Approximating M/G/1 Waiting Time Tail Probabilities

Abstract

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.     

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.     

A multivariate version of Gini's rank association coefficient

In this paper, we introduce a multivariate generalization of the population version of Gini's rank association coefficient, giving a response to this open question posed in [4]. We also study some properties of this version, present the corresponding results for the sample statistic, and provide several examples.     

On Christensen's conjecture

In this paper we give solution to a conjecture appearing in Christensen (1997, p. 360) in relation to the definitions of standardized residuals in loglinear models. Work done when he was visiting as the Distinguished Lukacs Professor on leave from the Complutense University of Madrid. An erratum to this article is available at .     

Two-stage Huber estimation

In this paper we propose a new robust estimator in the context of two-stage estimation methods directed towards the correction of endogeneity problems in linear models. Our estimator is a combination of Huber estimators for each of the two stages, with scale corrections implemented using preliminary median absolute deviation estimators. In this way we obtain a two-stage estimation procedure that is an interesting compromise between concerns of simplicity of calculation, robustness and efficiency. This method compares well with other possible estimators such as two-stage least-squares (2SLS) and two-stage least-absolute-deviations (2SLAD), asymptotically and in finite samples. It is notably interesting to deal with contamination affecting more heavily the distribution tails than a few outliers and not losing as much efficiency as other popular estimators in that case, e.g. under normality. An additional originality resides in the fact that we deal with random regressors and asymmetric errors, which is not often the case in the literature on robust estimators.     

Evaluation of multilevel decision trees

The evaluation of decision trees under uncertainty is difficult because of the required nested operations of maximizing and averaging. Pure maximizing (for deterministic decision trees) or pure averaging (for probability trees) are both relatively simple because the maximum of a maximum is a maximum, and the average of an average is an average. But when the two operators are mixed, no simplification is possible, and one must evaluate the maximization and averaging operations in a nested fashion, following the structure of the tree. Nested evaluation requires large sample sizes (for data collection) or long computation times (for simulations).     

Minimum contrast estimation of random processes based on information of second and third orders

This paper presents the results on consistency and asymptotic normality of a class of minimum contrast estimators for random processes with short- or long-range dependence based on the second- and third-order cumulant spectra. Asymptotic properties of sample spectral functionals of second and third orders, which are of independent interest in view of their possible use for nonparametric estimation of processes with short- or long-range dependence, are also provided.     

On the detection of changes in autoregressive time series I. Asymptotics

Various test statistics are discussed which can be used for detecting changes in the parameters of an autoregressive time series. In this first part of our study, the limiting behavior of the test statistics is derived under the null hypothesis of no change as well as under alternatives. In a forthcoming second part of our investigation, these asymptotic results will be compared to some corresponding bootstrap procedures, and a small simulation study will be conducted.     

Wilcoxon–Mann–Whitney test for stratified samples and Efron's paradox dice

Two-treatment multi-center clinical trials are the most common type of clinical trials in practice. The aim of this paper is to discuss a curious property of certain standard nonparametric procedures used in the analysis of such clinical trials. Different analyses of a simulated data example are presented, which lead to contrasting and surprising results. The source of the potentially misleading outcome is then explored while relating the simulated data with the concept of Efron's paradox dice and the notion of nontransitivity. With the root of the problem established, an alternate nonparametric method from the literature is shown to address the problem. Finally, pointing out an interpretational concern of using the alternate procedure, a modification to this procedure is also suggested and corresponding theoretical results are presented.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.