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18
On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models
 Annals of Applied Probability
, 1995
"... It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified ..."
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Cited by 223 (17 self)
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It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and firstbufferfirstserved preemptive resume discipline in a reentrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...
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 78 (31 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...
Drift transforms and Green function estimates for discontinuous processes
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2003
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Generalized Resolvents and Harris Recurrence of Markov Processes
, 1992
"... In this paper we consider a #irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedd ..."
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Cited by 24 (15 self)
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In this paper we consider a #irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedded chains with arbitrary sampling distributions. We show that the recurrence behavior of such generalized resolvents classifies the behavior of the continuous time process; from this we prove that hitting times on the small sets of a generalized resolvent chain provide criteria for, successively, (i) Harris recurrence of # (ii) the existence of an invariant probability measure # (or positive Harris recurrence of #) and (iii) the finiteness of #(f) for arbitrary f.
Gaugeability and Conditional Gaugeability
 TRANS. AMER. MATH. SOC
, 2001
"... New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing ..."
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Cited by 21 (5 self)
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New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. for transient Borel standard processes having strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.
General gauge and conditional gauge theorems
 Ann. Probab
, 2002
"... General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge an ..."
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Cited by 21 (13 self)
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General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge and conditional gauge theorems hold. These classes are larger than the classical Kato class when the process is Brownian motion in a bounded C 1,1 domain. 1. Introduction. Given a strong Markov process X and a potential q, the conditional expectation u(x, y) of the Feynman–Kac transform of X by q is called the conditional gauge function. (The precise definition will be given later.) The function u is important in studying the potential theory of the Schrödingertype operator L + q, as it is the ratio of the Green’s function of L + q and that
Network Adiabatic Theorem: An Efficient Randomized Protocol for Contention Resolution
"... The popularity of Aloha(like) algorithms for resolution of contention between multiple entities accessing common resources is due to their extreme simplicity and distributed nature. Example applications of such algorithms include Ethernet and recently emerging wireless multiaccess networks. Despit ..."
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Cited by 17 (5 self)
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The popularity of Aloha(like) algorithms for resolution of contention between multiple entities accessing common resources is due to their extreme simplicity and distributed nature. Example applications of such algorithms include Ethernet and recently emerging wireless multiaccess networks. Despite a long and exciting history of more than four decades, the question of designing an algorithm that is essentially as simple and distributed as Aloha while being efficient has remained unresolved. In this paper, we resolve this question successfully for a network of queues where contention is modeled through independentset constraints over the network graph. The work by Tassiulas and Ephremides (1992) suggests that an algorithm that schedules queues so that the summation of “weight ” of scheduled queues is maximized, subject to constraints, is efficient. However, implementing such an algorithm using Alohalike mechanism has remained a mystery. We design such an algorithm building upon a MetropolisHastings sampling mechanism along with selection of“weight” as an appropriate function of the queuesize. The key ingredient in establishing the efficiency of the algorithm is a novel adiabaticlike theorem for the underlying queueing network, which may be of general interest in the context of dynamical systems.
Conditional gauge theorem for nonlocal FeynmanKac transforms
 PROBAB. THEORY RELAT. FIELDS
, 2003
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G.O.Roberts, Subgeometric ergodicity of strong Markov processes. Ann.Appl.Probab
, 2005
"... We derive sufficient conditions for subgeometric fergodicity of strongly Markovian processes. We first propose a criterion based on modulated moment of some delayed returntime to a petite set. We then formulate a criterion for polynomial fergodicity in terms of a drift condition on the generator. ..."
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Cited by 6 (0 self)
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We derive sufficient conditions for subgeometric fergodicity of strongly Markovian processes. We first propose a criterion based on modulated moment of some delayed returntime to a petite set. We then formulate a criterion for polynomial fergodicity in terms of a drift condition on the generator. Applications to specific processes are considered, including Langevin tempered diffusions on R n and storage models.
Randomized Scheduling Algorithm for Queueing Networks. Arxiv preprint arXiv:0908.3670
, 2009
"... There has recently been considerable interest in design of lowcomplexity, myopic, distributed and stable scheduling policies for constrained queueing network models that arise in the context of emerging communication networks. Here, we consider two representative models. One, a model for the collect ..."
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Cited by 4 (3 self)
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There has recently been considerable interest in design of lowcomplexity, myopic, distributed and stable scheduling policies for constrained queueing network models that arise in the context of emerging communication networks. Here, we consider two representative models. One, a model for the collection of wireless nodes communicating through a shared medium, that represents randomly varying number of packets in the queues at the nodes of networks. Two, a buffered circuit switched network model for an optical core of future Internet, to capture the randomness in calls or flows present in the network. The maximum weight scheduling policy proposed by Tassiulas and Ephremide [32] leads to a myopic and stable policy for the packetlevel wireless network model. But computationally it is very expensive (NPhard) and centralized. It is not applicable to the buffered circuit switched network due to the requirement of nonpremption of the calls in the service. As the main contribution of this paper, we present a stable scheduling algorithm for both of these models. The algorithm is myopic, distributed and performs few logical operations at each node per unit time. 1. Introduction. The