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.     

On the approximation of empirical data for service system simulations

The difficulties associated with parameter estimation for phase-type approximations of empirical data distributions in queue modelling are well known. While significant progress has been achieved in improving such approximations, difficulties in parameter estimation still limit the extent to which queue modelling is applied in practice. This paper presents a simplified technique for approximating empirical data in service system simulations, based on the specialised Cox phase-type distribution. When utilised in simulation modelling of a service system, the specialised Cox distribution is shown to provide improved approximations to the combined waiting and service time distribution without the need for complex parameter estimation techniques. This approach should enable much greater flexibility in the application of queue modelling to service systems.     

Compartmental models with phase-type residence-time distributions

Phase-type (PH) probability distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where “individuals” or “particles” move between a series of compartments, so that phase-type compartmental residence-time distributions can be incorporated simply by increasing the size of the system. Examples of PH distributions covering a variety of behaviours are given, and an application involving data analysis is included.     

Probability masses fitting in the analysis of manufacturing flow lines

The analysis of manufacturing systems with finite capacity and with general service time distributions is made of two steps: the distributions have first to be transformed into tractable phase-type distributions, and then the modified system can be analytically modelled. In this paper, we propose a new alternative in order to build tractable phase-type distributions, and study its effects on the global modelling process. Called “probability masses fitting” (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic, refinable, it approximates distributions with finite support and it conserves the shape of the distribution. With the resulting discrete distributions, the evolution of the system is then exactly modelled by a Markov chain. Here, we focus on flow lines and show that the method allows us to compute upper and lower bounds on the throughput as well as good approximations of the cycle time distributions. Finally, the global modelling method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of a percent.     

服务系统中位相型分布的随机比较

本文运用应用概率中的随机占优研究位相型(PH)分布的随机比较问题,具体给出在一阶、二阶随机占优下比较两个离散PH分布或两个连续PH分布的充分条件及充分必要条件。研究表明,比较两个离散PH分布可变性的条件与比较两个连续PH分布可变性的条件不同,在二阶随机占优意义下比较两个连续PH分布的条件与均值无关,而比较两个离散PH分布的条件与均值有关。本文的结果可用于研究PH分布的最小变异系数问题和可变性问题,也可用于研究带有PH到达间隔或PH服务的排队系统中到达过程或服务时间可变性对系统队长或等待时间的影响。     

服务台休假的GI/Geom/1离散时间排队

本文研究了具有位相型休假、位相型启动和单重几何休假的离散时间排队,假定 顾客到达间隔服从一般分布,服务时间服从几何分布,运用矩阵解析方法我们得到了这 些排队系统中顾客在到达时刻稳态队长分布及其随机分解.     

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.     

Discrete Conditional Phase-type models utilising classification trees: Application to modelling health service capacities

Discrete Conditional Phase-type models (DC-Ph) consist of a process component (survival distribution) preceded by a set of related conditional discrete variables. This paper introduces a DC-Ph model where the conditional component is a classification tree. The approach is utilised for modelling health service capacities by better predicting service times, as captured by Coxian phase-type distributions, interfaced with results from a classification tree algorithm. To illustrate the approach, a case-study within the healthcare delivery domain is given, namely that of maternity services. The classification analysis is shown to give good predictors for complications during childbirth. Based on the classification tree predictions, the duration of childbirth on the labour ward is then modelled as either a two or three-phase Coxian distribution. The resulting DC-Ph model is used to calculate the number of patients and associated bed occupancies, patient turnover, and to model the consequences of changes to risk status.     

Convolutions of multivariate phase-type distributions

This paper is concerned with multivariate phase-type distributions introduced by Assaf et al. (1984). We show that the sum of two independent bivariate vectors each with a bivariate phase-type distribution is again bivariate phase-type and that this is no longer true for higher dimensions. Further, we show that the distribution of the sum over different components of a vector with multivariate phase-type distribution is not necessarily multivariate phase-type either, if the dimension of the components is two or larger.     

Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care?

We have previously used Markov models to describe movements of patients between hospital states; these may be actual or virtual and described by a phase-type distribution. Here we extend this approach to a Markov reward model for a healthcare system with Poisson admissions and an absorbing state, typically death. The distribution of costs is evaluated for any time and expressions derived for the mean and variances of costs. The average cost at any time is then determined for two scenarios: the Therapeutic and Prosthetic models, respectively. This example is used to illustrate the idea that keeping acute patients longer in hospital to ensure fitness for discharge, may reduce costs by decreasing the number of patients that become long-stay. In addition we develop a Markov Reward Model for a healthcare system including states, where the patient is in hospital, and states, where the patient is in the community. In each case, the length of stay is described by a phase-type distribution, thus enabling the representation of durations and costs in each phase within a Markov framework. The model can be used to determine costs for the entire system thus facilitating a systems approach to the planning of healthcare and a holistic approach to costing. Such models help us to assess the complex relationship between hospital and community care.     

Histogram based bounds and approximations for production lines

We present a modelling method for the analysis of production lines with generally distributed processing times and finite buffers. We consider the complete modelling process, from the data collection to the performance evaluation. First, the data about the processing times is supposed to be collected in the form of histograms. Second, tractable discrete phase-type distributions are built. Third, the evolution of the production line is described by a Markov chain, using a state model.     

Parameter Estimation of Discrete Multivariate Phase-Type Distributions

This paper considers parameter estimation of a class of discrete multi-variate phase-type distributions (DMPH). Such discrete phase-type distributions are based on discrete Markov chains with marked transitions introduced by He and Neuts (Stoch Process Appl 74(1):37–52, 1998) and is a generalization of the discrete univariate phase–type distributions. Properties of the DMPHs are presented. An EM-algorithm is developed for estimating the parameters for DMPHs. A number of numerical examples are provided to address some interesting parameter selection issues and to show possible applications of DMPHs.     

具有位相型修理的离散时间可修排队系统

本文研究了具有一般独立输入，位相型修理的离散时间可修排队系统，假定服务台对顾客的服务时间和服务台寿命服从几何分布，运用矩阵解析方法我们给出系统嵌入在到达时刻的稳态队长分布和等待时间分布，并证明这些分布均为离散位相型分布．我们也得到在广义服务时间内服务台发生故障次数的分布，证明它服从一个修正的几何分布．我们对离散时间可修排队与连续时间可修排队进行了比较，说明这两种排队系统在一些性能指标方面的区别之处．最后我们通过一些数值例子说明在这类系统中顾客的到达过程、服务时间和服务台的故障率之间的关系．     

Modelling and evaluation of time aspects in business processes

There is a need for modelling and performance evaluation techniques and tools for a fast and reliable design of workflow systems. This paper introduces a modelling methodology based on coloured stochastic Petri nets. It allows the integration of control flow, organizational, information related and timing aspects in one modelling framework. The processing delays include stochastic distributions in addition to deterministic times. Several workflows and the effects of constrained shared resources needed for different tasks can easily be described and analysed together. Control flow and organizational aspects are modelled separately in resource and workflow models. These models are automatically compiled into one model, which can then be used for qualitative analysis or performance evaluation. The proposed modelling and evaluation method is supported by the software tool TimeNET. An application example shows its use.     

Compartmental modelling of equipment subject to partial repair

A semi-Markov model for equipment that can be repaired and returned to service in a less than new state is developed, where several less than new states are possible before the equipment finally fails. A subclass of phase-type distributions is used to model the times spent in each of these states. Then, exploiting the phase-type structure, the composite distribution of the entire lifetime of the equipment can be constructed. Data from railway wagon wheel-sets are used to illustrate this modelling and data analysis.     

Jump diffusion processes and their applications in insurance and finance

For insurance risks, jump processes such as homogeneous/non-homogeneous compound Poisson processes and compound Cox processes have been used to model aggregate losses. If we consider the economic assumption of a positive interest to aggregate losses, Lévy processes have proven to be useful. Also in financial modelling, it has been observed that diffusion models are not robust enough to capture the appearance of jumps in underlying asset prices and interest rates. As a result, jump diffusion processes, which are, simply speaking, combinations of compound Poisson processes with Brownian motion, have gained popularity for modelling in insurance and finance. In this paper, considering a jump diffusion process, we obtain the explicit expression of the joint Laplace transform of the distribution of a jump diffusion process and its integrated process, assuming that jump size follows the mixture of two exponential distributions, which is a special case of phase-type distributions. Based on this Laplace transform, we derive the moments of the aggregate accumulated claim amounts of insurance risk. For a financial application, we concern non-defaultable zero-coupon bond pricing. We also provide several numerical examples for the moments of aggregate accumulated claims and default-free zero-coupon bond prices.     

Two shock and wear systems under repair standing a finite number of shocks

A shock and wear system standing a finite number of shocks and subject to two types of repairs is considered. The failure of the system can be due to wear or to a fatal shock. Associated to these failures there are two repair types: normal and severe. Repairs are as good as new. The shocks arrive following a Markovian arrival process, and the lifetime of the system follows a continuous phase-type distribution. The repair times follow different continuous phase-type distributions, depending on the type of failure. Under these assumptions, two systems are studied, depending on the finite number of shocks that the system can stand before a fatal failure that can be random or fixed. In the first case, the number of shocks is governed by a discrete phase-type distribution. After a finite (random or fixed) number of non-fatal shocks the system is repaired (severe repair). The repair due to wear is a normal repair. For these systems, general Markov models are constructed and the following elements are studied: the stationary probability vector; the transient rate of occurrence of failures; the renewal process associated to the repairs, including the distribution of the period between replacements and the number of non-fatal shocks in this period. Special cases of the model with random number of shocks are presented. An application illustrating the numerical calculations is given. The systems are studied in such a way that several particular cases can be deduced from the general ones straightaway. We apply the matrix-analytic methods for studying these models showing their versatility.     

An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

Regional development assessment: A structural equation approach

We propose a multivariate statistical framework for regional development assessment based on structural equation modelling with latent variables and show how such methods can be combined with non-parametric classification methods such as cluster analysis to obtain development grouping of territorial units. This approach is advantageous over the current approaches in the literature in that it takes account of distributional issues such as departures from normality in turn enabling application of more powerful inferential techniques; it enables modelling of structural relationships among latent development dimensions and subsequently formal statistical testing of model specification and testing of various hypothesis on the estimated parameters; it allows for complex structure of the factor loadings in the measurement models for the latent variables which can also be formally tested in the confirmatory framework; and enables computation of latent variable scores that take into account structural or causal relationships among latent variables and complex structure of the factor loadings in the measurement models. We apply these methods to regional development classification of Slovenia and Croatia.