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Fault diagnosis with imperfect tests
World Abstracts on Microelectronics and Reliability solutions can be derived in a much simpler and more efficient manner. This results in a resource allocation problem that can be solved by efficient techniques. The major assumption is that the original problem has only one associated constraining equation. However, this approach can be extended to problems with multiple constraints by applying the procedure in an iterative fashion within a penalty function framework. This approach applies to any system for which the reliability can be calculated. Each logical stage of the system under study can incorporate redundancy via a parallel arrangement, standby, or k-out-of-n :G.
Optimum replacement policies for a system composed of components. SHIGERU YAMADA and SHUNJI OSAKI. IEEE Trans. Reliab. R-3fi(3), 278 (1981). Most systems are composed of components which are classified into two groups from a viewpoint of system life. In one group, component replacement is made on failure or malfunction of a component. This group generally includes those components which are cheaper and are not essential to the system. The second group includes those components which are essential to the system or are more expensive, those for which we replace the whole system, either periodically, or at failure. The former are called nonessential units and the latter are called essential units. Assuming arbitrary failure distributions of nonessential and essential units, we analyze age and block replacement problems for the essential unit. We derive optimum replacement policies which minimize the total sexpected cost for replacement. Based on these policies, we also analyze an initial provisioning problem for non-essential units. Numerical examples illustrate these results.
Confidence estimates of reliability for complex systems. N. K. DEBNATH and K. SANTHAMMA.QR J. (India), 85 (May 1981). The Monte Carlo simulation is used for a complex system which comprises of electronic as well as non-electronic subsystems, for making a reliability estimate, in this paper, using a computer programme. A summary of optimal ordering policies. SHUNJI OSAKI, NAOTO KAIO and SHIGERU YAMADA. IEEE Trans. Reliab. R-30(3), 272 (1981). We consider a generalized ordering policy in which, prior to failure, we order a spare unit which can be delivered after a constant lead time. We are interested in the optimal ordering policy which minimizes the sexpected total discounted cost. The general ordering policy is reduced to two typical ordering policies, which are discussed thoroughly. We also discuss a similar ordering policy with minimal repair.
Exact statistical tolerance limits for sample variances from non-normal populations. H. R. SINGH and GAURI SHANKAR. QR J. (India), 93 (May 1981). Exact statistical tolerance limits on the population of sample variances and standard deviations arising from a normal population (/~,a 2) were derived by Tietjen and Johnson. They concluded that 3a limits are a bit too large as compared to the exact limits. Here the exact tolerance limits for a non-normal population represented by the first four terms of an Edgeworth Series have been derived. Tietjen and Johnson's conclusion is found true provided the form of the probability distribution is correctly specified, the condition not examined by them.
Quality deficiency: diagnosis and remedy--a case study. B. S. PRABHAKAR,R. SUBRAMANIANand A. K. CHAUDHURY.QR J. (India), 81 (May 1981). The paper reports work on an electrochemical instrument with defectives, in which stepwise pursuit with frequency and cause effect diagrams, the reasons were identified. By an experiment through the special device of mixup of "good" and "bad" instruments and analysis of vardance, correctives were established, whose implementation gave highly satisfactory results, preventing the need to change design. M.R.22/5--t
1041
An experimental comparison of the heuristic methods for solving reliability optimization problems. YuJl NAKAGAWA and SATOSHI MIYAZAKI. IEEE Trans. Reliab. R-30(2), 181 (1981). This paper compares three heuristic methods (Nakagawa - Nakashima, Gopal - Aggarwal - Gupta, Sharma-Venkateswaran) for solving reliability optimization problems with nonlinear constraints. Their effectiveness, measured in terms of computation time, optimality rate, and relative error, is evaluated on several sets of randomly generated test problems with nonlinear constraints for series systems. A FORTRAN program for generating the test problems and best known solutions (some of them are exact optimal) are included.
Surrogate constraints algorithm for reliability optimization problems with two constraints. YuJl NAKAGAWAand SATOSHI MIYAZAKI. IEEE Trans. Reliab. R-30(2), 175 (1981). This paper presents an algorithm (called the N & M algorithm) for solving pure-integer separable nonlinear programming problems with two constraints. The N & M algorithm uses surrogate constraints and solves a given problem by effectively generating values of surrogate multiplier (i.e., surrogate problems) and solving the surrogate" problems. Examples show that the N & M algorithm is superior to dynamic programming using the Lagrange multiplier in terms of number of iterations and narrowness of gap.
A generalized block-replacement policy. FRANK BEICHELT. I E E E Trans. Reliab. R-30(2), 171 (1981). T w o modes of system failure are possible: Mode 1 is removed by minimal repair, mode 2 by replacement. A generalized block replacement policy is introduced. The long-run cost rate is given. Integral equations of the renewal type for the basic reliability expressions are derived. A numerical example is presented. Availability of a 2-component dependent system. KAZU OKUMOTO. IEEE Trans. Reliab. R-30(2), 205 (1981). This paper investigates the effect of s-dependence among components on the availability of a 2-component repairable system with bivariate exponential failure time and repair time. Expressions are derived for state occupancy probabilities as well as system availability.
Parametric analysis of 2-unit redundant computer systems with corrective and preventive maintenance. JEAN-CLAUDE LAPRIE, ALAIN COSTES and CHRISTIAN LANDRAULT. IEEE Trans. Reliab. R-30(2), 139 (1981). This paper shows the span of results which can he obtained by modeling a system's behavior by stochastic processes and demonstrates practical rules for employing Markov and semi-Markov models. The introduction summarizes several methods for reliability analysis and gives the advantages and drawbacks of four methods: Markov processes, semi-Markov processes, supplementary variables, the method of stages. The remainder deals with reliability and availability modeling of a 2-unit redundant computer system. There are (a) two types of maintenance: corrective (c.m.) and preventive (p.m.), and (b) two system parameters: coverage, and an increased failure rate when one unit is under repair or inspection. Approximate expressions for reliability, mean time to failure, and asymptotic availability show the effects of the system parameters as well as of the shapes of the Cdrs of the times related to maintenance actions. For c.m., Markov modeling is a good approximation. For p.m., Markov modeling is a rough approximation; one can go to semi-Markov models or to the method of stages. Lastly, an approximate expression is given for the mean inspection interval which maximizes reliability and availability for p.m.
Fault diagnosis with imperfect tests. THEODORE J. SHESKIN. IEEE Trans. Reliab. R-30 (2), 156 (1981). This paper derives a minimum s-expected cost sequence of built-in-tests (BITs)
1042
World Abstracts on Microelectronics and Reliability
which will partition modular equipment into mutually exclusive groups of modules. After a fault in the equipment, one of these groups will be identified by a BIT diagnostic subsystem as the group which contains a faulty module. The BITs are imperfect in the sense only that they might not detect all of the possible faults in the equipment; they are perfect in the sense that fault indications are never false. The proportion of faults detectable by each BIT is known. Both the cost of a BIT and the probability that a BIT will pass or fail are functions of which modules are tested. A recursive algorithm is developed which determines a sequence of BITs with a m i n i m u m s-expected life-cycle cost. The recursive algorithm is applied to a 4-element numerical example. The algorithm has neither been proved nor implemented for a computer.
IEEE Trans. Reliab. R-30(2), 189 (1981). A scheme using multiple redundant computing devices with a majority voter improves the reliability of the output of the computing device. This paper analyzes some modular redundant systems with majority voters. An iterative structure is introduced for voters in order to improve the total reliability. The reliability of voters is introduced in several ways, Two models of imperfect voters are discussed in detail, that is, output-imperfect model and semiperfect model. The former is suitable for the case where voters are treated in the same manner as other modules. The latter is appropriate for the case where the characteristics of the voter are taken into account. Three theorems show the existence of optimal majorityvoted logic circuit for a given value of the reliability of each module under some reasonable assumptions.
Optimal replication for a specific system objective. J. M. GPOFFITH and D. M. RASMUSON. IEEE Trans. Reliab. R30 (2), 133 (1981). This paper defines component replication as the concept of using two or more i.i.d, components at a specific location in a system design. Both parallel and series physical connections are used to place replicate components into a system. The choice of parallel or series depends upon why the components are being made replicate and is complicated by the fact that a component might be used for several functions. A given physical connection can be logically in series for some situations and logically in parallel for others. Reducing system unavailability is one important use of component replication. The mathematical models relating replication and unavailability are often complex, and casual inspection does not reveal which components should be replicated to reduce unavailability. This paper develops a method for determining which components can be replicated (and how) for the greatest unavailability reduction. The method was used to study three system designs. For each design it was possible to identify several components which could be replicated with an associated reduction in unavailability at reasonable cost. In addition it was possible to demonstrate or confirm that replicating most components does not reduce unavailability. Applying this method helps the designer develop insight concerning the role of replication (in a system design) without complex calculations and interpretations.
Two repairable multistate devices with general repair-time distributions. MITSUO YAMASHIRO.IEEE Trans. Reliab. R-30 (2), 204 (1981). The system has two parallel redundant multistate devices with general repair-time distributions. The Laplace transforms of state probability and the mean time to system failure (MTSF) are obtained and a particular case is considered.
Bivariate survival model derived from a Weibull distribution. JOHN D. SPURRIER and D. R. WEIER. IEEE Trans. Reliab. R-30 (2), 194 (1981). A bivariate survival model is based on an underlying Weibull distribution and extends a bivariate exponential model considered by Freund. The model is motivated by a 2-component system which can function even if one of the components has failed. The components initially have a workload (inverse scale parameter) proportional to 2. U p o n the failure of one component, the workload of the remaining component becomes proportional to 02, where 0 > 0. The parameter 0 describes the a m o u n t of support or antagonism between the two components. The joint pdf of the first failure time and the time between the first and second failures is derived. The likelihood equation achieves its m a x i m u m in the interior of the parameter space, but the estimators do not have a closed form. A simulation study was performed to evaluate the performance of the m a x i m u m likelihood estimators.
Reliability analysis and optimal redundancy for majorityvoted logic circuits. HISASHI MINE and KAZUMI HATAYAMA.
Modified periodic replacement with minimal repair at failure. TOSHIO NAKAGAWA.IEEE Trans. Reliab. R-30 (2), 165 (1981). This paper summarizes four models of modified periodic replacement with minimal repair at failures when the scheduled replacement time is specified. If a failure occurs just before the replacement time, then three models are: (A) a unit remains as it is until the replacement time comes, (B) a unit is replaced by one of spares, (C) a unit is replaced by a new unit. Ira failure occurs well before replacement time then model D is: a unit is replaced at failure or at time T~. The sexpected costs for each model are obtained and the optimum policies are discussed.
Calculation of age-replacement with Weibull failure times. TOSHIO NAKAGAWAand KAZUMI YASUI. IEEE Trans, Reliab, R-30 (2), 163 (1981). An age-replacement policy with Weibull failure times is considered. It is troublesome to compute an optimum replacement time numerically, Upper and lower bounds of an optimum time are given in simple terms of replacement costs and parameters of a Weibull distribution. A numerical example shows that the approximation can be used when the ratio of the replacement cost for a failed unit to that for a non-failed unit is large. The approximation is best when the optimum age of replacement is small.
Analysis of performance-degradation data from accelerated tests. WAYNE NELSON. IEEE Trans. Reliab. R-30(2), 149 (1981). The performance of many products degrades as the product ages. Such degradation is usually slow but can be accelerated by a high "'stress". For example, the breakdown strength of electrical insulation depends on age and temperature. In some tests, the performance of a test unit is measured only once at a chosen age. For example, an insulation specimen yields only one breakdown measurement. A model and analyses for such data are described :.n this tutorial article. Also described is a method for estimating the distribution of time to failure, defined to be the age when performance degrades below a specified level.
Likelihood function of a debugging model for computer software reliability. BEy LITTLEWOODand JOHN L. VERRALL. IEEE Trans. Reliab. R-30(2), 145 (1981). A simple model for software reliability growth, originally suggested by Jelinski and Moranda, has been widely used but suffers from difficulties associated with parameter estimation. We show that a major reason for obtaining nonsensical results from the model is its application to data sets which exhibit decreasing reliability. We present a simple, necessary and sufficient condition for the m a x i m u m likelihood estimates to be finite and suggest that this condition be tested prior to using the model.
An efficient computational technique for evaluating the cut/tie sets and common-cause failures of complex systems. R. N.
Optimum replacement policies for a system composed of components. SHIGERU YAMADA and SHUNJI OSAKI. IEEE Trans. Reliab. R-3fi(3), 278 (1981). Most systems are composed of components which are classified into two groups from a viewpoint of system life. In one group, component replacement is made on failure or malfunction of a component. This group generally includes those components which are cheaper and are not essential to the system. The second group includes those components which are essential to the system or are more expensive, those for which we replace the whole system, either periodically, or at failure. The former are called nonessential units and the latter are called essential units. Assuming arbitrary failure distributions of nonessential and essential units, we analyze age and block replacement problems for the essential unit. We derive optimum replacement policies which minimize the total sexpected cost for replacement. Based on these policies, we also analyze an initial provisioning problem for non-essential units. Numerical examples illustrate these results.
Confidence estimates of reliability for complex systems. N. K. DEBNATH and K. SANTHAMMA.QR J. (India), 85 (May 1981). The Monte Carlo simulation is used for a complex system which comprises of electronic as well as non-electronic subsystems, for making a reliability estimate, in this paper, using a computer programme. A summary of optimal ordering policies. SHUNJI OSAKI, NAOTO KAIO and SHIGERU YAMADA. IEEE Trans. Reliab. R-30(3), 272 (1981). We consider a generalized ordering policy in which, prior to failure, we order a spare unit which can be delivered after a constant lead time. We are interested in the optimal ordering policy which minimizes the sexpected total discounted cost. The general ordering policy is reduced to two typical ordering policies, which are discussed thoroughly. We also discuss a similar ordering policy with minimal repair.
Exact statistical tolerance limits for sample variances from non-normal populations. H. R. SINGH and GAURI SHANKAR. QR J. (India), 93 (May 1981). Exact statistical tolerance limits on the population of sample variances and standard deviations arising from a normal population (/~,a 2) were derived by Tietjen and Johnson. They concluded that 3a limits are a bit too large as compared to the exact limits. Here the exact tolerance limits for a non-normal population represented by the first four terms of an Edgeworth Series have been derived. Tietjen and Johnson's conclusion is found true provided the form of the probability distribution is correctly specified, the condition not examined by them.
Quality deficiency: diagnosis and remedy--a case study. B. S. PRABHAKAR,R. SUBRAMANIANand A. K. CHAUDHURY.QR J. (India), 81 (May 1981). The paper reports work on an electrochemical instrument with defectives, in which stepwise pursuit with frequency and cause effect diagrams, the reasons were identified. By an experiment through the special device of mixup of "good" and "bad" instruments and analysis of vardance, correctives were established, whose implementation gave highly satisfactory results, preventing the need to change design. M.R.22/5--t
1041
An experimental comparison of the heuristic methods for solving reliability optimization problems. YuJl NAKAGAWA and SATOSHI MIYAZAKI. IEEE Trans. Reliab. R-30(2), 181 (1981). This paper compares three heuristic methods (Nakagawa - Nakashima, Gopal - Aggarwal - Gupta, Sharma-Venkateswaran) for solving reliability optimization problems with nonlinear constraints. Their effectiveness, measured in terms of computation time, optimality rate, and relative error, is evaluated on several sets of randomly generated test problems with nonlinear constraints for series systems. A FORTRAN program for generating the test problems and best known solutions (some of them are exact optimal) are included.
Surrogate constraints algorithm for reliability optimization problems with two constraints. YuJl NAKAGAWAand SATOSHI MIYAZAKI. IEEE Trans. Reliab. R-30(2), 175 (1981). This paper presents an algorithm (called the N & M algorithm) for solving pure-integer separable nonlinear programming problems with two constraints. The N & M algorithm uses surrogate constraints and solves a given problem by effectively generating values of surrogate multiplier (i.e., surrogate problems) and solving the surrogate" problems. Examples show that the N & M algorithm is superior to dynamic programming using the Lagrange multiplier in terms of number of iterations and narrowness of gap.
A generalized block-replacement policy. FRANK BEICHELT. I E E E Trans. Reliab. R-30(2), 171 (1981). T w o modes of system failure are possible: Mode 1 is removed by minimal repair, mode 2 by replacement. A generalized block replacement policy is introduced. The long-run cost rate is given. Integral equations of the renewal type for the basic reliability expressions are derived. A numerical example is presented. Availability of a 2-component dependent system. KAZU OKUMOTO. IEEE Trans. Reliab. R-30(2), 205 (1981). This paper investigates the effect of s-dependence among components on the availability of a 2-component repairable system with bivariate exponential failure time and repair time. Expressions are derived for state occupancy probabilities as well as system availability.
Parametric analysis of 2-unit redundant computer systems with corrective and preventive maintenance. JEAN-CLAUDE LAPRIE, ALAIN COSTES and CHRISTIAN LANDRAULT. IEEE Trans. Reliab. R-30(2), 139 (1981). This paper shows the span of results which can he obtained by modeling a system's behavior by stochastic processes and demonstrates practical rules for employing Markov and semi-Markov models. The introduction summarizes several methods for reliability analysis and gives the advantages and drawbacks of four methods: Markov processes, semi-Markov processes, supplementary variables, the method of stages. The remainder deals with reliability and availability modeling of a 2-unit redundant computer system. There are (a) two types of maintenance: corrective (c.m.) and preventive (p.m.), and (b) two system parameters: coverage, and an increased failure rate when one unit is under repair or inspection. Approximate expressions for reliability, mean time to failure, and asymptotic availability show the effects of the system parameters as well as of the shapes of the Cdrs of the times related to maintenance actions. For c.m., Markov modeling is a good approximation. For p.m., Markov modeling is a rough approximation; one can go to semi-Markov models or to the method of stages. Lastly, an approximate expression is given for the mean inspection interval which maximizes reliability and availability for p.m.
Fault diagnosis with imperfect tests. THEODORE J. SHESKIN. IEEE Trans. Reliab. R-30 (2), 156 (1981). This paper derives a minimum s-expected cost sequence of built-in-tests (BITs)
1042
World Abstracts on Microelectronics and Reliability
which will partition modular equipment into mutually exclusive groups of modules. After a fault in the equipment, one of these groups will be identified by a BIT diagnostic subsystem as the group which contains a faulty module. The BITs are imperfect in the sense only that they might not detect all of the possible faults in the equipment; they are perfect in the sense that fault indications are never false. The proportion of faults detectable by each BIT is known. Both the cost of a BIT and the probability that a BIT will pass or fail are functions of which modules are tested. A recursive algorithm is developed which determines a sequence of BITs with a m i n i m u m s-expected life-cycle cost. The recursive algorithm is applied to a 4-element numerical example. The algorithm has neither been proved nor implemented for a computer.
IEEE Trans. Reliab. R-30(2), 189 (1981). A scheme using multiple redundant computing devices with a majority voter improves the reliability of the output of the computing device. This paper analyzes some modular redundant systems with majority voters. An iterative structure is introduced for voters in order to improve the total reliability. The reliability of voters is introduced in several ways, Two models of imperfect voters are discussed in detail, that is, output-imperfect model and semiperfect model. The former is suitable for the case where voters are treated in the same manner as other modules. The latter is appropriate for the case where the characteristics of the voter are taken into account. Three theorems show the existence of optimal majorityvoted logic circuit for a given value of the reliability of each module under some reasonable assumptions.
Optimal replication for a specific system objective. J. M. GPOFFITH and D. M. RASMUSON. IEEE Trans. Reliab. R30 (2), 133 (1981). This paper defines component replication as the concept of using two or more i.i.d, components at a specific location in a system design. Both parallel and series physical connections are used to place replicate components into a system. The choice of parallel or series depends upon why the components are being made replicate and is complicated by the fact that a component might be used for several functions. A given physical connection can be logically in series for some situations and logically in parallel for others. Reducing system unavailability is one important use of component replication. The mathematical models relating replication and unavailability are often complex, and casual inspection does not reveal which components should be replicated to reduce unavailability. This paper develops a method for determining which components can be replicated (and how) for the greatest unavailability reduction. The method was used to study three system designs. For each design it was possible to identify several components which could be replicated with an associated reduction in unavailability at reasonable cost. In addition it was possible to demonstrate or confirm that replicating most components does not reduce unavailability. Applying this method helps the designer develop insight concerning the role of replication (in a system design) without complex calculations and interpretations.
Two repairable multistate devices with general repair-time distributions. MITSUO YAMASHIRO.IEEE Trans. Reliab. R-30 (2), 204 (1981). The system has two parallel redundant multistate devices with general repair-time distributions. The Laplace transforms of state probability and the mean time to system failure (MTSF) are obtained and a particular case is considered.
Bivariate survival model derived from a Weibull distribution. JOHN D. SPURRIER and D. R. WEIER. IEEE Trans. Reliab. R-30 (2), 194 (1981). A bivariate survival model is based on an underlying Weibull distribution and extends a bivariate exponential model considered by Freund. The model is motivated by a 2-component system which can function even if one of the components has failed. The components initially have a workload (inverse scale parameter) proportional to 2. U p o n the failure of one component, the workload of the remaining component becomes proportional to 02, where 0 > 0. The parameter 0 describes the a m o u n t of support or antagonism between the two components. The joint pdf of the first failure time and the time between the first and second failures is derived. The likelihood equation achieves its m a x i m u m in the interior of the parameter space, but the estimators do not have a closed form. A simulation study was performed to evaluate the performance of the m a x i m u m likelihood estimators.
Reliability analysis and optimal redundancy for majorityvoted logic circuits. HISASHI MINE and KAZUMI HATAYAMA.
Modified periodic replacement with minimal repair at failure. TOSHIO NAKAGAWA.IEEE Trans. Reliab. R-30 (2), 165 (1981). This paper summarizes four models of modified periodic replacement with minimal repair at failures when the scheduled replacement time is specified. If a failure occurs just before the replacement time, then three models are: (A) a unit remains as it is until the replacement time comes, (B) a unit is replaced by one of spares, (C) a unit is replaced by a new unit. Ira failure occurs well before replacement time then model D is: a unit is replaced at failure or at time T~. The sexpected costs for each model are obtained and the optimum policies are discussed.
Calculation of age-replacement with Weibull failure times. TOSHIO NAKAGAWAand KAZUMI YASUI. IEEE Trans, Reliab, R-30 (2), 163 (1981). An age-replacement policy with Weibull failure times is considered. It is troublesome to compute an optimum replacement time numerically, Upper and lower bounds of an optimum time are given in simple terms of replacement costs and parameters of a Weibull distribution. A numerical example shows that the approximation can be used when the ratio of the replacement cost for a failed unit to that for a non-failed unit is large. The approximation is best when the optimum age of replacement is small.
Analysis of performance-degradation data from accelerated tests. WAYNE NELSON. IEEE Trans. Reliab. R-30(2), 149 (1981). The performance of many products degrades as the product ages. Such degradation is usually slow but can be accelerated by a high "'stress". For example, the breakdown strength of electrical insulation depends on age and temperature. In some tests, the performance of a test unit is measured only once at a chosen age. For example, an insulation specimen yields only one breakdown measurement. A model and analyses for such data are described :.n this tutorial article. Also described is a method for estimating the distribution of time to failure, defined to be the age when performance degrades below a specified level.
Likelihood function of a debugging model for computer software reliability. BEy LITTLEWOODand JOHN L. VERRALL. IEEE Trans. Reliab. R-30(2), 145 (1981). A simple model for software reliability growth, originally suggested by Jelinski and Moranda, has been widely used but suffers from difficulties associated with parameter estimation. We show that a major reason for obtaining nonsensical results from the model is its application to data sets which exhibit decreasing reliability. We present a simple, necessary and sufficient condition for the m a x i m u m likelihood estimates to be finite and suggest that this condition be tested prior to using the model.
An efficient computational technique for evaluating the cut/tie sets and common-cause failures of complex systems. R. N.