Analysis of queues with hyperexponential arrival distributions

We study H₂/H₂/1, H₂/M/1, and M/H₂/1 queueing systems with hyperexponential arrival distributions for the purpose of finding a solution for the mean waiting time in the queue. To this end we use the spectral decomposition method for solving the Lindley integral equation. For practical application of the obtained results, we use the method of moments. Since the hyperexponential distribution law has three unknown parameters, it allows to approximate arbitrary distributions with respect to the first three moments. The choice of this distribution law is due to its simplicity and the fact that in the class of distributions with coefficients of variation greater than 1, such as log-normal, Weibull, etc., only the hyperexponential distribution makes it possible to obtain an analytical solution.     

A note on mixed exponential approximations for Gi/G/1 waiting times

A method is offered for the effective estimation of the stationary waiting-time distribution of the GI/G/1 queue by a (possibly nonconvex) mixed exponential CDF. The approach relies on obtaining a generalized exponential mixture as an approximation for the distribution of the service times. This is done by the adaptation of a nonlinear optimization algorithm previously developed for the maximum-likelihood estimation of parameters from mixed Weibull distributions. The approach is particularly well-suited for obtaining the delay distribution beginning from raw interarrivai and service-time data.     

Diffusion approximations for a single node accessed by congestion-controlled sources

We consider simple models of congestion control in high-speed networks, and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ON-OFF type with exponential ON and OFF times, then, under a certain scaling, the steady-state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steady-state distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service-time distributions that are phase type, and show the impact of these distributions on the steady-state distribution of the number of sources in the system. We also establish an insensitivity to service-time distribution when the arrival process is Poisson. We use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic.     

Testing for a class of bivariate exponential distributions

Bivariate and multivariate exponential distributions are widely applied in several areas such as reliability, queueing systems or hydrology. A frequently used bivariate exponential distribution is the Moran–Downton distribution. Because of this reason, this paper proposes a goodness-of-fit test for this distribution. The test statistic exploits the analytically convenient formula of its characteristic function. Large sample properties of the proposed test such as consistency against fixed and local alternatives are studied. The finite sample performance is numerically studied. Finally, an application of this distribution to hydrological data is presented.     

Mean-delay approximation for cyclic-service queueing systems

This paper investigates mean delays in a single-server queueing model with cyclic service. An approximate analysis is performed for an arbitrary number of queues, constant switch-over time, exhaustive service discipline, Poisson arrival processes, and general service-time distributions. Neither inter-arrival nor service-time distributions have to be identical for the different queues. A major value of our approximation lies in the simplicity of its numerical evaluation for an arbitrary number of queues and any traffic pattern. Extensive comparisons with simulation results show the high accuracy of the approximation over a wide range of parameters.     

On the probability distribution of faults in complex software systems

ContextThere are several empirical principles related to the distribution of faults in a software system (e.g. the Pareto principle) widely applied in practice and thoroughly studied in the software engineering research providing evidence in their favor. However, the knowledge of the underlying probability distribution of faults, that would enable a systematic approach and refinement of these principles, is still quite limited.ObjectiveIn this paper we study the probability distribution of faults detected during verification in four consecutive releases of a large-scale complex software system for the telecommunication exchanges. This is the first such study analyzing closed software system, replicating two previous studies for open source software.MethodWe take into consideration the Weibull, lognormal, double Pareto, Pareto, and Yule–Simon probability distributions, and investigate how well these distributions fit our empirical fault data using the non-linear regression.ResultsThe results indicate that the double Pareto distribution is the most likely choice for the underlying probability distribution. This is not consistent with the previous studies on open source software.ConclusionThe study shows that understanding the probability distribution of faults in complex software systems is more complicated than previously thought. Comparison with previous studies shows that the fault distribution strongly depends on the environment, and only further replications would make it possible to build up a general theory for a given context.     

A maximum entropy priority approximation for a stableG/G/1 queue

Summary The principle of maximum entropy is used under two different sets of mean value constraints to analyse a stableG/G/1 queue withR priority classes under preemptive-resume (PR) and non-preemptive head-of-line (HOL) scheduling disciplines. New one-step recursions for the maximum entropy state probabilities are established and closed form approximations for the marginal queue length distribution per priority class are derived. To expedite the utility of the maximum entropy solutions exact analysis, based on the generalised exponential (GE) distribution, is used to approximate the marginal mean queue length and idle state probability class constraints for both the PR and HOLG/G/1 priority queues. Moreover, these results are used as building blocks in order to provide new approximate formulae for the mean and coefficient of variation of the effective priority service-time and suggest a maximum entropy algorithm for general open queueing networks with priorities in the context of the reduced occupancy approximation (ROA) method. Numerical examples illustrate the accuracy of the proposed maximum entropy approximations in relation to simulations involving different interarrival-time and service-time distributions per class. Comments on the extension of the work to more complex types of queueing systems are included.This work is sponsored in part by the Science and Engineering Research Council (SERC), UK, under grant GR/D/12422 and in part by the Ministry of Higher Education of the Algerian Government     

Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring

The step-stress accelerated life tests allow the experimenter to increase the stress levels at fixed times during the experiment. The lifetime of a product at any level of stress is assumed to have an exponentiated distribution, whose baseline distribution is a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz, normal and logistic distributions. The scale parameter of the baseline distribution is assumed to be a log-linear function of the stress and a cumulative exposure model holds. Special attention is paid to an exponentiated exponential distribution. Based on type-I censoring, the maximum likelihood estimates of the parameters under consideration are obtained. A Monte Carlo simulation study is carried out to investigate the precision of the maximum likelihood estimates and to obtain the coverage probabilities of the bootstrap confidence intervals for the parameters involved. Finally, an example is presented to illustrate the two discussed methods of bootstrap confidence intervals.     

侧扫声呐图像统计特性研究

在分析侧扫声呐成像的散射模型基础上,给出了海底混响服从瑞利分布的理论依据.分析了其它典型概率分布的合理性并给出了各概率分布参数的估计方法.通过实验来检验比较几种典型的概率分布的最佳拟合.实验结果表明:总体来看,威布尔分布更为精确,但瑞利分布仍保持了较好的拟合特性.特别是当底质粗糙、混响较强时,使用瑞利分布和威布尔分布相当.这一研究结果可为侧扫声呐图像的底质分割提供重要依据.