Asymptotic optimality of RESTART estimators in highly dependable systems

Abstract We consider a wide class of models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multi-component systems in reliability settings. Repair times and component lifetimes are random variables that follow a general distribution, and the repair service adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly-dependable systems, the RESTART method is used for the estimation of steady-state unavailability and other reliability measures. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty involved in applying this method is finding a suitable function, called the importance function, to define the regions. In this paper we introduce an importance function which, for unbalanced systems, represents a great improvement over the importance function used in previous papers. We also demonstrate the asymptotic optimality of RESTART estimators in these models. Several examples are presented to show the effectiveness of the new approach, and probabilities up to the order of 10-42 are accurately estimated with little computational effort.

Variable Kernel estimates: On the impossibility of tuning the parameters

For the standard kernel density estimate, it is known that one can tune the bandwidth such that the expected L1 error is within a constant factor of the optimal L1 error (obtained when one is allowed to choose the bandwidth with knowledge of the density). In this paper, we pose the same problem for variable bandwidth kernel estimates where the bandwidths are allowed to depend upon the location. We show in particular that for positive kernels on the real line, for any data-based bandwidth, there exists a densityfor which the ratio of expected L1 error over optimal L1 error tends to infinity. Thus, the problem of tuning the variable bandwidth in an optimal manner is ``too hard''. Moreover, from the class of counterexamples exhibited in the paper, it appears thatplacing conditions on the densities (monotonicity, convexity, smoothness) does not help.

Almost sure testability of classes of densities

Let a class $\F$ of densities be given. We draw an i.i.d.\ sample from a density $f$ which may or may not be in $\F$. After every $n$, one must make a guess whether $f \in \F$ or not. A class is almost surely testable if there exists such a testing sequence such that for any $f$, we make finitely many errors almost surely. In this paper, several results are given that allowone to decide whether a class is almost surely testable. For example, continuity and square integrability are not testable, but unimodality, log-concavity, and boundedness by a given constant are.

Inequalities for a new data-based method for selecting nonparametric density estimates

We continue the development of a method for the selection of a bandwidth or a number of design parameters in density estimation. We provideexplicit non-asymptotic density-free inequalities that relate the $L_1$ error of the selected estimate with that of the best possible estimate,and study in particular the connection between the richness of the classof density estimates and the performance bound. For example, our methodallows one to pick the bandwidth and kernel order in the kernel estimatesimultaneously and still assure that for {\it all densities}, the $L_1$error of the corresponding kernel estimate is not larger than aboutthree times the error of the estimate with the optimal smoothing factor and kernel plus a constant times $\sqrt{\log n/n}$, where $n$ is the sample size, and the constant only depends on the complexity of the family of kernels used in the estimate. Further applications include multivariate kernel estimates, transformed kernel estimates, and variablekernel estimates.

SINGULARLY PERTURBED DISCOUNTED MARKOV CONTROL PROCESSES IN A GENERAL STATE SPACE

This paper studies the asymptotic optimality of discrete-time Markov decision processes (MDPs) with general state space and action space and having weak and strong interactions. By using a similar approach as developed by Liu, Zhang, and Yin [Appl. Math. Optim., 44 (2001), pp. 105-129], the idea in this paper is to consider an MDP with general state and action spaces and to reduce the dimension of the state space by considering an averaged model. This formulation is often described by introducing a small parameter epsilon > 0 in the definition of the transition kernel, leading to a singularly perturbed Markov model with two time scales. Our objective is twofold. First it is shown that the value function of the control problem for the perturbed system converges to the value function of a limit averaged control problem as epsilon goes to zero. In the second part of the paper, it is proved that a feedback control policy for the original control problem defined by using an optimal feedback policy for the limit problem is asymptotically optimal. Our work extends existing results of the literature in the following two directions: the underlying MDP is defined on general state and action spaces and we do not impose strong conditions on the recurrence structure of the MDP such as Doeblin's condition.

Tori embedded in S³ with dense asymptotic lines

In this paper are given examples of tori T² embedded in S³ with all their asymptotic lines dense.

LOCAL WELL POSEDNESS, ASYMPTOTIC BEHAVIOR AND ASYMPTOTIC BOOTSTRAPPING FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS OF THE SECOND ORDER IN TIME

A class of semilinear evolution equations of the second order in time of the form u(tt)+Au+mu Au(t)+Au(tt) = f(u) is considered, where -A is the Dirichlet Laplacian, 92 is a smooth bounded domain in R(N) and f is an element of C(1) (R, R). A local well posedness result is proved in the Banach spaces W(0)(1,p)(Omega)xW(0)(1,P)(Omega) when f satisfies appropriate critical growth conditions. In the Hilbert setting, if f satisfies all additional dissipativeness condition, the nonlinear Semigroup of global solutions is shown to possess a gradient-like attractor. Existence and regularity of the global attractor are also investigated following the unified semigroup approach, bootstrapping and the interpolation-extrapolation techniques.

On the asymptotic enumeration of accessible automata

We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.

The transient response of a CSTR containing porous catalyst pellets. Asymptotic solutions

Transient response of a CSTR containing porous catalyst pellets is analyzed theoretically using a matched asymptotic expansion technique. This singular perturbation technique leads directly to the conditions under which the minima of reservoir concentration occur. The existence of the minima may be used to estimate some inherent parameters of the catalyst pellet.

Asymptotic limits of a self-dual Ginzburg-Landau functional in bounded domains

In the author's joint paper [HJS] with Jest and Struwe, we discuss asymtotic limits of a self-dual Ginzburg-Landau functional involving a section of a line bundle over a closed Riemann surface and a connection on this bundle. In this paper, the author generalizes the above results [HJS] to the case of bounded domains.

Asymptotic solutions to the Gross-Pitaevskii gain equation: Growth of a Bose-Einstein condensate

We give an asymptotic analytic solution for the generic atom-laser system with gain in a D-dimensional trap, and show that this has a non-Thomas-Fermi behavior. The effect is due to Bose-enhanced condensate growth, which creates a local-density maximum and a corresponding outward momentum component. In addition, the solution predicts amplified center-of-mass oscillations, leading to enhanced center-of-mass temperature.

Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations

We consider the two-dimensional Navier-Stokes equations with a time-delayed convective term and a forcing term which contains some hereditary features. Some results on existence and uniqueness of solutions are established. We discuss the asymptotic behaviour of solutions and we also show the exponential stability of stationary solutions.

Strategic trade policy under uncertainty: Sufficient conditions for the optimality of ad valorem, specific and quadratic trade taxes

In most previous work on strategic trade policy the form of government intervention has been prescribed in advance. In this paper, we apply a solution concept discussed by Klemperer and Meyer for games in which the strategy space consists of the class of all (non state-contingent) price quantity schedules. We examine a series of specific assumptions on demand and supply conditions and derive the associated equilibrium trade policies. We derive welfare implications for all cases examined.