Queues in Reliability

Queueing models can be useful in solving many complex reliability problems. Component failures are usually interpreted as the arrival of customers and the repair or replacement of failed components is typically associated with the service facility. A distinctive characteristic of queues in reliability is that requests for service are usually generated by a finite customer population because, in general, there are a limited number of units, e.g. machines which can fail, and when they are all in the system, being repaired or waiting for repair, no more can arrive. Thus the arrivals do not form a renewal process as they may depend on the number of units in the system. This is an essential difference from typical queueing systems, where the population of potential arrivals can be considered to be effectively limitless. This article overviews the main queueing models used in reliability which are illustrated using the classical machine repairmen model. Some statistical methods to estimate the main quantities of interest in a queue are also discussed

An introduction to quadrature and other numerical integration techniques

The objective in numerical integration is the approximation of a definite integral using numerical techniques. There are a large number of numerical integration methods in the literature and this article overviews some of the most common ones, namely, the Newton-Cotes formulas, including the trapezoidal and Simpson's rules, and the Gaus- sian quadrature. Difeerent procedures are compared and illustrated with examples. Discussions about more advanced numerical integration procedures are also included

Bayesian estimation of the Gaussian mixture GARCH model

Bayesian inference and prediction for a generalized autoregressive conditional heteroskedastic (GARCH) model where the innovations are assumed to follow a mixture of two Gaussian distributions is performed. The mixture GARCH model can capture the patterns usually exhibited by many financial time series such as volatility clustering, large kurtosis and extreme observations. A Griddy–Gibbs sampler implementation is proposed for parameter estimation and volatility prediction. Bayesian prediction of the Value at Risk is also addressed providing point estimates and predictive intervals. The method is illustrated using the Swiss Market Index

Bayesian estimation of ruin probabilities with heterogeneous and heavy-tailed insurance claim size distribution

This paper describes a Bayesian approach to make inference for risk reserve processes with unknown claim size distribution. A flexible model based on mixtures of Erlang distributions is proposed to approximate the special features frequently observed in insurance claim sizes such as long tails and heterogeneity. A Bayesian density estimation approach for the claim sizes is implemented using reversible jump Markov Chain Monte Carlo methods. An advantage of the considered mixture model is that it belongs to the class of phase-type distributions and then, explicit evaluations of the ruin probabilities are possible. Furthermore, from a statistical point of view, the parametric structure of the mixtures of Erlang distribution others some advantages compared with the whole over-parameterized family of phase-type distributions. Given the observed claim arrivals and claim sizes, we show how to estimate the ruin probabilities, as a function of the initial capital, and predictive intervals which give a measure of the uncertainty in the estimations

Seasonal copula models for the analysis of glacier discharge at King George Island, Antarctica

Modelling glacier discharge is an important issue in hydrology and climate research. Glaciers represent a fundamental water resource when melting of snow contributes to runoff. Glaciers are also studied as natural global warming sensors. GLACKMA association has implemented one of their Pilot Experimental Watersheds at the King George Island in the Antarctica which records values of the liquid discharge from Collins glacier. In this paper, we propose the use of time-varying copula models for analyzing the relationship between air temperature and glacier discharge, which is clearly non constant and non linear through time. A seasonal copula model is defined where both the marginal and copula parameters vary periodically along time following a seasonal dynamic. Full Bayesian inference is performed such that the marginal and copula parameters are estimated in a one single step, in contrast with the usual twostep approach. Bayesian prediction and model selection is also carried out for the proposed model such that Bayesian credible intervals can be obtained for the conditional glacier discharge given a value of the temperature at any given time point. The proposed methodology is illustrated using the GLACKMA real data where there is, in addition, a hydrological year of missing discharge data which were not possible to measure accurately due to hard meteorological conditions

Bayesian prediction of the transient behaviour and busy period in short and long-tailed GI/G/1 queueing systems

Bayesian inference for the transient behavior and duration of a busy period in a single server queueing system with general, unknown distributions for the interarrival and service times is investigated. Both the interarrival and service time distributions are approximated using the dense family of Coxian distributions. A suitable reparameterization allows the definition of a non-informative prior and Bayesian inference is then undertaken using reversible jump Markov chain Monte Carlo methods. An advantage of the proposed procedure is that heavy tailed interarrival and service time distributions such as the Pareto can be well approximated. The proposed procedure for estimating the system measures is based on recent theoretical results for the Coxian/Coxian/1 system. A numerical technique is developed for every MCMC iteration so that the transient queue length and waiting time distributions and the duration of a busy period can be estimated. The approach is illustrated with both simulated and real data

Bayesian control of the number of servers in a GI/M/c queueing system

In this paper we consider the problem of designing a GI/M/c queueing system. Given arrival and service data, our objective is to choose the optimal number of servers so as to minimize an expected cost function which depends on quantities, such as the number of customers in the queue. A semiparametric approach based on Erlang mixture distributions is used to model the general interarrival time distribution. Given the sample data, Bayesian Markov Chain Monte Carlo methods are used to estimate the system parameters and the predictive distributions of the usual performance measures. We can then use these estimates to minimize the steady-state expected total cost rate as a function of the control parameter c. We provide a numerical example based on real data obtained from a bank in Madrid

Vine copula models for predicting water flow discharge at King George Island, Antarctica

[EN]In order to understand the behavior of the glaciers, their mass balance should be studied. The loss of water produced by melting, known as glacier discharge, is one of the components of this mass balance. In this paper, a vine copula structure is proposed to model the multivariate and nonlinear dependence among the glacier discharge and other related meteorological variables such as temperature, humidity, solar radiation and precipitation. The multivariate distribution of these variables is expressed as a mixture of four components according to the presence or not of positive discharge and/or positive precipitation. Then, each of the four subgroups is modelled with a vine copula. The conditional probability of zero discharge for given meteorological conditions is obtained from the proposed joint distribution. Moreover, the structure of the vine copula allows us to derive the conditional distribution of the glacier discharge for the given meteorological conditions. Three different prediction methods for the values of the discharge are used and compared. The proposed methodology is applied to a large database collected since 2002 by the GLACKMA association from a measurement station located in the King George Island in the Antarctica. Seasonal effects are included by using different parameters for each season. We have found that the proposed vine copula model outperforms a previous work where we only used the temperature to predict the glacier discharge using a time-varying bivariate copula

Seasonal copula models for the analysis of glacier discharge at King George Island, Antarctica

Modelling glacier discharge is an important issue in hydrology and climate research. Glaciers represent a fundamental water resource when melting of ice and snow contributes to runoff. Glaciers are also studied as natural global warming sensors. GLACKMA association has implemented one of their Pilot Experimental Catchment areas at the King George Island in the Antarctica which records values of the liquid discharge from Collins glacier. In this paper, we propose the use of time-varying copula models for analyzing the relationship between air temperature and glacier discharge, which is clearly non constant and non linear through time. A seasonal copula model is defined where both the marginal and copula parameters vary periodically along time following a seasonal dynamic. Full Bayesian inference is performed such that the marginal and copula parameters are estimated in a one single step, in contrast with the usual two-step approach. Bayesian prediction and model selection is also carried out for the proposed model such that Bayesian credible intervals can be obtained for the conditional glacier discharge given a value of the temperature at any given time point. The proposed methodology is illustrated using the GLACKMA real data where there is, in addition, a hydrological year of missing discharge data which were not possible to measure accurately due to problems in the sounding.We are very grateful to the GLACKMA association. The second author acknowledges financial support by UC3M-BS Institute of Financial Big Data at Universidad Carlos III de Madrid. The third author would like to thank the Russian, Argentinean, German, Uruguayan and Chilean Antarctic Programs for their continuous logistic support over the years. The crews of Bellingshausen, Artigas, and Carlini station as well as the Dallmann Laboratory provided a warm and pleasant environment during fieldwork. GLACKMA’s contribution was also partially financed by the European Science Foundation, ESF project IMCOAST (EUI2009-04068) and the Ministerio de Educación y Ciencia (CGL2007-65522-C02-01/ANT)