Circular Bernstein polynomial distributions

This paper introduces a new non-parametric approach to the modeling of circular data, based on the use of Bernstein polynomial densities which generalizes the standard Bernstein polynomial model to account for the specific characteristics of circular data. It is shown that the trigonometric moments of the proposed circular Bernstein polynomial distribution can all be derived in closed form. We comment on how to fit the Bernstein polynomial density approximation to a sample of data and illustrate our approach with a real data example.Circular data, Non-parametric modeling, Bernstein polynomials

Non-parametric methods for circular-circular and circular-linear

We present a non-parametric approach for the estimation of the bivariate distribution of two circular variables and the modelling of the joint distribution of a circular and a linear variable. We combine nonparametric estimates of the marginal densities of the circular and linear components with the use of class of nonparametric copulas, known as empirical Bernstein copulas, to model the dependence structure. We derive the necessary conditions to obtain continuous distributions defined on the cylinder for the circular-linear model and on the torus for the circular-circular model. We illustrate these two approaches with two sets of real environmental dataBernstein polynomials, Circular distributions, Circular-Circular data, Circular-linear data, Copulas, Non-parametric estimation

TRANSIENT BAYESIAN INFERENCE FOR SHORT AND LONG-TAILED GI/G/1 QUEUEING SYSTEMS

In this paper, we describe how to make Bayesian inference for the transient behaviour and busy period in a single server system with general and unknown distribution for the service and interarrival time. The dense family of Coxian distributions is used for the service and arrival process to the system. This distribution model is reparametrized such that it is possible to define a non-informative prior which allows for the approximation of heavytailed distributions. Reversible jump Markov chain Monte Carlo methods are used to estimate the predictive distribution of the interarrival and service time. Our procedure for estimating the system measures is based in recent results for known parameters which are frequently implemented by using symbolical packages. Alternatively, we propose a simple numerical technique that can be performed for every MCMC iteration so that we can estimate interesting measures, such as the transient queue length distribution. We illustrate our approach with simulated and real queues.

BAYESIAN CONTROL OF THE NUMBER OF SERVERS IN A GI/M/C QUEUING 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.

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