Bayesian analysis of Er/M/1 and Er/M/c queues

Bayesian inference and prediction tasks for Er/M/1 and Er/M/c queues are undertaken. Equilibrium probabilities of the queue size and waiting time distributions are estimated using conditional Monte-Carlo simulation methods. We illustrate that some standard queueing measures do not exist when independent priors are used for the arrival and service rates of a G/M/1 queue.     

Bayesian Inference and Prediction in an M/G/1 with Optional Second Service

In this article, we exploit the Bayesian inference and prediction for an M/G/1 queuing model with optional second re-service. In this model, a service unit attends customers arriving following a Poisson process and demanding service according to a general distribution and some of customers need to re-service with probability “p”. First, we introduce a mixture of truncated Normal distributions on interval (? ∞, 0) to approximate the service and re-service time densities. Then, given observations of the system, we propose a Bayesian procedure based on birth-death MCMC methodology to estimate some performance measures. Finally, we apply the theories in practice by providing a numerical example based on real data which have been obtained from a hospital.     

Fitting with Matrix-Exponential Distributions

Abstract

It is well known that general phase-type distributions are considerably overparameterized, that is, their representations often require many more parameters than is necessary to define the distributions. In addition, phase-type distributions, even those defined by a small number of parameters, may have representations of high order. These two problems have serious implications when using phase-type distributions to fit data. To address this issue we consider fitting data with the wider class of matrix-exponential distributions. Representations for matrix-exponential distributions do not need to have a simple probabilistic interpretation, and it is this relaxation which ensures that the problems of overparameterization and high order do not present themselves. However, when using matrix-exponential distributions to fit data, a problem arises because it is unknown, in general, when their representations actually correspond to a distribution. In this paper we develop a characterization for matrix-exponential distributions and use it in a method to fit data using maximum likelihood estimation. The fitting algorithm uses convex semi-infinite programming combined with a nonlinear search.     

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.     

Bayesian analysis of hierarchical linear mixed modeling using the multivariate t distribution

This article presents a fully Bayesian approach to modeling incomplete longitudinal data using the t linear mixed model with AR(p) dependence. Markov chain Monte Carlo (MCMC) techniques are implemented for computing posterior distributions of parameters. To facilitate the computation, two types of auxiliary indicator matrices are incorporated into the model. Meanwhile, the constraints on the parameter space arising from the stationarity conditions for the autoregressive parameters are handled by a reparametrization scheme. Bayesian predictive inferences for the future vector are also investigated. An application is illustrated through a real example from a multiple sclerosis clinical trial.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

A cold standby repairable system with the repairman having multiple vacations and operational,repair, and vacation times following phase-type distributions

ABSTRACT

This paper studies a cold standby repairable system with two identical components and one repairman having multiple vacations applying matrix-analytic methods. The lifetime of the component follows a phase-type distribution. The repair times and the vacation times of the repairman are governed by different phase-type distributions, respectively. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime, the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated, respectively. A numerical application is performed to illustrate the calculations.     

Predicting observables from a general class of distributions

A general class of distributions is proposed to be the underlying population model from which observables are to be predicted using the Bayesian approach. This class of distributions includes, among others, the Weibull, compound Weibull (or three-parameter Burr-type XII), Pareto, beta, Gompertz and compound Gompertz distributions. A proper general prior density function is suggested and the predictive density functions are obtained in the one- and two-sample cases. The informative sample is assumed to be a type II censored sample. Illustrative examples of Weibull (α,β), Burr-type XII (α,β), and Pareto (α,β) distributions are given and compared with the results obtained by previous researchers.     

Two useful distributions for Bayesian predictive procedures under normal models

The K-prime and K-square distributions, involved in the Bayesian predictive distributions of standard t and F tests are investigated. They generalize the classical noncentral t and noncentral F distributions and can receive different characterizations. Their moments and their probability density and distribution functions are made explicit.     

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT

We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.     

Semiparametric distribution forecasting

Given m time series regression models, linear or not, with additive noise components, it is shown how to estimate semiparametrically the predictive probability distribution of one of the time series conditional on past random covariate data. This is done by assuming that the distributions of the residual components associated with the regression models are tilted versions of a reference distribution.     

Bayesian inference for power law processes with applications in repairable systems

Statistical models for recurrent events are of great interest in repairable systems reliability and maintenance. The adopted model under minimal repair maintenance is frequently a nonhomogeneous Poisson process with the power law process (PLP) intensity function. Although inference for the PLP is generally based on maximum likelihood theory, some advantages of the Bayesian approach have been reported in the literature. In this paper it is proposed that the PLP intensity be reparametrized in terms of (β,η), where β is the elasticity of the mean number of events with respect to time and η is the mean number of events for the period in which the system was actually observed. It is shown that β and η are orthogonal and that the likelihood becomes proportional to a product of gamma densities. Therefore, the family of natural conjugate priors is also a product of gammas. The idea is extended to the case that several realizations of the same PLP are observed along overlapping periods of time. Some Monte Carlo simulations are provided to study the frequentist behavior of the Bayesian estimates and to compare them with the maximum likelihood estimates. The results are applied to a real problem concerning the determination of the optimal periodicity of preventive maintenance for a set of power transformers. Prior distributions are elicited for β and η based on their operational interpretation and engineering expertise.     

A Parametric Estimation Procedure for Relapse Time Distributions

In a relapse clinical trial patients who have recovered from some recurrent disease (e.g.,ulcer or cancer) are examined at a number of predetermined times. A relapse can be detected either at one of these planned inspections or at a spontaneous visit initiated by the patient because of symptoms. In the first case the observations of the time to relapse, X, is interval-censored by two predetermined time-points. In the second case the upper endpoint of the interval is an observation of the time to symptoms,Y . To model the progression of the disease we use a partially observable Markov process. This approach results in a bivariate phase-type distribution for the joint distribution of (X,Y). It is a flexible model which contains several natural distributions for X, and allows the conditional distributions of the marginals to smoothly depend on each other. To estimate the distributions involved we develop an EM-algorithm. The estimation procedure is evaluated and compared with a non-parametric method in a couple of examples based on simulated data.     

Bayesian inference in joint modelling of location and scale parameters of the t distribution for longitudinal data

This paper presents a fully Bayesian approach to multivariate t regression models whose mean vector and scale covariance matrix are modelled jointly for analyzing longitudinal data. The scale covariance structure is factorized in terms of unconstrained autoregressive and scale innovation parameters through a modified Cholesky decomposition. A computationally flexible data augmentation sampler coupled with the Metropolis-within-Gibbs scheme is developed for computing the posterior distributions of parameters. The Bayesian predictive inference for the future response vector is also investigated. The proposed methodologies are illustrated through a real example from a sleep dose–response study.     

Quantitative comparisons between finitary posterior distributions and Bayesian posterior distributions

The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to considering a finite horizon framework. However, assuming infinite exchangeability gives rise to fairly tractable a posteriori quantities, which is very attractive in applications. Hence, with a view to a reconciliation between these two aspects of the Bayesian way of reasoning, in this paper we provide quantitative comparisons between posterior distributions of finitary parameters and posterior distributions of allied parameters appearing in usual statistical models.     

Maximum penalized likelihood estimation for skew-normal and skew-t distributions

The skew-normal and the skew-t distributions are parametric families which are currently under intense investigation since they provide a more flexible formulation compared to the classical normal and t distributions by introducing a parameter which regulates their skewness. While these families enjoy attractive formal properties from the probability viewpoint, a practical problem with their usage in applications is the possibility that the maximum likelihood estimate of the parameter which regulates skewness diverges. This situation has vanishing probability for increasing sample size, but for finite samples it occurs with non-negligible probability, and its occurrence has unpleasant effects on the inferential process. Methods for overcoming this problem have been put forward both in the classical and in the Bayesian formulation, but their applicability is restricted to simple situations. We formulate a proposal based on the idea of penalized likelihood, which has connections with some of the existing methods, but it applies more generally, including the multivariate case.     

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).     

Bayesian analysis of incomplete time and cause of failure data

For series systems with k components it is assumed that the cause of failure is known to belong to one of the 2^k − 1 possible subsets of the failure-modes. The theoretical time to failure due to k causes are assumed to have independent Weibull distributions with equal shape parameters. After finding the MLEs and the observed information matrix of (λ₁, …, λ_k, β), a prior distribution is proposed for (λ₁, …, λ_k), which is shown to yield a scale-invariant noninformative prior as well. No particular structure is imposed on the prior of β. Methods to obtain the marginal posterior distributions of the parameters and other parametric functions of interest and their Bayesian point and interval estimates are discussed. The developed techniques are illustrated using a numerical example.     

Estimating parametric semi-Markov models from panel data using phase-type approximations

Inference for semi-Markov models under panel data presents considerable computational difficulties. In general the likelihood is intractable, but a tractable likelihood with the form of a hidden Markov model can be obtained if the sojourn times in each of the states are assumed to have phase-type distributions. However, using phase-type distributions directly may be undesirable as they require estimation of parameters which may be poorly identified. In this article, an approach to fitting semi-Markov models with standard parametric sojourn distributions is developed. The method involves establishing a family of Coxian phase-type distribution approximations to the parametric distribution and merging approximations for different states to obtain an approximate semi-Markov process with a tractable likelihood. Approximations are developed for Weibull and Gamma distributions and demonstrated on data relating to post-lung-transplantation patients.     

Matrix-analytic Models and their Analysis

We survey phase-type distributions and Markovian point processes, aspects of how to use such models in applied probability calculations and how to fit them to observed data. A phase-type distribution is defined as the time to absorption in a finite continuous time Markov process with one absorbing state. This class of distributions is dense and contains many standard examples like all combinations of exponential in series/parallel. A Markovian point process is governed by a finite continuous time Markov process (typically ergodic), such that points are generated at a Poisson intensity depending on the underlying state and at transitions; a main special case is a Markov-modulated Poisson process. In both cases, the analytic formulas typically contain matrix-exponentials, and the matrix formalism carried over when the models are used in applied probability calculations as in problems in renewal theory, random walks and queueing. The statistical analysis is typically based upon the EM algorithm, viewing the whole sample path of the background Markov process as the latent variable.