Reliability Analysis of Multi-state Systems with S-dependent
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failures of the affected components but results in an increase or decrease of deterioration of the affected components. In Type 1 failure dependence (IFD), when a component fails, it will either cause the affected components to fail with probability p or leave no effect on the affected components with probability 1- p. Further discussion on IFD can be found in [2]-[4]. Type 2 s-dependence implies that the failure of a component affects the failure rates of the affected components. Sun et al. [1] modeled the failure rate of an affected component by adding a portion of failure rate from the influencing component to its independent failure rate. Lai and Chen [5] studied a two-unit system with Type 2 failure dependence. The failure of unit 1 causes damage to unit 2 by increasing the failure rate of unit 2 by a certain amount corresponding to the number of failures experienced by unit 1. Rasmekonen and Parlikad [6] considered a system consisting of M parallel non-critical machines feeding a critical machine. Parallel elements are independent but the performance of the critical component decreases when a non-critical component fails. The failure and repair times of components were assumed to be deterministic. In practice, many systems and components can degrade and operate at reduced performance levels. As a component deteriorates, its performance may be in several states varying from perfect functioning to complete failure. Such a component is called multi-state component, and a system consisting of multi-state components is called multi-state system (MSS). MSS with s-independent components have been studied in the literature. General concepts and reliability analysis methods can be found in [7]. Recently, more efforts have been made on the analysis of MSS with dependent components. Levitin [8] studied the reliability evaluation problem for MSS with dependent components using the universal generating function (UGF) technique. He assumed that the conditional probability distributions of the affected components are deterministic and explicitly known. In addition, the detailed mechanism of s-dependence was not discussed. Type 1 s-dependence in multi-state systems can be treated in the same way as in binary systems, i.e. the failure of a
Reliability Analysis of Multi-state Systems with S-dependent Components
Cuong D. Dao, University of Alberta Ming J. Zuo, PhD, University of Alberta
Key Words: Multi-state systems, multi-state components, s-dependence, reliability, universal generating function, Monte Carlo simulation SUMMARY & CONCLUSIONS The assumption of stochastic independence between components is frequently made in studies of system reliability. However, in a specific system, the failure of a component can trigger the failure of other components, or the current health condition of a component may affect the performance of other components. Thus, the state of a component can affect the state and degradation of other components in a multicomponent system, that is, stochastic dependence (sdependence) may exist in real and complex systems. In this paper, reliability analysis of multi-state systems (MSS) with sdependent components will be considered. The MSS consists of 2 multi-state components in series, with each component possibly operating at different performance levels, varying from perfect functioning to complete failure. When the first component degrades to a lower performance level, it affects the state as well as the degradation of the other component in the system. A combined technique of stochastic process and modified universal generating function is used to evaluate the system reliability. The combined approach is then verified using the Monte Carlo Simulation method. An illustrative example on reliability analysis of MSS with dependent components is also provided. 1 INTRODUCTION In binary systems, s-dependence is referred to as “failure dependence” and it is often analyzed based on the nature of components’ failures or operating modes. The components are interconnected in a specific system design or subjected to a common stress such as operating environment, shocks, etc. Thus, the failure of a component may affect the operation of other components in a multi-component system. In [1], sdependence between components is referred to as “interactive failure” and it can be categorized into two types based on the consequences that arise when a component in the system fails. • Type 1 - Immediate failure dependence (IFD): whenonent) fails, it immediately causes the failures of other components (affected components). • Type 2 - Gradual degradation dependence: The failure of the influencing component does not immediately cause
multi-state component can immediately cause failures of the affected components. A study on reliability of MSS with Type 1 s-dependence, referred to as “propagated failure”, can be found in [9]. On the other hand, Type 2 s-dependence between binary components cannot be directly extended to the multistate context. It is necessary to investigate Type 2 sdependence between multi-state components. In this paper, we will first focus on modeling Type 2 sdependence between multi-state components in a 2-component series system. Since the components can operate at different performance levels, Type 2 s-dependence in MSS has to describe the “degradation dependence” between components. In general, both the state and the transition rates between states of the affected component can be affected by the influencing component. Thus, s-dependence in this paper is defined as “the state of a component can affect the state and the degradation of other components in a multi-state system”. A combined approach based on Markov analysis and the modified UGF technique will then be employed to evaluate the reliability of multi-state series systems. We also use the Monte Carlo simulation method to model the gradual degradation dependence between two components and estimate the system reliability. The remaining part of this paper is organized as follows. Section 2 models the MSS with Type 2 s-dependence. Section 3 represents two methods for reliability analysis of such systems: 1. Stochastic process and modified UGF technique and 2. Monte Carlo Simulation. Illustrative examples, results and discussion will be presented in section 4 and a general summary is provided in the last section. 2 MSS WITH S-DEPENDENT COMPONENTS 2.1 Multi-state systems The multi-state system in this study consists of 2 multistate components connected in series (Figure 1). Component 1 is an influencing component that causes Type 2 s-dependence to component 2, i.e. the state of component 1 has effect on the state and degradation of component 2. The components in the system can operate in one of K + 1 possible states, 0,1, 2,..., K , where K is the perfect functioning state and 0 is the state of complete failure. Other intermediate states, 1, 2,..., K − 1, are imperfect functioning states, i.e. the component can continue operating but at a reduced performance rate. The multi-state system in this study is a flow transmission system, where the output performance rate of the system and its components is defined as its productivity or capacity [10].
failures of the affected components but results in an increase or decrease of deterioration of the affected components. In Type 1 failure dependence (IFD), when a component fails, it will either cause the affected components to fail with probability p or leave no effect on the affected components with probability 1- p. Further discussion on IFD can be found in [2]-[4]. Type 2 s-dependence implies that the failure of a component affects the failure rates of the affected components. Sun et al. [1] modeled the failure rate of an affected component by adding a portion of failure rate from the influencing component to its independent failure rate. Lai and Chen [5] studied a two-unit system with Type 2 failure dependence. The failure of unit 1 causes damage to unit 2 by increasing the failure rate of unit 2 by a certain amount corresponding to the number of failures experienced by unit 1. Rasmekonen and Parlikad [6] considered a system consisting of M parallel non-critical machines feeding a critical machine. Parallel elements are independent but the performance of the critical component decreases when a non-critical component fails. The failure and repair times of components were assumed to be deterministic. In practice, many systems and components can degrade and operate at reduced performance levels. As a component deteriorates, its performance may be in several states varying from perfect functioning to complete failure. Such a component is called multi-state component, and a system consisting of multi-state components is called multi-state system (MSS). MSS with s-independent components have been studied in the literature. General concepts and reliability analysis methods can be found in [7]. Recently, more efforts have been made on the analysis of MSS with dependent components. Levitin [8] studied the reliability evaluation problem for MSS with dependent components using the universal generating function (UGF) technique. He assumed that the conditional probability distributions of the affected components are deterministic and explicitly known. In addition, the detailed mechanism of s-dependence was not discussed. Type 1 s-dependence in multi-state systems can be treated in the same way as in binary systems, i.e. the failure of a
Reliability Analysis of Multi-state Systems with S-dependent Components
Cuong D. Dao, University of Alberta Ming J. Zuo, PhD, University of Alberta
Key Words: Multi-state systems, multi-state components, s-dependence, reliability, universal generating function, Monte Carlo simulation SUMMARY & CONCLUSIONS The assumption of stochastic independence between components is frequently made in studies of system reliability. However, in a specific system, the failure of a component can trigger the failure of other components, or the current health condition of a component may affect the performance of other components. Thus, the state of a component can affect the state and degradation of other components in a multicomponent system, that is, stochastic dependence (sdependence) may exist in real and complex systems. In this paper, reliability analysis of multi-state systems (MSS) with sdependent components will be considered. The MSS consists of 2 multi-state components in series, with each component possibly operating at different performance levels, varying from perfect functioning to complete failure. When the first component degrades to a lower performance level, it affects the state as well as the degradation of the other component in the system. A combined technique of stochastic process and modified universal generating function is used to evaluate the system reliability. The combined approach is then verified using the Monte Carlo Simulation method. An illustrative example on reliability analysis of MSS with dependent components is also provided. 1 INTRODUCTION In binary systems, s-dependence is referred to as “failure dependence” and it is often analyzed based on the nature of components’ failures or operating modes. The components are interconnected in a specific system design or subjected to a common stress such as operating environment, shocks, etc. Thus, the failure of a component may affect the operation of other components in a multi-component system. In [1], sdependence between components is referred to as “interactive failure” and it can be categorized into two types based on the consequences that arise when a component in the system fails. • Type 1 - Immediate failure dependence (IFD): whenonent) fails, it immediately causes the failures of other components (affected components). • Type 2 - Gradual degradation dependence: The failure of the influencing component does not immediately cause
multi-state component can immediately cause failures of the affected components. A study on reliability of MSS with Type 1 s-dependence, referred to as “propagated failure”, can be found in [9]. On the other hand, Type 2 s-dependence between binary components cannot be directly extended to the multistate context. It is necessary to investigate Type 2 sdependence between multi-state components. In this paper, we will first focus on modeling Type 2 sdependence between multi-state components in a 2-component series system. Since the components can operate at different performance levels, Type 2 s-dependence in MSS has to describe the “degradation dependence” between components. In general, both the state and the transition rates between states of the affected component can be affected by the influencing component. Thus, s-dependence in this paper is defined as “the state of a component can affect the state and the degradation of other components in a multi-state system”. A combined approach based on Markov analysis and the modified UGF technique will then be employed to evaluate the reliability of multi-state series systems. We also use the Monte Carlo simulation method to model the gradual degradation dependence between two components and estimate the system reliability. The remaining part of this paper is organized as follows. Section 2 models the MSS with Type 2 s-dependence. Section 3 represents two methods for reliability analysis of such systems: 1. Stochastic process and modified UGF technique and 2. Monte Carlo Simulation. Illustrative examples, results and discussion will be presented in section 4 and a general summary is provided in the last section. 2 MSS WITH S-DEPENDENT COMPONENTS 2.1 Multi-state systems The multi-state system in this study consists of 2 multistate components connected in series (Figure 1). Component 1 is an influencing component that causes Type 2 s-dependence to component 2, i.e. the state of component 1 has effect on the state and degradation of component 2. The components in the system can operate in one of K + 1 possible states, 0,1, 2,..., K , where K is the perfect functioning state and 0 is the state of complete failure. Other intermediate states, 1, 2,..., K − 1, are imperfect functioning states, i.e. the component can continue operating but at a reduced performance rate. The multi-state system in this study is a flow transmission system, where the output performance rate of the system and its components is defined as its productivity or capacity [10].