Phase-type Distributions Research Articles

On a Dirichlet Process Mixture Representation of Phase-Type Distributions

An explicit representation of phase-type distributions as an infinite mixture of Erlang distributions is introduced. The representation unveils a novel and useful connection between a class of Bayesian nonparametric mixture models and phase-type distributions. In particular, this sheds some light on two hot topics, estimation techniques for phase-type distributions, and the availability of closed-form expressions for some functionals related to Dirichlet process mixture models. The power of this connection is illustrated via a posterior inference algorithm to estimate phase-type distributions, avoiding some difficulties with the simulation of latent Markov jump processes, commonly encountered in phase-type Bayesian inference. On the other hand, closed-form expressions for functionals of Dirichlet process mixture models are illustrated with density and renewal function estimation, related to the optimal salmon weight distribution of an aquaculture study.

Rates of convergence in the two-island and isolation-with-migration models

A number of powerful demographic inference methods have been developed in recent years, with the goal of fitting rich evolutionary models to genetic data obtained from many populations. In this paper we investigate the statistical performance of these methods in the specific case where there is continuous migration between populations. Compared with earlier work, migration significantly complicates the theoretical analysis and requires new techniques. We employ the theories of phase-type distributions and concentration of measure in order to study the two-island and isolation-with-migration models, resulting in both upper and lower bounds on rates of convergence for parametric estimators in migration models. For the upper bounds, we consider inferring rates of coalescent and migration on the basis of directly observing pairwise coalescent times, and, more realistically, when (conditionally) Poisson-distributed mutations dropped on latent trees are observed. We complement these upper bounds with information-theoretic lower bounds which establish a limit, in terms of sample size, below which inference is effectively impossible.

Mortality modeling and regression with matrix distributions

In this paper we investigate the flexibility of matrix distributions for the modeling of mortality. Starting from a simple Gompertz law, we show how the introduction of matrix-valued parameters via inhomogeneous phase-type distributions can lead to reasonably accurate and relatively parsimonious models for mortality curves across the entire lifespan. A particular feature of the proposed model framework is that it allows for a more direct interpretation of the implied underlying aging process than some previous approaches. Subsequently, towards applications of the approach for multi-population mortality modeling, we introduce regression via the concept of proportional intensities, which are more flexible than proportional hazard models, and we show that the two classes are asymptotically equivalent. We illustrate how the model parameters can be estimated from data by providing an adapted EM algorithm for which the likelihood increases at each iteration. The practical feasibility and competitiveness of the proposed approach, including the right-censored case, are illustrated by several sets of mortality and survival data.

Two-commodity queueing-inventory system with phase-type distribution of service times

Two-commodity queueing-inventory system with phase-type distribution of service times

Reliability Evaluation and Optimization of a System with Mixed Run Shock

In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or k1 consecutive external shocks, the size of at least d1 or k2 consecutive external shocks the size of at least d2 occur, where d1<d2, k1>k2. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy (L,N). The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

MMAPs to Model Complex Multi-State Systems with Vacation Policies in the Repair Facility

Two complex multi-state systems subject to multiple events are built in an algorithmic and computational way by considering phase-type distributions and Markovian arrival processes with marked arrivals. The internal performance of the system is composed of different degradation levels and internal repairable and non-repairable failures can occur. Also, the system is subject to external shocks that may provoke repairable or non-repairable failure. A multiple vacation policy is introduced in the system for the repairperson. Preventive maintenance is included in the system to improve the behaviour. Two types of task may be performed by the repairperson; corrective repair and preventive maintenance. The systems are modelled, the transient and stationary distributions are built and different performance measures are calculated in a matrix-algorithmic form. Cost and rewards are included in the model in a vector matrix way. Several economic measures are worked out and the net reward per unit of time is used to optimize the system. A numerical example shows that the system can be optimized according to the existence of preventive maintenance and the distribution of vacation time. The results have been implemented computationally with Matlab and R (packages: expm, optim).

Reliability assessment for a k-out-of-n: F system supported by a multi-state protective device in a shock environment

Protective devices can reduce system failure risks by protecting systems from external shocks and therefore play a critical role in the stable operations of safety–critical systems. Because of the importance of protective devices in real engineering, their performances attract extensive attentions, but existing research on protective devices are quite limited. Previous studies mainly focused on evaluating the reliability of single-unit systems equipped with binary-state protective devices by considering triggering mechanisms based on system state or shock numbers. To fill the research gaps, this paper proposes a k-out-of-n: F system with a multi-state protective device subject to external shocks, where the triggering of the protective device depends on the number of failed components in the system. Furthermore, the protective capacity of the multi-state protective device becomes weaker as the device state deteriorates due to valid shocks. After suffering a certain number of valid shocks, the protective device may transit into a lower state directly, or it can resist some additional valid shocks in the current state because of its resistance against shocks. A combination of the finite Markov chain imbedding approach and phase-type distributions is employed to analyze a series of probabilistic indices both for the system and the protective device. Finally, a case study based on a multi-cylinder diesel engine supported by a cooling system is provided to demonstrate the applicability of the proposed model.

Phase-type mixture-of-experts regression for loss severities

ABSTRACT The task of modeling claim severities is addressed when data is not consistent with the classical regression assumptions. This framework is common in several lines of business within insurance and reinsurance, where catastrophic losses or heterogeneous sub-populations result in data difficult to model. Their correct analysis is required for pricing insurance products, and some of the most prevalent recent specifications in this direction are mixture-of-experts models. This paper proposes a regression model that generalizes the latter approach to the phase-type distribution setting. More specifically, the concept of mixing is extended to the case where an entire Markov jump process is unobserved and where states can communicate with each other. The covariates then act on the initial probabilities of such underlying chain, which play the role of expert weights. The basic properties of such a model are computed in terms of matrix functionals, and denseness properties are derived, demonstrating their flexibility. An effective estimation procedure is proposed, based on the EM algorithm and multinomial logistic regression, and subsequently illustrated using simulated and real-world datasets. The increased flexibility of the proposed models does not come at a high computational cost, and the motivation and interpretation are equally transparent to simpler MoE models.

Reliability modeling for multi-component system subject to dependent competing failure processes with phase-type distribution considering multiple shock sources

In this article, a reliability model for the multi-component system subject to dependent competing failure processes considering multiple shock sources is presented. For a single component, soft failure occurs when the sum of natural degradation and instantaneous degradation increment is greater than the soft failure threshold. The soft failure reliability model is established using the degradation-threshold interference model. The hard failure model is developed using the phase-type distribution survival function. Then, system reliability is derived based on the history of shocks from all shock sources. Finally, the reliability model is demonstrated by a numerical example.

Continuous scaled phase-type distributions

Products between phase-type distributed random variables and any independent, positive and continuous random variable are studied. Their asymptotic properties are established, and an expectation-maximization algorithm for their effective statistical inference is derived and implemented using real-world datasets. In contrast to discrete scaling studied in earlier literature, in the present continuous case closed-form formulas for various functionals of the resulting distributions are obtained, which facilitates both their analysis and implementation. The resulting mixture distributions are very often heavy-tailed and yet retain various properties of phase-type distributions, such as being dense (in weak convergence) on the set of distributions with positive support.

Penalised likelihood methods for phase-type dimension selection

Abstract Phase-type distributions are dense in the class of distributions on the positive real line, and their flexibility and closed-form formulas in terms of matrix calculus allow fitting models to data in various application areas. However, the parameters are in general non-identifiable, and hence the dimension of two similar models may be very different. This paper proposes a new method for selecting the dimension of phase-type distributions via penalisation of the likelihood function. The penalties are in terms of the Green matrix, from which it is possible to extract the contributions of each state to the overall mean. Since representations with higher dimensions are penalised, a parsimony effect is obtained. We perform a numerical study with randomly generated phase-type samples to illustrate the effectiveness of the proposed procedure, and also apply the technique to the absolute log-returns of EURO STOXX 50 and Bitcoin prices.

A sequential direct hybrid algorithm to compute stationary distribution of continuous-time Markov chain

In modeling a wide variety of problems of the real world using Markov processes, we usually get large sparse Markov chains. In the case of non-Markovian process, the behavior of a stochastic phenomenon or system can be approximated by continuous-time Markov chains (CTMC) using Phase-type distributions. However, this substantially increases the number of states, which requires a large amount of computation time and operational memory resources. Even the most recent computer hardware is not enough to provide a solution in almost real time, unless a specific algorithm is designed.In this paper, a new hybrid algorithm for the computation of a stationary distribution of a large ergodic homogeneous Markov chains with a finite state space and continuous time is suggested. This algorithm would solve the aforementioned challenges. Depending on model size and sparsity, it significantly reduces a computational time when compared with other recent methods, such as the SparseLU solver from Eigen library (version 3.9.9) and the MUMPS solver (MUltifrontal Massively Parallel sparse direct Solver, version 5.4.0). It is important to note that the algorithm works well independently on the structure of the generator matrix of a continuous-time Markov chain. For the study of the developed hybrid algorithm, the generator matrix is constructed randomly using a special algorithm, based on combining two methods: the slightly modified Grassmann, Taskar and Heyman (GTH) algorithm and the Gauss elimination method. The new idea is to remove the majority of states in a near-optimal order by the GTH method and complete computations for the remaining states by the Gauss method. The robust approach to identify a switching position from one algorithm to another one has been proposed and tested.

Extended transit compartment model to describe tumor delay using Coxian distribution

The measured response of cell population is often delayed relative to drug injection, and individuals in a population have a specific age distribution. Common approaches for describing the delay are to apply transit compartment models (TCMs). This model reflects that all damaged cells caused by drugs suffer transition processes, resulting in death. In this study, we present an extended TCM using Coxian distribution, one of the phase-type distributions. The cell population attacked by a drug is described via age-structured models. The mortality rate of the damaged cells is expressed by a convolution of drug rate and age density. Then applying to Erlang and Coxian distribution, we derive Erlang TCM, representing the existing model, and Coxian TCMs, reflecting sudden death at all ages. From published data of drug and tumor, delays are compared after parameter estimations in both models. We investigate the dynamical changes according to the number of the compartments. Model robustness and equilibrium analysis are also performed for model validation. Coxian TCM is an extended model considering a realistic case and captures more diverse delays.

Exponential and gamma form for tail expansions of first-passage distributions in semi-markov processes

AbstractWe consider residue expansions for survival and density/mass functions of first-passage distributions in finite-state semi-Markov processes (SMPs) in continuous and integer time. Conditions are given which guarantee that the residue expansions for these functions have a dominant exponential/geometric term. The key condition assumes that the relevant states for first passage contain an irreducible class, thus ensuring the same sort of dominant exponential/geometric terms as one gets for phase-type distributions in Markov processes. Essentially, the presence of an irreducible class along with some other conditions ensures that the boundary singularity b for the moment generating function (MGF) of the first-passage-time distribution is a simple pole. In the continuous-time setting we prove that b is a dominant pole, in that the MGF has no other pole on the vertical line $\{\text{Re}(s)=b\}.$ In integer time we prove that b is dominant if all holding-time mass functions for the SMP are aperiodic and non-degenerate. The expansions and pole characterisations address first passage to a single new state or a subset of new states, and first return to the starting state. Numerical examples demonstrate that the residue expansions are considerably more accurate than saddlepoint approximations and can provide a substitute for exact computation above the 75th percentile.

Analysis of k-Out-of-N-Systems with Different Units under Simultaneous Failures: A Matrix-Analytic Approach

An N-system with different units submitted to shock and wear is studied. The shocks cause damage and, eventually, simultaneous failures of several units. The units can also fail due to internal failures. At random times, the system is inspected, and the down units are simultaneously replaced by identical ones. The arrival of shocks is governed by a Markovian arrival process. The operational times and the interarrival times between inspections follow phase-type distributions. The generator of the multidimensional Markov process modeling the system is constructed. This is performed introducing indicator functions for the different transition rates among the units using the algorithm of Kronecker. This is a general Markov process that can be applied for modeling different reliability systems depending on the structure of the units and how the systems operate. The general model is applied to the study of k-out-of-N systems, calculating the main performance measures. A practical example is presented showing the approximation of the model to a system with units following different Weibull distributions.

Conditional multivariate distributions of phase-type for a finite mixture of Markov jump processes given observations of sample path

This paper presents conditional joint probability distributions of first-exit times to absorbing sets of a finite mixture of right-continuous Markov jump processes given observations of its sample path. Each underlying Markov jump process comprising the mixture has its own intensity matrix and is defined on the same finite state space with overlapping absorbing sets. The main distinctive aspect of the mixture is that despite the underlying processes have the Markov property, the mixture itself in general does not. Conditional transition matrix and the joint probability distribution of the first-exit times form non-stationary functions of time with ability to capture path dependence when conditioning on available information (either full or partial) of the sample path. In particular, initial profile of the joint distribution forms a generalized mixture of the multivariate phase-type distributions of Assaf et al. (1984). When the underlying Markov processes have the same intensity matrix, in which case the mixture becomes a simple Markov jump process, the path dependence and non-stationary properties are removed from the transition matrix and the joint distribution, while the initial profile distribution reduces to Assaf et al. (1984). Some explicit and numerical examples are discussed to illustrate the results.

Performance evaluation of software‐defined wide area network based on queueing theory

Software-defined wide area network (SD-WAN) is part of wider technologies of software-defined networking (SDN). SDN is one of the pivotal network management approaches that abstracts the underlying network infrastructure away from its applications. SDN allows for the centralisation of network intelligence. That allows higher network automation, the simplification of operations, the provisioning, the monitoring, and the troubleshooting. These days, data volume that enterprises and users make is growing rapidly and continuously. As a result, wide area networks are growing increasingly and they are becoming very complicated to manage and monitor using traditional tools. So, many enterprises are interested in developing SD-WAN and benefiting from its advantages. To achieve this goal, a proposed model for the SD-WAN network will be presented based on the queueing theory; then, the proposed model will be analysed using the Quasi-birth–death approach to evaluate traffic behaviour of a proposed queueing model for Poisson distribution in the SD-WAN. Moreover, it compares Poisson to phase type distribution via calculating some parameters that influence traffic behaviour of those distributions in the SD-WAN. The analysis results indicate that the proposed queueing model displays a more accurate SD-WAN controller performance than the existing ones.

Covariate model-based fault tree analysis for risk assessment in chemical process industries: A case study of a chlorine manufacturing facility

Fault tree analysis (FTA) is a promising quantitative technique for risk analysis in chemical process industries (CPIs). In FTA, a certain sequence of basic events (causes) leads to one specific Top event (critical event of interest). However, the conventional fault tree analysis has the limitations of staticity and uncertainty. The staticity in conventional FTA arises due to its inability to accommodate time-dependent characteristics of the process system. Whereas uncertainty primarily lies in the failure probability data of basic events. This paper proposes an innovative methodology that uses a time-dependent covariate model to update the failure probability values of major contributing basic events in FTA. A novel subclass of the family of phase-type distributions is used to model the covariates corresponding to the basic events. The newly developed methodology is applied for a case study in a chlorine manufacturing facility to estimate the chlorine release probability. The blockage in the pipeline was identified as the significant reason for chlorine release from expert opinion and sensitivity analysis. The results of the proposed model of FTA are compared with that of conventional FTA.

Rare Event Simulation by Combining Trend and Probability: An Application to Naval Ship Critical Failures

Although the Rare event has a very wide application range, it has limitations in that it requires mathematical knowledge and the calculation process is complicated for the Bayesian approach, which was mainly used in previous studies. Therefore, rare events need to be easily analyzed. In this study, the critical failures of the ROK Naval ship were analyzed. Bayesian inference was not applied. The long-term trend (B-spline) and short-term probability (phase-type distribution) were fitted and combined. Phase-type distribution is a useful technique that can estimate the probability distribution of a desired shape with one parameter. This study is meaningful in that only an easy methodology was used, Phase-type distribution was applied to the analysis of rare events, and only a very small number of data were analyzed.

On the Representation of Correlated Exponential Distributions by Phase Type Distributions

In this paper we present results for bivariate exponential distributions which are represented by phase type distributions. The paper extends results from previous publications [3, 11] on this topic by introducing new representations that require a smaller number of phases to reach some correlation coefficient and introduces different ways to describe correlation between exponentially distributed random variables.