scholarly journals Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1752
Author(s):  
Alexander Zeifman ◽  
Yacov Satin ◽  
Alexander Sipin
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Continuous Time ◽  
Upper Bounds ◽  
Specific Class ◽  
Differential Inequalities ◽  
Method Of Differential Inequalities ◽  
Finite Markov Chains ◽  
Intensity Functions ◽  
Forward System

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

scholarly journals On the rate of convergence for a class of Markovian queues with group services

2020 ◽  
Vol 28 (3) ◽  
pp. 205-215
Author(s):  
Anastasia L. Kryukova
Keyword(s):  
Rate Of Convergence ◽  
Operator Function ◽  
Computer Network ◽  
Human Life ◽  
Queueing Models ◽  
Differential Inequalities ◽  
Queuing Systems ◽  
Stationary Model ◽  
Logarithmic Norm ◽  
Method Of Differential Inequalities

There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.

Stability and exponential convergence of continuous-time Markov chains

2003 ◽  
Vol 40 (04) ◽  
pp. 970-979 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Exponential Convergence ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
The Relationship

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

Conditions for strong ergodicity using intensity matrices

1988 ◽  
Vol 25 (1) ◽  
pp. 34-42 ◽  
Author(s):  
Jean Johnson ◽  
Dean Isaacson
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Limiting Behavior ◽  
Strong Ergodicity ◽  
Continuous Time Markov Chains

Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψn of Pn(ψ nPn = ψ n) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.

A Note on External Uniformization for Finite Markov Chains in Continuous Time

1992 ◽  
Vol 6 (1) ◽  
pp. 127-131 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Strong Law ◽  
Large Numbers ◽  
Finite Markov Chains

An external uniformization technique was developed by Ross [4] to obtain approximations of transition probabilities of finite Markov chains in continuous time. Yoon and Shanthikumar [7] then reported through extensive numerical experiments that this technique performs quite well compared to other existing methods. In this paper, we show that external uniformization results from the strong law of large numbers whose underlying distributions are exponential. Based on this observation, some remarks regarding properties of the approximation are given.

Spectral representations of the transition probability matrices for continuous time finite Markov chains

1996 ◽  
Vol 33 (1) ◽  
pp. 28-33 ◽  
Author(s):  
Nan Fu Peng
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Probability ◽  
Algebraic Method ◽  
Finite State ◽  
Transition Probability Matrices ◽  
Finite Markov Chains ◽  
Probability Matrices ◽  
Spectral Representations

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

Spectral representations of the transition probability matrices for continuous time finite Markov chains

1996 ◽  
Vol 33 (01) ◽  
pp. 28-33 ◽  
Author(s):  
Nan Fu Peng
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Probability ◽  
Algebraic Method ◽  
Finite State ◽  
Transition Probability Matrices ◽  
Finite Markov Chains ◽  
Probability Matrices ◽  
Spectral Representations

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

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