Discrete Phase-type Distributions Research Articles

Phase-type stress-strength reliability models under progressive type-II right censoring

. The study of stress-strength reliability estimation based on phase-type distribution helps to gather results on estimation of stress-strength reliability with any probability distribution that is defined on the non negative real numbers as any discrete or continuous probability distributions on the positive real line can be represented as phase-type. The matrix representation of the parameters of phase-type distributions helps in their flexible evaluation and easy manipulation. Also in many of the experimental studies, it is very convenient and useful to apply progressive type-II right censoring mechanism in the process of data collection. In this article, we consider the estimation of stress-strength reliability (R) based on phase-type distribution under progressive type-II right censoring mechanism. Both stress and strength random variables are assumed to follow either continuous phase-type or discrete phase-type distribution. We have developed the algorithm for computing Maximum likelihood estimate (MLE) of R based on the expectation maximization (EM) method and the Bayes estimate of R using Markov Chain Monte Carlo technique. A detailed numerical illustration using simulated data/ real data sets are carried out.

Optimizing Erlangization-based approximations for finite discrete distributions and discrete phase-type distributions

In He et al.[8], continuous phase-type (PH) distributions are constructed to approximate finite discrete probability distributions and discrete PH-distributions. The approximations are based on Erlangization with a fixed number of phases. In this article, we first introduce continuous PH approximations with Erlang distributions of different orders. Then we develop an algorithm to find the continuous PH approximation with the minimum variance, among all such PH approximations with the same total number of phases. Thus, the proposed continuous PH approximations lead to a smaller gap between the variances of the Erlangization-based approximations and the original discrete random variables, which is achieved without adding more phases. The new approximations are useful to mitigate the burden in computation caused by the large number of phases needed in Erlangization approximation. Stochastic dominance is shown between the original (discrete) distributions and the approximations, which leads to bounds on the quantities of original distributions and/or stochastic models (e.g., reliability models). The approximation method is applied to analyze reliability models and a COVID-19 isolation program.

Reliability of a mixed δ-shock model with a random change point in shock magnitude distribution and an optimal replacement policy

A mixed δ-shock model when there is a change in the distribution of the magnitudes of shocks is defined and studied. Such a model which is a combination of the δ-shock model and the extreme shock model with a random change point (studied by Eryilmaz and Kan, 2019), is useful in practice since a sudden change in environmental conditions may cause a larger shock. In particular, the reliability and the mean time to failure of the system are evaluated by assuming that the random change point has a discrete phase-type distribution. Analytical results for evaluating the reliability function of the system for several joint distributions of the interarrival times and the magnitudes of shocks, are also given. The optimal replacement policy that is based on a control limit is also proposed when the number of shocks until the change point follows geometric distribution. The results are illustrated by numerical examples.

Robust claim frequency modeling through phase-type mixture-of-experts regression

This paper addresses the problem of modeling loss frequency using regression when the counts have a non-standard distribution. We propose a novel approach based on mixture-of-experts specifications on discrete-phase type distributions. Compared to continuous phase-type counterparts, our approach offers fast estimation via expectation-maximization, making it more feasible for use in real-life scenarios. Our model is both robust and interpretable in terms of risk classes, and can be naturally extended to the multivariate case through two different constructions. This avoids the need for ad-hoc multivariate claim count modeling. Overall, our approach provides a more effective solution for modeling loss frequency in non-standard situations.

Bayesian Inference of the Phase-Type Stress-Strength Reliability Models

Stress-strength modeling has achieved considerable attention in recent years due to its applicability in various areas like engineering, quality control, psychology, biology, genetics, medicine etc. Phase-type distribution is a generalized class of distributions that is closed under several mathematical operations like maxima, minima, convolution, finite mixture etc and any discrete or continuous probability distributions on the positive real line can be represented as phase-type. Hence stress-strength reliability models based on phase-type distributions give a generalized structure for the stress-strength models. Moreover, matrix representation of the parameters helps in their flexible evaluation and easy manipulation. In Bayesian inference, we combine the prior knowledge with the information provided by the set of current observations to make more reliable inferences. Bayesian approach has the advantage of providing more meaningful inferences by making use of all available information. In this paper, we assume that the strength of the system and the stress imposed on the system are phase-type random variables, and the Bayesian inference of stress-strength reliability is discussed for the single-component systems and multi-component systems. The computation of the Bayes estimate of stress-strength reliability using the Markov Chain Monte Carlo method, based on continuous and discrete phase-type distributions are explained.

Optimal decisions in stochastic graphs with uncorrelated and correlated edge weights

Stochastic graph models and algorithms for the computation of different stochastic shortest path problems (SSPPs) continue to play an important role in various fields of computer science and operations research, with recent applications ranging from vehicle routing to social networks and decision problems in finance. In this paper, we first introduce stochastic graph models where weights to pass an edge in a graph are, possibly correlated, random variables. Depending on the application context and the interpretation of uncertain edge weights, random variables either model the time to pass an edge or describe a reward one gains by traversing an edge. For modeling the random variables discrete phase type distributions are used. Several algorithms for the computation of optimal policies which depend on the current weight upon arriving at a vertex while still exploiting weight dependencies among edges are developed. It will be shown that with correlated edge weights the stochastic shortest path problems become PSPACE-complete and the solution of instances with a bounded number of phases is NP-hard such that for an exact computation of the shortest path, algorithms from partially observable Markov decision processes (POMDPs) or mixed integer linear programming (MILP) have to be applied. Alternatively, an efficient heuristic algorithm can be used which often yields good or even optimal results with a small effort.

Parisian ruin in a discrete-time Markov-modulated dual risk model

In this paper, we investigate the Parisian ruin problems in a discrete-time Markov-modulated dual risk model, wherein the gain process is governed by the underlying Markov process with a finite state space. By using the strong Markov property of the risk process, we derive recursive expressions for the conditional probability generating functions of the classical ruin time and the Parisian ruin time. From this, we not only obtain the infinite-time ruin probabilities but also compute the finite-time ruin probabilities by using numerical inversion. In addition, for the case in which the gain amounts have discrete phase-type distributions, we obtain specialized expressions for the probability generating functions of the classical and Parisian ruin times, which can be used to reduce the computational effort needed for the numerical computation of the ruin probabilities. Finally, we present numerical examples for the computation of the finite- and infinite-time ruin probabilities.

Continuous approximations of discrete phase-type distributions and their applications to reliability models

Using Erlangization, we construct two types of continuous phase-type (PH) random variables that approximate discrete PH-random variables and finite discrete random variables. The key idea of the method is to use Erlang random variables to approximate constants. The approximations have (i) explicit closed form PH-representations; (ii) a small set of parameters, (iii) the same mean as the original random variable; (iv) moments and variances that can be arbitrarily close to that of the original random variable; (v) distribution functions that converge to that of the original random variable; and (vi) distribution functions that decreasing in convex order. The approximations are utilized in the analysis of two basic reliability structures: the serial model and the parallel model. Some bounds on such basic stochastic models are obtained. Numerical examples are presented to demonstrate the effectiveness of the approximations.

Multivariate phase-type theory for the site frequency spectrum.

Linear functions of the site frequency spectrum (SFS) play a major role for understanding and investigating genetic diversity. Estimators of the mutation rate (e.g. based on the total number of segregating sites or average of the pairwise differences) and tests for neutrality (e.g. Tajima'sD) are perhaps the most well-known examples. The distribution of linear functions of the SFS is important for constructing confidence intervals for the estimators, and to determine significance thresholds for neutrality tests. These distributions are often approximated using simulation procedures. In this paper we use multivariate phase-type theory to specify, characterize and calculate the distribution of linear functions of the site frequency spectrum. In particular, we show that many of the classical estimators of the mutation rate are distributed according to a discrete phase-type distribution. Neutrality tests, however, are generally not discrete phase-type distributed. For neutrality tests we derive the probability generating function using continuous multivariate phase-type theory, and numerically invert the function to obtain the distribution. A main result is an analytically tractable formula for the probability generating function of the SFS. Software implementation of the phase-type methodology is available in the R package PhaseTypeR, and R code for the reproduction of our results is available as an accompanying vignette.

Reliability modeling for dependent competing failure processes with phase-type distribution considering changing degradation rate

In this paper, a system reliability model subject to Dependent Competing Failure Processes (DCFP) with phase-type (PH) distribution considering changing degradation rate is proposed. When the sum of continuous degradation and sudden degradation exceeds the soft failure threshold, soft failure occurs. The interarrival time between two successive shocks and total number of shocks before hard failure occurring follow the continuous PH distribution and discrete PH distribution, respectively. The hard failure reliability is calculated using the PH distribution survival function. Due to the shock on soft failure process, the degradation rate of soft failure will increase. When the number of shocks reaches a specific value, degradation rate changes. The hard failure is calculated by the extreme shock model, cumulative shock model, and run shock model, respectively. The closed-form reliability function is derived combining with the hard and soft failure reliability model. Finally, a Micro-Electro-Mechanical System (MEMS) demonstrates the effectiveness of the proposed model

Stationary joint distribution of a discrete-time group-arrival and batch-size-dependent service queue with single and multiple vacation

Discrete-time queueing systems have widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the joint probabilities in terms of roots of the characteristic equation. We also find the joint distribution at arbitrary and pre-arrival slot. Discrete phase type distribution plays a noteworthy role in order to regulate the high transmission error through a particular channel. In view of this, numerical illustrations include the service time distribution to follow the discrete phase type distribution.

On the joint distribution of an infinite-buffer discrete-time batch-size-dependent service queue with single and multiple vacations

ABSTRACT Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacations where customers/messages arrive according to the Bernoulli process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at the service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at an arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. During the transmission of data through a particular communication channel, high transmission error may take place due to several factors. For this reason, one may skip the transmission through that particular channel. However the discrete phase-type distribution plays a noteworthy role to control this transmission error which eventually motivates us to include a numerical example where service time distribution follows discrete phase-type distribution. A comparison between batch-size-dependent and -independent services has been drawn through the graphical representation of some performance measures and total system cost.

Revisiting discrete time age replacement policy for phase-type lifetime distributions

For a system (or unit) whose lifetime is measured by the number cycles, according to the discrete time age replacement policy, it is replaced preventively after n cycles or correctively at failure, whichever occurs first. In this paper, discrete time age replacement policy is revisited when the lifetime of the system is modeled by a discrete phase-type distribution. In particular, the necessary conditions for the unique and finite replacement cycle which minimizes the expected cost per unit of time are obtained. The necessary conditions are mainly based on the behavior of the hazard rate. The results are illustrated for some special discrete phase-type lifetime distributions. Computational results are also presented for the optimal replacement cycle under specific real life setups.

Discrete time dynamic reliability modeling for systems with multistate components

This study proposes modeling techniques for the exact dynamic reliability analyses of systems with the lifetimes of all components following independent and nonidentical discrete phase-type (DPH) distributions. The systems may have series, parallel, standby, K-out-of-N, and bridge structures with any combinations of them. The models produce numerical formulas and algorithms for generating system reliability and hazard functions; thus, they are applicable to the dynamic reliability analysis of systems, including networks. The approach is by showing that the system lifetime follows a DPH distribution. For network reliability analysis, the DPH distribution is generalized into a matrix-geometric (MG) distribution. The use of the DPH distribution makes the models suitable for systems with multistate components and simplifies the calculations of the system reliability measures. Its effectiveness is illustrated using results from complex structure systems.

Reliability assessment of a discrete time cold standby repairable system

This paper is concerned with the study of a discrete time repairable system consisting of one active and one standby component. The lifetime and repair time are assumed to have discrete phase-type distributions. The system’s lifetime is represented as a compound random variable. A matrix-based expression for the probability generating function of the system’s lifetime is obtained based on the phase characteristics of lifetime and repair time distributions. The probability generating function is then used to obtain the distribution of the system’s lifetime. Reliability and hazard rate functions are computed and evaluated for some particular choices of lifetime and repair time distributions. The limiting behavior of the hazard rates is also investigated.

Estimation of stress-strength reliability using discrete phase type distribution

In this paper, the stress-strength reliability of single and multi-component systems are estimated assuming discrete phase type distribution for stress and strength components. The systems with strength following mixture of discrete phase type distributions is also considered. Matrix based expressions are obtained for stress-strength reliability and its maximum likelihood estimate is obtained using EM algorithm. The numerical illustration using various special cases of discrete phase type distribution like geometric, negative binomial, generalized negative binomial and different mixtures of discrete distributions are also carried out.

On the decision support model for the patient admission scheduling problem with random arrivals and departures: A solution approach

The focus of this work is the numerical application of a stochastic decision support model for the patient admission scheduling problem with random arrivals and departures. Here, we discuss the methodology for applying our model to real-world problems. We outline a solution approach for efficient computation, provide a numerical analysis of the model, and illustrate the methodology with examples. A key component of the model is an integer linear program which formulates the patient admission scheduling problem as an optimization of the total expected cost accumulated over a finite planning horizon. We rewrite some of the components of this integer linear program in order to improve numerical efficiency. We use Poisson processes and discrete phase-type distributions to model the random arrivals and departures, respectively. We argue that this stochastic component is essential for an accurate treatment of real-world problems which are stochastic in nature. We support our claim with simple numerical examples, and show that the optimal solutions obtained from deterministic models are inadequate when compared with the solutions of our stochastic model. We also construct more complex numerical examples for large-scale problems using heuristics that approximate the objective function, in order to demonstrate that our model can be efficiently applied to real-world problems, which typically involve large data sets.

Fitting discrete phase-type distribution from censored and truncated observations with pre-specified hazard sequence

Phase-type distribution allows approximation of non-Markovian models, which permits to analyze complex systems under Markovian deterioration. In addition, reliability data is often composed of truncated and censored observations. This paper presents a novel approach that fits a restricted class of discrete phase-type distribution through pre-specified hazard sequence from incomplete observations. Numerical results are shown using Balakrishnan’s mimicked power transformers dataset. Furthermore, it can be used to fit transition probabilities of maintenance optimization’s Markov decision process models from incomplete reliability data.

A Reliability Modeling Method Based on Phase-Type Distribution Aiming at Shock Model

The traditional shock model generally describes the magnitude of the cumulative damage caused by a random shock sequence and compares the magnitude with a predetermined threshold to obtain the failure time of a component. There are two limitations in this kind of models in practice: First, the statistical characteristics of the damage due to a single shock may be difficult to obtain, which means the magnitude of the damage may not be described by an appropriate distribution; Second, the cumulative shock magnitude may be difficult to measure, or it may be difficult for a failure mode to be described by a threshold, meaning that the magnitude of the damage and the threshold may not be compared with each other. Considering both failure and censored samples, a reliability modeling method is proposed in this work to address the above problems. The shock model is first established by using both continuous and discrete phase-type (PH) distributions. Then the parameter estimation method of the shock model is derived based on EM method and the identifiability of the parameters in PH distributions is also given. Finally, the adaptability of the model is analyzed using three different types of simulation cases.

Classification and properties of acyclic discrete phase-type distributions based on geometric and shifted geometric distributions

Acyclic phase-type distributions form a versatile model, serving as approximations to many probability distributions in various circumstances. They exhibit special properties and characteristics that usually make their applications attractive. Compared to acyclic continuous phase-type (ACPH) distributions, acyclic discrete phase-type (ADPH) distributions and their subclasses (ADPH family) have received less attention in the literature. In this paper, we present the definition, properties, characteristics and PH representations of ADPH distributions and their subclasses with finite state space. Based on the definitions of geometric and shifted geometric distributions, we propose a distinct classification for the ADPH subclasses analogous to ACPH family. We develop the PH representation for each ADPH subclass and prove them through their closure properties. The advantage of our proposed classifications is in applying precise representations of each subclass and preventing miscalculation of the probability mass function, by computing the ADPH family based on geometric and shifted geometric distributions.