POLLING SYSTEMS WITH SIMULTANEOUS BATCH ARRIVALS

We study the delay in polling systems with simultaneous batch arrivals. Arrival epochs are generated according to a Poisson process. At any arrival epoch, batches of customers may arrive simultaneously at the different queues, according to a general joint batch-size distribution. The server visits the queues in cyclic order, the service times and the switch-over times are generally distributed, and the service disciplines are general mixtures of gated and exhaustive service. We derive closed-form expressions for the expected delay at each of the queues when the load tends to unity (under proper scalings), in a general parameter setting. The results are strikingly simple and reveal explicitly how the expected delay depends on the system parameters, and in particular, on the batch-size distributions and the simultaneity of the batch arrivals. Moreover, the results suggest simple and fast-to-evaluate approximations for the expected delay in heavily loaded polling systems. Numerical experiments demonstrate that the approximations are highly accurate in medium and heavily loaded systems.     

Matrix Product-Form Solutions for LCFs Preemptive Service Single-Server Queues with Batch Markovian Arrival Streams

ABSTRACT

This paper considers three variants of last-come first-served (LCFS) preemptive service single-server queues, where customers are served under the LCFS preemptive resume (LCFS-PR), preemptive repeat-different (LCFS-PD), and preemptive repeat-identical (LCFS-PI) disciplines, respectively. These LCFS queues are fed by multiple batch Markovian arrival streams. Service times of customers from each arrival stream are generally distributed and their distributions may differ among different streams. For each of LCFS-PR, LCFS-PD, and LCFS-PI queues, we show that the stationary distribution of the queue string representing enough information to keep track of queueing dynamics has a matrix product-form solution. Further, this paper discusses the stability of LCFS-PD and LCFS-PI queues based on the busy cycle. Finally, by numerical experiment, we examine the impact of the variation of the service time distribution on the mean queue lengths for the three variants of LCFS queues.     

A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.     

Explicit formulae and convergence rate for the system M α /G/1/N as N→∞

The paper deals with the system M ^α /G/1/N with a finite number of waiting places in which arrivals can occur in a group. The number of customers in the line and the virtual waiting time are studied both in the transient and in the stationary regime. Special attention is paid to the stationary distributions of these functionals as N→∞. The number of customers lost during a busy period is considered as well.     

Joint tests of non-nested models and general error specifications

This paper is concerned with joint tests of non-nested models and simultaneous departures from homoskedasticity, serial independence and normality of the disturbance terms. Locally equivalent alternative models are used to construct joint tests since they provide a convenient way to incorporate more than one type of departure from the classical conditions. The joint tests represent a simple asymptotic solution to the “pre-testing” problem in the context of non-nested linear regression models. Our simulation results indicate that the proposed tests have good finite sample properties.     

ON THE MAP/G/1 QUEUE WITH LEBESGUE-DOMINATED SERVICE TIME DISTRIBUTION AND LCFS PREEMPTIVE REPEAT SERVICE DISCIPLINE

The present paper contains an analysis of the MAP/G/1 queue with last come first served (LCFS) preemptive repeat service discipline and Lebesgue-dominated service time distribution. The transient distribution is given in terms of a recursive formula. The stationary distribution as well as the stability condition are obtained by means of Markov renewal theory via a QBD representation of the embedded Markov chain at departures and arrivals.     

Multivariate lagrange distributions and a subfamiliy

Second order moments about its means, i.e. the variances and covari-ances for multivariate Lagrange distributions are derived in a matrix form. A subfamily of multivariate Lagrange distributions which can be characterized as the distributions of customers served in a busy period in queues with some conditions are considered. Theorems about their probability functions, one of which is a multivariate generalization of a formula by Takà cs(1989). are given and the means and second order moments about its means are considered. As an example, a multivariate Borel-Tanner distribution is derived.     

A variant of the Geo/G/1 queues with disasters and general repair times

Abstract

This paper deals with Geo/G/1 queues with a repairable server. The server is subject to failure due to a disaster arrival, which can occur while the server is turned on and not only when it is busy. At a failure instant, the server is turned off and its repair process begins. During the repair process, two models are considered. For both models, we present the PGF and the expected number of clients in the system in the steady state.     

Distance-based approach in univariate longitudinal data analysis

In this paper, we propose a methodology to analyze longitudinal data through distances between pairs of observations (or individuals) with regard to the explanatory variables used to fit continuous response variables. Restricted maximum-likelihood and generalized least squares are used to estimate the parameters in the model. We applied this new approach to study the effect of gender and exposure on the deviant behavior variable with respect to tolerance for a group of youths studied over a period of 5 years. Were performed simulations where we compared our distance-based method with classic longitudinal analysis with both AR(1) and compound symmetry correlation structures. We compared them under Akaike and Bayesian information criterions, and the relative efficiency of the generalized variance of the errors of each model. We found small gains in the proposed model fit with regard to the classical methodology, particularly in small samples, regardless of variance, correlation, autocorrelation structure and number of time measurements.     

Alternative approaches to testing non-nested models with autocorrelated disturbances

Since departures from the classical assumptions regarding the disturbances in a linear tegression model arise frequently in empirical application, deveral computationally Straightforward procedutes are presented in this paper for testiog non-nested models when the disturbances of these models follow first- or higher-order autoregressive processes. Anempirical example is used to illustrate how the procedures may be used to test competing Keynesian and New Classical non-nested models of unemployment for the U.S using annual time series data for 1955-85.     

A note on predictive inference for multivariate elliptically contoured distributions

The prediction problem is considered for the multivariate regression model with an elliptically contoured error distribution. We show that the predictive distribution under elliptical errors assumption is the same as that obtained under normally distributed error in both the Bayesian approach using an im-proper prior and the classical approach. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution.     

Testing for departures from nominal dispersion in generalized nonlinear models with varying dispersion and/or additive random effects

This paper discusses the tests for departures from nominal dispersion in the framework of generalized nonlinear models with varying dispersion and/or additive random effects. We consider two classes of exponential family distributions. The first is discrete exponential family distributions, such as Poisson, binomial, and negative binomial distributions. The second is continuous exponential family distributions, such as normal, gamma, and inverse Gaussian distributions. Correspondingly, we develop a unifying approach and propose several tests for testing for departures from nominal dispersion in two classes of generalized nonlinear models. The score test statistics are constructed and expressed in simple, easy to use, matrix formulas, so that the tests can easily be implemented using existing statistical software. The properties of test statistics are investigated through Monte Carlo simulations.     

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).     

Comparison of confidence intervals of traffic intensity forM/E k/1 queueing systems

This paper compares the width of confidence intervals for the traffic intensity inM/E _k/1 queueing systems with three methods-Lilliefors' method, a second method based on the number of arrivals during the service period, and a third method using censored observations, in which only those service periods with no arrivals are observed. It is shown that the width of confidence intervals, based on the number of arrivals during the service period, is narrower than for other methods.     

On a test for exponentiality against Laplace order dominance

Surles and Padgett recently introduced two-parameter Burr Type X distribution, which can also be described as the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both the distributions can be used quite effectively to analyze skewed data set. For a given data set the problem of selecting either generalized Rayleigh or log-normal distribution is discussed in this paper. The ratio of maximized likelihood (RML) is used in discriminating between the two distributing functions. Asymptotic distributions of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between these two families of distributions for a used specified probability of correct selection and the tolerance limit.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

An experiment in primary health care provision

Missing data and, more generally, imperfections in implementing a study design are an endemic problem in large scale studies involving human subjects. We present an analysis of an experiment in the interaction between general practitioners and their patients, in which the issue of missing data is addressed by a sensitivity analysis using multiple imputation. Instead of specifying a model for missingness we explore certain extreme ways of departing from the assumption of data missing at random and establish the largest extent of such departures which would still fail to supplant the evidence about the studied effect. An important advantage of the approach is that the algorithm intended for the complete data, to fit generalized linear models with random effects, is used without any alteration.     

Bayesian hierarchical models in manufacturing bulk service queues

In this paper, Queueing Theory and Bayesian statistical tools are used to analyze the congestion of various manufacturing bulk service queues with the same characteristics that are working independently of one another and in equilibrium. Hierarchical models are discussed in order to develop the whole inferential process for the parameters governing the system. Markov Chain Monte Carlo methods and numerical inversion of transforms are addressed to compute the posterior predictive distributions of the usual measures of performance in practice.     

A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.