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.

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.

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.

scholarly journals A Combinatorial Derivation of the PASEP Stationary State

10.37236/1134 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Richard Brak ◽  
Sylvie Corteel ◽  
John Essam ◽  
Robert Parviainen ◽  
Andrew Rechnitzer
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Exclusion Process ◽  
Direct Proof ◽  
Asymmetric Exclusion Process ◽  
Original Process ◽  
Combinatorial Derivation ◽  
Path Models ◽  
Asymmetric Exclusion

We give a combinatorial derivation and interpretation of the weights associated with the stationary distribution of the partially asymmetric exclusion process. We define a set of weight equations, which the stationary distribution satisfies. These allow us to find explicit expressions for the stationary distribution and normalisation using both recurrences and path models. To show that the stationary distribution satisfies the weight equations, we construct a Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original process and that the stationary distribution on this subset is simply related to that of the original chain. We also provide a direct proof of the validity of the weight equations.

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.

On a Single Server Queue

1965 ◽  
Vol 14 (3-4) ◽  
pp. 163-166 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Stationary State ◽  
Waiting Time ◽  
Queueing System ◽  
Alternative Proof ◽  
Single Server ◽  
Single Server Queue ◽  
State Distribution ◽  
Simple Method ◽  
Server Queue

Summary: In this note all alternative proof of Lindley's (1952) theorem for the existence of the stationary state distribution of the waiting time of a customer in the queueing system GI/G/1 is given. Further, a direct and simple method is given for deriving the stationary state distribution of the waiting time in the queueing system GI/M/1.

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.

scholarly journals Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

2019 ◽  
Vol 29 (3) ◽  
pp. 375-391
Author(s):  
Lala Alem ◽  
Mohamed Boualem ◽  
Djamil Aissani
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Performance Measures ◽  
Time Distribution ◽  
Retrial Queue ◽  
Main Idea ◽  
Embedded Markov Chain ◽  
Stochastic Comparison ◽  
Service Time Distribution ◽  
Comparison Approach

In this article we analyze the M=G=1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M=M=1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

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