Phase-type Model Research Articles

Joint discrete and continuous matrix distribution modeling

In this paper, we introduce a bivariate distribution on arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.

Phase-type models for competing risks, with emphasis on identifiability issues

We first review some main results for phase-type distributions, including a discussion of Coxian distributions and their canonical representations. We then consider the extension of phase-type modeling to cover competing risks. This extension involves the consideration of finite state Markov chains with more than one absorbing state, letting each absorbing state correspond to a particular risk. The non-uniqueness of Markov chain representations of phase-type distributions is well known. In the paper we study corresponding issues for the competing risks case with the aim of obtaining identifiable parameterizations. Statistical inference for the Coxian competing risks model is briefly discussed and some real data are analyzed for illustration.

Queue hurdle Coxian phase-type model for two-stage process of population-based cancer screening

The quality assurance of two-stage population-based cancer screening program is determined by arrival rate (attending screening), positive rate (determined by the criteria of screening test), the compliance and the waiting time (WT) for confirmatory diagnosis in those screened as positive. These parameters were correlated between the process of screening procedures and the effectiveness of screening program. To capture such an inter-dependence of these parameters and quantify the effectiveness of program, we proposed a Queue hurdle Coxian phase-type (QH-CPH) model to estimate the arrival rate of screenees with the Poisson Queue process and the compliance rate of confirmatory diagnosis with the hurdle model, and also to identify the hidden states of WT that is affected by the capacity of health care and relevant covariates (such as demographic features and geographic areas) with the Coxian phase-type (CPH) process. We applied the proposed QH-CPH model to Taiwanese nationwide colorectal cancer screening program data for estimating the arrival rate and the probability of not complying with colonoscopy and classifying the compliers into two hidden states, short-waiting phase and long-waiting phase for colonoscopy. Significant covariates responsible for three processes were also identified by using the proportional hazards regression forms. A simulation study was further performed to assess the joint effect of these parameters on WT through a series of scenarios. The proposed QH-CPH model can provide an insight into the optimal and the practical design on population-based cancer screening for health policy-makers given the limited health care resources and capacity.

The Co-Production of Service: Modeling Service Times in Contact Centers Using Hawkes Processes

In customer support contact centers, a successful service interaction involves a messaging dialogue between a customer and an agent. Both parties depend on one another for information and problem solving, and this interaction defines a co-produced service process. In this paper, we propose, develop, and compare new stochastic models for service co-production in a contact center. A key observation is that the customer and agent’s co-produced service has cross- and self-exciting dynamics within each conversation. The cross-excitation stems from the two parties responding to one another, and the self-excitation captures one party sending follow-ups to their own prior message. Hence, messages beget messages, and we capture this phenomenon by introducing Hawkes point process models of the conversational services. These models distinguish between the role of the customer and of the agent, reflect the service process's dynamic evolution over time based on its own history, and include additional behavioral and operational aspects, including the agent's number of simultaneous assignments and measures of the amount of information and sentiment each message contains. To evaluate our service co-production models, we apply them to an industry contact center dataset containing nearly 5 million messages. We show that the Hawkes models better represent the service dynamics than do the classic Poisson and phase-type models. Indeed, we find that service interactions are characterized by strong agent-customer dependency and the centrality of the process's cross- and self-excitation attributes. Finally, we use the proposed models to improve upon routing algorithms used in contact centers. We show how an activity-based dynamic routing based on predicted information easily computed from our models can outperform well-known and widely used concurrency-based routing rules and substantially reduce customer waiting time, demonstrating how these history-dependent stochastic models can improve operational decision making in practice.

Ruin probabilities for the phase-type dual model perturbed by diffusion

In risk theory, ruin probabilities and dividend strategies have drawn lots of attentions. The dual risk model assumes that the surplus process decreases at a constant rate over time and gains by means of random jumps at random times, which is in accordance with real-life scenes such as life insurance. The dual model is widely used in describing security portfolio, pension funds, profits of enterprises whose income-generating depends on inventions or found, etc. In this paper, we consider a phase-type dual model perturbed by diffusion, whose inter-claim times are continuously distributed in phase-type distribution. We derive integro-differential equations, boundary conditions for ruin probability and obtain an explicit expression of the ruin probability when the phase-type distribution degenerates into the generalized Erlang (n) distribution. Finally, we obtain the integro-differential equations for the expected discounted dividend function under dividend payment with a threshold strategy.

Modeling the overdetection of screen-identified cancers in population-based cancer screening with the Coxian phase-type Markov process.

Modeling overdetection resulting from screening often uses the conventional competing risk model. This model assigns screen-detected cases dying from other causes as overdetection modeled by a one-jump process, which may not be true for the censored overdetected cases. To relax this restrictive assumption, accommodate a finite Markov process for overdetection, and dispense with long-term follow-up until death, we propose a generalized Coxian phase-type Markov process to distinguish the progressive latent multistate pathway from the nonprogressive (overdetected) latent multistate pathway. Various new likelihood functions were developed to estimate the transition parameters with the available data accrued at the time of diagnosis. The proportion of overdetected cancers by the cured model was further estimated by using parameters with and without distinguishing between the two latent pathways. While perturbation analyses were conducted by changing their parameters to assess their effects on overdetection, the results, including of asymptotic analyses, were very robust for an overdetection rate higher than 20% but not for low overdetection rates. These two scenarios were demonstrated by applying the Coxian phase-type model to prostate cancer and breast cancer screening, yielding a substantial proportion of overdetected prostate cancer (60%) attributed to the prostate specific antigen test and a small fraction of overdetected breast cancer (3%) detected by mammography. This kind of variation in overdetection elucidated by the Coxian phase-type Markov process provides new insights into the quantitative mechanisms producing overdetection, which is informative for evaluating the benefits and risks of various types of population-based cancer screening programs.

Phase-Type Modeling Approaches for Software Reliability Modeling with Debugging Process

Reflecting the software fault debugging procedure or environment of testing activities on software reliability models is often discussed as the approaches for improving assessment accuracy for model-based reliability assessment. We discuss a modeling approach reflecting software debugging procedure based on phase-type modeling scheme and propose probability models for software reliability measurement. Further, we give brief consideration for the usefulness of this modeling approach by using a few data sets.

Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

Moments of Phase-type Aging Modeling for Health Dependent Costs

Moments of Phase-type Aging Modeling for Health Dependent Costs

Extended Phase-type models for multistate competing risk systems

Two extended Phase-type models with competing risks for failures of Markov repairable systems with absorbing states are presented in this article. We divide the states into three subsets: perfect, imperfect and failure (absorption) for a Markov repairable system. Two models are developed in terms of the proposed failure criteria of the repairable system. For model 1, the system failure criteria of the repairable system are, whichever occurs first, (1) when the system goes into the failure state (absorption state), or (2) when the system transfers from imperfect states to other states, and before that time the transitions from perfect to imperfect states reached a specified number. For model 2, there still are two criteria whichever occurs first. The first one is the same as failure criterion (1) in model 1, but failure criterion (2) is replaced by (3) which is when the sojourn time in imperfect states exceeds a given threshold. Under two models, two distributions are proposed, which are extensions of the well-known Phase-type distribution. Some reliability indexes under two models, such as the distributions of lifetimes, the point-wise availabilities, various interval availabilities, are given. Finally, some numerical examples are presented to illustrate the results obtained in this article.

Phase-type distributions for studying variability in resistive memories

A new statistical approach has been developed to analyze Resistive Random Access Memory (RRAM) variability. The stochastic nature of the physical processes behind the operation of resistive memories makes variability one of the key issues to solve from the industrial viewpoint of these new devices. The statistical features of variability have been usually studied making use of Weibull distribution. However, this probability distribution does not work correctly for some resistive memories, in particular for those based on the Ni/HfO 2 /Si structure that has been employed in this work. A completely new approach based on phase-type modeling is proposed in this paper to characterize the randomness of resistive memories operation. An in-depth comparison with experimental results shows that the fitted phase-type distribution works better than the Weibull distribution and also helps to understand the physics of the resistive memories. • Reliability analysis of switching parameters in Resistive Random Access Memories (RRAMS) is developed. • The lack of fit of the Weibull model is shown with data of voltage up to the conductive filament failure (Vreset). • A new statistical modeling of Vreset based on phase-type distributions (PHDs) is introduced. • Estimation and selection of parameters of the PHD via EM algorithm provide the Erlang distribution (ED) as the best fit. • The ED has two parameters with a clear physical interpretation.

Understanding Dynamic Status Change of Hospital Stay and Cost Accumulation via Combining Continuous and Finitely Jumped Processes.

The Coxian phase-type models and the joint models of longitudinal and event time have been extensively used in the studies of medical outcome data. Coxian phase-type models have the finite-jump property while the joint models usually assume a continuous variation. The gap between continuity and discreteness makes the two models rarely used together. In this paper, a partition-based approach is proposed to jointly model the charge accumulation process and the time to discharge. The key construction of our new approach is a set of partition cells with their boundaries determined by a family of differential equations. Using the cells, our new approach makes it possible to incorporate finite jumps induced by a Coxian phase-type model into the charge accumulation process, therefore taking advantage of both the Coxian phase-type models and joint models. As a benefit, a couple of measures of the “cost” of staying in each medical stage (identified with phases of a Coxian phase-type model) are derived, which cannot be approached without considering the joint models and the Coxian phase-type models together. A two-step procedure is provided to generate consistent estimation of model parameters, which is applied to a subsample drawn from a well-known medical cost database.

Modelling mortality and discharge of hospitalized stroke patients using a phase-type recovery model

We model the length of in-patient hospital stays due to stroke and the mode of discharge using a phase-type stroke recovery model. The model allows for three different types of stroke: haemorrhagic (the most severe, caused by ruptured blood vessels that cause brain bleeding), cerebral infarction (less severe, caused by blood clots) and transient ischemic attack or TIA (the least severe, a mini-stroke caused by a temporary blood clot). A four-phase recovery process is used, where the initial phase depends on the type of stroke, and transition from one phase to the next depends on the age of the patient. There are three differing modes of absorption for this phase-type model: from a typical recovery phase, a patient may die (mode 1), be transferred to a nursing home (mode 2) or be discharged to the individual’s usual residence (mode 3). The first recovery phase is characterized by a very high rate of mortality and very low rates of discharge by the other two modes. The next two recovery phases have progressively lower mortality rates and higher mode 2 and 3 discharge rates. The fourth recovery phase is visited only by those who experience a very mild TIA, and they are discharged to home after a short stay. The novelty of our approach to phase representation is two-fold: first, it aligns the phases with labelled diagnosis states, representing stages of illness severity; second, the model allows us to obtain expressions for Key Performance Indicators that are of use to healthcare professionals. This allows us to use a backward estimation process where we leverage the fact that we know the phase of admission (the diagnosis), but not which phases are subsequently entered or when this happens; this strategy improves both computational efficiency and accuracy. The model has clear practical value as it yields length of stay distributions by age and type of stroke, which are useful in resource planning. Also, inclusion of the three modes of discharge permits analyses of outcomes.

Physiological Age, Health Costs, and Their Interrelation

ABSTRACTWe demonstrate the impact of an observable health-related quantity on the evaluation of individual physiological ages by extending the phase-type aging model proposed by Lin and Liu in 2007. In their model, an individual of a given calendar age has a determinable distribution of his or her physiological age. In our article, we use observable information to refine this distribution, thereby better connecting physiological age with the health of the individual. We illustrate our model using health cost data, and we investigate the impact of an observed health cost on the distribution of an individual’s physiological age. We also explore the impact on the expected present value of future health costs.

Phase type delay distribution with PU return and sensing error in CR networks

Many studies have been done in cognitive radios to maximize the network efficiency. There is always a trade-off between sensing and transmission time which has been addressed fully in the literature. There is also some missed idle times during the waiting phase when secondary user finds the channel busy. Therefore, the longer the transmission time is, the higher the successfully delivered bits would be and the higher the missed idle times in the waiting stage would be expected. In this work, we formulate these missed idle times to addressed this trade-off. Furthermore, interference due to PU re-occupancy is modelled in successful transmitted time and in collision probability. Moreover, we derive secondary user’s packet delay distribution using phase type model at which retransmission due to collision results from both sensing error and PU re-occupancy is considered. Thanks to derived delay distribution, any statistical moments of delay could be found as a closed form mathematical expression. These moments can be used as constraints for applications with delay sensitive packets. Numerical results are given to clarify the network metrics. Finally, the optimized values for sensing and transmission time are computed using genetic algorithm in order to maximize access efficiency.

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.

Dynamic assessment of multi-state systems using phase-type modeling

Multi-state systems have attracted great attention due to their wide applications in engineering. They have been effectively used in modeling various systems such as power supply systems and transportation systems. In this paper, phase type modeling is proposed for dynamic assessment of nonrepairable multi-state systems when the system degrades according to a Markov process. The utility of phase type modeling is demonstrated in the computation of mean lifetimes, mean residual lifetimes, and derivation of survival functions of series and parallel systems. A stochastic comparison result between two systems is also obtained using phase representations of survival functions. Extensive numerical results are presented to illustrate the applicability of the approach.

Phase-Type Model Spectrum Sensing for Cognitive Radios

ABSTRACTSpectrum sensing is one of the main functionalities of cognitive radios to find transmission opportunities without interfering primary users’ transmission. The more accurate and efficient the spectrum sensing is, the higher the throughput of secondary and primary networks is achieved. This paper presents adaptive spectrum sensing method based on phase-type modelling that is computationally efficient for secondary users to conclude about the channel state (idle or busy) under collision constraint. The parameters of phase-type model can be adjusted based on the desired operating point of the receiver sensor in its receiver operating characteristic (ROC) curve. The presented approach can run a simple trade-off between sensing time and the two error probabilities of the receiver sensor, i.e. false-alarm and miss-detection, the trade-off that cannot be easily achieved in other sensing method.

Phase Type Viewpoint Spectrum Sensing for Cognitive Radios

AbstractSpectrum sensing is one of the main functionalities of cognitive radios to find transmission opportunities without interfering primary users’ transmission. The more accurate and efficient the spectrum sensing is, the higher the throughput of secondary and primary networks is achieved. This paper presents adaptive spectrum sensing method based on phase type modelling that is computationally efficient for secondary users to conclude about the channel state (idle or busy) under collision constraint. The parameters of phase type model can be adjusted based on the desired operating point of the receiver sensor in its ROC curve. The presented approach can run a simple trade off between sensing time and the two error probabilities of the receiver sensor i.e. False alarm and Miss-detection, the trade off that cannot be easily achieved in other sensing method.

Costing Mixed Coxian Phase-type Systems with Poisson Arrivals

Previously, we developed a modeling framework which classifies individuals with respect to their length of stay (LOS) in the transient states of a continuous-time Markov model with a single absorbing state; phase-type models are used for each class of the Markov model. We here add costs and obtain results for moments of total costs in (0, t], for an individual, a cohort arriving at time zero and when arrivals are Poisson. Based on stroke patient data from the Belfast City Hospital we use the overall modelling framework to obtain results for total cost in a given time interval.