Phase-type Distributions Research Articles

Static Output Energy-to-Peak Control for Semi-Markov Jump Linear Systems With Phase-Type Distributions and Hidden Modes

Static Output Energy-to-Peak Control for Semi-Markov Jump Linear Systems With Phase-Type Distributions and Hidden Modes

Quantitative and Qualitative Analysis of Aircraft Round-Trip Times Using Phase Type Distributions

One of the major issues facing commercial airlines is the time that it takes to board passengers. Further, most airlines wish to increase the number of trips that an aircraft can make between two or more cities. Thus, reducing the overall boarding times by a few minutes will have a significant impact on the number of trips made by an aircraft, as well as enabling improvements in key measures such as the median and 75th and 95th percentiles. Looking at such measures other than the mean is critical as it is well known that the mean can under- or overestimate the performance of any model. While there is considerable literature on the study of strategies to decrease boarding times, the same cannot be said about the study of the boarding time given a particular strategy for boarding. Thus, the focus of this paper is to study analytically (using suitable stochastic models) and numerically the impact of reducing the average time on the key measures to help the system to plan accordingly. This is achieved using a well-known probability distribution, namely the phase type distribution, to model various events involved in the boarding process. Illustrative numerical results show a reduction in the percentile values when the average boarding times are decreased. Understanding the percentiles of the boarding times, as opposed to relying only on the average boarding times, will help management to adopt a better boarding strategy that in turn will lead to an increase in the number of trips that an aircraft can make.

Reliability Evaluation of Multi-State Solar Energy Generating System with Inverters Considering Common Cause Failures

To ensure the efficient functioning of solar energy generation systems, it is crucial to have dependable designs and regular maintenance. However, when these systems or their components operate at multiple working levels, optimizing reliability becomes a complex task for models and analyses. In the context of reliability modeling in solar energy generation systems, researchers often assume that random variables follow an exponential distribution (binary-state representation) as a simplification, although this may not always hold true for real-world engineering systems. In the present paper, a multi-state solar energy generating system with inverters in series configuration is investigated, in which unreliable by-pass changeover switches, common cause failures (CCFs), and multiple repairman vacations are also considered. Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs), and the lifetimes and repair times of inverters and by-pass changeover switches and the repairman vacation time in the system have different phase-type (PH) distributions. Therefore, the behavior of the system is represented using a Markov process methodology, and reliability measures for the proposed system are derived utilizing aggregated stochastic process theory. Finally, a numerical example and a comparison analysis are presented to demonstrate the findings.

Phase-type distribution models for performance evaluation of condition-based maintenance

ABSTRACT Condition-based maintenance (CBM) is gaining attention due to sensor and cloud-based analytics advancements, but research on its impact on system-level performance is limited. Insufficient understanding during CBM implementation can lead to confidence issues and failures. This study introduces a class of models using phase-type distribution to assess three maintenance strategises: run-to-failure (RTF), time-based preventive maintenance (TBM), and CBM. Employing machine health-index, the framework characterizes production performance by estimating effective process times. The model demonstrates how adjusting CBM thresholds influences process time variations and assesses the impact of changing maintenance frequency for TBM. Applied to a smart cellular manufacturing system, the model shows CBM’s early-stage implementation. Findings indicate CBM with optimized thresholds boosts maximum throughput by 6.77%. Further, CBM achieves an additional 6.84% increase assuming corrective maintenance time can be reduced by 20%. This approach can help manufacturing become smarter through smarter maintenance in the Industry 4.0 era and beyond.

Microstructure and strengthening mechanism of TIG welded joint of AZ31 alloy based on FSP technique

Magnesium alloys, renowned for their outstanding properties, present promising applications within the realm of advanced manufacturing. Addressing the challenges of coarse grains and inferior properties in the fusion welding of magnesium alloys, a novel strengthening approach for fusion-welded joints is proposed in this study. This method employs fine, uniform, and dense grains as the matrix, with particle-reinforced structures serving as the strengthening backbone. By utilizing powder feeding in conjunction with fusion welding and friction stir processing, ideal microstructures were successfully fabricated, and the mechanisms influencing grain morphology and joint properties were explored. Following the strengthening treatment, the grain size of the magnesium alloy molten weld was reduced from 193.5 μm to 15.9 μm, resulting in a grain refinement efficacy of 91.8 %. Concurrently, the tensile strength was augmented by 70.2 MPa, and the elongation experienced a 136 % enhancement. The grain morphology, eutectic phase structure, texture strength, precipitated phase type and dislocation distribution at each processing stage of the welded joint were observed. Research shows that nano-enhanced particles adhere to the eutectic structure of grain boundaries, effectively inhibiting the diffusion of solute elements and significantly reducing the dendrite growth rate. It is the main mechanism by which reinforced particles lead to grain refinement. Discontinuous recrystallization during FSP treatment leads to grain refinement, work hardening and enhanced pinning of particles at grain boundaries, which are the main reasons for the strength improvement.

Assessing dengue risk globally using non-Markovian models

Dengue is a vector-borne disease transmitted by Aedes mosquitoes. The worldwide spread of these mosquitoes and the increasing disease burden have emphasized the need for a spatio-temporal risk map capable of assessing dengue outbreak conditions and quantifying the outbreak risk. Given that the life cycle of Aedes mosquitoes is strongly influenced by habitat temperature, numerous studies have utilized temperature-dependent development rates of these mosquitoes to construct virus transmission and outbreak risk models. In this study, we contribute to existing research by developing a mechanistic model for the mosquito life cycle that accurately captures its non-Markovian nature. Beginning with integral equations to track the mosquito population across different life cycle stages, we demonstrate how to derive the corresponding differential equations using phase-type distributions. This approach can be further applied to similar non-Markovian processes that are currently described with less accurate Markovian models. By fitting the model to data on human dengue cases, we estimate several model parameters, allowing the development of a global spatiotemporal dengue risk map. This risk model employs temperature and precipitation data to assess the environmental suitability for dengue outbreaks in a given area.

A queueing-based approach for integrated routing and appointment scheduling

This paper aims to address the integrated routing and appointment scheduling (RAS) problem for a single service provider. The RAS problem is an operational challenge faced by operators that provide services requiring home attendance, such as grocery delivery, home healthcare, or maintenance services. While considering the inherently random nature of service and travel times, the goal is to minimize a weighted sum of the operator’s travel times and idle time, and the client’s waiting times. To handle the complex search space of routing and appointment scheduling decisions, we propose a queueing-based approach to effectively deal with the appointment scheduling decisions. We use two well-known approximations from queueing theory: first, we use an approach based on phase-type distributions to accurately approximate the objective function, and second, we use an heavy-traffic approximation to derive an efficient procedure to obtain good appointment schedules. Combining these two approaches results in a fast and sufficiently accurate hybrid approximation, thus essentially reducing RAS to a routing problem. Moreover, we propose the use of a simple yet effective large neighborhood search metaheuristic to explore the space of routing decisions. The effectiveness of our proposed methodology is tested on benchmark instances with up to 40 clients, demonstrating an efficient and accurate methodology for integrated routing and appointment scheduling.

Automated phase-type distribution fitting via expectation maximization

Automated phase-type distribution fitting via expectation maximization

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.

Analysis of single server repairable queueing system with many kinds of service, single working vacation and standby server

We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.

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.