Renewal Reward Research Articles

System-level Probabilistic Remaining Useful Life Prognostics and Predictive Inspection Planning for Wind Turbines

Wind energy plays a crucial role in the energy transition. However, it is often seen as an unreliable source of energy, with many production peaks and lows. Some of the drivers of uncertainty in energy production are the unexpected wind turbine (WT) failures and associated unscheduled maintenance. To support an effective health management and maintenance planning of WTs, we propose an integrated data-driven framework for Remaining Useful Life (RUL) prognostics and inspection planning of WTs. We propose a Long-short term memory (LSTM) neural network with Monte Carlo dropout to estimate the distribution of the RUL of WTs, i.e. we develop probabilistic prognostics. Different from existing studies focused on prognostics for single components, we consider the simultaneous health-monitoring of multiple components of the WTs, thus seeing the turbine as an integrated system. The obtained prognostics are further included into a stochastic planning model which determines optimal moments for inspections. For this, we pose the problem of WT inspections as a renewal reward process. We illustrate our framework for four offshore WTs which are continuously monitored by Supervisory Control and Data Acquisition (SCADA) systems. The results show that LSTMs are able to estimate well the RUL of the WTs, even in the early phase of their usage. We also show that the prognostics are informative for maintenance planning and are conducive to conservative inspections.

Quenched large deviations in renewal theory

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

Farm-specific estimates of forage variability result in larger optimal control limits for forage quality monitoring

Variation in forage composition decreases the accuracy of diets delivered to dairy cows. However, variability of forages can be managed using a renewal reward model (RRM) and genetic algorithm (GA) to optimize sampling and monitoring practices for farm conditions. Specifically, use of quality-control-charts to monitor forage composition can identify changes in composition for which adjustment in the formulated diet will result in a better match of the nutrients delivered to cows. The objectives of this study were 1) assess the use of a clustering algorithm to estimate the mean time the process is stable or in-control (d) (TStable) and the magnitude of the change in forage composition between stable periods (ΔForage) for corn silage and alfalfa-grass silage which are input parameters for the RRM; 2) compare optimized farm-specific sampling practices (number of samples (n), sampling interval (TSample) and control limits (ΔLimit) using previously proposed defaults and our estimates for the TStable and ΔForage input parameters; and 3) conduct a simulation study to compare the number of recommended diet changes costs of quality control under the proposed sampling and monitoring protocols. We estimated the TStable and ΔForage parameters for corn silage NDF and starch and alfalfa-grass silage NDF and CP using a k-means clustering approach applied to forage samples collected from 8 farms, 3x/week during a 16-week period. We compared 4 sampling and monitoring protocols that resulted from the 2 methods for estimating TStable and ΔForage (default values and our proposed method) and either optimizing only the control limit (Optim1) or optimizing the control limits, the number of samples, and the number of days between sampling (Optim2). We simulated the outcomes of implementing the optimized monitoring protocols using a quality control chart for corn silage and alfalfa-grass silage of each farm. Estimates of T^Stable and Δ^Forage from the k-means clustering analysis were, respectively, shorter and larger than previously proposed default values. In the simulated quality control monitoring, larger Δ^Forage estimates increased the optimized ΔLimit resulting in fewer detected shifts in composition of forages and a lower frequency of false alarms and a lower quality control cost ($/d). Recommended diet reformulation intervals from the simulated quality control analysis were specific for the type of forage and farm management practices. The median of the diet reformulation intervals for all farms using our optimal protocols was 14 d (Q1 = 8, Q3 = 26) for corn silage and 16 d (Q1 = 8, Q3 = 26) for alfalfa-grass silage.

Optimal Bivariate Order-Replacement Policies for a Multi-State Repairable System with Reliability Constraint

A bivariate order-replacement policy for a multi-state repairable system with imperfect repair is put forward in this paper, where the decisions on when to order a spare part and when to place a replacement are based on the number of failures. The geometric processes are generalized to depict the characteristics that the successive working times are shorter and shorter while the consecutive repair times are longer and longer. According to the renewal reward theorem, a constrained optimization model for the order-replacement policy is formulated, in which the aim is to minimize the expected long-run cost rate (ELRCR) under the constraint that system availability is not less than a predetermined availability requirement. General representations of the ELRCR and system availability are derived and an approximation result of the optimization model is given under certain assumptions. Illustrative examples are provided to validate the applicability of the proposed order-replacement model and show the advantage of the order-replacement policy for the multi-state repairable system.

Residual life modeling and maintenance planning for repairable systems

Abstract Accepted by: Phil Scarf This paper formulates a state-dependent mean residual lifetime model for a repairable system operating in a dynamic environment. The problem is addressed by means of a two-state damage process reflecting the effect of operating environment on the system and a repair process associated with the damage process. As the damage process shifts to a higher state, to maintain a minimum level of performance, the decision maker repairs the system at times that arise according to a point process with a constant intensity. We demonstrate the generality of the proposed model and show how existing models emerge as specific cases. Our approach stimulates further research on the determination of two types of maintenance policies: maintenance policy based on the number of imperfect repairs (Model I) and time-based maintenance policy (Model II). In both cases, using the renewal reward theorem argument, we aim at minimizing the long-run average maintenance cost per unit time by determining optimal replacement policies and the optimal level of imperfect repairs. We illustrate the proposed models and carry out a comparative analysis of maintenance policies through numerical examples. The main conclusions drawn are that repair and maintenance policies depend on the failure mechanism, repair frequency and the level of costs involved. Also, numerical comparison shows that the maintenance modelling based on the number of imperfect repairs (model I) outperforms the time-based replacement model (model II) and two baseline models ignoring the effect of operating environment or whose attention is restricted to perfect repair.

Evaluating cyber loss in star-ring and star-bus hybrid networks based on the bond percolation model

The rise of cyber risk as a significant concern for businesses and companies has prompted increased attention towards evaluating the loss of cyber risk in networks vulnerable to contagious cyber attacks. Expanding on recent research focusing on standard network topologies, this study investigates two hybrid topologies: star-ring and star-bus. We propose random star-ring and star-bus hybrid network models that incorporate heterogeneous node types and a contagion model based on the bond percolation model to describe the contagion of a cyber attack on them. We derive exact expressions for the mean, variance, and covariance of compromise sizes for different node types. Utilizing these expressions, we further compute the mean and variance of the aggregate cyber loss of the networks, employing a dynamic framework of loss based on the renewal reward process. Moreover, through a numerical study and analysis of mathematical properties, we gain valuable insights into the impact of parameters on the mean, variance, and correlation of compromise sizes.

Reliability modelling with uncertain threshold and fractional differential degenerate equation in uncertain environment

A dependent competitive failure model of Caputo uncertain fractional differential equation (UFDE) degradation with uncertain threshold is proposed. There are two main reasons for the failure of an equipment or system, long-term natural wear and the impact of external loads. Each shock will cause a sharp increase in wear. In the absence of fault data, the natural wear process is driven by a Caputo type of UFDE, and the external load is an uncertain renewal reward process. Three different shock models are considered and calculated the belief reliability of device according to uncertainty theory, taking the jet pipe servo control valve as an example, the proposed model is simulated, and the results show that the proposed model is effective.

Moment-based approximations for stochastic control model of type (s, S)

In this study, we propose an approximation for a renewal reward process that describes a stochastic control model of type (s, S) based on the first three moments of demand random variables. Various asymptotic expansions for this model exist in the literature. All these studies rely on the condition of knowing the distribution function of demand random variables and require obtaining the asymptotic expansion of the renewal function produced by them. However, obtaining a renewal function can be challenging for certain distribution families, and in some cases, the mathematical structure of the renewal function is difficult to apply. Therefore, in this study, simple and compact approximations are presented for the stochastic control model of type (s, S). The findings of this study rely on Kambo’s method, through which we obtain approximations for the ergodic distribution, and the n t h order ergodic moments of this process. To conclude the study, the accuracy of the proposed approximate formulas are examined through a specialized illustrative example. Moreover, it has been noted that the proposed approximation is more accurate than the approximations existing in the literature.

An approach to modeling residual life of a renewal process for reliability analysis and maintenance planning

This paper extends and relaxes the structure of a mean remaining time to renewal model for a failure prone system subject to minimal repairs. Compared to existing related works, the contributions of our model are threefold. It not only accounts for changes in operating states and age factor, but also can adapt itself to minimal repair process history. It is implemented through the use of a damage process X(t) coupled with a supplementary variable N(t) counting the number of failures. The bivariate approach allows a wide class of models to be considered including age-based models and those ignoring the repair effect. Another feature enhancing the domain of its applicability is to predict failures and quantify some reliability indexes. These impose a challenge when coping with models either characterized by inter-correlated failure times or developed under varying operating conditions. Further, the structure of the devised model called state-dependent mean remaining time to renewal (in short SD-MRTR) enables the implementation of maintenance policy on the basis of renewal–reward arguments. In present setting, a renewal is defined with respect to the supplementary variable. That means, a renewal (perfect repair) is carried out as soon as the number of failures reaches a give value j. Using the renewal–reward theorem argument and the explored SD-MRTR, the practical contribution of the model is enhanced by proposing a j-based preventive replacement policy achieving minimum repair costs. The proposed model and the behavior of the optimal solution as the model parameters change are illustrated through numerical examples.

Investigation the ergodic distribution of a semi-Markovian inventory model of type (s,S) with intuitive approximation approach

This paper concerns a stochastic process expressing (s,S) type inventory system with intuitive approximation approach. The stock level in the system is modeled as a semi-Markovian renewal reward process X(t). Therefore, the ergodic distributions of this process can be analyzed with the help of the renewal function. Obtaining explicit formula for renewal function U(x) is difficult from a practical standpoint. Mitov and Omey recently present some intuitive approximations in literature for renewal function which cover a large number of existing results. Using their approach we were able to establish asymptotic approximations for ergodic distribution of a stochastic process X(t). Obtained results can be used in many situations where demand random variables have different distributions from different classes such as Γ(g) class.

On a cost and availability analysis for software systems via phase type non-homogeneous Poisson process

To build highly reliable software products, software testing is required. The software engineer has a problem determining whether to complete the testing process and when to release the software system to the market. As the focus on high software increases, project managers must consider the cost of testing, testing availability time, and release time strategy. This research article utilizes a phase type non-homogeneous Poisson process to investigate the cost analysis and operational availability of a software reliability model. The software failure inter-arrival time is considered to follow a phase type distribution. The phase type software reliability model is valuable in reducing the time and effort required to select the appropriate models for software reliability evaluation. An explicit expression for the expected, bi-criterion cost analysis and operational availability is derived using standard results of the renewal reward theorem. The optimal software release policy T* is obtained analytically. Numerical illustrations and sensitivity analysis have been presented to illustrate the cost and availability analysis’s efficiency and demonstrate conformity with the study’s observations.

Tree Representation, Growth Rate of Blockchain and Reward Allocation in Ethereum With Multiple Mining Pools

It is interesting but difficult and challenging to study Ethereum with multiple mining pools. One of the main difficulties comes from not only how to represent such a general tree with multiple block branches (or sub-chains) related to the multiple mining pools, but also how to analyze a multi-dimensional stochastic system due to the mining competition among the multiple mining pools. In this paper, we first set up a mathematical representation for the tree with multiple block branches. Then we provide a block classification of Ethereum: Regular blocks (in the main chain), orphan blocks, uncle blocks, stale blocks, and nephew blocks, and give some key ratios and probabilities of generating the different types of blocks by applying the law of large numbers. Based on this, we further discuss the growth rate of blockchain and the reward allocation among the multiple mining pools through applying the renewal reward theorem. Finally, we use some simulation experiments to verify our theoretical results, and show that the approximate computation approaches developed, such as the key ratios and probabilities, the long-term growth rate of blockchain, and the long-term reward allocation (rate) among the multiple mining pools, can have a faster convergence. Therefore, we provide a powerful tool for observing and understanding the influence of the selfish mining attacks on the performance of Ethereum with multiple mining pools. We believe that the methodology and results developed in this paper will shed light on the study of Ethereum with multiple mining pools, such that a series of promising research can be inspired potentially.

Hybrid uncertainty quantification of dependent competing failure process with chance theory

Dependent competing failure process (DCFP) has been widely studied in recent years. When evaluating the reliability of DCFP system, the aleatory uncertainty due to randomness and the epistemic uncertainty due to lack of data and knowledge should be considered. In this paper, the hybrid uncertainty of DCFP system is analyzed and quantified with chance theory. The uncertainty random renewal reward process and the arithmetic Liu process are used to describe the shock and degradation process. The quantification of uncertainty propagation is realized by decoupling DCFP from the physical correlation aspect. Reliability of DCFP system considering two kinds of unilateral dependencies and mutual dependency is modeled and a combinational algorithm is proposed to solve it. Finally, the DCFP effect in control system of an aircraft is studied as a case, aleatory uncertainty and epistemic uncertainty are modeled, and the system reliability is evaluated. Meanwhile, the effects of key parameters on system reliability are discussed. Results show that the chance theory is an appropriate method to qualify the hybrid uncertainty for engineering systems when data is insufficient. The mutual dependency of soft failure and hard failure will reduce the system reliability. The smaller the parameter of hard failure, the greater the influence of mutual dependency on the results.

An optimal ordering inventory policy when the purchase price follows a discrete-time Markov chain

Most of the time retailers are offered discount prices that are occurring at random points in time. One such scenario is the supplier offering discounts to the retailers to increase market share, cash flow, and to reduce the inventory of specific items. Surprisingly no models exist in the literature survey to model the randomly occurring discount price under discrete time. The primary objective of this article is to develop an optimal inventory policy for a retailer when the discount price arrives randomly and the time variable is discrete. Under such a scenario, the optimal inventory policy can be obtained by considering the variations in the purchase price as a discrete-time Markov chain. The renewal reward theorem is used to derive the expected profit per unit of time. To illustrate the application of the developed model a real case study is considered. The optimal solution of the developed model is compared with the EOQ policy and found that the developed model solution provides better profit. This justifies the significance of the developed model. The inventory manager can use the developed model as a tool to obtain the optimal solution for any two price problems that repeat randomly.

Large deviation principles for renewal–reward processes

We establish a sharp large deviation principle for renewal–reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cramér’s theory. We also exhibit sufficient conditions for exponential tightness of renewal–reward processes, which leads to a full large deviation principle.

A systematic review of uncertainty theory with the use of scientometrical method

Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems.

Implementing a bivariate ordering and replacement policy for deteriorating systems with two failure types

AbstractFor plenty of existing maintenance policies in literature, it is always assumed that the spare unit for replacement is available at any time if needed, while this perquisite may be unrealistic or even impossible. In this paper, we investigate an ordering and replacement modeling framework for a deteriorating repairable system with two failure types, involving a repairable failure (Type I failure) and a nonrepairable failure (Type II failure). A regular order and an expedited order are both incorporated into the spare unit ordering policy while a preventive replacement and a corrective replacement are both considered into the replacement policy, of which the ordering and replacement policy is scheduled in regard to the number of repairable failures. The expected profit per unit of time over an infinite time span is maximized to seek the optimal solutions in terms of the renewal reward theorem with a recursive algorithm procedure. An illustrative example is designed to validate the applicability and effectiveness of the developed ordering and replacement policy. In addition, the sensitivity of the optimal ordering‐replacement policy is studied and the effect of some key characteristics on the optimal decisions is discussed.

Joint Production and Maintenance Optimization of a Series–Parallel System with Quality-Contingent Demand

Making a reasonable and effective production plan is always an essential and challenging task in industrial production. A joint optimization model of production and maintenance is proposed in this paper, which considers the structural relationship between production units and the influence of the unit state on demand. A three-unit series–parallel system is selected to calculate the steady-state probability density function of the system, and the model is established by dividing different maintenance situations in one cycle. By analyzing the composition of expected cost and expected time in each situation, the expected cost rate is calculated by using renewal reward theory. The objective function of the model is to minimize the expected cost rate. The genetic algorithm is improved according to the model characteristics. The application of the model is illustrated by a case, and the sensitivity analysis is set to show the influence of different parameters on the decision-making results of the system, providing ideas for decision-makers. Finally, the contrast experiments show the advantages of the proposed model and method.

Quantity-based emergency shipment policies

Either structurally or in an ad hoc manner, emergency shipments are widely utilized for preventing the potentially negative impacts of stock-outs. In addition to having policies for their regular orders, companies need to decide when and for how many units to place emergency orders. In this article, we consider the periodic-review problem of a retailer who uses a quantity-based emergency shipment policy. Under this policy an emergency order is triggered if the inventory level within the review period falls below a certain level. We consider the base-stock policy for the regular replenishment orders. The goal is to determine the optimal base-stock level, the optimal period length as well as the optimal size and threshold value of emergency orders such that the total expected cost rate is minimized. We use renewal reward theory to derive the expressions of operating characteristics. The expected cost rate’s properties are analyzed to develop an optimization algorithm for finding the optimal policy parameters. We compare our policy with the time-based and hybrid emergency shipment policies and conclude that the quantity-based policy can bring substantial benefits compared to other policies.

The Decision of Production Systems with Quality-Contingent Demand and Condition-Based Maintenance

To make a production plan fit with the actual situation better, we focus on the production system with equipment, and design a joint optimization strategy combining the economic production quantity (EPQ) model with condition-based maintenance. In this strategy, different maintenance operations are carried out when the state of the equipment exceeds different thresholds. The numerical relationship between product demand rate and equipment state is established, and the average cost rate is calculated by using the renewal reward theory. An optimization model is proposed, which takes the lowest average cost rate as the objective function with the economic production quantity and condition-based maintenance threshold are taken as the decision variables. An improved genetic algorithm with an elite strategy is used to solve the model. The results shows that the cost of the proposed model is lower and the sensitivity analysis can describe the relationship between the various elements of the production system clearly, understand the system state quickly, and demonstrate the proposed model.