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.     

A Markov model for analyzing the evolution of bladder carcinoma

A Markovian approach to analyze different states of the superficial vesical carcinoma is considered, taking into account up to two recurrences and the possibility of progression. So, three transient states are considered: free of disease, first, and second recurrence; and an absorbent state, the progression. A methodology based in phase-type distributions is also used, that allows the usual quantities of interest in survival studies to be expressed in a well-structured form. This type of distribution has shown its utility in queue theory, and has the advantage that mathematical expressions can be presented in a closed form that allows algebraic treatment.     

Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option

A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented.     

Replacement times and costs in a degrading system with several types of failure: The case of phase-type holding times

A reliability system submitted to external and internal failures, that can be repairable or non-repairable, with degradation levels, and with sojourn times phase-type distributed, is considered. Repair is not as good as new, and the repair of internal failure follows policy N, that is, after N completed repairs the system is replaced by a new one to the following failure, repairable or not. For this system, a Markov model is constructed, and the stationary probability vector is calculated. It is shown that the distribution of the time between two consecutive replacements follows a phase-type distribution, whose representation is determined. The costs of these periods are calculated. An optimization problem involving the costs, the availability, and the number of internal repairs is illustrated by a numerical example.     

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。     

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.     

Optimal inspection and replacement policies for multi-state deteriorating systems

In this paper, a phase-type approach is proposed to derive optimal inspection and replacement policies for semi-Markovian deteriorating systems. In this approach, the general sojourn time distributions of a semi-Markovian maintenance model are approximated by acyclic phase-type distributions. Using the approximation, a semi-Markovian maintenance model can be transformed into a Markovian maintenance model such that the analytical tractability of Markov processes can be preserved. Based on the Markovian model, algorithms are provided to derive the optimal state-dependent and state-age-dependent inspection and replacement policies such that the expected long-run cost rate is minimized. Furthermore, procedures are developed to implement the optimal policies on semi-Markovian deteriorating systems. The implementation of the optimal policies are illustrated by numerical examples.     

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.     

A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K

A reliability system subject to shocks producing damage and failure is considered. The source of shocks producing failures is governed by a Markovian arrival process. All the shocks produce deterioration and some of them failures, which can be repairable or non-repairable. Repair times are governed by a phase-type distribution. The number of deteriorating shocks that the system can stand is fixed. After a fatal failure the system is replaced by another identical one. For this model the availability, the reliability, and the rate of occurrence of the different types of failures are calculated. It is shown that this model extends other previously published in the literature.     

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.     

Interval reliability for aggregated Markov repairable system with repair time omission

In this paper, Markov models of repairable systems with repair time omission are considered whose finite state space is grouped into two sets, the set of working states, W, and the set of failed states, F. If the system enters failed states from a working state at any instance, and sojourns at the failed states F less than a given nonnegative critical value τ, then the repair interval can be omitted from downtime records. Otherwise, If the system enters failed states from a working state at any instance, and sojourns at the failed states F more than the given nonnegative critical value τ, then the repair interval cannot be omitted from downtime records. In terms of the assumption, a new model is developed. The focus of attention is the new model’s availability, interval reliability and interval unreliability. Several results are derived for these reliability indexes for the new model. Some special cases and numerical examples are given to illustrate the results obtained by using Maple software in the paper.     

A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations

This paper analyzes a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. The repairman may mean here the human beings who are used to repair the failed device. When the device functions smoothly, the repairman leaves the system for a vacation, the duration of which is an exponentially distributed random variable. In vacation period, the repairman can perform other secondary jobs to make some extra profits for the system. The lifetimes and the repair times of the device are governed by phase-type distributions (PH distributions), and the condition of device following repair is not “as good as new”. After a prefixed number of repairs, the device is replaced by a new and identical one. The spare device for replacement is available only by an order and the procurement lead time for delivering the spare device also follows a PH distribution. Under these assumptions, the vector-valued Markov process governing the system is constructed, and several important performance measures are studied in transient and stationary regimes. Furthermore, employing the standard results in renewal reward process, the explicit expression of the long-run average profit rate for the system is derived. Meanwhile, the optimal maintenance policy is also numerically determined.     

Performance evaluation of transfer lines with general repair times and multiple failure modes

The paper proposes a decomposition method for evaluating the performance of transfer lines where machines can fail in multiple modes and can be repaired with non-exponential times. Indeed, while times to machine failure can be often modeled using exponential distributions with acceptable accuracy, times to repair are very rarely observed to be exponentially distributed in actual systems. This feature limits the applicability of existing approximate analytical methods to real production lines. In this paper, the discrete acyclic phase-type distribution is used to model the repair process, for each failure mode of the machines composing the system. The exact analysis of the two-machine system is used as a building block for the decomposition method, proposed to study multi-stage lines. Numerical results show the high accuracy of the developed method in estimating the average throughput and buffer levels.     

A multi-phase DES modelling framework for patient-centred care

We provide a framework for simulating the entire patient journey across different phases (such as diagnosis, treatment, rehabilitation and long-term care) and different sectors (such as GP, hospital, social and community services), with the aim of providing better understanding of such processes and facilitating evaluation of alternative clinical and care strategies. A phase-type modelling approach is used to promote better modelling and management of the specific elements of a patient pathway, using performance measures such as clinical outcomes, patient quality of life, and cost. The approach is illustrated using stroke disease. Approximately 5% of the United Kingdom National Health Service budget is spent treating stroke disease each year. There is an urgent need to assess whether existing services are cost-effective or new interventions could increase efficiency. This assessment can be made using models across primary and secondary care; in particular we evaluate the cost-effectiveness of thrombolysis (clot busting therapy), using discrete event simulation. Using our model, patient quality of life and the costs of thrombolysis are compared under different regimes. In addition, our simulation framework is used to illustrate the impact of internal discharge queues, which can develop while patients are awaiting placement. Probabilistic Sensitivity Analysis of the value parameters is also carried out.     

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.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

Threshold policies for controlled retrial queues with heterogeneous servers

Retrial queues are an important stochastic model for many telecommunication systems. In order to construct competitive networks it is necessary to investigate ways for optimal control. This paper considers K -server retrial systems with arrivals governed by Neut' Markovian arrival process, and heterogeneous service time distributions of general phase-type. We show that the optimal policy which minimizes the number of customers in the system is of a threshold type with threshold levels depending on the states of the arrival and service processes. An algorithm for the numerical evaluation of an optimal control is proposed on the basis of Howar's iteration algorithm. Finally, some numerical results will be given in order to illustrate the system dynamics. AMS subject classification: 60K25 93E20     

A semi-Markov model with holdout transshipment policy and phase-type exponential lead time

In this paper, a semi-Markov decision model of a two-location inventory system with holdout transshipment policy is reviewed under the assumption of phase-type exponential replenishment lead time rather than exponential lead time. The phase-type exponential lead time more closely approximates fixed lead time as the number of phases increases. Unlike past research in this area which has concentrated on the simple transshipment policies of complete pooling or no pooling, the research presented in this paper endeavors to develop an understanding of a more general class of transshipment policy. In addition, we propose an effective method to approximate the dynamic holdout transshipment policy.     

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.     

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.