Shift Techniques and Canonical Factorizations in the Solution of M/G/1-Type Markov Chains

ABSTRACT

By using properties of canonical factorizations, we prove that under very mild assumptions, the shifted cyclic reduction method (SCR) can be applied for solving QBD problems with no breakdown and that it always converges. For general M/G/1 type Markov chains we prove that SCR always converges if no breakdown is encountered. Numerical experiments showing the acceleration provided by SCR versus cyclic reduction are presented.     

An Exact Analysis of the Multi‐class M/M/k Priority Queue with Partial Blocking

Abstract

We consider a multi‐server queuing model with two priority classes that consist of multiple customer types. The customers belonging to one priority class customers are lost if they cannot be served immediately upon arrival. Each customer type has its own Poisson arrival and exponential service rate. We derive an exact method to calculate the steady state probabilities for both preemptive and nonpreemptive priority disciplines. Based on these probabilities, we can derive exact expressions for a wide range of relevant performance characteristics for each customer type, such as the moments of the number of customers in the queue and in the system, the expected postponement time and the blocking probability. We illustrate our method with some numerical examples.     

On the Restricted Liu Estimator in the Gauss–Markov Model

In this paper, we compare two estimators, the RLE (restricted Liu estimator) and the RLSE (restricted least squares estimator) of parameters in linear models under Gauss–Markov models. Using generalized inverse of matrices, we found some equivalency conditions for the superiority of the RLE with respect to the MSE criterion.     

Bayesian Estimation of Stochastic Frontier Models with Multivariate Skew t Error Terms

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.     

Estimation of the Entropy Rate of a Countable Markov Chain

We consider here ergodic homogeneous Markov chains with countable state spaces. The entropy rate of the chain is an explicit function of its transition and stationary distributions. We construct estimators for this entropy rate and for the entropy of the stationary distribution of the chain, in the parametric and nonparametric cases. We study estimation from one sample with long length and from many independent samples with given length. In the parametric case, the estimators are deduced by plug-in from the maximum likelihood estimator of the parameter. In the nonparametric case, the estimators are deduced by plug-in from the empirical estimators of the transition and stationary distributions. They are proven to have good asymptotic properties.     

Filtering and the EM-Algorithm for the Markovian Arrival Process

In this article, we deal with the so-called Markovian Arrival Process (MAP). An MAP is thought of as a partially observed Markov process, so that the Expectation-Maximization (EM) algorithm is a natural way to estimate its parameters. Then, nonlinear filters of basic statistics related to the MAP must be computed. The forward–backward principle is the basic way to do it. Here, bearing in mind a filter-based formulation of the EM-algorithm proposed by Elliott, these filters are shown to be the solution of nonlinear stochastic differential equations (SDEs) which allows a recursive computation. This is well suited for processing large data sets. We also derive linear SDEs or Zakai equations for the so-called unnormalized filters.     

Bayesian Inference Using Artificial Augmenting Regressions

In this article, it is shown that many intractable problems of Bayesian inference can be cast in a form called “artificial augmenting regression” in which application of Markov Chain Monte Carlo techniques, especially Gibbs sampling with data augmentation, is rather convenient. The new techniques are illustrated using several challenging statistical problems and numerical results are presented.     

The Variable Sampling Interval (VSI) in an Analysis of Taguchi's On-line Quality Monitoring Procedure for Variables

The procedure of on-line process control for variables proposed by Taguchi consists of inspecting the mth item (a single item) of every m items produced and deciding, at each inspection, whether the mean value is increased or not. If the value of the monitored statistic is outside of the control limits, one decides the process is out-of-control and the production is stopped for adjustment; otherwise, it continues. In this article, a variable sampling interval (with a longer L and a shorter m ≤ L) chart with two set of limits is used. These limits are the warning (±W) and the control (±C), where W ≤ C. The process is stopped for adjustment when an observation falls outside of the control limits or a sequence of h observations falls between the warning limits and the control limits. The longer sample interval is used after an adjustment or when an observation falls inside the warning limits; otherwise, the short sampling interval is used. The properties of an ergodic Markov chain are used to evaluate the time (in units) that the process remains in-control and out-of-control, with the aim of building an economic–statistical model. The parameters (the sampling intervals m and L, the control limits W and C and the length of run h) are optimized by minimizing the cost function with constraints on the average run lengths (ARLs) and the conformity fraction. The performance of the current proposal is more economical than the decision taken based on a sequence of length h = 1, L = m, and W = C, which is the model employed in earlier studies. A numerical example illustrates the proposed procedure.