.

Renewal-type behavior of absorption times in Markov chains

1994 ◽  
Vol 26 (04) ◽  
pp. 988-1005 ◽  
Author(s):  
Bernard Van Cutsem ◽  
Bernard Ycart
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Limit Theorem ◽  
Markov Chains ◽  
Transition Matrix ◽  
Renewal Theory ◽  
Stochastic Comparison ◽  
Absorption Time ◽  
Starting Point ◽  
Lower Triangular

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

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