Recently, orbits of two-dimensional Markov chains have been used to generate computer images. These chains evolve according to products of i.i.d. affine maps. We deal with mixing models, whereby one mixes together several of these Markov chains, so as to create a mixed image. These mixtures involve starting one Markov chain off at the stationary distribution of another, and then running it for a geometrically distributed number of steps. We use this to analyze various mixing scenarios.

The mathematical theory of evidence (Shafer et al. [9]) has recently found much interest as an approach to treat uncertainty in expert and knowledge-based systems. Although the theory is very promising, there are not yet many practical applications. Modeling practice has still to be developed. This is a crucial task in view of facilitating the application of evidential modeling. It is the aim of this paper to discuss an important element of evidential modeling–conditional belief–within the scope of the mathematical theory of evidence.

Generalized semi-Markov processes and stochastic Petri nets have been proposed as general frameworks for a discrete event simulation on a countable state space. The two formal systems differ, however, with respect to the clock setting (event scheduling) mechanism, the state transition mechanism, and the form of the state space. We obtain conditions under which the marking process of a stochastic Petri net “mimics” a generalized semi-Markov process in the sense that the two processes (and their underlying general state-space Markov chains) have the same finite dimensional distributions. The results imply that stochastic Petri nets have at least the modeling power of generalized semiMarkov processes for discrete event simulation.

There are three general methods for system reliability evaluation, namely; (1) inclusion–exclusion, (2) sum of disjoint products, and (3) pivoting. Of these, only pivoting can be applied directly to a logic tree or network graph representation without first finding minimal path (or cut) sets. Domination theory provides the basis for selecting optimal pivoting strategies. Simple proofs of domination-theory results for coherent systems are given, based on the reliability polynomial. These results are related to the problem of finding efficient strategies for computing coherent system reliability. The original results for undirected networks are due to Satyanarayana and Chang [5] (cf. [1]). Many of the original set theoretic results are due to Huseby [3]. However, he does not use the reliability polynomial to prove his results.

Recently, Ross [1] proposed an elegant method of approximating transition probabilities and mean occupation times in continuous-time Markov chains based upon recursively inspecting the process at exponential times. The method turned out to be amazingly efficient for the examples investigated. However, no formal rough error bound was provided. Any error bound even though robust is of practical interest in engineering (e.g., for determining truncation criteria or setting up an experiment). This note primarily aims to show that by a simple and standard comparison relation a rough error bound of the method is secured. Also, some alternative approximations are inspected.

Interconnections between occurrence times of nonhomogeneous Poisson processes, record values, minimal repair times, and the relevation transform are explained. A number of properties of the distributions of occurrence times and interoccurrence times of a nonhomogeneous Poisson process are proved when the mean-value function of the process is convex, starshaped, or superadditive. The same results hold for upper record values of independently identically distributed random variables from IFR, IFRA, and NBU distributions.

An infinite dam with discrete inputs arriving at times 0 = T0 < T1 <…< Tn <… is considered, where {Tn;n = 0,1,2,…} forms a renewal process. There is a random release from the dam during the interrenewal times which depends on the content as well as on the length of the time after the last input. A relation is derived that connects the content distribution at any time t to that when t is a renewal point. This relation is used to obtain the content distribution in the limiting case and in the transient case. The probability of emptiness is obtained and an integral equation is derived for the probability of first emptiness in a special case.

The materials-loss problem is examined under the assumption that one wishes to minimize the maximum expected amount of material diverted. A solution is obtained in the case of constant diversion under a simple sequential probability ratio test (SPRT). Extensions to the cases of correlated errors, random diversion schemes, and the Page and Shirayev-Roberts tests are examined. A thought-provoking application to the regulation of strategic special nuclear materials is given.