Transience, Recurrence, and Harris Recurrence

2018 ◽  
pp. 221-239
Author(s):  
Randal Douc ◽  
Eric Moulines ◽  
Pierre Priouret ◽  
Philippe Soulier
Keyword(s):  
Harris Recurrence

scholarly journals Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains

2006 ◽  
Vol 16 (4) ◽  
pp. 2123-2139 ◽  
Author(s):  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal
Keyword(s):  
Markov Chains ◽  
Harris Recurrence

scholarly journals On the saturation rule for the stability of queues

1995 ◽  
Vol 32 (02) ◽  
pp. 494-507 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss
Keyword(s):  
Queueing Networks ◽  
Stability Region ◽  
Queueing Systems ◽  
Main Tool ◽  
Stationary Regime ◽  
Arrival Process ◽  
State Variables ◽  
Arrival Times ◽  
Harris Recurrence ◽  
The Stability

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.

On non-singular Markov renewal processes with an application to a growth–catastrophe model

1985 ◽  
Vol 22 (02) ◽  
pp. 253-266
Author(s):  
Seppo Niemi
Keyword(s):  
Markov Process ◽  
Total Variation ◽  
Population Model ◽  
Limit Theorems ◽  
Renewal Processes ◽  
Regenerative Processes ◽  
Catastrophe Model ◽  
Markov Renewal Processes ◽  
Harris Recurrence ◽  
Forward Process

The paper is concerned with Markov renewal processes satisfying a certain non-singularity condition. The relation of this condition to irreducibility, Harris recurrence and regularity of the associated forward Markov process is studied. This enables one to prove limit theorems of a total variation type for Markov renewal processes and semi-regenerative processes by applying Orey's theorem to the forward process. The results are applied to a GI/G/1 queue and a growth-catastrophe population model.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

1995 ◽  
Vol 27 (2) ◽  
pp. 567-583 ◽  
Author(s):  
John S. Sadowsky
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Waiting Times ◽  
General Setting ◽  
Batch Arrival ◽  
Positive Recurrence ◽  
Waiting Time Distributions ◽  
Harris Recurrence ◽  
Stationary Queue Length ◽  
Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

1993 ◽  
Vol 25 (3) ◽  
pp. 518-548 ◽  
Author(s):  
Sean P. Meyn ◽  
R. L. Tweedie
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Linear Feedback ◽  
Test Function ◽  
Markovian Processes ◽  
Harris Recurrence ◽  
Lyapunov Inequalities ◽  
Continuous Time Processes ◽  
Risk Processes ◽  
Stability Concepts

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator.Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula.We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case.We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

scholarly journals Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels

2019 ◽  
Vol 23 ◽  
pp. 770-796 ◽  
Author(s):  
Aline Duarte ◽  
Eva Löcherbach ◽  
Guilherme Ost
Keyword(s):  
Wasserstein Distance ◽  
Convergence To Equilibrium ◽  
Thinning Algorithm ◽  
Hawkes Processes ◽  
Lipschitz Continuous ◽  
Piecewise Deterministic Markov Processes ◽  
Harris Recurrence ◽  
Non Linear ◽  
Rate Functions ◽  
With Memory

Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

1993 ◽  
Vol 25 (03) ◽  
pp. 518-548 ◽  
Author(s):  
Sean P. Meyn ◽  
R. L. Tweedie
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Linear Feedback ◽  
Test Function ◽  
Markovian Processes ◽  
Harris Recurrence ◽  
Lyapunov Inequalities ◽  
Continuous Time Processes ◽  
Risk Processes ◽  
Stability Concepts

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

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