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152
Stability of Markovian processes III: FosterLyapunov criteria for continuoustime processes
 Adv. Appl. Prob
, 1993
"... In Part I we developed stability concepts for discrete chains, together with FosterLyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuoustime processes. In this paper we develop criteria for these forms of stability for continuousparameter ..."
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Cited by 145 (23 self)
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In Part I we developed stability concepts for discrete chains, together with FosterLyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuoustime processes. In this paper we develop criteria for these forms of stability for continuousparameter Markovian processes on general state spaces, based on FosterLyapunov inequalities for the extended generator. Such test function criteria are found for nonexplosivity, nonevanescence, 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 continuoustime processes. In particular we are able to show that the test function approach provides a criterion for fnorm 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 nonlinear feedback, workmodulated queues, general release storage processes and risk processes.
Stability and Convergence of Moments for Multiclass Queueing Networks via Fluid Limit Models
 IEEE Transactions on Automatic Control
, 1995
"... The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at ..."
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Cited by 113 (37 self)
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The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including reentrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications. Keywords: Multiclass queueing networks, ergodicity, general state space Markov processes, polling models, generalized Jackson networks, stability, performance analysis. 1 Introduction The subject of this paper is open multiclass queueing networks, which are models of complex systems such as wafer fabri...
The stochastic randomcluster process and the uniqueness of randomcluster measures
, 1995
"... The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated phy ..."
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Cited by 98 (14 self)
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The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey randomcluster measures from the probabilist’s point of view, giving clear statements of some of the many open problems. Secondly, we present new results for such measures, as follows. We discuss the relationship between weak limits of randomcluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of randomcluster measures for all but (at most) countably many values of the parameter p. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way, in part of these arguments. In the second part of this paper is constructed a Markov process whose levelsets are reversible Markov processes with randomcluster measures as unique equilibrium measures. This construction enables a coupling of randomcluster measures for all values of p. Furthermore it leads to a proof of the semicontinuity of the percolation probability, and provides a heuristic probabilistic justification for the widely held belief that there is a firstorder phase transition if and only if the clusterweighting factor q is sufficiently large.
Estimates on Green functions and Poisson kernels of symmetric stable processes
, 1998
"... for symmetric stable processes ..."
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Exponential and Uniform Ergodicity of Markov Processes
 Ann. Probab
, 1995
"... Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially ..."
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Cited by 78 (12 self)
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Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially bounded hitting times on one and then all suitably "small" sets; (b) the existence of "FosterLyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; (c) the existence of drift conditions on the extended generator e A of the process. We use the identity e AR fi = fi(R fi \Gamma I) connecting the extended generator and the resolvent kernels R fi , to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of (a)(c). These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state ...
A Lyapunov Bound for Solutions of Poisson's Equation
 Ann. Probab
, 1996
"... In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which ..."
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Cited by 67 (28 self)
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In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the solution is square integrable. These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to Poisson's equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes. AMS subject classifications: 68M20, 60J10 Running head: Poisson's Equation Keywords: Markov chain, Markov process, Poisson's equation, Lyapunov f...
Drift transforms and Green function estimates for discontinuous processes
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2003
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Fast Equilibrium Selection by Rational Players Living in a Changing World
 Econometrica
, 1996
"... We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 ..."
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Cited by 52 (9 self)
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We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 \Theta 2 game. The payoff matrix changes over time according to some Brownian motion. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. We solve the game using only backwards induction. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk dominant equilibrium. History dependence emerges in some cases when frictions remain positive. As an application we show how frictions and aggregate cost shocks can lead to the selecti...
Harnack inequality for some classes of Markov processes
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. ..."
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Cited by 46 (14 self)
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In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.
Stability of queueing networks
 Lecture Notes in Mathematics
, 2006
"... Milan Paris Tokyo2 This manuscript will appear as Springer Lecture Notes in Mathematics 1950, École d ’ Été de Probabilités de SaintFlour XXXVI2006. ..."
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Cited by 45 (1 self)
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Milan Paris Tokyo2 This manuscript will appear as Springer Lecture Notes in Mathematics 1950, École d ’ Été de Probabilités de SaintFlour XXXVI2006.