We present empirical evidence that the distribution of effort required to solve CSPs randomly generated at the 50% satisfiable point, when using a backtracking algorithm, can be approximated by two standard families of continuous probability distribution functions. Solvable problems can be modelled by the Weibull distribution, and unsolvable problems by the lognormal distribution. These distributions fit equally well over a variety of backtracking based algorithms. 1. Introduction Several key developments in the 1990's have contributed to the advancement of empirical research on CSP algorithms, to the extent that the field may even be called an experimental science. Striking increases in computer power and decreases in cost, coupled with the general adoption of C as the programming language of choice, have made it possible for the developer of a new algorithm or heuristic to test it on large numbers of random instances. Another important advance was the recognition of the "50% satisfi...

High reliability systems generally require individual system components having extremely high reliabilityover long periods of time. Short product development times require reliability tests to be conducted with severe time constraints. Frequently few or no failures occur during such tests, even with acceleration. Thus, it is difficult to assess reliabilitywith traditional life tests that record only failure times. For some components, degradation measures can be taken over time. A relationship between component failure and amountof degradation makes it possible to use degradation models and data to make inferences and predictions about a failure-time distribution. This paper describes degradation reliability models that correspond to physical-failure mechanisms. We explain the connection between degradation reliability models and failuretime reliabilitymodels. Acceleration is modeled byhaving an acceleration model that describes the effect that temperature (or another accelerating vari...

. We analyze the distribution of computational effort required by backtracking algorithms on unsatisfiable CSPs, using analogies with reliability models, where lifetime of a specimen before failure corresponds to the runtime of backtracking on unsatisfiable CSPs. We extend the results of [7] by showing empirically that the lognormal distribution is a good approximation of the backtracking effort on unsolvable CSPs not only at the 50% satisfiable point, but in a relatively wide region. We also show how the law of proportionate effect [9] commonly used to derive the lognormal distribution can be applied to modeling the number of nodes expanded in a search tree. Moreover, for certain intervals of C=N , where N is the number of variables, and C is the number of constraints, the parameters of the corresponding lognormal distribution can be approximated by the linear lognormal model [11] where mean log(deadends) is linear in C=N , and variance of log(deadends) is close to constant. The line...

Accelerated life tests are used to obtain timely information on the durability and reliability of materials. Test units are subjected to higher than usual levels of “stress ” and a model is used to estimate life at use conditions. Although it is desirable to use a physically-based model to justify the required extrapolation, in many practical situations, no such model is available or the physical basis for extrapolation is uncertain. In such situations, extrapolation is based on an empirical model. Sensitivity analysis tools then become important to assess the effect of model departures and to allow engineers to make safe design decisions. This paper presents models, methods, and a description of software tools for performing systematic sensitivity analysis to assess potential model error. These methods are illustrated by studying the results of an experiment that was conducted to determine if the fatigue life of a spring would meet a given specification.

Accelerated life tests (ALTs) provide timely assessments of the reliability of materials, components, and subsystems. ALTs can be run at any of these levels or at the full-system level. Sometimes ALTs generate multiple failure modes. A frequently asked question, coming near to the end of an ALT program, is “What do these test results say about field performance? ” ALTs are carefully controlled whereas the field environment is highly variable. Products in the field see, for example, different average use rates across the product population. With good characterization of field use conditions, it may be possible to use ALT results to predict the failure time distribution in the field. When such information is not available but both life test data and field data (e.g., from warranty returns) are available, it may be possible to find a model to relate the two data sets. Under a reasonable set of practical assumptions, this model can then be used to predict the failure time distribution for a future component or product operating in the same use environment. This paper describes a model and methods for such situations. The methods will be illustrated by an example to predict the failure time distribution of a newly designed product with two failure modes.

Engineers in the manufacturing industries have used accelerated test (AT) experiments for many decades. The purpose of AT experiments is to acquire reliability information quickly. Test units of a material, component, subsystem, or entire systems are subjected to higher-than-usual levels of one or more accelerating variables such as temperature or stress. Then the AT results are used to predict life of the units at use conditions. The extrapolation is typically justified (correctly or incorrectly) on the basis of physically motivated models or a combination of empirical model fitting with a sufficient amount of previous experience in testing similar units. The need to extrapolate in both time and the accelerating variables generally necessitates the use of fully parametric models. Statisticians have made important contributions in the development of appropriate stochastic models for AT data (typically a distribution for the response and regression relationships between the parameters of this distribution and the accelerating variable(s)), statistical methods for AT planning (choice of accelerating variable levels and allocation of available test units to those levels), and methods of estimation of suitable reliability metrics. This paper provides a review of many of the AT models that have been use successfully in this area.

Today's manufacturers need accelerated test (AT) methods that can usefully predict service life in a timely manner. For example, automobile manufacturers would like to develop a three-month test to predict 10-year field reliability of a coating system (an acceleration factor of 40). Developing a methodology to simulate outdoor weathering is a particularly challenging task and most previous attempts to establish an adequate correlation between laboratory tests and field experience has met with failure. Difficulties arise, for example, because the intensity and the frequency spectrum of ultraviolet (UV) radiation from the Sun are highly variable, both temporally and spatially and because there is often little understanding of how environmental variables affect chemical degradation processes. This paper describes the statistical aspects of a cooperative project being conducted at the U.S. National Institute of Standards and Technology (NIST) to generate necessary experimental data and the development of a model relating cumulative damage to environmental variables like UV spectrum and intensity, as well as temperature and relative humidity. The parameters of the cumulative damage are estimated from the laboratory data. The adequacy of the model predictions are assessed by comparing with specimens tested in an outdoor environment for which the environmental variables were carefully measured.

Today's manufacturers need to develop newer, higher technology products in record time while improving productivity, reliability, and quality. This requires improved accelerated test (AT) methods that can usefully predict service life. For example, automobile manufacturers would like to develop a 3-month test to predict 5 or 10-year field reliability of a coating system. Such estimation/prediction from ATs involves extrapolation. Seriously inadequate predictions will result unless adequate models and methods are used. This paper describes a general framework within which one can use laboratory test results to predict product field service life performance of certain products in a highly-variable environment. Introduction Difficulty establishing correlation between laboratory tests and outdoor weathering tests for paints and coatings Manufacturers of paints and coatings, for example, have had difficulty in establishing adequate correlation between their laboratory tests and field expe...

Some importan t reliability problems in volve estimatin g a life distribution when failure is due to chemical degradation of materials or products that are exposed to the outdoor en viron men t. There is a growin gn eed to obtain timely prediction s of such degradation behaviors on the basis of accelerated laboratory tests. Laboratory life tests provide in formation about degradation processes. Historical weather data are used to characterize the stochastic outdoor en viron men t over time. A physical/chemical model for degradation rate is used as a basis for usin g these data to produce reliability estimates. We proposean d illustrate the use ofan evaluation /estimation method that in volves time series modelin g. The method is illustrated with an examplein volvin g the degradation of a solar-reflector material. We will also show how to con struct approximate con5# n e in tervals for importan t reliability metrics. Key words: Service Life Prediction , Time Series, Accelerated Testin g, Risk Assessmen t. 1 1

Accelerated Destructive Degradation Tests (ADDTs) provide reliability information quickly. An ADDT plan specifies factor level combinations of an accelerating variable (e.g., temperature) and evaluation time and the allocations of test units to these combinations. This paper describes methods to find good ADDT plans for an important class of destructive degradation models. First, a collection of optimum plans is derived. These plans minimize the large sample approximate variance of the maximum likelihood (ML) estimator of a specified failure-time quantile. The General Equivalence Theorem (GET) is used to verify the optimality of these plans. Because an optimum plan is not robust to the model specification and the planning information used in deriving the plan, a more robust and useful compromise plan is proposed. Sensitivity analyses show the effects that changes in sample size, time duration of the experiment, levels of the accelerating variable, and misspecification of the planning information have on the precision of the ML estimator of a quantile of the failure-time distribution. Monte Carlo simulations are used to evaluate the statistical characteristics of the ADDT plans. The methods are illustrated with an application for an adhesive bond.