MoMA Algorithm: A Bottom-Up Modeling Procedure for a Modular System under Environmental Conditions

The functioning of complex systems relies on subsystems (modules) that in turn are composed of multiple units. In this paper, we focus on modular systems that might fail due to wear on their units or environmental conditions (shocks). The lifetimes of the units follow a phase-type distribution, while shocks follow a Markovian Arrival Process. The use of Matrix-Analytic methods and a bottom-up approach for constructing the system generator is proposed. The use of modular structures, as well as its implementation by the Modular Matrix-Analytic (MoMA) algorithm, make our methodology flexible in adapting to physical changes in the system, e.g., incorporation of new modules into the current model. After the model for the system is built, the modules are seen as a ‘black box’, i.e., only the contribution of the module as a whole to system performance is considered. However, if required, our method is able to keep track of the events within the module, making it possible to identify the state of individual units. Compact expressions for different reliability measures are obtained with the proposed description, optimal maintenance strategies based on critical operative states are suggested, and a numerical application based on a k-out-of-n structure is developed.Spanish Ministry of Science and Innovation-State Research Agency PID2020-120217RB-I00 PID2021-123737NB-I00Junta de Andalucia B-FQM-284-UGR20 CEX2020-001105-/AEI/10.13039/50110001103

Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units

A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model

A complex multi-state k-out-of-n: G system with preventive maintenance and loss of units

In this study, a multi-state k-out-of-n: G system subject to multiple events is modeled through a Markovian Arrival Process with marked arrivals. The system is composed initially of n units and is active when at least k units are operational. Each unit is multi-state, each of which is classified as minor or major according to the level of degradation presented. Each operational unit may undergo internal repairable or non-repairable failures, external shocks and/or random inspections. An external shock can provoke extreme failure, while cumulative external damage can deteriorate internal performance. This situation can produce repairable and non-repairable failures. When a repairable failure occurs the unit is sent to a repair facility for corrective repair. If the failure is non-repairable, the unit is removed. When the system has insufficient units with which to operate, it is restarted. Preventive maintenance is employed in response to random inspection. The system is modeled in an algorithmic and computational form. Several interesting measures of performance are considered. Costs and rewards are included in the system. All measures are obtained for transient and stationary regimes. A numerical example is analyzed to determine whether preventive maintenance is profitable, financially and in terms of performance.Junta de Andalucía (Spain) FQM-307Ministerio de Economía y Competitividad (España) MTM2017-88708-PEuropean Regional Development Fund (ERDF

A discrete MMAP for analysing the behaviour of a multi-state complex dynamic system subject to multiple events.

A complex multi-state system subject to different types of failures, repairable and/or nonrepairable, external shocks and preventive maintenance is modelled by considering a discrete Markovian arrival process with marked arrivals (D-MMAP). The internal performance of the system is composed of several degradation states partitioned into minor and major damage states according to the risk of failure. Random external events can produce failures throughout the system. If an external shock occurs, there may be an aggravation of the internal degradation, cumulative external damage or extreme external failure. The internal performance and the cumulative external damage are observed by random inspection. If major degradation is observed, the unit goes to the repair facility for preventive maintenance. If a repairable failure occurs then the system goes to corrective repair with different time distributions depending on the failure state. Time distributions for corrective repair and preventive maintenance depend on the failure state. Rewards and costs depending on the state at which the device failed or was inspected are introduced. The system is modelled and several measures of interest are built into transient and stationary regimes. A preventive maintenance policy is shown to determine the effectiveness of preventive maintenance and the optimum state of internal and cumulative external damage at which preventive maintenance should be taken into account. A numerical example is presented, revealing the efficacy of the model. Correlations between the numbers of different events over time and in non-overlapping intervals are calculated. The results are expressed in algorithmic-matrix form and are implemented computationally with Matlab.Junta de Andalucía, Spain, under the grant FQM307Ministerio de Economía y Competitividad, España, MTM2017-88708-PEuropean Regional Development Fund (ERDF

Optimal Periodic Inspection of a Stochastically Degrading System

This thesis develops and analyzes a procedure to determine the optimal inspection interval that maximizes the limiting average availability of a stochastically degrading component operating in a randomly evolving environment. The component is inspected periodically, and if the total observed cumulative degradation exceeds a fixed threshold value, the component is instantly replaced with a new, statistically identical component. Degradation is due to a combination of continuous wear caused by the component\u27s random operating environment, as well as damage due to randomly occurring shocks of random magnitude. In order to compute an optimal inspection interval and corresponding limiting average availability, a nonlinear program is formulated and solved using a direct search algorithm in conjunction with numerical Laplace transform inversion. Techniques are developed to significantly decrease the time required to compute the approximate optimal solutions. The mathematical programming formulation and solution techniques are illustrated through a series of increasingly complex example problems

Reliability and Condition-Based Maintenance Analysis of Deteriorating Systems Subject to Generalized Mixed Shock Model

For successful commercialization of evolving devices (e.g., micro-electro-mechanical systems, and biomedical devices), there must be new research focusing on reliability models and analysis tools that can assist manufacturing and maintenance of these devices. These advanced systems may experience multiple failure processes that compete against each other. Two major failure processes are identified to be deteriorating or degradation processes (e.g., wear, fatigue, erosion, corrosion) and random shocks. When these failure processes are dependent, it is a challenging problem to predict reliability of complex systems. This research aims to develop reliability models by exploring new aspects of dependency between competing risks of degradation-based and shock-based failure considering a generalized mixed shock model, and to develop new and effective condition-based maintenance policies based on the developed reliability models. In this research, different aspects of dependency are explored to accurately estimate the reliability of complex systems. When the degradation rate is accelerated as a result of withstanding a particular shock pattern, we develop reliability models with a changing degradation rate for four different shock patterns. When the hard failure threshold reduces due to changes in degradation, we investigate reliability models considering the dependence of the hard failure threshold on the degradation level for two different scenarios. More generally, when the degradation rate and the hard failure threshold can simultaneously transition multiple times, we propose a rich reliability model for a new generalized mixed shock model that is a combination of extreme shock model, δ-shock model and run shock model. This general assumption reflects complex behaviors associated with modern systems and structures that experience multiple sources of external shocks. Based on the developed reliability models, we introduce new condition-based maintenance strategies by including various maintenance actions (e.g., corrective replacement, preventive replacement, and imperfect repair) to minimize the expected long-run average maintenance cost rate. The decisions for maintenance actions are made based on the health condition of systems that can be observed through periodic inspection. The reliability and maintenance models developed in this research can provide timely and effective tools for decision-makers in manufacturing to economically optimize operational decisions for improving reliability, quality and productivity.Industrial Engineering, Department o

Analytical Results for a Single-Unit System Subject To Markovian Wear and Shocks

This thesis develops and analyzers a mathematical model for the reliability measures of a single-unit system subject to continuous wear due to its operating environment and randomly occurring shocks that inflict a random amount of damage to the unit. Assuming a Markovian operating environment and shock arrival mechanism, Laplace-Stieltjes transform expressions are obtained for the failure time distribution and all of its moments. Moreover, an analytical expression is derived for the long-run availability of the single-unit system when it is subject to an inspect-and-replace maintenance policy. The analytical results are illustrated, and their results compared with those of Monte Carlo-simulated failure data. The numerical results indicate that the reliability measures may be accurately computed via numerical inversion of the transform expressions in a straightforward manner when the input parameters are known a priori. In stark contrast to the simulation model which requires several hours to obtain the reliability measures, the analytical procedure computes the same measures in only a few seconds

Age Replacement and Service Rate Control of Stochastically Degrading Queues

This thesis considers the problem of optimally selecting a periodic replacement time for a multiserver queueing system in which each server is subject to degradation as a function of the mean service rate and a stochastic and dynamic environment. Also considered is the problem of optimal service rate selection for such a system. In both cases, the performance metric is the long-run average cost rate. Analytical expressions are obtained, in terms of Laplace transforms, for the nonlinear objective functions, necessitating the use of numerical Laplace transform inversion to evaluate candidate solutions in conjunction with standard numerical algorithms. Due to the convexity of the objective function, the optimal replacement time is computed using a hybrid bisection-secant method which yields globally optimal solutions. The optimal service rates are obtained via gradient search methods but are only guaranteed to provide locally optimal solutions. The analytical results are implemented on three notional examples that demonstrate the benefits of dynamically adjusting service rates under the described maintenance policy

Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields

This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners