Estimation in M/Er /1 queueing systems

In this paper, the maximum likelihood estimates of the parameters for the M/E_r /1 queueing model are derived when the queue size at each departure point is observed. A numerical example is generated by simulating a finite Markov chain to illustrate the methodology for estimating the parameters with variable Erlang service time distribution. The problem of hypothesis testing and simultaneous Confidence regions of the parameter is also investigated.0     

Workload Process,Waiting Times,and Sojourn Times in a Discrete Time MMAP[K]/SM[K]/1/FCFS Queue

Abstract

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.     

Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n ? 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.     

Semiparametric estimation for the dispersion parameter in the analysis of over- or underdispersed count data

This paper investigates several semiparametric estimators of the dispersion parameter in the analysis of over- or underdispersed count data when there is no likelihood available. In the context of estimating the dispersion parameter, we consider the double-extended quasi-likelihood (DEQL), the pseudo-likelihood and the optimal quadratic estimating (OQE) equations method and compare them with the maximum likelihood method, the method of moments and the extended quasi-likelihood through simulation study. The simulation study shows that the estimator based on the DEQL has superior bias and efficiency property for moderate and large sample size, and for small sample size the estimator based on the OQE equations outperforms the other estimators. Three real-life data sets arising in biostatistical practices are analyzed, and the findings from these analyses are quite similar to what are found from the simulation study.     

Matrix–geometric analysis of the discrete time Gi/G/1 system

In this paper, we show that the discrete GI/G/1 system can be analysed as a QBD process with infinite blocks. Most importantly, we show that Matrix–geometric method can be used for analyzing this general queue system including establishing its stability criterion and for obtaining the explicit stationary probability and the waiting time distributions. This also settles the unwritten myth that Matrix–geometric method is limited to cases with at least one Markov based characterizing parameter, i.e. either interarrival or service times, in the case of queueing systems.     

An M/G/1 clearing queueing system with setup time and multiple vacations for an unreliable server

This paper studies an M/G/1 clearing queueing system with setup time and multiple vacations, in which all present customers in the system are served simultaneously and breakdowns may occur in busy or setup period. We investigate the stationary distribution of system size and the Laplace–Stieltjes transform of sojourn time. In addition, various performance measures are discussed, such as the mean system size at arbitrary time and the mean length of a vacation circle. Moreover, a cost analysis is carried out for this queueing system. Numerical results are presented to study the sensitivity of the system parameters on the expected cost function and system performances.     

Estimation of Traffic Intensity Based on Queue Length in a Single M/M/1 Queue

In this article, maximum likelihood estimator (MLE) as well as Bayes estimator of traffic intensity (ρ) in an M/M/1/∞ queueing model in equilibrium based on number of customers present in the queue at successive departure epochs have been worked out. Estimates of some functions of ρ which provide measures of effectiveness of the queue have also been derived. A comprehensive simulation study starting with the transition probability matrix has been carried out in the last section.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

Estimators for a Poisson parameter using ranked set sampling

Using ranked set sampling, a viable BLUE estimator is obtained for estimating the mean of a Poisson distribution. Its properties, such as efficiency relative to the ranked set sample mean and to the maximum likelihood estimator, have been calculated for different sample sizes and values of the Poisson parameter. The estimator (termed the normal modified r.s.s. estimator is more efficient than both the ranked set sample mean and the MLE. It is recommended as a reasonable estimator of the Poisson mean ( λ) to be used in a ranked set sampling environment.     

Semi-parametric modelling and likelihood estimation with estimating equations

This paper proposes a semi-parametric modelling and estimating method for analysing censored survival data. The proposed method uses the empirical likelihood function to describe the information in data, and formulates estimating equations to incorporate knowledge of the underlying distribution and regression structure. The method is more flexible than the traditional methods such as the parametric maximum likelihood estimation (MLE), Cox's (1972) proportional hazards model, accelerated life test model, quasi-likelihood (Wedderburn, 1974) and generalized estimating equations (Liang & Zeger, 1986). This paper shows the existence and uniqueness of the proposed semi-parametric maximum likelihood estimates (SMLE) with estimating equations. The method is validated with known cases studied in the literature. Several finite sample simulation and large sample efficiency studies indicate that when the sample size is larger than 100 the SMLE is compatible with the parametric MLE; and in all case studies, the SMLE is about 15% better than the parametric MLE with a mis-specified underlying distribution.     

Filtering recursions for calculating likelihoods for queues based on inter-departure time data

We consider inference for queues based on inter-departure time data. Calculating the likelihood for such models is difficult, as the likelihood involves summing up over the (exponentially-large) space of realisations of the arrival process. We demonstrate how a likelihood recursion can be used to calculate this likelihood efficiently for the specific cases of M/G/1 and Er/G/1 queues. We compare the sampling properties of the mles to the sampling properties of estimators, based on indirect inference, which have previously been suggested for this problem.     

A transient analysis of M/G/1 queues with N-policy

We consider an M/G/1 queueing model with N-policy operating. This means, that the server will start up only if a queue of a prescribed length has built up. For this model the time dependent distribution of the queue length is given by simple renewal arguments without resorting to integral transform techniques.     

Comparison of methods of estimation of parameters of the weibull distribution

The generalised least squares, maximum likelihood, Bain-Antle 1 and 2, and two mixed methods of estimating the parameters of the two-parameter Weibull distribution are compared. The comparison is made using (a) the observed relative efficiency of parameter estimates and (b) themean squared relative error in estimated quantiles, to summarize the results of 1000 simulated samples of sizes 10 and 25. The results are that: generalised least squares is the best method of estimating the shape parameter ß the best method of estimating the scale parameter a depends onthe size of ß for quantile estimation maximum likelihood is best Bain-Antle 2 is uniformly the worst of the methods.     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

Practical Point Estimation from Higher-Order Pivots

Point estimators for a scalar parameter of interest in the presence of nuisance parameters can be defined as zero-level confidence intervals as explained in Skovgaard (1989). A natural implementation of this approach is based on estimating equations obtained from higher-order pivots for the parameter of interest. In this paper, generalising the results in Pace and Salvan (1999) outside exponential families, we take as an estimating function the modified directed likelihood. This is a higher-order pivotal quantity that can be easily computed in practice for a wide range of models, using recent advances in higher-order asymptotics (HOA, 2000). The estimators obtained from these estimating equations are a refinement of the maximum likelihood estimators, improving their small sample properties and keeping equivariance under reparameterisation. Simple explicit approximate versions of these estimators are also derived and have the form of the maximum likelihood estimator plus a function of derivatives of the loglikelihood function. Some examples and simulation studies are discussed for widely-used model classes.     

C335. Some numerology for the mass ratio of proton to electron,and a partially Bayesian evaluation

Recent small sample studies of estimators for the shape parameter a of the negative binomial distribution (NBD) tend to indicate that the choice of estimator can be reduced to a choice between the method of moments estimator, maximum likelihood estimator (MLE), maximum quasi-likelihood estimator and the conditional likelihood estimator (CLE). In this paper the results of a comprehensive simulation study are reported to assist with the choice from these four estimators. The study includes a traditional procedure for assessing estimators for the shape parameter of the NBD and in addition introduces an alternative assessment procedure. Based on the traditional approach the CLE is considered to perform the best overall for the range of parameter values and sample sizes considered. The alternative assessment procedure indicates that the MLE is the preferred estimator.     

Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

A Random Coefficient Degradation Model With Ramdom Sample Size

In testing product reliability, there is often a critical cutoff level that determines whether a specimen is classified as failed. One consequence is that the number of degradation data collected varies from specimen to specimen. The information of random sample size should be included in the model, and our study shows that it can be influential in estimating model parameters. Two-stage least squares (LS) and maximum modified likelihood (MML) estimation, which both assume fixed sample sizes, are commonly used for estimating parameters in the repeated measurements models typically applied to degradation data. However, the LS estimate is not consistent in the case of random sample sizes. This article derives the likelihood for the random sample size model and suggests using maximum likelihood (ML) for parameter estimation. Our simulation studies show that ML estimates have smaller biases and variances compared to the LS and MML estimates. All estimation methods can be greatly improved if the number of specimens increases from 5 to 10. A data set from a semiconductor application is used to illustrate our methods.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.