Insensitive bounds for the stationary distribution of non-reversible Markov chains

1988 ◽  
Vol 25 (01) ◽  
pp. 9-20 ◽  
Author(s):  
Arie Hordijk ◽  
AD Ridder
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Service Time ◽  
Form Solution ◽  
Tree Form ◽  
Standard Product ◽  
Reversible Markov Chains ◽  
Stationary Probabilities ◽  
Networks Of Queues ◽  
General Method

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

1988 ◽  
Vol 25 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Arie Hordijk ◽  
AD Ridder
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Service Time ◽  
Form Solution ◽  
Tree Form ◽  
Standard Product ◽  
Reversible Markov Chains ◽  
Stationary Probabilities ◽  
Networks Of Queues ◽  
General Method

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

A Note on Queueing Networks with Signals and Random Triggering Times

1994 ◽  
Vol 8 (2) ◽  
pp. 213-219 ◽  
Author(s):  
Xiuli Chao
Keyword(s):  
Stationary Distribution ◽  
Service Time ◽  
Queueing Networks ◽  
Time Distribution ◽  
Network Models ◽  
Form Solution ◽  
Service Time Distribution ◽  
Signal Changes ◽  
Networks Of Queues ◽  
Random Amount

There is a growing interest in networks of queues with customers and signals. The signals in these models carry commands to the service nodes and trigger customers to move instantaneously within the network. In this note we consider networks of queues with signals and random triggering times; that is, when a signal arrives at a node, it takes a random amount of time to trigger a customer to move with distribution depending on the source of the signal. By appropriately choosing the triggering times, we can obtain network models such that a signal changes a customer's service time distribution – for example, the signal increases or decreases the service time of a customer. We show that the stationary distribution of this model has a product form solution.

scholarly journals On Approximating the Stationary Distribution of Time-Reversible Markov Chains

2019 ◽  
Vol 64 (3) ◽  
pp. 444-466 ◽  
Author(s):  
Marco Bressan ◽  
Enoch Peserico ◽  
Luca Pretto
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Reversible Markov Chains

The insensitivity of stationary probabilities in networks of queues

1978 ◽  
Vol 10 (04) ◽  
pp. 906-912 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Stochastic Processes ◽  
Service Time ◽  
General Class ◽  
Special Cases ◽  
Stationary Probabilities ◽  
Networks Of Queues

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.

The insensitivity of stationary probabilities in networks of queues

1978 ◽  
Vol 10 (4) ◽  
pp. 906-912 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Stochastic Processes ◽  
Service Time ◽  
General Class ◽  
Special Cases ◽  
Stationary Probabilities ◽  
Networks Of Queues

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.

Geometric Approaches to the Estimation of the Spectral Gap of Reversible Markov Chains

1993 ◽  
Vol 2 (3) ◽  
pp. 301-323 ◽  
Author(s):  
Salvatore Ingrassia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Spectral Gap ◽  
Systematic Treatment ◽  
Reversible Markov Chain ◽  
Underlying Graph ◽  
Reversible Markov Chains

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohamed Boualem ◽  
Natalia Djellab ◽  
Djamil Aïssani
Keyword(s):  
Markov Chains ◽  
Service Time ◽  
Retrial Queue ◽  
Stochastic Ordering ◽  
Stochastic Comparison ◽  
Single Server ◽  
Stochastic Approximations ◽  
Bernoulli Feedback ◽  
Monotonicity Results

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for anM/G/1retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

Geometric Markov chains

1995 ◽  
Vol 32 (02) ◽  
pp. 349-374
Author(s):  
William Rising
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Nearest Neighbor ◽  
First Passage Times ◽  
First Passage ◽  
Death Rates ◽  
Passage Times ◽  
Birth Death

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.

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