scholarly journals Ergodicity conditions and invariant probability measure for an embedded Markov chain in a controlled bulk queueing system with a bilevel service delay discipline part I

1992 ◽  
Vol 5 (4) ◽  
pp. 25-27 ◽  
Author(s):  
Lev Abolnikov ◽  
Jewengi H. Dshalalow
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Queueing System ◽  
Invariant Probability Measure ◽  
Embedded Markov Chain ◽  
Service Delay

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters

2004 ◽  
Vol 41 (02) ◽  
pp. 547-556 ◽  
Author(s):  
Alexander Dudin ◽  
Olga Semenova
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Queueing System ◽  
Embedded Markov Chain ◽  
State Distribution ◽  
Stable Algorithm ◽  
Distribution Calculation

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

GEOMETRIC-FORM BOUNDS FOR THE GIX/M/1 QUEUEING SYSTEM

1999 ◽  
Vol 13 (4) ◽  
pp. 509-520 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Queueing System ◽  
Embedded Markov Chain ◽  
Geometric Form ◽  
Stationary Distributions ◽  
Discrete Time Markov Chain ◽  
Numerical Studies ◽  
Simple Product

The GI/M/1 queueing system was long ago studied by considering the embedded discrete-time Markov chain at arrival epochs and was proved to have remarkably simple product-form stationary distributions both at arrival epochs and in continuous time. Although this method works well also in several variants of this system, it breaks down when customers arrive in batches. The resulting GIX/M/1 system has no tractable stationary distribution. In this paper we use a recent result of Miyazawa and Taylor (1997) to obtain a stochastic upper bound for the GIX/M/1 system. We also introduce a class of continuous-time Markov chains which are related to the original GIX/M/1 embedded Markov chain that are shown to have modified geometric stationary distributions. We use them to obtain easily computed stochastic lower bounds for the GIX/M/1 system. Numerical studies demonstrate the quality of these bounds.

A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters

2004 ◽  
Vol 41 (2) ◽  
pp. 547-556 ◽  
Author(s):  
Alexander Dudin ◽  
Olga Semenova
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Queueing System ◽  
Embedded Markov Chain ◽  
State Distribution ◽  
Stable Algorithm ◽  
Distribution Calculation

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

scholarly journals Queueing system with passive servers

1996 ◽  
Vol 9 (2) ◽  
pp. 185-204 ◽  
Author(s):  
Alexander N. Dudin ◽  
Valentina I. Klimenok
Keyword(s):  
Functional Equation ◽  
Markov Chain ◽  
Response Time ◽  
Stationary Distribution ◽  
Random Environment ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Embedded Markov Chain ◽  
Matrix Analog

In this paper the authors introduce systems in which customers are served by one active server and a group of passive servers. The calculation of response time for such systems is rendered by analyzing a special kind of queueing system in a synchronized random environment. For an embedded Markov chain, sufficient conditions for the existence of a stationary distribution are proved. A formula for the corresponding vector generating function is obtained. It is a matrix analog of the Pollaczek-Khinchin formula and is simultaneously a matrix functional equation. A method for solving this equation is proposed.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (1) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (01) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

An Unreliable Quorum Queueing System with Single and Multiple Vacations

2017 ◽  
Vol 34 (06) ◽  
pp. 1750036
Author(s):  
Lotfi Tadj
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Queueing System ◽  
System Size ◽  
Embedded Markov Chain ◽  
Queueing Models ◽  
Single Server ◽  
Multiple Vacations ◽  
Service Completion

This paper contributes to the literature of single server queueing models with a vacationing server. We have incorporated many features for a better control over the system. The server implements the N-policy, takes both single and multiple vacations, and is subject to breakdowns. The embedded Markov chain technique is used to obtain the pgf of the system size at a service completion epoch in the steady-state. The semi-regenerative technique is used to obtain the pgf of the system size at an arbitrary instant of time in the steady-state.

scholarly journals A non-Markovian queueing system with a variable number of channels

2003 ◽  
Vol 16 (4) ◽  
pp. 375-395
Author(s):  
Hong-Tham T. Rosson ◽  
Jewgeni H. Dshalalow
Keyword(s):  
Queueing System ◽  
Invariant Probability Measure ◽  
Variable Number ◽  
Time Parameter ◽  
Embedded Markov Chain ◽  
Supplementary Variable ◽  
Variable Technique ◽  
Queueing Process ◽  
Variable Approach ◽  
Parallel Channels

In this paper we study a queueing model of type GI/M/m˜a/∞ with m parallel channels, some of which may suspend their service at specified random moments of time. Whether or not this phenomenon occurs depends on the queue length. The queueing process, which we target, turns out to be semi-regenerative, and we fully explore this utilizing semi-regenerative techniques. This is contrary to the more traditional supplementary variable approach and the less popular approach of combination semi-regenerative and supplementary variable technique. We pass to the limiting distribution of the continuous time parameter process through the embedded Markov chain for which we find the invariant probability measure. All formulas are analytically tractable.

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