Phase-type Distributions Research Articles

Lifetime Distribution for a Mixed Redundant System with Imperfect Switch and Components Having Phase–Type Time-to-Failure Distribution

Recently, a mixed redundancy was introduced among the redundant design strategies to achieve a more reliable system within the equivalent resources. This study deals with a lifetime distribution for a mixed redundant system with an imperfect fault detector/switch. The lifetime distribution model was formulated using a structured continuous Markov chain (CTMC) and considers the time-to-failure (TTF) distribution of a component as a phase-type distribution (PHD). The model’s versatility and practicality are enhanced because the PHD can represent diverse degradation patterns of the components exposed to varied operating environments. The model provides accurate reliability for a mixed redundant system by advancing the approximate reliability function suggested in previous studies. Furthermore, the model would be useful in system design and management because it provides information such as the nth moment of the system’s lifetime distribution. In numerical experiments on some examples, the mixed redundancy was confirmed to devise a more reliable system than the existing active and standby redundancies, and the improvement effect increased as the number of redundant components decreased. The optimal structure for maximizing the expected lifetime of the system changes depends on the reliability of the components and fault detector/switch.

Analysis of Tandem Queue with Multi-Server Stages and Group Service at the Second Stage

In this paper, we consider a tandem dual queuing system consisting of multi-server stages. Stage 1 is characterized by an infinite buffer, one-by-one service of customers, and an exponential distribution of service times. Stage 2 is characterized by a finite buffer and a phase-type distribution of service times. Service at Stage 2 is provided to groups of customers. The service time of a group depends on the size of the group. The size is restricted by two thresholds. The waiting time of a customer at each stage is limited by a random variable with an exponential distribution, with the parameter depending on the stage. After service at Stage 1, a customer can depart from the system or try to enter Stage 2. If the buffer at this stage is full, the customer is either lost or returns for service at Stage 1. Customer arrivals are described by the versatile Markov arrival process. The system is studied via consideration of a multi-dimensional continuous-time Markov chain. Numerical examples, which highlight the influence of the thresholds on the system performance measures, are presented. The possibility of solving optimization problems is illustrated.

Availability and maintenance strategy under time-varying environments for redundant repairable systems with PH distributions

Systems commonly operate in time-varying environments which have essential impact on system performance, thus reliability modeling and maintenance strategy are faced with new challenges. In this study, a cold standby repairable system in a time-varying environment is investigated based on different operational levels and maintenance strategies, in which repairman's vacation and maintenance rules vary with environments. The evolution of dynamic environments is governed by a Markov process, and the successive vacation time is governed by Markovian arrival processes (MAPs). Using Phase-type (PH) distribution not only increases the flexibility of analytical properties because it can approximate to any probability distribution defined on a nonnegative real line, but also could effectively ensure the applicability of the model in engineering practice, specially all the lifetimes of components and maintenance time are subject to disparate PH distributions in diverse environments. By employing the matrix-analytic method, Kronecker operator theory and aggregated stochastic processes, some reliability indexes and the system average benefit are obtained respectively. Finally, an example of a redundant power supply is given to illustrate the validity and applicability of the proposed model.

Product form solution for a queueing-inventory system with lost sales and phase type distribution of the vacation time

ABSTRACT In this paper, a continuous review s , S queueing-inventory system with lost sales is investigated. Customers arrive according to a Poisson process, and each arriving customer needs exactly one item from inventory for service. The server takes a single vacation once the inventory is depleted. It is assumed that the vacation time follows a phase type distribution, and the service time and the lead time follow exponential distributions. We obtain the product form solution for the steady-state distribution of the joint process of the queue length, the on-hand inventory level and the server’s status. Furthermore, the conditional distributions of the on-hand inventory level when the server is turned off due to a vacation and when it is turned on are derived, respectively. Some performance measures of the system are computed by using the steady-state distribution of system. Based on some performance measures, we develop a cost model and determine a total expected cost under an appropriate cost structure by using the genetic algorithm. The effects of various parameters on the optimal inventory policy and the optimal expected cost are investigated by numerical examples.

Investigation of the Fork–Join System with Markovian Arrival Process Arrivals and Phase-Type Service Time Distribution Using Machine Learning Methods

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K≥2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (βk, Sk), 1≤k≤K. An analytical solution was derived for the case of K=2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

Analyzing the multi-state system under a run shock model

Abstract A system experiences random shocks over time, with two critical levels, d1 and d2, where $d_{1} \lt d_{2}$ . k consecutive shocks with magnitudes between d1 and d2 partially damaging the system, causing it to transition to a lower, partially working state. Shocks with magnitudes above d2 have a catastrophic effect, resulting in complete failure. This theoretical framework gives rise to a multi-state system characterized by an indeterminate quantity of states. When the time between successive shocks follows a phase-type distribution, a detailed analysis of the system’s dynamic reliability properties such as the lifetime of the system, the time it spends in perfect functioning, as well as the total time it spends in partially working states are discussed.

Energy-efficient production control of a make-to-stock system with buffer- and time-based policies

Increasing energy efficiency in manufacturing has significant environmental and cost benefits. Turning on or off a machine dynamically while considering the production rate requirements can offer substantial energy savings. In this work, we examine the optimal policies to control production and turn on and off a machine that operates in working, idle, off, and warmup modes for the case where demand inter-arrival, production, and warmup times have phase-type distributions. The optimal control problem that minimises the expected costs associated with the energy usage in different energy modes and the inventory and backlog costs is solved using a linear program associated with the underlying Markov Decision Process. We also present a matrix-geometric method to evaluate the steady-state performance of the system under a given threshold control policy. We show that when the inter-arrival time distribution is not exponential, the optimal control policy depends on both the current phase of the inter-arrival time and inventory position. The phase-dependent policy implemented by estimating the current phase based on the time elapsed since the last arrival yields a buffer- and time-based policy to control the energy mode and production. We show that policies that only use the inventory position information can be effective if the control parameters are chosen appropriately. However, the control policies that use both the inventory and time information further improve the performance.

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.

Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies

We discuss two queueing-inventory systems with catastrophes in the warehouse. Catastrophes occur according to the Poisson process and instantly destroy all items in the inventory. The arrivals of the consumer customers follow a Markovian arrival process and they can be queued in an infinite buffer. The service time of a consumer customer follows a phase-type distribution. The system receives negative customers which have Poisson flows and as soon as a negative customer comes into the system, he causes a consumer customer to leave the system, if any. One of two inventory policies is used in the systems: either (s,S) or (s,Q). If the inventory level is zero when a consumer customer arrives, then this customer is either lost (lost sale) or joins the queue (backorder sale). The system is formulated by a four-dimensional continuous-time Markov chain. Ergodicity condition for both systems is established and steady-state distribution is obtained using the matrix-geometric method. By numerical studies, the influence of the distributions of the arrival process and the service time and the system parameters on performance measures are deeply analyzed. Finally, an optimization study is presented in which the criterion is the minimization of expected total costs and the controlled parameter is warehouse capacity.

A phase-type maintenance model considering condition-based inspections and maintenance delays

This paper proposes a maintenance model for systems subject to discrete-state deterioration processes. The deterioration of the system is described by the phase-type (PH) expansion of a multi-state Markov model, where the sojourn times at different states follow general distributions. In contrast to the periodic inspections and exponentially distributed repairs in most works, this study considers condition-based inspections and significant delays from planning a maintenance task until its actual implementation. These delays are usually due to logistic reasons and are considered close to deterministic. This allows the decision-makers to evaluate more flexible inspection strategies and move towards more cost-effective solutions. A case study about road bridge maintenance is presented to demonstrate the modelling process and the value of introducing PH distributions and condition-based inspections. With the proposed model, a less conservative maintenance strategy with fewer inspections and postponed repairs can be derived, bringing significant cost reductions.

Aggregate Markov models in life insurance: estimation via the EM algorithm

In this paper, we consider statistical estimation of time–inhomogeneous aggregate Markov models. Unaggregated models, which corresponds to Markov chains, are commonly used in multi–state life insurance to model the biometric states of an insured. By aggregating microstates to each biometric state, we are able to model dependencies between transitions of the biometric states as well as the distribution of occupancy in these. This allows for non–Markovian modelling in general. Since only paths of the macrostates are observed, we develop an expectation–maximisation (EM) algorithm to obtain maximum likelihood estimates of transition intensities on the micro level. Special attention is given to a semi-Markovian case, known as the reset property, which leads to simplified estimation procedures where EM algorithms for inhomogeneous phase–type distributions can be used as building blocks. We provide a numerical example of the latter in combination with piecewise constant transition rates in a three–state disability model with data simulated from a time–inhomogeneous semi–Markov model. Comparisons of our fits with more classic GLM-based fits as well as true and empirical distributions are provided to relate our model to existing models and their tools.

Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System

In this paper, we consider a multi-server queue with a finite buffer. Request arrivals are defined by the Markov arrival process. Service is provided to groups of requests. The minimal and maximal group sizes are fixed. The service time of a group has a phase-type distribution with an irreducible representation depending on the size of the group. The requests are impatient. The patience time for an arbitrary request has an exponential distribution. After this time expires, the request is lost if all servers are busy or, if some server is idle, with a certain probability, all requests staying in the buffer start their service even if their number is below the required minimum. The behavior of the system is described by a multi-dimensional continuous-time Markov chain that does not belong to the class of level-independent quasi-birth-and-death processes. The algorithm for the computation of the stationary distribution of this chain is presented, and expressions for the computation of the queuing system’s performance characteristics are derived. The description of a delivery system operation in terms of the analyzed queuing model is given, and the problem of the optimization of its operation is numerically solved. Multi-server queues with a phase-type distribution for the group service time that are dependent on the size of the group, the account of request impatience, and the correlated arrival process have not previously been analyzed in the existing literature. However, they represent a precise model of many real-world objects, including delivery systems.

Robust scheduling in a two-machine re-entrant flow shop to minimise the value-at-risk of the makespan: branch-and-bound and heuristic algorithms based on Markovian activity networks and phase-type distributions

Robust scheduling in a two-machine re-entrant flow shop to minimise the value-at-risk of the makespan: branch-and-bound and heuristic algorithms based on Markovian activity networks and phase-type distributions

Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers

A queuing system having two different servers is under study. Demands enter the system according to a Markov arrival process. Service times have phase-type distribution. Service of demands is possible only if the fixed number of energy units, probably different for two servers, is available in the system at the potential service beginning moment. Energy units arrive in the system also according to a Markov arrival process and are stored in a stock (battery) of a finite capacity. Leakage of energy units from the stock can occur. Demands waiting in the infinite buffer are impatient and can leave the buffer after an exponentially distributed waiting time. One server is the main one and permanently provides service when the buffer is not empty and the required number of energy units is available. The second server is the assistant server and is switched on or off depending on the availability of energy units and queue length according to the hysteresis strategy defined by two thresholds. The assistant server is switched on when the queue length is not less than the greater threshold and is switched off when the queue length becomes smaller than the smaller threshold. The use of the assistant server has to be paid. Thus, the problem of the optimal selection of the thresholds defining the control strategy naturally arises. To solve this problem, the study of the behavior of the system under any fixed values of the parameters of the control strategy is necessary. Such a study is given in this paper. Numerical results are presented. They illustrate the feasibility of computer realization of the developed algorithms for computation of the stationary distribution of the system states and the main key performance indicators as well as the result of solving one of the possible optimization tasks.

Phase-type distributions for product return data with two-layer censoring

Abstract Product return data, such as warranty claims, are usually subject to two layers of right censoring. The first layer, called warranty censoring, applies to the product lifetime due to a fixed warranty limit. The second layer, called end-of-study censoring, applies to the sum of the sales lag and the lifetime due to the end-of-study date for the data collection. The two-layer censoring in the product return data renders traditional nonparametric methods for right-censored data inapplicable. This study develops a generic method for the two-layer censored data using acyclic phase-type distributions (APHDs) in the canonical form. The APHD estimators can be regarded as nonparametric sieve estimators since the family of APHDs is dense in the field of all positive-valued distributions. Based on the property that the class of APHDs is closed under convolution, a dedicated expectation-maximisation algorithm is proposed to compute the maximum likelihood estimators. Comprehensive simulations are conducted to evaluate the performance of the proposed method and compare it with the inverse probability of censoring weighted approach, which is applicable in the absence of warranty censoring. Two real examples from production-delivery supply chains are analysed by the proposed method to provide guidance on warranty reserve management for the manufacturers.

Phase-Type Distributions of Animal Trajectories with Random Walks

Animal monitoring systems often rely on expensive and challenging GPS-based systems to obtain accurate trajectories. However, an alternative approach is to generate synthetic trajectories that exhibit similar statistical properties to real trajectories. These synthetic trajectories can be used effectively in the design of surveillance systems such as wireless sensor networks and drone-based techniques, which aid in data collection and the delineation of areas for animal conservation and reintroduction efforts. In this study, we propose a data generation method that utilizes simple phase-type distributions to produce synthetic animal trajectories. By employing probability distribution functions based on the exponential distribution, we achieve highly accurate approximations of the movement patterns of four distinct animal species. This approach significantly reduces processing time and complexity. The research primarily focuses on generating animal trajectories for four endangered species, comprising two terrestrial and two flying species, in order to demonstrate the efficacy of the proposed method.

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.

Analysis of a production inventory system with MAP arrivals, phase-type services and vacation to production facility

In this paper, we discuss a production inventory system with service times. Customers arrive in the system according to a Markovian arrival process. The service times follow a phase-type distribution. We assume that there is an infinite waiting space for customers. Arriving customers demand only one unit of item from the inventory. The production facility produces items according to an (s, S)-policy. Once the inventory level becomes the maximum level S, the production facility goes on a vacation of random duration. When the production facility returns from the vacation, if the inventory level depletes to the fixed level s, it is immediately switched on and starts production until the inventory level becomes S. Otherwise, if the inventory level is greater than s on return from the vacation, it takes another vacation. The vacation times are exponentially distributed. The production inventory system in the steady-state is analyzed by using the matrix-geometric method. A numerical study is performed on the system performance measures. Besides, an optimization study is discussed for the inventory policy.

Randomized Threshold Strategy for Providing Flexible Priority in Multi-Server Queueing System with a Marked Markov Arrival Process and Phase-Type Distribution of Service Time

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account.

A branch-and-bound approach to minimise the value-at-risk of the makespan in a stochastic two-machine flow shop

Planning and scheduling approaches in real manufacturing environments entail the need to cope with random attributes and variables to match the characteristics of real scheduling problems where uncertain events are frequent. Moreover, the capability of devising robust schedules, which are less sensitive to the disruptive effects of unexpected events, is a major request in real applications. In this paper, a branch-and-bound approach is proposed to solve the two-machine permutation flow shop scheduling problem with stochastic processing times. The objective is the minimisation of the value-at-risk of the makespan, to support decision-makers in the trade-off between the expected performance and the mitigation of the impact of extreme scenarios. A Markovian Activity Network (MAN) model is adopted to estimate the distribution of the makespan and assess the value-at-risk for both partial and complete schedules. Phase-type distributions are used to enable general distributions for processing times while maintaining the capability to exploit a Markovian approach. The effectiveness and performance of the proposed approach are demonstrated through a set of computational experiments.