Approximating Unreliable Queueing Networks Under the Assumption of Exponentiality

This paper deals with the approximate analysis and application of an unreliable closed queueing network to model the performance of a flexible manufacturing system. The service stations in the network are subject to breakdown and repair when active. An exact equivalent network is represented with two-stage Coxian service stations. We approximate the equivalent network by assuming exponentiality for the service completion time, which has a two-stage Coxian distribution. We validate the approximation for a wide range of model parameters and demonstrate the robustness of queueing network formulae to the assumption of exponentiality.     

Computational learning of the conditional phase-type (C-Ph) distribution

This paper presents a new algorithm for learning the structure of a special type of Bayesian network. The conditional phase-type (C-Ph) distribution is a Bayesian network that models the probabilistic causal relationships between a skewed continuous variable, modelled by the Coxian phase-type distribution, a special type of Markov model, and a set of interacting discrete variables. The algorithm takes a data set as input and produces the structure, parameters and graphical representations of the fit of the C-Ph distribution as output. The algorithm, which uses a greedy-search technique and has been implemented in MATLAB, is evaluated using a simulated data set consisting of 20,000 cases. The results show that the original C-Ph distribution is recaptured and the fit of the network to the data is discussed.     

Mixture Modeling for Longitudinal Data

In this article, we propose an unbiased estimating equation approach for a two-component mixture model with correlated response data. We adapt the mixture-of-experts model and a generalized linear model for component distribution and mixing proportion, respectively. The new approach only requires marginal distributions of both component densities and latent variables. We use serial correlations from subjects’ subgroup memberships, which improves estimation efficiency and classification accuracy, and show that estimation consistency does not depend on the choice of the working correlation matrix. The proposed estimating equation is solved by an expectation-estimating-equation (EEE) algorithm. In the E-step of the EEE algorithm, we propose a joint imputation based on the conditional linear property for the multivariate Bernoulli distribution. In addition, we establish asymptotic properties for the proposed estimators and the convergence property using the EEE algorithm. Our method is compared to an existing competitive mixture model approach in both simulation studies and an election data application. Supplementary materials for this article are available online.     

Queueing models for a flexible machining station Part II: The method of Coxian phases

A work station of a flexible manufacturing system (fms) is modeled as a multi-server queue with finite waiting room. The interarrival/service time distributions have squared coefficients of variations not less than 0.5 and are modeled as Coxian laws of two phases. A recursive scheme is developed to calculate the equilibrium queue length distribution. The model, together with the diffusion approximation model of Part I, can be used to aid the design of fms work stations.     

Experimenting with the Coxian Phase-Type Distribution to Uncover Suitable Fits

In the past few decades, Coxian phase-type distributions have become increasingly more popular as a means of representing survival times. In healthcare, they are considered suitable for modelling the length of stay of patients in hospital and more recently for modelling the patient waiting times in Accident and Emergency Departments. The Coxian phase-type distribution has not only been shown to provide a good representation of real survival data, but its interpretation seems reasonably initiative to the medical experts. The drawback, however, is fitting the distribution to the data. There have been many attempts at accurately estimating the Coxian phase-type parameters. This paper wishes to examine the most promising of the approaches reported in the literature to determine the most accurate. Three performance measures are introduced to assess the fitting process of the algorithms along with the likelihood values and AIC to examine the goodness of fit and complexity of the model. Previous research suggests that the fitting process is strongly influenced by the initial parameter estimates and the data itself being quite variable. To overcome this, one experiment in this research paper will use the same initial parameter values for each estimation and perform the fits on the data simulated from a Coxian phase-type distribution with known parameters.     

Modelling activities at a neurological rehabilitation unit

A queuing model of a specialist neurological rehabilitation unit is studied. The application is to the Neurological Rehabilitation Centre at Rookwood Hospital (Cardiff, UK), the national rehabilitation unit for Wales. Due to high demand this 21-bed inpatient facility is nearly always at maximum occupancy, and with a significant bed-cost per day this makes it a prime candidate for mathematical modelling. Central to this study is the concept that treatment intensity has an effect on patient length of stay. The model is constructed in four stages. First, appropriate patient groups are determined based on a number of patient-related attributes. Second, a purpose-built scheduling program is used to deduce typical levels of treatment to patients of each group. These are then used to estimate the mean length of stay for each patient group. Finally, the queuing model is constructed. This consists of a number of disconnected homogeneous server queuing systems; one for each patient group. A Coxian phase-type distribution is fitted to the length of time from admission until discharge readiness and an exponential distribution models the remainder of time until discharge. Some hypothetical scenarios suggested by senior management are then considered and compared on the grounds of a number of performance measures and cost implications.     

Derivation of the N-step interdeparture time distribution in GI/G/1 queueing systems

The departure process of a queueing system has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this paper, we derive a closed form expression for the distribution of interdeparture time in a GI/G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general K_M/G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.     

Heteroscedastic mixture transition distribution (HMTD) model

The paper further studies the heteroscedastic mixture transition distribution (HMTD) model introduced by Berchtold. Both the expectation and the standard deviation of each component are written as functions of the past of the process. The stationarity conditions are derived. An expectation conditional maximization (ECM) algorithm is used and shown to work well for estimation, the model selection problem is addressed, and the formulaes for computing the observed information matrix are derived. The shape changing feature of conditional distributions makes the model capable of modelling time series with asymmetric or multimodal distribution. The model is applied to several simulated and real datasets with satisfactory results.     

贝叶斯向量自回归分析方法及其应用

由于经济环境的多变,使得经济预测面临数据量少的建模难题,贝叶斯方法对小样本数据建模问题具有明显优势。本文在共轭条件似然函数"矩阵正态-Wishart分布"意义下,首先讨论了向量自回归模型的贝叶斯分析方法,得到了模型参数的后验分布与一步预测分布。其次,给出了分量方程的对应结果,说明了模型阶数的推断方法。最后,列出了计算步骤,并作为应用,对上海房地产价格指数数据进行预测建模,取得了较好效果。     

Approximate sojourn time distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system scheduled by the discriminatory processor sharing discipline. Poisson arrivals of customers and exponential amounts of service requirements are assumed. At any moment of being served, a customer can leave the system without completion of its service. In the asymptotic regime, where the ratio of the time scales of the two-class customers is infinite, we obtain the conditional sojourn time distribution of each class customers. Numerical experiments show that the time scale decomposition approach developed in this paper gives a good approximation to the conditional sojourn time distribution as well as the expectation of it.     

A robust-CVaR optimization approach with application to breast cancer therapy

We present a framework to optimize the conditional value-at-risk (CVaR) of a loss distribution under uncertainty. Our model assumes that the loss distribution is dependent on the state of some system and the fraction of time spent in each state is uncertain. We develop and compare two robust-CVaR formulations that take into account this type of uncertainty. We motivate and demonstrate our approach using radiation therapy treatment planning of breast cancer, where the uncertainty is in the patient’s breathing motion and the states of the system are the phases of the patient’s breathing cycle. We use a CVaR representation of the tails of the dose distribution to the points in the body and account for uncertainty in the patient’s breathing pattern that affects the overall dose distribution.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

Learning Bayesian network classifiers by risk minimization

Bayesian networks (BNs) provide a powerful graphical model for encoding the probabilistic relationships among a set of variables, and hence can naturally be used for classification. However, Bayesian network classifiers (BNCs) learned in the common way using likelihood scores usually tend to achieve only mediocre classification accuracy because these scores are less specific to classification, but rather suit a general inference problem. We propose risk minimization by cross validation (RMCV) using the 0/1 loss function, which is a classification-oriented score for unrestricted BNCs. RMCV is an extension of classification-oriented scores commonly used in learning restricted BNCs and non-BN classifiers. Using small real and synthetic problems, allowing for learning all possible graphs, we empirically demonstrate RMCV superiority to marginal and class-conditional likelihood-based scores with respect to classification accuracy. Experiments using twenty-two real-world datasets show that BNCs learned using an RMCV-based algorithm significantly outperform the naive Bayesian classifier (NBC), tree augmented NBC (TAN), and other BNCs learned using marginal or conditional likelihood scores and are on par with non-BN state of the art classifiers, such as support vector machine, neural network, and classification tree. These experiments also show that an optimized version of RMCV is faster than all unrestricted BNCs and comparable with the neural network with respect to run-time. The main conclusion from our experiments is that unrestricted BNCs, when learned properly, can be a good alternative to restricted BNCs and traditional machine-learning classifiers with respect to both accuracy and efficiency.     

LTC: A latent tree approach to classification

Latent tree models were proposed as a class of models for unsupervised learning, and have been applied to various problems such as clustering and density estimation. In this paper, we study the usefulness of latent tree models in another paradigm, namely supervised learning. We propose a novel generative classifier called latent tree classifier (LTC). An LTC represents each class-conditional distribution of attributes using a latent tree model, and uses Bayes rule to make prediction. Latent tree models can capture complex relationship among attributes. Therefore, LTC is able to approximate the true distribution behind data well and thus achieves good classification accuracy. We present an algorithm for learning LTC and empirically evaluate it on an extensive collection of UCI data. The results show that LTC compares favorably to the state-of-the-art in terms of classification accuracy. We also demonstrate that LTC can reveal underlying concepts and discover interesting subgroups within each class.     

A single server queue with gated processor-sharing discipline

In this paper we consider a single server queue in which arrivals occur according to a Poisson process and each customer's service time is exponentially distributed. The server works according to the gated process-sharing discipline. In this discipline, the server provides service to a batch of at mostm customers at a time. Once a batch of customers begins service, no other waiting customer can receive service until all members of the batch have completed their service. For this queue, we derive performance characteristics, such as waiting time distribution, queue length distribution etc. For this queue, it is possible to obtain the mean conditional response time for a customer whose service time is known. This conditional response time is a nonlinear function (as opposed to the linear case for the ordinary processor-sharing queue). A special case of the queue (wherem=) has an interesting and unusual solution. For this special case, the size of the batch for service is a Markov chain whose steady state distribution can be explicitly written down. Apart from the contribution to the theory of Markov chains and queues, the model may be applicable to scheduling of computer and communication systems.     

A dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models

We develop a highly efficient MC method for computing plain vanilla European option prices and hedging parameters under a very general jump-diffusion option pricing model which includes stochastic variance and multi-factor Gaussian interest short rate(s). The focus of our MC approach is variance reduction via dimension reduction. More specifically, the option price is expressed as an expectation of a unique solution to a conditional Partial Integro-Differential Equation (PIDE), which is then solved using a Fourier transform technique. Important features of our approach are (1) the analytical tractability of the conditional PIDE is fully determined by that of the Black–Scholes–Merton model augmented with the same jump component as in our model, and (2) the variances associated with all the interest rate factors are completely removed when evaluating the expectation via iterated conditioning applied to only the Brownian motion associated with the variance factor. For certain cases when numerical methods are either needed or preferred, we propose a discrete fast Fourier transform method to numerically solve the conditional PIDE efficiently. Our method can also effectively compute hedging parameters. Numerical results show that the proposed method is highly efficient.     

On asymptotic tests in three-factor factorial designs with no replications

We revisit the problems of testing three-factor classification models with a single observation per cell. A common approach in analyzing such nonreplicated data is to omit the highest order interaction and regard it as error. This paper discusses the use of a multiplicative model (See and Smith, 1996 and 1998) which is applied on residuals in order to separate the variability due to three-factor interaction from what is counted as random error. In particular, to test the significance of the interaction term, we derived an approximated distribution of the likelihood ratio test statistic based on the quadrilinear model known as Tucker’s three-mode principal component model. The derivation utilizes the distribution of the eigenvalues of the Wishart matrix.     

A Foreground–Background queueing model with speed or capacity modulation

Two models involving a foreground and a background queue are studied in the steady state. Service is provided either by a single server whose speed depends on the total number of jobs present, or by several parallel servers whose number may be controlled dynamically. Job service times have a two-phase Coxian distribution. Incoming jobs join the foreground queue where they execute phase 1, and then possibly move to the background queue for the second phase at lower priority. The trade-offs between holding and energy consumption costs are examined by means of a suitable cost function. Two different two-dimensional Markov processes are solved exactly. The solutions are used in several numerical experiments, aimed at illustrating different aspects of system behaviour.     

A Predictive Approach to Tail Probability Estimation

One of the issues contributing to the success of any extreme value modeling is the choice of the number of upper order statistics used for inference, or equivalently, the selection of an appropriate threshold. In this paper we propose a Bayesian predictive approach to the peaks over threshold method with the purpose of estimating extreme quantiles beyond the range of the data. In the peaks over threshold (POT) method, we assume that the threshold identifies a model with a specified prior probability, from a set of possible models. For each model, the predictive distribution of a future excess over the corresponding threshold is computed, as well as a conditional estimate for the corresponding tail probability. The unconditional tail probability for a given future extreme observation from the unknown distribution is then obtained as an average of the conditional tail estimates with weights given by the posterior probability of each model.