An approximation method to obtain the reliability of a 2-dimensional consecutive k-out-of-n: F system
is discussed. Although analysis to obtain exact reliability requires many calculation resources for a
system with a large number of components, the proposed approximation method obtains the reliability
easily by giving an assumption on the maximum number of failed components in an operable system.
This approximated reliability is exact when the total number of failed components is less than the
assumed maximum number. The accuracy of the new method is confirmed by numerical examples.
Modeling Inefficiencies in a Reliability System Using Stochastic Frontier Regression
For some reliability systems, it is possible to have the system reliability smaller than the reliability obtained using the configuration of the components. This may be due to the inefficiency of the system. By inefficiency, we mean any tendency or attribute that will bring down the performance of the system from the level the configuration is capable of or expected to provide or designed for. This sets a maximum limit (or frontier) for the performance of the system. Therefore, deviation of the observed level from this limit would then be an indicator of the inefficiency. In this paper, we have made an attempt to model inefficiencies in the working of a reliability system, and to define an inefficiency index. The paper discusses the practical estimation of the coefficient of inefficiency in the system performance. The stochastic frontier regression methods are used to estimate the inefficiency. The validity of the methodology has been assessed for an exponential model, using a limited simulation study. The inefficiency indices proposed in this paper are simple, as they must be to be useful to engineers. We found that the suggested indices & their estimation procedures work well.
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An O(k^2·log(n)) algorithm for computing the reliability of consecutive-k-out-of-n: F systems
This study presents an O(k2·log(n)) algorithm for computing the reliability of a linear as well as a circular consecutive-k-out-of-n: F system. The proposed algorithm is more efficient and much simpler than the O(k3·log(n/k)) algorithm of Hwang & Wright.
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Competing causes of failure and reliability tests for Weibull lifetimes under type I progressive censoring
For many high reliability products where very few items are expected to fail during the test period, testing undernormal conditions is not feasible. Further, the requirement for high reliability increases the need for test procedures which yield valuable degradation and other useful information for improving product reliability. Thus in some manufacturing and other experiments, various types of failure censored and accelerated life tests are commonly employed for life testing. In this paper we discuss Type I progressively censored variable-sampling plans for Weibull lifetime distributions under competing causes of failure. The proposed procedure is attractive as it yields useful degradation-related information for improving product quality. In addition, the procedure is useful when a test is conducted under severe time constraint and/or when the experimenter wishes to save costly specimens or scarce test facilities for other use.
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Discounted warranty cost of minimally repaired series systems
Many factors should be considered in modeling DWC (discounted warranty cost) of repairable systems or products including system structure, components failure processes, methods of discounting as well as the warranty policy itself. In this paper, we present DWC models for repairable series systems. In particular, a free repair warranty policy and a pro-rata warranty policy are studied. The impact of repair actions on components failure times is assumed to be minimal, hence NHPPs are used to describe the failure processes. Two types of discounting methods are considered in this paper: a continuous discount function and a discrete discount function. Expressions for both the expected value and variance of DWC are derived. The applications of our findings can be seen in warranty design, warranty reserve determination and risk analysis. Our approach incorporates the information of system structure, the value of time and the impact of repair actions, which are of great importance to warranty cost prediction and evaluation, but have not been sufficiently studied in the literature of warranty analysis.
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Generalized multi-state k-out-of-n:G systems
In a binary k-out-of-n:G system, k is the minimum number of components that must work for the system to work. Let 1 represent the working state and 0 the failure state, k then indicates the minimum number of components that must be in state 1 for the system to be in state 1. This paper defines the multi-statek-out-of-n:G system: each component and the system can be in 1 of M+1 possible states: 0, 1, ..., M. In Case I, the system is in state ⩾j iff at least kj components are in state ⩾j. The value of kj I 1 can be different for different required minimum system-state level j. Examples illustrate applications of this definition. Algorithms for reliability evaluation of such systems are presented
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What designers of microelectronic systems should know about arrays spared by rows and columns
Perhaps the most common fault tolerant architecture configures a nominal t×a·t array using b·t dedicated sparerows and c·t dedicated spare columns. Despite an extensive literature, two problems about row-column sparing appear unresolved: how to minimize the area of the layout and how to minimize the maximum wirelength. This paper answers these questions, consolidates results, and describes the implications for the designer. An outstanding conjecture is counterexampled by using a graph-theoretic procedure to lay out arrays spared bydedicated rows and columns, in area proportional to the number of array elements. However, dedicated sparing is somewhat more costly than homogeneous extraction of at×a·t array from a (1+b)·t×(a+c). T array. Complementing our results for layout, we quantify the worst-case and probabilistic fault tolerance for both dedicated and homogeneous sparing, as a function of the nominal aspect ratio a⩾1, the redundancy parameters b, c, and the scale parameter t. In the process, we contribute to the solution to an open question in extremal graph theory, the problem of Zarnnkiewicz: what is the least integer Z(t; a, b, c) such that every (1+b)·t×(a+c)·t binary array with Z ones contains a t×a·t subarray having no zeros? Whereas the mathematical literature traditionally focuses on subarrays possessing a constant number of rows or columns, we are interested in scalable constructions for microelectronics. Reflecting this priority, we derive exact formulae for Z(t; a, b, c) when the extracted subarray, grows in proportion to embedding array
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Reliability evaluation of combined k-out-of-n:F, consecutive-k-out-of-n:F and linear connected-(r, s)-out-of-(m, n):F system structures
Based on a real industrial application, three new system reliability models are proposed: combined k-out-of-n:F and consecutive-k c-out-of-n:F system; combined k-out-of-m·n:F and linear connected-(r,s)-out-of-(m,n):F system; and combined k-out-of-m·n:F consecutive-kc-out-of-n:F and linear connected-(r,s)-out-of-(m,n):F system. Reliability evaluation algorithms are provided for these models. The computation times of the algorithms for these models are, respectively: O(n·k), O(k·n·2 m·sm-r+2), O(k·n·(2kc )m·sm-r+1). The algorithms are used for system reliability evaluation of furnace systems. The concept of the combined k-out-of-n:F and 1-dimensional and 2-dimensional consecutive-k-out-of-n:F systems can be extended to other variations of the consecutive-k-out-of-n:F systems, e.g., the consecutive-k-out-of-n:G system and 1-dimensional and 2-dimensional r-within-k-out-of-n:F systems. The concept of Markov chain imbeddable (MIS) systems is another excellent tool that can be used for analysis of such combined system structures
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Common-Mode Failures in Redundant VLSI Systems: A Survey
Abstract—This paper presents a survey of CMF (common-mode
failures) in redundant systems with emphasis on VLSI (very large
scale integration) systems. The paper discusses CMF in redundant
systems, their possible causes, and techniques to analyze reliability
of redundant systems in the presence ofCMF. Current practice and
recent results on the use of design diversity techniques forCMFare
reviewed. By revisiting the CMF problem in the context of VLSI
systems, this paper augments earlier surveys on CMF in nuclear
and power-supply systems. The need for quantifiable metrics and
effective models for CMF in VLSI systems is re-emphasized. These
metrics and models are extremely useful in designing reliable systems.
For example, using these metrics and models, system designers
and synthesis tools can incorporate diversity in redundant
systems to maximize protection against CMF.
Index Terms—Common-mode failures, concurrent error detection,
data integrity, design diversity, redundancy.
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failures) in redundant systems with emphasis on VLSI (very large
scale integration) systems. The paper discusses CMF in redundant
systems, their possible causes, and techniques to analyze reliability
of redundant systems in the presence ofCMF. Current practice and
recent results on the use of design diversity techniques forCMFare
reviewed. By revisiting the CMF problem in the context of VLSI
systems, this paper augments earlier surveys on CMF in nuclear
and power-supply systems. The need for quantifiable metrics and
effective models for CMF in VLSI systems is re-emphasized. These
metrics and models are extremely useful in designing reliable systems.
For example, using these metrics and models, system designers
and synthesis tools can incorporate diversity in redundant
systems to maximize protection against CMF.
Index Terms—Common-mode failures, concurrent error detection,
data integrity, design diversity, redundancy.
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Software-Implemented EDAC Protection Against SEUs
Abstract—In many computer systems, the contents of memory
are protected by an error detection and correction (EDAC) code.
Bit-flips caused by single event upsets (SEU) are a well-known
problem in memory chips; EDAC codes have been an effective
solution to this problem. These codes are usually implemented
in hardware using extra memory bits and encoding/decoding
circuitry. In systems where EDAC hardware is not available, the
reliability of the system can be improved by providing protection
through software. Codes and techniques that can be used for
software implementation of EDAC are discussed and compared.
The implementation requirements and issues are discussed, and
some solutions are presented. The paper discusses in detail how
system-level and chip-level structures relate to multiple error correction.
A simple solution is presented to make the EDAC scheme
independent of these structures.
The technique in this paper was implemented and used effectively
in an actual space experiment. We have observed that SEU
corrupt the operating system or programs of a computer system
that does not have any EDAC for memory, forcing the system
to be reset frequently. Protecting the entire memory (code and
data) might not be practical in software. However, this paper
demonstrates that software-implemented EDAC is a low-cost
solution that provides protection for code segments and can
appreciably enhance the system availability in a low-radiation
space environment. This reliability improvement is demonstrated
through both a satellite experiment and analytic estimates which
are based on parameter values that closely match the environment
of the satellite experiment.
For applications where read and write operations are done in
blocks of words, such as secondary storage systems made of solidstate
memories (RAM discs), software-implemented EDAC can be
a better choice than hardware EDAC, because it can be used with
a simple memory system and it provides the flexibility of implementing
more complex coding schemes.
Index Terms—Commercial off-the-shelf (COTS) in space,
error-correcting codes (ECC), error detection and correction
(EDAC), low-cost fault tolerance, memory-bit-flips,
memory protection, single-event upset (SEU), soft errors,
software-implemented, transient error.
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are protected by an error detection and correction (EDAC) code.
Bit-flips caused by single event upsets (SEU) are a well-known
problem in memory chips; EDAC codes have been an effective
solution to this problem. These codes are usually implemented
in hardware using extra memory bits and encoding/decoding
circuitry. In systems where EDAC hardware is not available, the
reliability of the system can be improved by providing protection
through software. Codes and techniques that can be used for
software implementation of EDAC are discussed and compared.
The implementation requirements and issues are discussed, and
some solutions are presented. The paper discusses in detail how
system-level and chip-level structures relate to multiple error correction.
A simple solution is presented to make the EDAC scheme
independent of these structures.
The technique in this paper was implemented and used effectively
in an actual space experiment. We have observed that SEU
corrupt the operating system or programs of a computer system
that does not have any EDAC for memory, forcing the system
to be reset frequently. Protecting the entire memory (code and
data) might not be practical in software. However, this paper
demonstrates that software-implemented EDAC is a low-cost
solution that provides protection for code segments and can
appreciably enhance the system availability in a low-radiation
space environment. This reliability improvement is demonstrated
through both a satellite experiment and analytic estimates which
are based on parameter values that closely match the environment
of the satellite experiment.
For applications where read and write operations are done in
blocks of words, such as secondary storage systems made of solidstate
memories (RAM discs), software-implemented EDAC can be
a better choice than hardware EDAC, because it can be used with
a simple memory system and it provides the flexibility of implementing
more complex coding schemes.
Index Terms—Commercial off-the-shelf (COTS) in space,
error-correcting codes (ECC), error detection and correction
(EDAC), low-cost fault tolerance, memory-bit-flips,
memory protection, single-event upset (SEU), soft errors,
software-implemented, transient error.
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Stochastic Ordering Results for Consecutive k-out-of-n : F Systems
Abstract—A linear (circular) consecutive k-out-of-n Fsystem
is a system of n linearly (circularly) ordered components which fails
if and only if at least k consecutive components fail.We use recursive
relationships on the reliability of such systems with independent
identically distributed components to show that for any fixed k
,the lifetime of a (linear or circular) consecutive k-out-of-n : F
system is stochastically decreasing in n . This result also holds for
linear systemswhenthe components are independent and not necessarily
identically distributed, but not in general for circular systems.
Index Terms—Linear and circular consecutive -out-of- systems,
stochastic order, system reliability.
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is a system of n linearly (circularly) ordered components which fails
if and only if at least k consecutive components fail.We use recursive
relationships on the reliability of such systems with independent
identically distributed components to show that for any fixed k
,the lifetime of a (linear or circular) consecutive k-out-of-n : F
system is stochastically decreasing in n . This result also holds for
linear systemswhenthe components are independent and not necessarily
identically distributed, but not in general for circular systems.
Index Terms—Linear and circular consecutive -out-of- systems,
stochastic order, system reliability.
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An Inspection Model with Minimal and Major Maintenance for a System with Deterioration and Poisson Failures
Abstract—A new condition-based maintenance model for a
system, subject to deterioration-failures and to Poisson-failures,
is presented. After an inspection, based on the degree of deterioration,
a minimal maintenance or a major maintenance
is performed, or no action is taken. Deterioration failures are
restored by major repair; Poisson failures are restored by minimal
repair. Major maintenance or major repair restores the system to
“good as new,” while minimal maintenance restores the system
one stage. Generalized stochastic Petri Nets are used to represent
and analyze the model, which represents a condition-based
maintenance strategy. Based on maximization of the system
throughput, an optimal inspection policy within this strategy
and optimal inter-inspection time are obtained. The effects of
inspection, maintenance, and repair parameters are investigated.
For a given inspection parameter, a 3-region diagram identifies
the effectiveness of an inspection policy based on minimal maintenance,
major maintenance, and major repair parameters.
Index Terms—Condition-based maintenance, deterioration
failure, generalized stochastic Petri nets, flexible manufacturing
cell, inspection, major maintenance, major repair, minimal
maintenance, poisson failure.
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system, subject to deterioration-failures and to Poisson-failures,
is presented. After an inspection, based on the degree of deterioration,
a minimal maintenance or a major maintenance
is performed, or no action is taken. Deterioration failures are
restored by major repair; Poisson failures are restored by minimal
repair. Major maintenance or major repair restores the system to
“good as new,” while minimal maintenance restores the system
one stage. Generalized stochastic Petri Nets are used to represent
and analyze the model, which represents a condition-based
maintenance strategy. Based on maximization of the system
throughput, an optimal inspection policy within this strategy
and optimal inter-inspection time are obtained. The effects of
inspection, maintenance, and repair parameters are investigated.
For a given inspection parameter, a 3-region diagram identifies
the effectiveness of an inspection policy based on minimal maintenance,
major maintenance, and major repair parameters.
Index Terms—Condition-based maintenance, deterioration
failure, generalized stochastic Petri nets, flexible manufacturing
cell, inspection, major maintenance, major repair, minimal
maintenance, poisson failure.
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Generic Rules to Evaluate System-Failure Frequency
Abstract—Frequency of failure of a system with -independent
components can be obtained from the system availability (unavailability)
expression and failure and repair rates of the components.
Although, Grouped Variable Inversion is an efficient technique to
find the system availability, there is no convenient method to convert
the “availability expression obtained by this technique” into
an “expression for system-failure frequency.” This paper present
generic rules to find system-failure frequency, particularly, when
the availability or unavailability expression of a system is obtained
using this technique. The rules are straightforward, and produce
appreciably shorter expressions for system-failure frequency. Examples
illustrate the simplicity and efficiency of the proposed rules.
Index Terms—Disjoint product, sum of disjoint products, system
availability, system-failure frequency.
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components can be obtained from the system availability (unavailability)
expression and failure and repair rates of the components.
Although, Grouped Variable Inversion is an efficient technique to
find the system availability, there is no convenient method to convert
the “availability expression obtained by this technique” into
an “expression for system-failure frequency.” This paper present
generic rules to find system-failure frequency, particularly, when
the availability or unavailability expression of a system is obtained
using this technique. The rules are straightforward, and produce
appreciably shorter expressions for system-failure frequency. Examples
illustrate the simplicity and efficiency of the proposed rules.
Index Terms—Disjoint product, sum of disjoint products, system
availability, system-failure frequency.
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Stochastic Models for Random-Request Availability
Abstract—The system receives a stream of tasks which arrive
according to some random processes. A new availability measure,
random-request availability, is proposed. This stochastic model
provides closed-form mathematical expressions which incorporate
the 3 basic elements:
• random task arrivals,
• system state,
• operational requirements of the system.
An analytic model is derived and illustrated by numerical examples.
Index Terms—Operational requirements of the system, randomrequest
availability, random task arrival, system state.
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according to some random processes. A new availability measure,
random-request availability, is proposed. This stochastic model
provides closed-form mathematical expressions which incorporate
the 3 basic elements:
• random task arrivals,
• system state,
• operational requirements of the system.
An analytic model is derived and illustrated by numerical examples.
Index Terms—Operational requirements of the system, randomrequest
availability, random task arrival, system state.
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Multistate Series-Parallel System Expansion-Scheduling Subject to Availability Constraints
Abstract—This paper addresses the multistage expansion
problem for multistate series-parallel systems. The study period is
divided into several stages. At each stage the demand distribution
is predicted in the form of a cumulative demand curve. The
additional elements chosen from a list of available products can
be included into any system-component at any stage to increase
the total system capacity and/or reliability. Each element is
characterized by its capacity (productivity), availability, and cost.
The objective is to minimize the sum of costs of the investments
over the study period while satisfying reliability constraints at
each stage.
To solve the problem, a genetic algorithm is used as an optimization
tool. The solution encoding technique allows the genetic algorithm
to manipulate integer strings representing multistage expansion
planes. A solution quality index comprises both reliability
& cost estimations. The procedure based on the universal generating
function is used for evaluating the availability of multistate
series-parallel systems. An example illustrates finding the optimal
expansion plan for a coal-transportation system of a power station.
Index Terms—Expansion planning, genetic algorithm, redundancy
optimization, universal generating function.
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problem for multistate series-parallel systems. The study period is
divided into several stages. At each stage the demand distribution
is predicted in the form of a cumulative demand curve. The
additional elements chosen from a list of available products can
be included into any system-component at any stage to increase
the total system capacity and/or reliability. Each element is
characterized by its capacity (productivity), availability, and cost.
The objective is to minimize the sum of costs of the investments
over the study period while satisfying reliability constraints at
each stage.
To solve the problem, a genetic algorithm is used as an optimization
tool. The solution encoding technique allows the genetic algorithm
to manipulate integer strings representing multistage expansion
planes. A solution quality index comprises both reliability
& cost estimations. The procedure based on the universal generating
function is used for evaluating the availability of multistate
series-parallel systems. An example illustrates finding the optimal
expansion plan for a coal-transportation system of a power station.
Index Terms—Expansion planning, genetic algorithm, redundancy
optimization, universal generating function.
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Using Simulation for Assessing the Real Impact of Test-Coverage on Defect-Coverage
Abstract—The use of test-coverage measures (e.g., block-coverage)
to control the software test process has become an
increasingly common practice. This is justified by the assumption
that higher test-coverage helps achieve higher defect-coverage
and therefore improves software quality. In practice, data often
show that defect-coverage and test-coverage grow over time, as
additional testing is performed. However, it is unclear whether this
phenomenon of concurrent growth can be attributed to a causal
dependency, or if it is coincidental, simply due to the cumulative
nature of both measures. Answering such a question is important
as it determines whether a given test-coverage measure should be
monitored for quality control and used to drive testing.
Although it is no general answer to this problem, a procedure is
proposed to investigate whether any test-coverage criterion has a
genuine additional impact on defect-coverage when compared to
the impact of just running additional test cases. This procedure
applies in typical testing conditions where
• the software is tested once, according to a given strategy,
• coverage measures are collected as well as defect data.
This procedure is tested on published data, and the results are compared
with the original findings. The study outcomes do not support
the assumption of a causal dependency between test-coverage
and defect-coverage, a result for which several plausible explanations
are provided.
Index Terms—Defect-coverage, Monte Carlo simulation, Software
test, Test-coverage, Test intensity.
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to control the software test process has become an
increasingly common practice. This is justified by the assumption
that higher test-coverage helps achieve higher defect-coverage
and therefore improves software quality. In practice, data often
show that defect-coverage and test-coverage grow over time, as
additional testing is performed. However, it is unclear whether this
phenomenon of concurrent growth can be attributed to a causal
dependency, or if it is coincidental, simply due to the cumulative
nature of both measures. Answering such a question is important
as it determines whether a given test-coverage measure should be
monitored for quality control and used to drive testing.
Although it is no general answer to this problem, a procedure is
proposed to investigate whether any test-coverage criterion has a
genuine additional impact on defect-coverage when compared to
the impact of just running additional test cases. This procedure
applies in typical testing conditions where
• the software is tested once, according to a given strategy,
• coverage measures are collected as well as defect data.
This procedure is tested on published data, and the results are compared
with the original findings. The study outcomes do not support
the assumption of a causal dependency between test-coverage
and defect-coverage, a result for which several plausible explanations
are provided.
Index Terms—Defect-coverage, Monte Carlo simulation, Software
test, Test-coverage, Test intensity.
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Failure Correlation in Software Reliability Models
Abstract—Perhaps the most stringent restriction in most software
reliability models is the assumption of statistical independence
among successive software failures. Our research was motivated
by the fact that although there are practical situations in
which this assumption could be easily violated, much of the published
literature on software reliability modeling does not seriously
address this issue.
The research work in this paper is devoted to developing the software
reliability modeling framework that can consider the phenomena
of failure correlation and to study its effects on the software
reliability measures. The important property of the developed
Markov renewal modeling approach is its flexibility. It allows construction
of the software reliability model in both discrete time and
continuous time, and (depending on the goals) to base the analysis
either on Markov chain theory or on renewal process theory. Thus,
our modeling approach is an important step toward more consistent
and realistic modeling of software reliability. It can be related
to existing software reliability growth models. Many input-domain
and time-domain models can be derived as special cases under the
assumption of failure -independence.
This paper aims at showing that the classical software reliability
theory can be extended to consider a sequence of possibly -dependent
software runs, viz, failure correlation. It does not deal with inference
nor with predictions, per se. For the model to be fully specified
and applied to estimations and predictions in real software development
projects, we need to address many research issues, e.g.,
the
• detailed assumptions about the nature of the overall reliability
growth,
• way modeling-parameters change as a result of the fault-removal
attempts.
Index Terms—Failure correlation, Markov renewal process, sequence
of dependent software runs, software reliability.
reliability models is the assumption of statistical independence
among successive software failures. Our research was motivated
by the fact that although there are practical situations in
which this assumption could be easily violated, much of the published
literature on software reliability modeling does not seriously
address this issue.
The research work in this paper is devoted to developing the software
reliability modeling framework that can consider the phenomena
of failure correlation and to study its effects on the software
reliability measures. The important property of the developed
Markov renewal modeling approach is its flexibility. It allows construction
of the software reliability model in both discrete time and
continuous time, and (depending on the goals) to base the analysis
either on Markov chain theory or on renewal process theory. Thus,
our modeling approach is an important step toward more consistent
and realistic modeling of software reliability. It can be related
to existing software reliability growth models. Many input-domain
and time-domain models can be derived as special cases under the
assumption of failure -independence.
This paper aims at showing that the classical software reliability
theory can be extended to consider a sequence of possibly -dependent
software runs, viz, failure correlation. It does not deal with inference
nor with predictions, per se. For the model to be fully specified
and applied to estimations and predictions in real software development
projects, we need to address many research issues, e.g.,
the
• detailed assumptions about the nature of the overall reliability
growth,
• way modeling-parameters change as a result of the fault-removal
attempts.
Index Terms—Failure correlation, Markov renewal process, sequence
of dependent software runs, software reliability.
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