Constant Repair Rates Research Articles

Reliability and availability models for ageing safety-related systems

This paper proposes reliability and availability models for ageing systems subject to failures detected on-line and (hidden) failures only revealed by proof tests, such as safety-related systems. Both types of failures are modelled by Weibull distributions with three parameters: shape, scale, and (virtual) age. For on-line detected failures (e.g. self-revealed or quickly detected by frequent automatic diagnostic tests), a corrective maintenance is assumed with a constant repair rate. For the other failures, a preventive maintenance consists in periodic proof tests that reveal the failures and, if applicable, a corrective maintenance is also performed. While the elements are assumed as bad as old (ABAO) after corrective maintenance, arithmetic reduction of age (ARA) with memory is considered for preventive maintenance. Analytical formulas are proposed for the time-dependent (virtual) age, failure rate, and unavailability of an element, for each type of failure. These formulas are implemented in a fault tree analysis tool for application to different architectures of systems with elements subject to different types of failures. For safety-related systems, results show that the use of the proposed models may prevent underestimating the unavailability of ageing systems (and therefore the risks) and enables definition of more appropriate maintenance policies.

Finite difference solution to stochastic partial differential equations in reliability

The paper presents a numerical method to solve for the reliability measures of a Markovian system. The model is obtained from the field of reliability engineering for systems having constant failure and repair rates. Generally, the steady state behaviour of the system is studied due to some constraint on obtaining the transition state solution. The proposed method helps to govern the transition state probability values, by utilising a finite difference method in concurrence with the results of integral appearing in stochastic differential equation obtained using the supplementary variable technique. Results thus achieved are found to be effective for studying the transition state behaviour of the system.

Availability evaluation of manufacturing systems using Semi-Markov model

Availability is a key measure of performance of systems in general and manufacturing systems are no exception. The elements of complex manufacturing systems such as Computer Integrated Manufacturing (CIM) follow variable, i.e., decreasing or increasing failure and repair rates. This creates difficulty in their availability evaluation. Conventional methods, which predict availability based on past experience, are inadequate for such systems. Existing analytical or simulation methods provide inaccurate availability estimate as these assume constant failure and repair rates, which follow exponential distribution. To address this concern, this paper suggests a methodology for availability evaluation of manufacturing systems using Semi-Markov model, which considers variable failure or repair rates. This is carried out by first understanding the system structure by identifying its elements; subsystem, assembly and component at its hierarchical levels and subsequently identifying their states; failed, operating, etc. Based on this, Semi-Markov models are developed for each element at various hierarchical levels of the system. The unique feature of this approach is that it does assume Weibull distribution for variable failure times and lognormal for variable repair times. The developed models are solved using an analytical solution method to obtain steady-state probabilities of their states and hence, the system steady-state availability. The methodology is illustrated by means of a case study conducted on a Vertical Machining Center (VMC) that is used in an automotive production plant.

On determination of the preventive maintenance interval guaranteeing system availability under a periodic maintenance policy

In order for an engineering system to maintain a certain level of availability, preventive maintenance actions are commonly performed. This paper deals with the analysis for determining the optimum frequency to perform preventive maintenance actions for a system exhibiting a linearly increasing hazard rate and a constant repair rate. Under a periodic maintenance policy, as long as a predetermined availability is guaranteed, the time interval between preventive maintenance actions incurring the minimal maintenance cost is selected. Considering both the maintenance cost and system availability, guidelines for determining the maintenance time interval are presented. Simple examples are also provided to illustrate these ideas.

An Availability System with General Repair Distribution: Statistical Inference

This article we study the statistical inferences of an availability system with imperfect coverage. The time-to-failure and time-to-repair of the active and standby components are assumed to be exponential and general distribution, respectively. Assume that the coverage factor is the same for an active-component failure as that for a standby-component failure. Firstly, we propose a consistent and asymptotically normal (CAN) estimator of availability for such repairable system. Based on the CAN estimator of the system availability, interval estimation and testing (hypothesis) are performed. To implement the simulation inference for the system availability, we adopt two repair-time distributions, such as lognormal and Weibull distribution, in which three types of Weibull distribution are considered according to the shape parameter β. The component holds the decreasing repair rate (DRR), constant repair rate (CRR), and increasing repair rate (IRR) if β < 1, β = 1, and β > 1, respectively. Finally, all simulation results are displayed by appropriate tables and curves for understanding performance of the statistical inference procedures presented in this article.

SOFTWARE FAILURE INTENSITY, RELIABILITY AND OPTIMAL STOPPING TIME INCORPORATING REPAIR POLICIES

Reliability of a software application, its failure intensity and the residual number of faults are three important metrics that provide a quantitative assessment of the failure characteristics of an application. Ultimately, it is also necessary, based on these metrics, to determine an optimal release time at which costs justify the stop test decision. Typically, one of the many stochastic models known as software reliability growth models (SRGMs) is used to characterize the failure behavior of an application to provide estimates of the failure intensity, residual number of faults, reliability, and optimal release time and cost. To ensure analytical tractability, SRGMs assume instantaneous repair and thus the estimates of these metrics obtained using SRGMs tend to be optimistic. In practice, repair activity consumes a non trivial amount of time and resources. Also, repair may be conducted according to many policies which reflect the schedule and budget constraints of a project. A few efforts which incorporate repair into SRGMs are restrictive, since they consider only some SRGMs, model the repair process using a constant repair rate, and provide an estimate of only the residual number of faults. These efforts do not address the issue of estimating the failure intensity, reliability and optimal release time and cost in the presence of repair. In this paper we present a generic framework based on the rate-based simulation technique to incorporate repair policies into finite failure non homogeneous Poisson process (NHPP) class of SRGMs. We describe a methodology to compute the failure intensity and reliability in the presence of repair, and apply it to four popular finite failure NHPP models. We also present an economic cost model which considers explicit repair in providing estimates of optimal release time and cost. We illustrate the potential of the framework to quantify the impact of the parameters of the repair policies on the above metrics using examples. Through these examples we discuss how the framework could be used to guide the allocation of resources to achieve the desired reliability target in a cost-effective manner.

Optimization of the preventive maintenance plan of a series components system

One of the most critical problems in preventive maintenance is the determination of the optimum frequency to perform preventive maintenance in equipment, in order to ensure its availability. In this paper, we propose an algorithm to solve the previous problem for equipment that exhibit linearly increasing hazard rate and constant repair rate. Based on this algorithm, we have developed another one to solve the problem of maintenance management of a series system based on preventive maintenance over the different system components. We assume that all components of the system still exhibit linearly increasing hazard rate and constant repair rate and that preventive maintenance would bring the system to the as good as new condition. We define a cost function for maintenance tasks (preventive and corrective) for the system. The algorithm calculates the interval of time between preventive maintenance actions for each component, minimizing the costs, and in such a way that the total downtime, in a certain period of time, does not exceed a predetermined value.

Interaction of UV Radiation and Inorganic Carbon Supply in the Inhibition of Photosynthesis: Spectral and Temporal Responses of Two Marine Picoplankters¶

ABSTRACTThe effect of ultraviolet radiation (UVR) on inhibition of photosynthesis was studied in two species of marine picoplankton with different carbon concentration mechanisms: Nannochloropsis gaditana Lubián possesses a bicarbonate uptake system and Nannochloris atomus Butcher a CO2 active transport system. Biological weighting functions (BWFs) for inhibition of photosynthesis by UVR and photosynthesis vs irradiance (PI) curves for photosynthetically active radiation (PAR) were estimated for both species grown with an enriched CO2 supply (high dissolved inorganic carbon [DIC]: 1% CO2 in air) and in atmospheric CO2 levels (low DIC: 0.03% CO2). The response to UVR and PAR exposures was different in each species depending on the DIC treatment. Under PAR exposure, rates of maximum photosynthesis were similar between treatments in N. gaditana. However, the cultures growing in high DIC had lower sensitivity to UVR than the low DIC cultures. In contrast, N. atomus had higher rates of photosynthesis under PAR exposure with high DIC, but the BWFs were not significantly different between treatments. The results suggest that one or more processes in N. gaditana associated with HCO3− transport are target(s) for UV photodamage because there was relatively less UV inhibition of the high DIC‐grown cultures in which inorganic carbon fixation is supplied by passive CO2 diffusion. Time courses of photochemical efficiency in PAR, during UV exposure and during subsequent recovery in PAR, were determined using a pulse amplitude modulated fluorometer. The results were consistent with the BWFs. In all time courses, a steady state was obtained after an initial decrease, consistent with a dynamic balance between damage and repair as found for other phytoplankton. However, the relationship of response to exposure showed a steep decline in activity that is consistent with a constant rate of repair. A novel feature of a model developed from a constant repair rate is an explicit threshold for photosynthetic response to UV.

Optimal control for a class of manufacturing systems with production rate dependent failure rates

In this paper we consider a failure prone, single part type, single machine manufacturing system. The failure repair process of the machine is characterized by a non-homogeneous Markov chain, with a constant repair rate and a failure rate which is a non decreasing function qd(u) of the production rate. The stability issue is discussed with respect to the failure rate function considered. For a piecewise constant, two value function qd(u), the optimal policy is characterized and it is completely derived for a set of examples. This optimal policy is not a hedging point policy, which is known to be optimal only for particular failure rate functions qd(u).

Interaction of UV-Radiation and Inorganic Carbon Supply in the Inhibition of Photosynthesis: Spectral and Temporal Responses of Two Marine Picoplankters

The effect of ultraviolet radiation (UVR) on inhibition of photosynthesis was studied in two species of marine picoplankton with different carbon concentration mechanisms: Nannochloropsis gaditana Lubian possesses a bicarbonate uptake system and Nannochloris atomus Butcher a CO2 active transport system. Biological weighting functions (BWFs) for inhibition of photosynthesis by UVR and photosynthesis vs irradiance (PI) curves for photosynthetically active radiation (PAR) were estimated for both species grown with an enriched CO2 supply (high dissolved inorganic carbon [DIC]: 1% CO2 in air) and in atmospheric CO2 levels (low DIC: 0.03% CO2). The response to UVR and PAR exposures was different in each species depending on the DIC treatment. Under PAR exposure, rates of maximum photosynthesis were similar between treatments in N. gaditana. However, the cultures growing in high DIC had lower sensitivity to UVR than the low DIC cultures. In contrast, N. atomus had higher rates of photosynthesis under PAR exposure with high DIC, but the BWFs were not significantly different between treatments. The results suggest that one or more processes in N. gaditana associated with HCO3- transport are target(s) for UV photodamage because there was relatively less UV inhibition of the high DIC-grown cultures in which inorganic carbon fixation is supplied by passive CO2 diffusion. Time courses of photochemical efficiency in PAR, during UV exposure and during subsequent recovery in PAR, were determined using a pulse amplitude modulated fluorometer. The results were consistent with the BWFs. In all time courses, a steady state was obtained after an initial decrease, consistent with a dynamic balance between damage and repair as found for other phytoplankton. However, the relationship of response to exposure showed a steep decline in activity that is consistent with a constant rate of repair. A novel feature of a model developed from a constant repair rate is an explicit threshold for photosynthetic response to UV.

Steady state behaviour and maintenance planning of a desulphurization system in a urea fertilizer plant

This paper presents the steady state behaviour and maintenance planning of the desulphurization process in the fertilizer industry. The process consists of four subsystems, A, B, C and D in series with three states; good, reduced and failed. One standby unit is provided for each pump. Taking constant failure and repair rates for each subsystem, mathematical modelling is done using the Chapman-Kolmogorov birth-death process. An expression for steady state availability is given. Based on the data available from a medium sized ammonia production process, the behaviour of each working unit in the process has been analysed. The computed results are discussed with the concerned plant personnel which is helpful to the management for implementing any future plan regarding design modification of the system/processes.

Some large availability models: computation and bounds

A dynamic analytic solution is described for a 2/sup N/ state general availability model with N components having constant failure and repair rates. From this model, a family of models is developed using truncation and/or attenuation of transition rates. Expressions are derived for steady-state solutions. Then spread-sheet programs are: (1) given for obtaining these solutions, and (2) compared with BASIC programs yielding the same results. State probabilities of these truncation and level-attenuation models are either greater than or less than comparable states in the general model. Thus the states of the general model become either lower bounds or upper bounds for states in these two model types. Other bounds can be constructed from single exponentials based on steady-state probabilities. From this family of models, bounds should exist on state probabilities in models of similar structure but different constraints on failure and repair rates. A specific model is pursued where failures are restricted to any 2 components; and the failure rate of one component is assumed to change on second level of failure. Under these conditions, dynamic bounds on state-probabilities of the initial-state and some, but not all, steady state bounds on the other state probabilities can be found. Examples illustrate various bounds.

Maximizing mean-time to failure in k-resilient systems with repair

A k-resilient system with N components can tolerate up to k component failures and still function correctly. We consider k-resilient systems where the number of component failures is a constant fraction of the total number of components, that is k=N/c and c is a constant such that 2/spl les/c</spl infin/. Under a Markovian assumption of constant failure and repair rates, we compute the system size N/sub max/ at which the mean-time to failure (MTTF) for such a system is maximized. Our results indicate that N/sub max/ can be expressed in terms of constant c and parameter /spl rho/ as N/sub max/=K(c,/spl rho/)//spl rho/, where /spl rho/=/spl lambda///spl mu/ and K(c, /spl rho/) is a function of c,/spl rho/. In addition, we have found that the variation of N/sub max/ over the whole range of c is remarkably small, and as a result, even if the resilience k of a system as a function of N varies widely, the system size at which the MTTF is maximized is within the range 0.36//spl rho/ and 0.5//spl rho/. We validate our results through event-driven simulation, and, in addition, examine the behavior of systems with Weibull distributed failure times.

Reliability of imperfect switching of cold stanby systems with multiple non-critical and critical errors

Reliability and availability analysis of having k active, N cold standby units with repair facilities and multiple non-critical and critical errors while the switching mechanism subjected to failure is presented. Failed (active and/or by any one of the multiple non-critical errors) units will be repaired at a constant repair rate. The system is in a failed state when any one of the multiple critical errors has occurred, ( N + 1) units have failed or there is a failure of switching mechanism. A failed system will be repaired with repair times arbitrarily distributed. The expressions for reliability and steady-state availability are given.

Stochastic analysis of k-out-of-N:G redundant systems with repair and multiple critical and non-critical errors

Probabilistic analysis of k-out-of- N:G redundant systems with repair facilities and multiple critical and non-critical errors is presented. Failed unit (active and/or by any one of the multiple non-critical errors) will be repaired with the same constant repair rate. The system is in a failed state when any one of the multiple critical errors has occurred or ( N − k + 1) units have failed. Failed system will be repaired with repair times arbitrarily distributed. The formulas for reliability and steady-state availability are given.

Reliability and availability analysis of cold standby system with repair and multiple non-critical and critical errors

Reliability and availability analysis of systems having k active, N cold standby identical units with repair facilities and multiple non-critical and critical errors is presented. Failed unit (active and/or any one of the multiple non-critical errors) will be repaired with same constant repair rate. The system is in a failed state when any one of the multiple critical errors has occurred or when (N+1) units (active and/or any one of the multiple non-critical errors) have failed. Failed system will be repaired with repair times which are arbitrarily distributed. The formulas for reliability and steady-state availability are given.

Reliability analysis of a cold standby system with multiple critical errors

A reliability and availability analysis of k active and N cold standby identical units of r repair facilities with multiple critical errors is presented. The system is in a failed system when (N+1) units have failed or any one of the multiple errors has occurred. Failed units will be repaired with constant repair rate. Failed system will also be repaired with same repair rate. The expressions for reliability, availability and steady-state availability are given.

Reliability and availability analysis of a k-out-of-N:G redundant system with repair in the presence of chance of multiple critical errors

The paper presents a reliability and availability analysis of a k-out-of-N:G redundant system with repair facilities in the presence of chance of multiple critical errors. The system is in a failed state when N−k+1 units have failed or any one of the multiple critical errors has occurred. Failed units and failed system will be repaired with constant repair rate to state with N−k+1 failed units. Laplace transforms of the state probabilities, the reliability and the availability of the system are derived. The system steady-state availability is also given.

Stochastic analysis of a warm standby system with repair in the presence of chance of multiple critical errors

Abstract The paper presents a stochastic analysis of a k active and N warm standby system with r repair facilities in the presence of chance of multiple critical errors. The system is in a failed state when N+1 (active and/or warm) units have failed or any one of the multiple critical errors has occurred. Failed units will be repaired with constant repair rate. The failed system due to critical errors will be repaired with constant repair rate to state with N+1 failed units. Laplace transforms of the reliability and the availability and the steady-state availability of the system are given.

Reliability and availability analysis of warm standby systems with repair and multiple critical errors

Abstract Reliability and availability analysis of systems having k active, N warm standby units with repair facilities and multiple critical errors is presented. Failed (active and/or warm standby) unit will be repaired with same constant repair rate. The system is in a failed state when any one of the multiple critical errors has occurred or when (N+1) units have failed. Failed system will be repaired with repair times arbitrarily distributed. The formulas for reliability and steady-state availability are given.