Poisson Rates Research Articles

A queueing model in an alternating random environment

In this paper, we consider a single server queueing model with alternately varying Poisson arrival rates. Such arrival processes are usually modelled using the Doubly Stochastic Poisson (DSP) processes whose Poisson rates vary alternately and remains the same during an exponential time duration called a switching time. Queues with the DSP arrival processes are often employed to describe production, computer and communication systems. A DSP process includes both renewal and nonrenewal point processes. Since performance measures of a single server queue with a DSP input are difficult to evaluate, we approximate a DSP arrival process by a renewal process. We then use parameters so developed in an expression for the expected waiting time in a GI/ G/1. To demonstrate the quality of our approximations, we also present exact results. Our approximations have applications in production, inventory, insurance and maintenance systems.

The RIPP density estimate: an alternative method for the estimation of peri-stimulus spike density

Cross-correlation between stimuli and neuronal discharges determines the peri-stimulus spike density. This density function is usually estimated by the peri-stimulus time histogram (PSTH). In this paper an alternative method for spike-density estimation is considered that employs estimation of the rate of an inhomogeneous Poisson process by Jth waiting times. This procedure is called the RIPP density estimate. It involves sorting the spike times for all trials and obtaining the Poisson rates for successive groups of spikes. By application of this procedure to simulated action-potential sequences from the leaky-integrator model subject to a realistic input, it is shown that the RIPP density estimate reveals much more information from a given set of spike train data than PSTH. An application of the RIPP density estimate to experimental spike train data from a human hand muscle motoneuron subject to a low-threshold-afferent volley is also presented.

A Technique for Estimating Marginal Posterior Densities in Hierarchical Models Using Mixtures of Conditional Densities

Abstract A technique called quantile integration is proposed for the estimation of marginal posterior densities arising in Bayesian models having hierarchical representations. The method is based on approximating marginal densities as mixtures of conditional densities, where the conditioning variables are selected deterministically from the mixing distributions. The form of the approximation makes it easy to implement, and the resulting approximations are computationally efficient to obtain. The technique leads to particularly simple approximations for the predictive and posterior densities in Kalman filter or state-space models, and specific formulas are provided for the special case in which innovations belong to location-scale families. Other applications include a hierarchical empirical Bayes model for Poisson rates and a hierarchical linear model with exchangeable regression parameters and unknown variance components.

A Technique for Estimating Marginal Posterior Densities in Hierarchical Models Using Mixtures of Conditional Densities

Abstract A technique called quantile integration is proposed for the estimation of marginal posterior densities arising in Bayesian models having hierarchical representations. The method is based on approximating marginal densities as mixtures of conditional densities, where the conditioning variables are selected deterministically from the mixing distributions. The form of the approximation makes it easy to implement, and the resulting approximations are computationally efficient to obtain. The technique leads to particularly simple approximations for the predictive and posterior densities in Kalman filter or state-space models, and specific formulas are provided for the special case in which innovations belong to location-scale families. Other applications include a hierarchical empirical Bayes model for Poisson rates and a hierarchical linear model with exchangeable regression parameters and unknown variance components.

On a 2n-valued telegraph signal and the related integrated process

We consider the stochastic process describing the position of a particle whose velocity changes in sign and magnitude at the occurrences of two independent Poisson processes We decompose the probability law of the process into n components, which jointly yield a solution of a system of n telegraph equations. In the particular case n = 2, the fourth-order equation governing the probability law is presented and its explicit expression is obtained when the Poisson rates are suitably connected

Regression analysis of hierarchical poisson-like event rate data: superpopulation model effect on predictions

This paper studies prediction of future failure (rates) by hierarchical empirical Bayes (EB) Poisson regression methodologies. Both a gamma distributed superpopulation as well as a more robust (long-tailed) log student-t superpopulation are considered. Simulation results are reported concerning predicted Poisson rates. The results tentatively suggest that a hierarchical model with gamma superpopulation can effectively adapt to data coming from a log-Student-t superpopulation particularly if the additional computation involved with estimation for the log-Student-t hierarchical model is burdensome.

Sensitivity Analysis for Simulations via Likelihood Ratios

We present a simple method of estimating the sensitivity of quantities obtained from simulation with respect to a class of parameters. Here sensitivity means the derivative of an expectation with respect to a parameter. The class of parameters includes, for example, Poisson rates, discrete probabilities, and the mean and variance of a Normal distribution. The method is extremely well suited to regenerative simulation, and can be implemented on extant simulations with little effort, increase in running time, or memory requirements. It is based on some change-of-measure formulas derived from likelihood ratios. In addition to the theorems that underly the technique, we present some numerical examples.

Unidirectional multiple comparisons of Poisson rates.

This paper presents an exact significance test procedure for unidirectional multiple comparisons of Poisson variables. It is sufficiently general for application to situations where the 'denominator', which might be the amount of follow-up time in cohort studies or the expected number of events in studies that use standardized rates, differs from one group to the other, and where a group-specific alternative hypothesis is of interest. We use unidirectional Z statistics, of the form proposed by Dunnett, along with the conditional distribution needed in computing the exact attained significance levels, to address three distinct unidirectional hypotheses. In one case, we compare one exposed with several reference groups, while in the other two cases, we compare several exposed groups with one reference group. We also provide a multiple stage testing approach. Data from studies in cancer and trauma epidemiology illustrate the procedures.