Transient Analysis of Averaged Queue Length in Markovian Queues

Abstract

We consider a Markovian queue and its associated exponentially averaged length. The set of partial differential equations satisfied by the joint distribution of the queue and the averaged queue length is given. We obtain a recursive expression for the moments of the averaged queue length, and develop a stable algorithm to compute them. These results are illustrated through numerical examples.     

Erlangized Fluid Queues with Application To Uncontrolled Fire Perimeter

Abstract

The present paper develops an “Erlangization” method for fluid queues. It then applies the results we derive to a forestry problem: the evolution of an uncontrolled fire perimeter over time. Specifically, we focus on the probability of containing a fire prior to reaching a randomly distributed, finite time horizon. Transitions to lower non-zero levels are also investigated. A preliminary model is introduced to demonstrate the potential of the application, and numerical results are given for illustrative purposes.     

A Tree-Structured Markovian Model of the Shipment Consolidation Process

This article studies the dispatch of consolidated shipments. Orders, following a batch Markovian arrival process, are received in discrete quantities by a depot at discrete time epochs. Instead of immediate dispatch, all outstanding orders are consolidated and shipped together at a later time. The decision of when to send out the consolidated shipment is made based on a “dispatch policy,” which is a function of the system state and/or the costs associated with that state. First, a tree structured Markov chain is constructed to record specific information about the consolidation process; the effectiveness of any dispatch policy can then be assessed by a set of long-run performance measures. Next, the effect on shipment consolidation of varying the order-arrival process is demonstrated through numerical examples and proved mathematically under some conditions. Finally, a heuristic algorithm is developed to determine a favorable parameter of a special set of dispatch policies, and the algorithm is proved to yield the overall optimal policy under certain conditions.     

Filtering and the EM-Algorithm for the Markovian Arrival Process

In this article, we deal with the so-called Markovian Arrival Process (MAP). An MAP is thought of as a partially observed Markov process, so that the Expectation-Maximization (EM) algorithm is a natural way to estimate its parameters. Then, nonlinear filters of basic statistics related to the MAP must be computed. The forward–backward principle is the basic way to do it. Here, bearing in mind a filter-based formulation of the EM-algorithm proposed by Elliott, these filters are shown to be the solution of nonlinear stochastic differential equations (SDEs) which allows a recursive computation. This is well suited for processing large data sets. We also derive linear SDEs or Zakai equations for the so-called unnormalized filters.     

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.     

On-Line Learning for the Infinite Hidden Markov Model

We develop a sequential Monte Carlo algorithm for the infinite hidden Markov model (iHMM) that allows us to perform on-line inferences on both system states and structural (static) parameters. The algorithm described here provides a natural alternative to Markov chain Monte Carlo samplers previously developed for the iHMM, and is particularly helpful in applications where data is collected sequentially and model parameters need to be continuously updated. We illustrate our approach in the context of both a simulation study and a financial application.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.     

Analysis of a discrete-time queue with load dependent service under discrete-time Markovian arrival process

In the study of normal queueing systems, the server’s average service times are generally assumed to be constant. However, in numerous applications this assumption may not be valid. To prevent congestion in overload control telecommunication networks, the transmission rates vary depending on the number of packets waiting in the queue. As traffics in telecommunication networks are of bursty nature and correlated, we assume that arrivals follow the discrete-time Markovian arrival process. This paper analyzes a queueing model in which the server changes its service times (rates) only at the beginning of service depending on the number of customers waiting in the queue. We obtain the steady-state probabilities at various epochs and some performance measures. In addition, varieties of numerical results are discussed to display the effect of the system parameters on the performance measures.     

A Simulation Study Comparing Modeling Approaches in an Illness-Death Multi-State Model

In many medical studies, there are covariates that change their values over time and their analysis is most often modeled using the Cox regression model. However, many of these time-dependent covariates can be expressed as an intermediate event, which can be modeled using a multi-state model. Using the relationship of time-dependent (discrete) covariates and multi-state models, we compare (via simulation studies) the Cox model with time-dependent covariates with the most frequently used multi-state regression models. This article also details the procedures for generating survival data arising from all approaches, including the Cox model with time-dependent covariates.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

基于MixedLogit模型的道路交通统计生命价值评价研究

道路交通统计生命价值是道路交通安全项目经济评价的重要指标。运用意愿选择法和正交试验法设计了出行路径选择的调查问卷。假定时间、死亡风险为常数,费用服从对数正态分布,建立了基于Mixed Logit模型的统计生命价值评价模型。以大连市私家车出行者为调查对象获得调查数据,利用Monte Carlo方法并借助GAUSS软件对模型进行了150次仿真实验。研究表明：模型各参数的估计值具有较强集中性,t检验值具有较强显著性,模型优度比较高。统计生命价值服从参数为0．922和0．814的对数正态分布,其数学期望是35万元,可以作为道路交通安全项目经济评价的参考数据。     

Transient Analysis of the M/M/k/N/N Queue using a Continuous Time Homogeneous Markov System with Finite State Size Capacity

In this article, the M/M/k/N/N queue is modeled as a continuous-time homogeneous Markov system with finite state size capacity (HMS/c_s). In order to examine the behavior of the queue a continuous-time homogeneous Markov system (HMS) constituted of two states is used. The first state of this HMS corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the buffer state which represents the system's queue. In order to examine the variability of the state sizes formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence, the pmf of each state size can be evaluated for any t ∈ ?⁺. The theoretical results are illustrated by a numerical example.     

On semi-markov compartmental models with branching particles under the influence of disasters

A semi-Markov multi-compart mental system in which particles reproduce similar particles as a Markov branching process and being subjected to disasters is studied. Expressions for the mean number of particles alive at time t in each compartment are obtained. The results concerning irreversible, mammillarian and catenary compartmental systems have been discussed.     

A Stochastic Geometry Model for Functional Magnetic Resonance Images

In functional magnetic resonance imaging, spatial activation patterns are commonly estimated using a non-parametric smoothing approach. Significant peaks or clusters in the smoothed image are subsequently identified by testing the null hypothesis of lack of activation in every volume element of the scans. A weakness of this approach is the lack of a model for the activation pattern; this makes it difficult to determine the variance of estimates, to test specific neuroscientific hypotheses or to incorporate prior information about the brain area under study in the analysis. These issues may be addressed by formulating explicit spatial models for the activation and using simulation methods for inference. We present one such approach, based on a marked point process prior. Informally, one may think of the points as centres of activation, and the marks as parameters describing the shape and area of the surrounding cluster. We present an MCMC algorithm for making inference in the model and compare the approach with a traditional non-parametric method, using both simulated and visual stimulation data. Finally we discuss extensions of the model and the inferential framework to account for non-stationary responses and spatio-temporal correlation.     

Unobserved heterogeneity in Markovian analysis of the size distortion of unit root tests

We confirm that units root tests can exhibit substantial size distortion when breaks in mean are generated by a first-order Markov chain, but unlike previous literature, we find augmentation largely remedies this situation. However, considerable heterogeneity is evident in the size properties of the tests when faced with breaks in mean varying in duration, in number and in position within the sample. This heterogeneity will be hidden when a Markov chain is employed. For instance, when the transition probabilities generate single period outliers, rejection frequencies (RFs) rise substantially with the number of outliers, but augmentation results in approximately nominal RFs. Qualitatively similar results hold when a number of structural breaks are allocated randomly in the central section of the sample. Interestingly, very different behaviour is exposed by a design exploring the impact on the tests of two breaks imposed at a range of fixed intervals, RFs rising when break occur in the extremities of the sample, a situation unaffected by augmentation.     

Application of a Markovian process to the calculation of mean time equilibrium in a genetic drift model

The most common phenomena in the evolution process are natural selection and genetic drift. In this article, we propose a probabilistic method to calculate the mean and variance time for random genetic drift equilibrium, measured as number of generations, based on Markov process and a complex probabilistic model. We studied the case of a constant, panmictic population of diploid organisms, which had a demonstrated lack of mutation, selection or migration for a determined autonomic locus, and two possible alleles, H and h. The calculations presented in this article were based on a Markov process. They explain how genetic and genotypic frequencies changed in different generations and how the heterozygote alleles became extinguished after many generations. This calculation could be used in more evolutionary applications. Finally, some simulations are presented to illustrate the theoretical calculations presented using different basal situations.     

The Multi-scale Marked Area-interaction Point Process: A Model for the Spatial Pattern of Trees

Abstract.  The spatial pattern of trees in forests often combines different types of structure (regularity, clustering or randomness) at different scales. Taking species or size into account leads to marked patterns. The question addressed is to model such multi-scale marked patterns using a single process. Within the category of Markov processes, the area-interaction process has the advantage of being locally stable, whether it is attractive or repulsive. This process was originally defined as a one-scale non-marked process. We propose an extension as a multi-scale marked process. Three examples are presented to show the adequacy of this process to model tree patterns: 1. A pine pattern showing anisotropic regularity and clustering at different scales. 2. A bivariate (adult/juvenile) kimboto pattern in French Guiana, showing regularity for one type, clustering for the other and repulsion between the two. 3. A marked pattern in Gabon where the mark is tree diameter.     

A Class of Markov Chain Models for Average Run Length Computations for Autocorrelated Processes

The average run length (ARL) of conventional control charts is typically computed assuming temporal independence. However, this assumption is frequently violated in practical applications. Alternative ARL computations have often been conducted via time consuming and yet not necessarily very accurate simulations. In this article, we develop a class of Markov chain models for evaluating the run length performance of traditional control charts for autocorrelated processes. We show extensions from the univariate AR(1) model to the general multivariate VARMA(p, q) time series. The results of the proposed method are highly comparable to those of simulations and with significantly less computational overhead.