Gamma Process Research Articles

Reliability analysis of corroded pipelines considering 3D defect growth

Integrity management of in-service corroded pipelines depends on the applied stochastic corrosion growth model. 3D corrosion growth in axial, circumferential, and radial directions of the pipeline is modeled using dependent stochastic gamma processes. The unknown model variables are inferred from the results of in-line inspections. The model is embedded in structural reliability analysis to estimate the time-dependent probability of leak and rupture failures of corrosion in pipelines. The results show that the effect of dependent corrosion growth in the three directions has a significant impact on the pipeline's reliability and safe remaining lifetime.

Generalized Mittag-Leffler Lévy process and its connections to first passage times of Lévy subordinators

In this article, we introduce generalized Mittag-Leffler Lévy (GMLL) process. GMLL distribution is represented as a general Lévy subordinator delayed by a gamma process. We show various properties of this new process like it’s corresponding Lévy density function and the so-called long-range dependence property. We provide also an explicit representation for the cumulative density function (CDF) for such process which can be an inspiration for construction of various versions of the GMLL processes. Moreover we establish Fokker-Planck type equation for its one dimensional probability density functions (PDF). Our construction will allow us to link GMLL process to the first passage time of the Lévy subordinator.

A time-discrete and zero-adjusted gamma process model with application to degradation analysis

The standard gamma process is widely used for degradation modelling when the degradation paths are always positive and strictly increasing. Nevertheless, the standard gamma process is considered inappropriate if the observed sample paths are horizontal over several time intervals. Motivated by this phenomenon, we develop a time-discrete and zero-adjusted gamma process model. Statistical inference method of this model is then proposed. To illustrate our proposed model, both simulation study and real data analysis are conducted, whose results demonstrate that the proposed model has satisfactory fit performance and predictive performance.

Infinite Mixtures of Infinite Factor Analysers

Factor-analytic Gaussian mixture models are often employed as a model-based approach to clustering high-dimensional data. Typically, the numbers of clusters and latent factors must be specified in advance of model fitting, and remain fixed. The pair which optimises some model selection criterion is then chosen. For computational reasons, models in which the number of latent factors differ across clusters are rarely considered. Here the infinite mixture of infinite factor analysers (IMIFA) model is introduced. IMIFA employs a Pitman-Yor process prior to facilitate automatic inference of the number of clusters using the stick-breaking construction and a slice sampler. Furthermore, IMIFA employs multiplicative gamma process shrinkage priors to allow cluster-specific numbers of factors, automatically inferred via an adaptive Gibbs sampler. IMIFA is presented as the flagship of a family of factor-analytic mixture models, providing flexible approaches to clustering high-dimensional data. Applications to a benchmark data set, metabolomic spectral data, and a manifold learning handwritten digit example illustrate the IMIFA model and its advantageous features. These include obviating the need for model selection criteria, reducing the computational burden associated with the search of the model space, improving clustering performance by allowing cluster-specific numbers of factors, and quantifying uncertainty in the numbers of clusters and cluster-specific factors.

A root location-based framework for degradation modeling of dynamic systems with predictive maintenance perspective

This paper considers dynamic systems widely used in industry for which the behavior can be approximated to a second order differential equation. The components of such a system suffer from random faults and failures due to wear, age or usage. These events impact the dynamic behavior which is interpreted as a modification of the initial differential equation with random coefficients. At given times, the system is solicited and its output – the only source of information – is measured to infer the position of equation roots in the complex plane. The Euclidian distance between the current and initial positions of a root is proposed as a new indicator reflecting the gradual deterioration of system performance. Such an indicator presents stochastic trajectories in time according to the random evolution of the root location in complex plane. More especially, these trajectories can be modeled by an univariate non-linear diffusion process if underlying degradation sources are assumed to be homogeneous Gamma processes. Based on this model, the system remaining useful lifetime is assessed. Two predictive maintenance policies are also designed showing the feasibility to easily maintain dynamic systems solely on the system output.

Joint optimization of condition-based maintenance and inventory control for a k-out-of-n:F system of multi-state degrading components

Maintaining a system of multiple degrading components requires having a supply of spare components to replace deteriorated ones in a timely manner. Decisions on whether to replace the deteriorated components are made based on the statuses of components and the system and are further restricted by the inventory level of spares. Inventory replenishment decisions themselves depend both on system status and maintenance policy. Both maintenance and replenishment decisions affect the system reliability and system maintenance and spare inventory cost. This paper considers the joint optimization of periodic condition-based replacement of components and inventory control for a k-out-of-n:F system of non-repairable degrading components under various system statuses. For modeling the degradation of components, both the Wiener process and gamma process are considered. We model the maintenance and inventory policy using the number of components in each discretized degradation state rather than the traditional approach, which uses the state of each component. Our approach considerably reduces the solution space. Based on that, we address the joint optimization using Markov decision process along with dynamic programming; we then analyze the complexity of the algorithm. We provide a numerical study on a 2-out-of-3:F system of components in three states to illustrate structural insights regarding the proposed model, including a sensitivity analysis on the length of inspection interval, system downtime cost, and inventory holding cost. We demonstrate the efficiency of the model and solution method by testing k-out-of-n:F systems with different numbers of components and discretized degradation states.

Probabilistic impacts of corrosion on structural failure and performance limits of reinforced concrete beams

Concrete structures in corrosive conditions have different types of uncertainties in the corrosion phases of their lifetime, which should be considered in the reliability analysis with an appropriate probabilistic model. Although corrosion is an inevitable phenomenon, it can be delayed, and the maintenance costs can be minimized by choosing the suitable methods and using the results of life cycle cost analyses. In this paper, reliability analyses were performed on the performance of concrete structures in two limit states of serviceability and structural capacity. Then, the flexural capacity obtained from modeling by the finite element method was considered as a random variable in the calculation of capacity reduction probability in the corroded beams with different corrosion percentages; this was done by defining the gamma process. Furthermore, the costs caused by the failure in the life cycle of reinforced concrete beams were evaluated and optimized using renewal process analysis in the capacity limit state. In another section, the lifetime of reinforced concrete structures subject to corrosion was investigated in the serviceability limit state, and using the performance-based destruction model, the penetration of aggressive agents of chloride ions into the structure was modeled theoretically by explaining the diffusion laws. In the next stage, the corrosion of bars produced a considerable volume of iron oxide, and the increase in stress in the concrete around the bars caused by iron oxide resulted in the cracking of cover. Probabilities were presented in an algorithmic form with the Monte Carlo method to predict the beginning time of corrosion and cracking propagation until failure. Afterward, the cost caused by failure was calculated using a performance-based life cycle cost management (LCCM) method.

Modeling basketball games by inverse Gaussian processes

The scoring processes of home and away team in basketball games are modeled by two dependent inverse Gaussian processes with a team-specific parameter that measures the team strength. A common latent variable that measures the game pace is designed to characterize the dependence. A moment estimation method combined with maximum likelihood estimation is proposed to fit the parameters and a Bayesian method is applied to update the estimation and make in-game predictions. It is shown that the proposed model can obtain the same performance as the benchmark model, Gamma process model, in outcome prediction, point spread betting and model gambling.

Post-prognostics decision making for a two-stacks fuel cell system based on a load-dependent deterioration model

Multi-stacks proton exchange membrane fuel cell (PEMFC) system has been applied to combined heat and power system (CHP), and serves as an alternative energy device due to its high efficiency and zero emission. Owing to the limited durability and larger power supply demand, the management of multi-stacks PEMFC system to obtain a longer service time has received recently growing attention. From the prognostics and health management (PHM) point of view, a post-prognostics decision making for multi-stacks PEMFC system is addressed in this work. Firstly, a load-dependent stochastic deterioration model is proposed for PEMFC. The overall ohmic resistance is chosen as the health indicator of PEMFC. Then the resistance is modeled using a Gamma process whose shape parameter is taken as a function of the current load applied to the stack. Finally, for the post-prognostics decision making phase, a decision probability based load repartition criterion is built to identify the optimal load split between the two stacks. The decision probability is calculated based on the system lifetime results (EoL) in each decision step. The EoL results of the decision phase are further compared with the system EoL that calculated without decision making strategy. The comparison result shows that extended service time can be achieved using the proposed post-prognostics decision making method.

A stochastic process model for resistance deterioration of aging bridges

The serviceability and safety level of bridges often deteriorates due to the environmental or operational conditions. A stochastic deterioration model is of essential importance for describing the time-variation of bridge performance (e.g., stiffness and strength), and for use in the estimate of bridge reliability under a probabilistic framework. In this context, the Gamma process has been widely used to model the resistance deterioration of aging bridges, yet suffers from the deficiency that the statistical characteristics of the process (namely mean value, variance and autocorrelation) are not mutually independent. This paper presents a new stochastic model for bridge resistance deterioration. As a modified version of the Gamma process-based one, the proposed model includes a new parameter that can release the dependence of the process autocorrelation and variance on the mean value. The impact of the new deterioration model on the time-dependent reliability of aging bridges is studied.

Sharing and Avatar-Based Innovation Tools on Digital Economy Perspectives Using Levy Processes Simulation

This article is the Enhancement of the Mkrttchian and Vertakova article “Digital Sharing Economy” published in the International Journal of Innovation in Digital Economy (IJIDE, Volume 10, issue 2) and the chapter “Avatar-Based Innovation Tools for Managerial Perspectives on Digital Sharing Economy” in the book “Avatar-Based Models, Tools, and Innovation in the Digital Economy,” focused on an entirely new area relevant to the scope of IJIDE. The article discusses the capabilities of the R language for modeling Levy processes - processes that currently closely correspond to the nature of the evolution of stock price movements. The efficient algorithm of the CGMY process simulation as a difference of the tempered stable independent Levy is processed and programmed at R language. The efficient algorithm of variance gamma process simulation using variance gamma random variables is processed and programmed at R language, as Modelling in the Digital Globalization Era.

Metoda modelowania degradacji obrotowego uszczelnienia wargowego w oparciu o analizę mechanizmu uszkodzenia i proces stochastyczny

Rotary lip seal is widely used in aircraft and its performance affects the safety of the aircraft. Hence, it is necessary to estimate useful lifetime and reliability of the seal. Degradation of rotary lip seal is always with random effects, which cannot be considered by theoretical failure mechanism analysis. Hence, in order to consider the random effects of rotary lip seal degradation, stochastic processes are applied. Furthermore, considering the monotonic degradation of the seal, Gamma process and inverse Gaussian process are selected as the candidate processes. To combine the candidate processes, Bayesian model averaging is introduced. Based on the failure mechanism analysis and numerical simulation, the theoretical wear path is predicted and corresponding linearization method is proposed. The measured degradation data is converted and the seal wear process is transformed to a linear degradation process. The model parameters and model probabilities are evaluated by fully Bayesian inference method. The effectiveness of the proposed method is verified by comparing the predicting degradation and experimental observations. The proposed method can be used to evaluate reliability and useful lifetime of rotary lip seal. According to sensitivity analysis, an effective way to improve lifetime and reliability of the seal is to increase the wear depth threshold.

Misspecification analysis of two‐phase gamma‐Wiener degradation models

AbstractA degradation model is used to analyze degradation data and product reliability. Owing to model uncertainty, the assumed degradation model may be improper, which can create a difference between the evaluated and actual product reliabilities. Thus, model misspecification analysis is essential. Recent studies on misspecification analysis have mainly considered single‐phase degradation models; two‐phase degradation models have not been investigated intensively. In this study, an approach to model misspecification analysis was developed for a two‐phase gamma‐Wiener process, where the change point was assumed to be known. The gamma and Wiener processes in a two‐phase degradation model may be incorrectly assumed, so five misspecified degradation models were considered: the two‐phase Wiener process, two‐phase gamma process, two‐phase Wiener‐gamma process, one‐phase Wiener process, and one‐phase gamma process. A Monte Carlo simulation was performed to examine the effects of model misspecification on the relative bias and relative variability. A case study was performed to examine the effects of model misspecification on the evaluated mean time to failure and median lifetime of a product for different sample sizes, termination time, and change points. The results clearly demonstrated the effects of model misspecification on the evaluated reliability of a product.

Reinforcement learning for dynamic condition-based maintenance of a system with individually repairable components

In this article, a reinforcement learning approach is used to develop a new dynamic maintenance policy for multi-component systems with individually repairable components, where each component experiences two competing failure processes of degradation and random shocks. The gamma process is used to model the degradation path of each component in the system. It is assumed that each incoming shock causes damage to the degradation path of all the components. A combination of component degradation is then used to define the system states. The optimal maintenance action for each component at each specific state is found by modeling the problem as Markov decision process and solving it by using a Q-learning algorithm. Using a reinforcement learning approach provides a more time-efficient and cost-effective method compared to the traditional maintenance optimization solutions, and it can also provide a dynamic maintenance policy for each specific degradation state of the system which is more useful and beneficial compared to the fixed or stationary maintenance plan which is often proposed by previous studies. A numerical example shows how the reinforcement learning can be used to find the optimal maintenance action for systems with different system configurations.

SOME PRICING TOOLS FOR THE VARIANCE GAMMA MODEL

We establish several closed pricing formulas for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Particular focus is made on the symmetric process, but extension to the asymmetric process is also provided. Speed of convergence and comparison with numerical methods (Fourier transform, quadrature approximations, Monte Carlo simulations) are also discussed; notable feature is the accelerated convergence of the series for short-term options, which constitutes an interesting improvement of numerical Fourier inversion techniques.

Optimizing the Gamma Process.

This article will discuss opportunities to improve the efficiency of cobalt-60 (Co-60) utilization within a gamma irradiator. It will show how redistributing the Co-60 within the source rack may lead to improved throughput or dose uniformity within a product. It presents examples of modifications to the equipment within the source pass; these include reduction in the carrier wall thickness and changes to the product stack size. It will discuss the process of scheduling and present ideas of how to optimize both the order of the products and transitions between the products to maximize process efficiencies.

An integrated assessment of safety and efficiency of aircraft maintenance strategies using agent-based modelling and stochastic Petri nets

Aircraft maintenance is key for safe and efficient aircraft operations. While most studies propose cost-efficient maintenance strategies, the safety and efficiency of these strategies need to be quantified. This paper proposes a formal framework to assess the safety and efficiency of maintenance strategies by means of agent-based modelling, stochastically and dynamically coloured Petri nets, and Monte Carlo simulation. We model an end-to-end aircraft maintenance process, considering several maintenance stakeholders. We apply our framework for aircraft landing gear brakes, and use a Gamma process to model the degradation trends of the brakes. The numerical results show that applying data-driven strategies reduces the number of inspections by 36%, while maintaining the same level of safety as in the case of traditional time-based maintenance strategies. Furthermore, in order to discuss the possibility to substitute all inspections by sensor monitoring, an advanced data-driven strategy using prognostics is considered. Overall, our proposed framework is generic and can readily be applied to assess the safety and efficiency of the maintenance of other aircraft components and maintenance strategies.

Optimal Investment and Reinsurance Under the Gamma Process

In this paper, the insurance company invests its wealth in a capital market composed of a riskless asset and a risky asset. The aggregate claim process of the insurance company is modeled by the Gamma process so as to make it closer to the reality. In practice, the insurance company provides not only those policies with large lose coverings but also policies with small ones. The Gamma process can describe this characteristic better than the compound Poisson process. It is assumed that the insurance company can purchase proportional reinsurance or excess-of-loss reinsurance. Two kinds of optimization problems are considered: maximizing the expected utility of the terminal wealth and minimizing the probability of ruin. For the problem of maximizing the expected utility of the terminal wealth, the explicit optimal value functions and optimal strategies are obtained. For the problem of minimizing the ruin probability, a sufficient condition for the optimal value function and the optimal strategy is obtained.

Optimization of Condition-Based Maintenance of Wood Utility Pole Network Subjected to Hurricane Hazard and Climate Change

Electric power distribution systems that link the bulk power grid (generation and transmission systems) to customers are the leading cause of power outages due to their vulnerability to extreme wind events, especially hurricanes. The strength of the wood poles that typically support the distribution lines also deteriorate over time. The vulnerability of the poles is expected to increase due to the potential impact of climate change on both hurricane hazard and wood decay rate. As such, an effective maintenance planning method is required for the vast number of poles supporting distribution lines. This paper presents a framework to optimize the maintenance of a network of wood utility poles. Corrective replacement due to failure caused by hurricanes and preventive replacement due to excessive decay are considered. The objective is to find the optimal inspection interval for the preventive replacement to minimize the long-term maintenance cost. To solve the optimization problem, the decay of the poles is modeled as a stationary gamma process. The impact of climate change on the rate of pole failure and replacement is also investigated. Two locations are considered as case studies: Miami, Florida, and New York City, New York. The period from 2010 to 2099 is considered for the study. The results of the case study show that the optimal inspection/replacement cycle determined using the developed framework results in lower total maintenance costs compared to current typical utility practice. Based on the inspection and replacement costs used in the study, the results show that adopting a periodic preventive maintenance policy decreases the failure rate of the poles but increases the total maintenance cost. However, only the cost of replacing the poles is considered here. Other considerations, such as indirect costs due to power outages and the impact of pole failure on system reliability, can render the adoption of a preventive replacement policy cost-effective. The results also show that climate change can increase the total maintenance cost. Based on current typical utility maintenance practice, climate change can increase the total maintenance cost by up to 8% in Miami and 6% in NYC, depending on the emission scenario considered.

Reliability estimation considering multi-stress monotonic degradation test data with non-constant scale parameter

The accelerated degradation testing (ADT) has been proven to be effective in quickly assessing the reliability of a highly reliable product, especially during product design and development stage. However, the modeling of ADT data to estimate reliability at normal operating conditions is still a challenging task. It becomes even more complex when there are more than one accelerating factors. In this study, a nonstationary gamma process is considered to model the degradation behavior assuming that the nature of product deterioration is strictly monotonic and non-negative. It further assumes that both gamma process parameters are stress-dependent while considering multiple stresses and their interaction effects. To deal with a problem of computational complexity caused by the consideration of multiple stresses and parameter dependency, a maximum likelihood method has been used for the model parameter estimation. The lifetime and reliability parameters under normal operating conditions are estimated using the acceleration models. A Monte Carlo simulation with the Bayesian updating method has been incorporated to update the gamma parameters when new degradation data become available. A carbon-film resistor degradation data is employed to demonstrate the efficacy of the proposed reliability assessment methodology.