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

1999 â—½  
Vol 13 (4) â—½  
pp. 509-520 â—½  
Author(s):  
Antonis Economou
Keyword(s):  
Markov Chain â—½  
Continuous Time â—½  
Queueing System â—½  
Embedded Markov Chain â—½  
Geometric Form â—½  
Stationary Distributions â—½  
Discrete Time Markov Chain â—½  
Numerical Studies â—½  
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 â—½  
Vol 41 (02) â—½  
pp. 547-556 â—½  
Author(s):  
Alexander Dudin â—½  
Olga Semenova
Keyword(s):  
Markov Chain â—½  
Stationary State â—½  
Stationary Distribution â—½  
Queueing System â—½  
Embedded Markov Chain â—½  
State Distribution â—½  
Stable Algorithm â—½  
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 Analysis of Multiserver Retrial Queueing System with Varying Capacity and Parameters

10.1155/2015/180481 â—½  
2015 â—½  
Vol 2015 â—½  
pp. 1-12 â—½  
Author(s):  
Alexander N. Dudin â—½  
Olga S. Dudina
Keyword(s):  
Markov Chain â—½  
Continuous Time â—½  
Queueing System â—½  
Optimization Problems â—½  
The State â—½  
Arrival Process â—½  
Continuous Time Markov Chain â—½  
Retrial Queueing System â—½  
Retrial Queueing â—½  
System States

A multiserver queueing system, the dynamics of which depends on the state of some external continuous-time Markov chain (random environment, RE), is considered. Change of the state of the RE may cause variation of the parameters of the arrival process, the service process, the number of available servers, and the available buffer capacity, as well as the behavior of customers. Evolution of the system states is described by the multidimensional continuous-time Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for the key performance measures are given. Numerical results illustrating the behavior of the system and showing possibility of formulation and solution of optimization problems are provided. The importance of the account of correlation in the arrival processes is numerically illustrated.

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

2004 â—½  
Vol 41 (2) â—½  
pp. 547-556 â—½  
Author(s):  
Alexander Dudin â—½  
Olga Semenova
Keyword(s):  
Markov Chain â—½  
Stationary State â—½  
Stationary Distribution â—½  
Queueing System â—½  
Embedded Markov Chain â—½  
State Distribution â—½  
Stable Algorithm â—½  
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.

The Setting Method of Project Buffer in Critical Chain Management of Mould Manufacturing Project Based Reworking

2011 â—½  
Vol 317-319 â—½  
pp. 418-422 â—½  
Author(s):  
Chang Wei Hu â—½  
Xin Du Chen â—½  
Li Hua Wu
Keyword(s):  
Markov Chain â—½  
Discrete Time â—½  
Classical Method â—½  
Discrete Time Markov Chain â—½  
Chain Management â—½  
Large Numbers â—½  
Quality Of Products â—½  
Critical Chain â—½  
Customer Demand Change

In the mould manufacturing projects, there are large numbers of uncertainties, such as change of customer demand, change of drawing, bad quality of products etc., and they often lead to numerous task reworked. Because of the random task reworking, the complete time of the project is uncertainty. Based on the critical chain project with task reworking, the paper carried out the research on setting of project buffer. We presented a new idea that the project buffer was calculated based on safety time and possible reworking time, and proposed a method to calculate the buffer of project reworking based on Discrete-time Markov chain. The problem that the classical method to setting critical chain project buffer is infeasible concerning task reworking was solved efficiently. The proposed method can set the project buffer more reasonable.

An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains by Sparse Transition Structure

1994 â—½  
Vol 26 (01) â—½  
pp. 68-79
Author(s):  
P. K. Pollett â—½  
D. E. Stewart
Keyword(s):  
Markov Chain â—½  
Continuous Time â—½  
Computational Procedure â—½  
Transition Rates â—½  
Continuous Time Markov Chain â—½  
Transition Structure â—½  
Stationary Distributions â—½  
Efficient Procedure â—½  
The Matrix â—½  
Quasi Stationary Distribution

We describe a computational procedure for evaluating the quasi-stationary distributions of a continuous-time Markov chain. Our method, which is an ‘iterative version' of Arnoldi's algorithm, is appropriate for dealing with cases where the matrix of transition rates is large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. We illustrate the method with reference to an epidemic model and we compare the computed quasi-stationary distribution with an appropriate diffusion approximation.

An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains by Sparse Transition Structure

10.2307/1427580 â—½  
1994 â—½  
Vol 26 (1) â—½  
pp. 68-79 â—½  
Author(s):  
P. K. Pollett â—½  
D. E. Stewart
Keyword(s):  
Markov Chain â—½  
Continuous Time â—½  
Computational Procedure â—½  
Transition Rates â—½  
Continuous Time Markov Chain â—½  
Transition Structure â—½  
Stationary Distributions â—½  
Efficient Procedure â—½  
The Matrix â—½  
Quasi Stationary Distribution

We describe a computational procedure for evaluating the quasi-stationary distributions of a continuous-time Markov chain. Our method, which is an ‘iterative version' of Arnoldi's algorithm, is appropriate for dealing with cases where the matrix of transition rates is large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. We illustrate the method with reference to an epidemic model and we compare the computed quasi-stationary distribution with an appropriate diffusion approximation.

scholarly journals Queueing system with passive servers

1996 â—½  
Vol 9 (2) â—½  
pp. 185-204 â—½  
Author(s):  
Alexander N. Dudin â—½  
Valentina I. Klimenok
Keyword(s):  
Functional Equation â—½  
Markov Chain â—½  
Response Time â—½  
Stationary Distribution â—½  
Random Environment â—½  
Queueing System â—½  
Sufficient Conditions â—½  
Special Kind â—½  
Embedded Markov Chain â—½  
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 Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

10.1017/apr.2021.20 â—½  
2022 â—½  
pp. 1-47
Author(s):  
Amarjit Budhiraja â—½  
Nicolas Fraiman â—½  
Adam Waterbury
Keyword(s):  
Markov Chain â—½  
Dynamical Systems â—½  
Markov Chains â—½  
State Space â—½  
Continuous Time â—½  
Stochastic Dynamical Systems â—½  
Stationary Distributions â—½  
Small Noise â—½  
Positive Orthant â—½  
Limit Points

Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

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