Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. Hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors; the analysis of agricultural field experiments, with adjustment for fertility gradients; and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling.

Traffic modeling of today's communication networks is a prime example of the role statistical inference methods for stochastic processes play in such classical areas of applied probability as queueing theory or performance analysis. In practice, however, statistics and applied probability have failed to interface. As a result, traffic modeling and performance analysis rely heavily on subjective arguments; hence, debates concerning the validity of a proposed model and its predicted performance abound. In this paper, we show how a careful statistical analysis of large sets of actual traffic measurements can reveal new features of network traffic that have gone unnoticed by the literature and, yet, seem to have serious implications for predicted network performance. We use hundreds of millions of high-quality traffic measurements from an Ethernet local area network to demonstrate that Ethernet traffic is statistically self-similar and that this property clearly distinguishes between currently used models for packet traffic and our measured data. We also indicate how such a unique data set (in terms of size and quality) (i) can be used to illustrate a number of different statistical inference methods for self-similar processes, (ii) gives rise to new and challenging problems in statistics, statistical computing and probabilistic modeling and (iii) opens up new areas of mathematical research in queueing theory and performance analysis of future high-speed networks.

This expository paper is an overview of a relatively new class of failure models, both univariate and multivariate, that are suitable for describing the lifelength of items that operate in dynamic environments. Many of the currently used failure models are developed under the premise that the operating environment is static: these models turn out to be special cases of the new models that are overviewed here. These new models are derived by describing the underlying failure-causing mechanisms, such as degradation and wear, using suitable stochastic processes: this is the underlying mathematical theme that derives their development. Because of their generality, the new models should provide improved descriptions of failure data and assessments of item survivability. Furthermore, they may signal a new philosophy of life-testing experiments wherein one also monitors the environmental factors that govern the tests. This overview categorizes the models by the various strategies used for their development, outlines the salient assumptions underlying them and provides a convenient road map to the pertinent references, which are scattered among the applied probability, engineering, reliability and survival analysis literatures. It is our hope that this paper sets a tone for the direction of future work in the development of models for survival wherein the physics of failure and the characteristics of the operating environment play a key role.