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.

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.

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.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Insensitive Bounds for the Stationary Distribution of a Single Server Retrial Queue with Server Subject to Active Breakdowns

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Mohamed Boualem
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Retrial Queue ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Waiting Room ◽  
Monotonicity Properties

The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results.

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.

scholarly journals M/M/1 Model with Unreliable Service

2017 ◽  
Vol 7 (1) ◽  
pp. 125
Author(s):  
Joshua Patterson, Andrzej Korzeniowski
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Queue Length ◽  
Sufficient Conditions ◽  
Probability Generating Function ◽  
Embedded Markov Chain ◽  
Queueing Models ◽  
Stieltjes Transform ◽  
Stationary Queue Length ◽  
Stationary Waiting Time

We define a new term ”unreliable service” and construct the corresponding embedded Markov Chain to an M/M/1 queue with so defined protocol. Sufficient conditions for positive recurrence and closed form of stationary distribution are provided. Furthermore, we compute the probability generating function of the stationary queue length and Laplace-Stieltjes transform of the stationary waiting time. In the course of the analysis an interesting decomposition of both the queue length and waiting time has emerged. A number of queueing models can be recovered from our work by taking limits of certain parameters.

scholarly journals Phase Enlargement of Semi-Markov Systems without Determining Stationary Distribution of Embedded Markov Chain

2020 ◽  
Vol 19 (3) ◽  
pp. 539-563
Author(s):  
Vadim Kopp ◽  
Mikhail  Zamoryonov ◽  
Nikita Chalenkov ◽  
Ivan Skatkov
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Stationary Distribution ◽  
Phase State ◽  
Random Variables ◽  
Embedded Markov Chain ◽  
Discrete State ◽  
Markov Systems ◽  
The Difference

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space.  Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (01) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

Continuous-time Markov chains in a random environment, with applications to ion channel modelling

1994 ◽  
Vol 26 (04) ◽  
pp. 919-946 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Ion Channel ◽  
Random Environment ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
The State ◽  
Continuous Time Markov Chain ◽  
Channel Process ◽  
Markov Reward

We study a bivariate stochastic process {X(t)} = Z(t))}, where {XE (t)} is a continuous-time Markov chain describing the environment and {Z(t)} is the process of interest. In the context which motivated this study, {Z(t)} models the gating behaviour of a single ion channel. It is assumed that given {XE (t)}, the channel process {Z(t)} is a continuous-time Markov chain with infinitesimal generator at time t dependent on XE (t), and that the environment process {XE {t)} is not dependent on {Z(t)}. We derive necessary and sufficient conditions for {X(t)} to be time reversible, showing that then its equilibrium distribution has a product form which reflects independence of the state of the environment and the state of the channel. In the special case when the environment controls the speed of the channel process, we derive transition probabilities and sojourn time distributions for {Z(t)} by exploiting connections with Markov reward processes. Some of these results are extended to a stationary environment. Applications to problems arising in modelling multiple ion channel systems are discussed. In particular, we present ways in which a multichannel model in a random environment does and does not exhibit behaviour identical to a corresponding model based on independent and identically distributed channels.

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