Stochastic Models, an Algorithmic Approach , by Henk C. Tijms (Chichester: Wiley, 1994), 375 pages, paperback.

We review basic modeling approaches for failure and mainte- nance data from repairable systems. In particular we consider imperfect re- pair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the ob- served events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.

On the statistical modeling and analysis of repairable systems

We investigate Laha–Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a \(q\)-Gausian variable.

/pdf/characterizations-of-some-free-random-variables-by-25videz8ge.pdf

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

Tandem queues with impatient customers

Block-accelerated aggregation multigrid for Markov chains with application to PageRank problems

It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework The main aim of the paper is to expose the mean value parametrization for possible statistical applications The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation Some classes of polynomial variance functions are revisited for illustration The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution

On the mean value parametrization of natural exponential families — a revisited review

We study a blood testing procedure for detecting viruses like HIV, HBV and HCV. In this procedure, blood samples go through two screening steps. The first test is ELISA (antibody Enzyme Linked Immuno-Sorbent Assay). The portions of blood which are found not contaminated in this first phase are tested in groups through PCR (Polymerase Chain Reaction). The ELISA test is less sensitive than the PCR test and the PCR tests are considerably more expensive. We model the two test phases of blood samples as services in two queues in series; service in the second queue is in batches, as PCR tests are done in groups. The fact that blood can only be used for transfusions until a certain expiration date leads, in the tandem queue, to the feature of customer impatience. Since the first queue basically is an infinite server queue, we mainly focus on the second queue, which in its most general form is an S-server M=G[k;K]=S + G queue, with batches of sizes which are bounded by k and K. Our objective is to maximize the expected profit of the system, which is composed of the amount earned for items which pass the test (and before their patience runs out), minus costs. This is done by an appropriate choice of the decision variables, namely, the batch sizes and the number of servers at the second service station. As will be seen, even the simplest version of the batch queue, the M=M[k;K]=1 + M queue, already gives rise to serious analytical complications for any batch size larger than 1. These complications are discussed in detail. In view of the fact that we aim to solve realistic optimization problems for blood screening procedures, these analytical complications force us to take recourse to either a numerical approach or approximations. We present a numerical solution for the queue length distribution in theM=M[k;K]=S+M queue and then formulate and solve several optimization problems. The power-series algorithm, which is a numerical-analytic method, is also discussed.

Tandem Queues with Impatient Customers for Blood Screening Procedures

This paper is devoted to the performance analysis and optimization of blood testing procedures. We present a queueing model of two queues in series, representing the two stages of a blood-testing procedure. Service (testing) in stage 1 is performed in batches, whereas it is done individually in stage 2. Since particular elements of blood can only be stored and used within a finite time window, the sojourn time of blood units in the system of two queues in series is an important performance measure, which we study in detail. We also introduce a profit objective function, taking into account blood acquisition and screening costs as well as profits for blood units, which were found uncontaminated and were tested fast enough. We optimize that profit objective function w.r.t. the batch size and the length of the time window.

/pdf/analysis-and-optimization-of-blood-testing-procedures-3nyivyov1o.pdf

Analysis and optimization of blood-testing procedures.

Characterizations of a distribution by zero (or constant) regression properties of arbitrary degree polynomial statistics on the sample mean are discussed. Various practical steps collected from the relevant literature are put together in this framework into a comprehensive guideline for constructing such characterizations. Applications are provided for natural exponential families (NEF’s). In particular, two reciprocal NEF’s associated with the continuous time symmetric Bernoulli random walk are characterized using this guideline. Moreover, a class of infinitely divisible NEF’s having some polynomial variance function structure is discussed in this framework.

Methods of Constructing Characterizations by Constancy of Regression on the Sample Mean and Related Problems for NEF's

A polynomial Q = Q(X1, …, Xn) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α1(F), …, αm(F)), where q(u1, …, um) is a polynomial in u1, …, um and αj(F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X1 + … + Xn) = 0 holds if and only if q(α1(θ), …, αm(θ)) ≡ 0, where αj(θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X1 + … + Xn is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions Fθ(x) = ∫−∞xeθu−k(θ)dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.

S.K. Bar-Lev

Papers

On the mean value parametrization of natural exponential families — a revisited review

Tandem Queues with Impatient Customers for Blood Screening Procedures

Analysis and optimization of blood-testing procedures.

Methods of Constructing Characterizations by Constancy of Regression on the Sample Mean and Related Problems for NEF's

Regression of polynomial statistics on the sample mean and natural exponential families