Availability and reliability of k-out-of-(M+N):G warm standby systems

Redundancy or standby is a technique that has been widely applied to improving system reliability and availability in system design. In most cases, components in standby system are assumed to be statistically identical and independent. However, in many practical applications, not all components in standby can be treated as identical because they have different failure and repair rates. In this paper, one kind of such systems with two categories of components is studied, which is named k-out-of-(M+N):G warm standby system. In the system, one category of the components is of type 1 and the other type 2. There are M type 1 components and N type 2 components. Components of type 1 have a lower failure rate and are preferably repaired if there is one failed. There are r repair facilities available. By using Markov model, the system state transition process can be clearly illustrated, and furthermore, the solutions of system availability and reliability are obtained based on this. An example representing a power-generator and transmission system is given to illustrate the solutions of the system availability and reliability.     

On the steady state unavailability of standby systems

In this paper we study the steady state unavailability of standby systems comprising n identical components of which n − 1 are normally operating and one is in standby, with one repair facility. The components are assumed to have constant failure rates, but arbitrary repair time distribution. Attention is given to the case that the mean time to repair MTTR is relatively small compared to the mean time to failure MTTF, which is the most common situation in practice. Simple approximation formulae are presented and compared with the standard Markov expressions. It is demonstrated that the Markov expressions produce relatively large errors when the coefficient of variation of the repair time distribution is not close to 1. Some easily computed error bounds of the approximation formulae are established.     

A multi-objective discrete reliability optimization problem for dissimilar-unit standby systems

A new methodology for the reliability optimization of a k dissimilar-unit nonrepairable cold-standby redundant system is introduced in this paper. Each unit is composed of a number of independent components with generalized Erlang distributions of lifetimes arranged in a series–parallel configuration. We also propose an approximate technique to extend the model to the general types of nonconstant hazard functions. To evaluate the system reliability, we apply the shortest path technique in stochastic networks. The purchase cost of each component is assumed to be an increasing function of its expected lifetime. There are multiple component choices with different distribution parameters available for replacement with each component of the system. The objective of the reliability optimization problem is to select the best components, from the set of available components, to be placed in the standby system to minimize the initial purchase cost of the system, maximize the system mean time to failure, minimize the system variance of time to failure, and also maximize the system reliability at the mission time. The goal attainment method is used to solve a discrete time approximation of the original problem.      

An optimal replacement policy for a repairable system based on its repairman having vacations

This paper studies a cold standby repairable system with two different components and one repairman who can take multiple vacations. If there is a component which fails and the repairman is on vacation, the failed component will wait for repair until the repairman is available. In the system, assume that component 1 has priority in use. After repair, component 1 follows a geometric process repair, while component 2 can be repaired as good as new after failures. Under these assumptions, a replacement policy N based on the failed times of component 1 is studied. The system will be replaced if the failure times of component 1 reach N. The explicit expression of the expected cost rate is given, so that the optimal replacement time N^? is determined. Finally, a numerical example is given to illustrate the theoretical results of the model.     

Analysis of a class of nonconservative systems reducible to pseudoconservative ones and their energy relations

This paper analyzes a class of nonconservative systems, whose Lagrangian equations can be reduced to Euler–Lagrangian equations by introducing a new Lagrangian, which is equal to a product of some function of time f(t) and the primary Lagrangian. These equations formally have the same form as for the systems with potential forces, while the influence of nonconservative forces is contained in the factor f(t), and such systems are called pseudoconservative. It is further shown that the requirement for a nonconservative system to be considered as a pseudoconservative is the existence of at least one particular solution of a system of differential equations with unknown function f(t), or their linear combination with suitably chosen multipliers. Further on, the energy relations and corresponding conservation laws of those systems are analyzed from two aspects: directly, on the basis of the corresponding Lagrangian equations and via modified Emmy Noether’s theorem. So, it has been shown, even in two different ways, that there are two types of the integrals of motion, in the form of the product of an exponential factor and the sum of the generalized energy (energy function) and an additional term. For the existence of these integrals of motion, it is necessary and sufficient that there exists at least one particular solution of a partial differential equation, which is in accordance with the Lagrangian equations for the observed problem. The obtained results are equivalent to so-called energy-like conservation laws, obtained via Vujanovi?-Djuki?’s generalized Noether’s theorem for nonconservative systems (Vujanovi? and Jones in: Variational Methods in Nonconservative Phenomena (monograph). Acad. Press, Boston, 1989).     

Applicability condition of time–temperature superposition principle (TTSP) to a multi-phase system

The applicability condition of the time–temperature superposition principle (TTSP) to a multi-phase system is analytically discussed assuming a mixture law. It was concluded that the TTSP does not hold for a multi-phase system in general but does hold for a multi-component system in which some components have the same temperature dependence and the others have no temperature dependence. On the basis of the results, the application of the TTSP to plant materials such as wood and bamboo was examined using a mixture law and a stretched-exponential function having a characteristic relaxation time τ ₀ and a stretching parameter β. Wood can be treated as a multi-phase system consisting of a framework (f) and matrix (m). In this case, it was expected that the TTSP holds for the matrix in the shorter time region t?τ _0f under T<T _gf , while the TTSP holds for the framework in the longer time region t?τ _0m under T>T _gm , where t and T _g is a measurement time and the glass transition temperature, respectively.     

Relationship Between System Failure Rate and Component Failure Rates

A simple sufficient condition is given for a system to have an increasing failure rate when the identical components comprising it have an increasing failure rate. Systems which function if and only if at least k of the n components function (“k out of n” systems) satisfy this condition. For systems of non-identical components, upper and lower bounds on failure rate are obtained in terms of component failure rates. These bounds are increasing functions of time for “k out of n” structures having components with increasing failure rates.     

Analysis for a two-dissimilar-component cold standby repairable system with repair priority

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. Assume that working time distributions and repair time distributions of the two components are both exponential, and Component 1 has repair priority when both components are broken down. After repair, Component 1 follows a geometric process repair while Component 2 obeys a perfect repair. Under these assumptions, using the perfect repair model, the geometric process repair model and the supplementary variable technique, we not only study some important reliability indices, but also consider a replacement policy T, under which the system is replaced when the working age of Component 1 reaches T. Our problem is to determine an optimal policy T^? such that the long-run average loss per unit time (i.e. average loss rate) of the system is minimized. The explicit expression for the average loss rate of the system is derived, and the corresponding optimal replacement policy T^? can be found numerically. Finally, a numerical example for replacement policy T is given to illustrate some theoretical results and the model's applicability.     

The up and down time distribution in a 1 out of n system with general life time and repair time distributions

In repairable systems with redundancy, failed units can be replaced by spare units in order to reduce system down time. The failed units are sent to a repair shop or manufacturer for maintenance and are subsequently returned for re-use.In this paper we consider a 1 out of n system with cold standby and we assume that repaired machines are as ‘good-as-new’. We derive an approximation method for the distribution of the up time and the down time of the system in case of a general failure distribution and a general repair time distribution. This allows the model to be used in a wide variety of applications.     

Modeling safety instrumented systems with MooN voting architectures addressing system reconfiguration for testing

This paper addresses the modeling of probability of dangerous failure on demand and spurious trip rate of safety instrumented systems that include MooN voting redundancies in their architecture. MooN systems are a special case of k-out-of-n systems. The first part of the article is devoted to the development of a time-dependent probability of dangerous failure on demand model with capability of handling MooN systems. The model is able to model explicitly common cause failure and diagnostic coverage, as well as different test frequencies and strategies. It includes quantification of both detected and undetected failures, and puts emphasis on the quantification of common cause failure to the system probability of dangerous failure on demand as an additional component. In order to be able to accommodate changes in testing strategies, special treatment is devoted to the analysis of system reconfiguration (including common cause failure) during test of one of its components, what is then included in the model. Another model for spurious trip rate is also analyzed and extended under the same methodology in order to empower it with similar capabilities. These two models are powerful enough, but at the same time simple, to be suitable for handling of dependability measures in multi-objective optimization of both system design and test strategies for safety instrumented systems. The level of modeling detail considered permits compliance with the requirements of the standard IEC 61508. The two models are applied to brief case studies to demonstrate their effectiveness. The results obtained demonstrated that the first model is adequate to quantify time-dependent PFD of MooN systems during different system states (i.e. full operation, test and repair) and different MooN configurations, which values are averaged to obtain the PFD_avg. Also, it was demonstrated that the second model is adequate to quantify STR including spurious trips induced by internal component failure and by test itself. Both models were tested for different architectures with 1≤N≤5 and 2≤M≤5 subject to uniform staggered test. The results obtained also showed the effects that modifying M and N has on both PFD_avg and STR, and also demonstrated the conflicting nature of these two measures with respect to one another.     

M out of n inspected systems subject to shocks in random environment

Consider a parallel redundant system, consisting n components, that is subject to shocks. The shocks cause the components to fail with certain probabilities. Shocks arrival rate and components’ failure probabilities may depend on an external Markovian environment. We consider warm and cold stand-by systems. Systems’ failures are silent. The system is maintained through inspection and repair/replacement. We propose several state-dependent maintenance policies and derive system availability and cost function.     

Lifetime and reliability estimation of repairable redundant system subject to periodic alternation

Lifetime distribution and reliability are analyzed for redundant systems consisting of units that alternate between operating and standby states periodically to inspect and detect failures of standby units. It is assumed that when a unit fails, a minimal repair with negligible repair time is performed. A cumulative exposure model is used to describe the failure time distribution. The method of maximum likelihood is used to estimate the parameters, and specific formulas for Weibull model are obtained. A simulation model is then developed to assess the system reliability based on the estimated lifetime distribution and the method is applied to a high pressure pump system.     

On the inspection policy of a two-component parallel system with failure interaction

In this paper we study a two-component standby system which can successfully operate upon a demand if at least one component is not failed. We assume that failures can be detected only by periodic inspections. We consider that the failure of one component can modify the (conditional) failure probability of the component still alive with probability p and do not interact with probability 1−p. For that failure interaction scheme we obtain the system reliability function for the case of staggered inspections. We compare staggered and non-staggered inspections through numerical examples considering constant hazard rates.     

Reliability analysis of standby systems with multi‐state elements subject to constant transition rates

Standby redundancy has been extensively applied to critical engineering systems to enhance system reliability. Researches on reliability evaluation for standby systems focus more on systems with binary‐state elements. However, multi‐state elements with different performances have played a significant role in engineering systems. This paper presents an approach for reliability analysis of standby systems composed of multi‐state elements with constant state transition rates and absorbing failure states. The approach allows modelling different standby systems beyond cold, warm and hot ones by taking into account differences in possible maintenance of elements in standby and operation modes and dependence of elements' operational behavior on their initial state at the time of activation. An iterative algorithm for reliability evaluation based on element state probabilities is suggested. Illustrating examples of evaluating reliability of different types of homogeneous and heterogeneous standby systems are demonstrated.     

Unavailability equations for k-out-of-n systems

This paper updates and extends earlier published analytical results for the unavailabilities (probabilities of failure on random demand) of redundant standby systems with k-out-of-n logic. Such systems are mostly used in safety and protection systems and are subject to latent and detectable failures. Latent failures are detected by periodic tests and repaired immediately after discovery. Many potential failure and error modes are included in the formalism. Interesting relations between two approaches are pointed out. Both consecutive and staggered testing schemes are evaluated, and methods for including certain common cause failures in the analyses are pointed out.     

Reliability analysis of a complex standby redundant systems

In any redundant system, the state of the standby unit is usually taken to be hot, warm or cold. In this paper, we present a new model of a two unit standby system wherein the standby unit is put in cold state for a certain amount of time before it is allowed to become warm. Upon failure of the online unit, the standby unit, if in warm state, instantaneously starts operating online; if it is in cold state, an emergency switching is made which takes it to warm state (and hence online) either instantaneously or non-instantaneously—each with some probability; if it is under repair, the system breaks down. Assuming all the associated distributions to be general except that of the life time of the standby unit in the warm state, various reliability characteristics that are of interest to reliability engineers and system designers are derived. A comprehensive cost function is also constructed and is then optimized with respect to three different control parameters numerically. In addition numerical results are presented to illustrate the behaviour of the various reliability charcateristics derived.     

Competition between cracking and peeling in composites: a simulation in 2-d

When a material coating is left to dry on a substrate, cracking and peeling may set in which leads to the ultimate degeneration of the coating. Parameters like the elastic properties of the constituents of the coating, desiccation rate, role of substrate and the volume fraction of the constituent components of the coating affect the failure mechanism. We simulate desiccation of a 2-D vertical section of a layer of a two component (A,B) composite on a substrate to study the dependence of cracking and peeling on these parameters. For a given composite and substrate, it is observed that cracking or peeling depends only on the percentage of the constituents present in the composite. In our simulation, p is the volume fraction of the constituent having greater cohesion (B). The characteristic time of complete failure t _crack is found to increase linearly with decrease in desiccation rate. This rate of increase in t _crack is the highest for peeling and least for horizontal cracking. When adhesion between substrate and layer is stronger than the cohesion (BB) and adhesion (AB) of the constituents of the composite, peeling is least favoured. Failure by horizontal cracking gives way to vertical cracking as the percentage of the stronger component (B) increases. If adhesion between substrate and layer is weaker than that between the constituent components of the composite, peeling is dominant. Mechanical strength is found to be always greater for a composite than a pure material. If one component has low cohesive strength (AA), the strength of the composite can be greatly increased if the other component (B) is chosen such that adhesion (AB) becomes comparable to cohesion (BB). For certain combinations of the components, the composite is found to completely resist failure.     

Roughness evolution in Ga doped ZnO films deposited by pulsed laser deposition

We analyze the morphology evolution of the Ga doped ZnO(GZO) films deposited on quartz substrates by a laser deposition system. The surface morphologies of the film samples grown with different times are measured by the atomic force microscope, and they are analyzed quantitatively by using the image data. In the initial stage of the growth time shorter than 8 min, our analysis shows that the GZO surface morphologies are influenced by such factors as the random fluctuations, the smoothening effects in the deposition, the lateral strain and the substrate. The interface width uw(t) and the lateral correlation length ξ(t) at first decrease with deposition time t. For the growth time larger than 8 min, w(t) and ξ(t) increase with time and it indicates the roughening of the surface and the surface morphology exhibits the fractal characteristics. By fitting data of the roughness w(t) versus deposition time t larger than 4 min to the power-law function, we obtain the growth exponent β is 0.3; and by the height-height correlation functions of the samples to that of the self-affine fractal model, we obtain the value of roughness exponent α about 0.84 for all samples with different growth time t.     

Cyclic crack tip deformation and its relation to fatigue crack growth

Characteristics and mechanism of fatigue crack growth in mild steel have been investigated taking account of crack tip deformations: crack tip opening displacement δt and the size of highly deformed zone ahead of crack tip Rx0.2 (the size of the zone with accumulated strain above 0.2). δt was measured directly at midsection of the specimen with a profile projector. Rx0.2 was obtained from strain distribution ahead of crack tip determined by the use of the recrystallizalion phenomenon.It is revealed that crack growth rates ranging from 0.02 μm/cycle to 200 μm/cycle are expressed well by a second power function of both δt and Rx0.2. Abrupt increases in μt and Rx0.2 occured at a growth rate of about 1 μm/cycle. At this growth rate, fracture appearance changed from striation to dimple. These transitions are due to the transition of stress state. It is also shown that fatigue fracture strain is constant independently of crack growth rate and is equal to the ductile fracture strain in monotonic loading. The constant fracture strain is the criterion for fatigue failure of ductile steels.     

Analysis of on-line maintenance strategies for -out-of- standby safety systems

The objective of this paper is to compare the performance of three on-line test and maintenance strategies (corrective maintenance, preventive maintenance and predictive maintenance) for standby k-out-of-n safety systems. Each channel of the k-out-of-n system is modelled by an age-dependent unavailability model to reflect the effect of maintenance on the aging process. The system unavailability, the probability of spurious operation and the overall cost under the above maintenance strategies are analyzed and compared to obtain the optimal maintenance strategy. Sensitivity analyses are performed to reveal the effect of different model parameters on the system performance. A standby safety system in Canadian Deuterium–Uranium (CANDU) Nuclear Power Plants (NPPs), the Shutdown System Number One (SDS1), is used to illustrate the proposed analysis and the procedure. It is concluded that maintenance should neither be performed too frequently nor too rarely. When the system deteriorates very slowly, the corrective maintenance is more preferable than the preventive and predictive maintenance. When the failure rate of the system is high, the preventive maintenance results in the best system performance.