On ergodicity and recurrence properties of a Markov chain by an application to an open jackson network

1992 ◽  
Vol 24 (2) ◽  
pp. 343-376 ◽  
Author(s):  
Arie Hordijk ◽  
Flora Spieksma
Keyword(s):  
Markov Chain ◽  
Product Form ◽  
Geometric Ergodicity ◽  
Two Dimensional ◽  
Marginal Distributions ◽  
Jackson Network ◽  
Strong Ergodicity ◽  
Key Theorem ◽  
Stieltjes Transforms

This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are called μ -geometric ergodicity and μ -geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows that μ -geometric ergodicity is equivalent to weak μ -geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain is μ -geometrically and geometrically ergodic, but not strongly ergodic. A consequence of μ -geometric ergodicity with μ of product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.

On ergodicity and recurrence properties of a Markov chain by an application to an open jackson network

1992 ◽  
Vol 24 (02) ◽  
pp. 343-376 ◽  
Author(s):  
Arie Hordijk ◽  
Flora Spieksma
Keyword(s):  
Markov Chain ◽  
Product Form ◽  
Geometric Ergodicity ◽  
Two Dimensional ◽  
Marginal Distributions ◽  
Jackson Network ◽  
Strong Ergodicity ◽  
Key Theorem ◽  
Stieltjes Transforms

This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are calledμ-geometric ergodicity andμ-geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows thatμ-geometric ergodicity is equivalent to weakμ-geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain isμ-geometrically and geometrically ergodic, but not strongly ergodic. A consequence ofμ-geometric ergodicity withμof product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.

scholarly journals Jackson networks with unlimited supply of work

2005 ◽  
Vol 42 (03) ◽  
pp. 879-882 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Joint Distribution ◽  
Product Form ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Marginal Distributions ◽  
Flow Rates ◽  
Jackson Network ◽  
Jackson Networks ◽  
Unlimited Supply ◽  
Infinite Supply

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

Geometric Ergodicity of the ALOHA-system and a Coupled Processors Model

1991 ◽  
Vol 5 (1) ◽  
pp. 15-42 ◽  
Author(s):  
F. M. Spieksma
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Geometric Ergodicity ◽  
Two Dimensional ◽  
Necessary And Sufficient ◽  
Stieltjes Transforms ◽  
Coupled Processors

μ-Geometric ergodicity of two-dimensional versions of the ALOHA and coupled processors models is verified by checking μ-geometric recurrence. Ergodicity and convergence of the Laplace-Stieltjes transforms in a neighborhood of 0 are necessary and sufficient conditions for the first model. The second model is exponential, for which ergodicity suffices to establish the required results.

scholarly journals Jackson networks with unlimited supply of work

2005 ◽  
Vol 42 (3) ◽  
pp. 879-882 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Joint Distribution ◽  
Product Form ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Marginal Distributions ◽  
Flow Rates ◽  
Jackson Network ◽  
Jackson Networks ◽  
Unlimited Supply ◽  
Infinite Supply

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

Strong ergodicity for continuous-time, non-homogeneous Markov chains

1982 ◽  
Vol 19 (3) ◽  
pp. 692-694 ◽  
Author(s):  
Mark Scott ◽  
Barry C. Arnold ◽  
Dean L. Isaacson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Homogeneous Markov Chain ◽  
Strong Ergodicity

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

Matrix product-form solutions for Markov chains with a tree structure

1994 ◽  
Vol 26 (04) ◽  
pp. 965-987 ◽  
Author(s):  
Raymond W. Yeung ◽  
Bhaskar Sengupta
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Closed Form Solution ◽  
Steady State Solution ◽  
Form Solution ◽  
Product Form ◽  
Matrix Product ◽  
Preemptive Resume ◽  
Finite Set ◽  
Markov Modulated

We have two aims in this paper. First, we generalize the well-known theory of matrix-geometric methods of Neuts to more complicated Markov chains. Second, we use the theory to solve a last-come-first-served queue with a generalized preemptive resume (LCFS-GPR) discipline. The structure of the Markov chain considered in this paper is one in which one of the variables can take values in a countable set, which is arranged in the form of a tree. The other variable takes values from a finite set. Each node of the tree can branch out into d other nodes. The steady-state solution of this Markov chain has a matrix product-form, which can be expressed as a function of d matrices Rl,· ··, Rd. We then use this theory to solve a multiclass LCFS-GPR queue, in which the service times have PH-distributions and arrivals are according to the Markov modulated Poisson process. In this discipline, when a customer's service is preempted in phase j (due to a new arrival), the resumption of service at a later time could take place in a phase which depends on j. We also obtain a closed form solution for the stationary distribution of an LCFS-GPR queue when the arrivals are Poisson. This result generalizes the known result on a LCFS preemptive resume queue, which can be obtained from Kelly's symmetric queue.

Two-dimensional Markov Chain Simulation of Soil Type Spatial Distribution

2004 ◽  
Vol 68 (5) ◽  
pp. 1479-1490 ◽  
Author(s):  
Weidong Li ◽  
Chuanrong Zhang ◽  
James E. Burt ◽  
A.-Xing Zhu ◽  
Jan Feyen
Keyword(s):  
Spatial Distribution ◽  
Markov Chain ◽  
Soil Type ◽  
Two Dimensional

Full Two-Dimensional Markov Chain Analysis of Thermal Soft Errors in Subthreshold Nanoscale CMOS Devices

2011 ◽  
Vol 11 (1) ◽  
pp. 50-59 ◽  
Author(s):  
Pooya Jannaty ◽  
Florian Cosmin Sabou ◽  
R. Iris Bahar ◽  
Joseph Mundy ◽  
William R. Patterson ◽  
...  
Keyword(s):  
Markov Chain ◽  
Soft Errors ◽  
Two Dimensional ◽  
Markov Chain Analysis ◽  
Nanoscale Cmos ◽  
Chain Analysis

Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/n queues

2004 ◽  
Vol 41 (03) ◽  
pp. 778-790
Author(s):  
Zhenting Hou ◽  
Yuanyuan Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Lyapunov Function ◽  
Markov Chains ◽  
Usual Method ◽  
Geometric Ergodicity ◽  
Return Probability ◽  
Drift Condition ◽  
Polynomial Ergodicity ◽  
Explicit Criteria

This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.

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