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57
Fast Simulation of SteadyState Availability in NonMarkovian Systems
, 1992
"... . This paper considers efficient simulation techniques for estimating steadystate quantities in models of highly dependable computing systems with general component failure and repair time distributions. Earlier approaches in this application setting for steadystate estimation rely on the regenera ..."
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Cited by 11 (2 self)
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. This paper considers efficient simulation techniques for estimating steadystate quantities in models of highly dependable computing systems with general component failure and repair time distributions. Earlier approaches in this application setting for steadystate estimation rely on the regenerative method of simulation, which can be used when the failure time distributions are exponentially distributed. However, when the failure times are generally distributed the regenerative structure is lost and a new approach must be taken. The approach we take is to exploit a ratio representation for steadystate quantities in terms of cycles that are no longer independent and identically distributed. A "splitting" technique is used in which importance sampling is used to speed up the simulation of rare system failure events during a cycle, and standard simulation is used to estimate the expected cycle length. Experimental results show that the method is effective in practice. Keywords. Relia...
Techniques for the Fast Simulation of Models of Highly Dependable Systems
 IEEE Transactions on Reliability
, 2001
"... this paper, we review some of the importancesampling techniques that have been developed in recent years to e#ciently estimate dependability measures in Markovian and nonMarkovian models of highly dependable systems. 1 Acronyms MTTF Mean time to failure. MTBF Mean time between failures. CTMC ..."
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Cited by 9 (0 self)
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this paper, we review some of the importancesampling techniques that have been developed in recent years to e#ciently estimate dependability measures in Markovian and nonMarkovian models of highly dependable systems. 1 Acronyms MTTF Mean time to failure. MTBF Mean time between failures. CTMC Continuoustime Markov chain. DTMC Discretetime Markov chain. GSMP Generalized semiMarkov process. SAVE System AVailability Estimator. CLT Central limit theorem. VRR Variance reduction ratio. TRR Total e#ort reduction ratio. MSDIS Measurespecific dynamic importance sampling. BLBLR Balance over links balanced likelihood ratio. BLBLRC Balance over links balanced likelihood ratio with cuts. 1 INTRODUCTION High dependability requirements of today's critical and/or commercial systems often lead to complicated and costly designs. The ability to predict relevant dependability measures for such complex systems is essential, not only to guarantee hig
Microeconomics as an Experimental
 Science,” American Economic Review
, 1982
"... We show that far from capturing a formally new phenomenon, informational herding is really a special case of singleperson experimentation — and ‘bad herds’ the typical failure of complete learning. We then analyze the analogous team equilibrium, where individuals maximize the present discounted wel ..."
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Cited by 8 (0 self)
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We show that far from capturing a formally new phenomenon, informational herding is really a special case of singleperson experimentation — and ‘bad herds’ the typical failure of complete learning. We then analyze the analogous team equilibrium, where individuals maximize the present discounted welfare of posterity. To do so, we generalize Gittins indices to our nonbandit learning problem, and thereby characterize when contrarian behaviour arises: (i) While herds are still constrained efficient, they arise for a strictly smaller belief set. (ii) A logconcave loglikelihood ratio density robustly ensures that individuals should lean more against their myopic preference for an action the more popular it becomes. We thank Patrick Bolton and three referees in guiding this radical revision, as well as Abhijit Banerjee, Katya Malinova, Meg Meyer, Christopher Wallace, and seminar participants at the MIT theory lunch, the
QUEUES WITH SERVER VACATIONS AND LEVY PROCESSES WITH SECONDARY JUMP INPUT
, 1990
"... Motivated by models of queues with server vacations, we consider a Levy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Le vy ´ process so that the overall Levy process with secondary jump input, can have a proper limiting distribution ..."
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Cited by 7 (7 self)
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Motivated by models of queues with server vacations, we consider a Levy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Le vy ´ process so that the overall Levy process with secondary jump input, can have a proper limiting distribution. For example, the workload process in an M/G/1 queue with a server vacation each time the server finds an empty system is such a Le vy ´ process with secondary jump input. We show that a certain functional of a Le vy ´ process with secondary jump input is a martingale and we apply this martingale to characterize the steadystate distribution. We establish stochastic decomposition results for the case in which the Levy process has no negative jumps, which extend and unify previous decomposition results for the workload process in the M/G/1 queue with server vacations and Brownian motion with secondary jump input. We also apply martingales to provide a new proof of the known simple form of the steadystate distribution of the associated reflected Levy process when the Levy process has no negative jumps (the generalized PollaczekKhinchine formula).
Singular vector autoregressions with deterministic terms: Strong consistency and lag order determination
, 2009
"... A vector autoregression is singular when explosive characteristic roots have geometric multiplicity larger than one. The singular component is a mixingale. Martingale decompositions are constructed for sample moments involving the singular component. This permits weak and strong analysis in the case ..."
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Cited by 7 (3 self)
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A vector autoregression is singular when explosive characteristic roots have geometric multiplicity larger than one. The singular component is a mixingale. Martingale decompositions are constructed for sample moments involving the singular component. This permits weak and strong analysis in the case of martingale difference innovations. While least squares estimators are shown to be inconsistent in the singular case, procedures for lag length determination are shown to have the same asymptotic properties in regular and singular cases.
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 7 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
Local time flow related to skew Brownian motion
, 2001
"... We define a local time flow of skew Brownian motions, that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the ..."
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Cited by 6 (4 self)
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We define a local time flow of skew Brownian motions, that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the Ray–Knight theorem on local times. In our case, however, the local time process viewed as a function of the spatial variable is a pure jump Markov process rather than a diffusion.
Duality and Other Results for M/G/1 and GI/M/1 Queues, via a New Ballot Theorem
 Mathematics of Operations Research
, 1989
"... We generalize the classical ballot theorem and use it to obtain direct probabilistic derivations of some wellknown and some new results relating to busy and idle periods and waiting times in M/G/1 and GI/M/1 queues. In particular, we uncover a duality relation between the joint distribution of seve ..."
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Cited by 6 (4 self)
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We generalize the classical ballot theorem and use it to obtain direct probabilistic derivations of some wellknown and some new results relating to busy and idle periods and waiting times in M/G/1 and GI/M/1 queues. In particular, we uncover a duality relation between the joint distribution of several variables associated with the busy cycle in M/G/1 and the corresponding joint distribution in GI/M/1. In contrast with the classical derivations of queueing theory, our arguments avoid the use of transforms, and thereby provide insight and termbyterm “explanations ” for the remarkable forms of some of these results.
On the Constructions of the Skew Brownian Motion
, 2006
"... This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applicat ..."
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Cited by 5 (1 self)
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This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.
Continuous time threshold autoregressive models, Statistica Sinica
 J. Appl. Prob
, 1991
"... This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stocha ..."
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Cited by 4 (2 self)
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This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stochastic differential equations. These processes are wellunderstood and there is a large body of literature devoted to their study. I summarise some of the relevant material and develop some further results. In particular, I propose a new and very fast method of estimation using an approach analogous to the Yule–Walker estimates for discrete time autoregressive processes. The models so estimated may be used for preliminary analysis of the appropriate model structure and as a starting point for maximum likelihood estimation. A natural extension of CAR processes is the class of continuous time threshold autoregressive (CTAR) processes defined by piecewise linear stochastic differential