Modeling for Reliability Analysis: Markov Modeling for Reliability, Maintainability, Safety, and Supportability Analyses of Complex Systems / Edition 1

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"Markov modeling has long been accepted as a fundamental and powerful technique for the fault tolerance analysis of mission-critical applications. However, the elaborate computations required have often made Markov modeling too time-consuming to be of practical use on these complex systems. With this hands-on tool, designers can use the Markov modeling technique to analyze safety, reliability, maintainability, and cost-effectiveness factors in the full range of complex systems in use today.

Featuring ground-breaking simulation software and a comprehensive reference manual, MARKOV MODELING FOR RELIABILITY ANALYSIS helps system designers surmount the mathematical computations that have previously prevented effective reliability analysis. The text and software compose a valuable self-study tool that is complete with detailed explanations, examples, and a library of Markov models that can be used for experiments and as derivations for new simulation models. The book details how these analyses are conducted, while providing hands-on instruction on how to develop reliability models for the full range of system configurations.

Computer-Aided Rate Modeling and Simulation (CARMS) software is an integrated modeling tool that includes a diagram-based environment for model setup, a spreadsheet like interface for data entry, an expert system link for automatic model construction, and an interactive graphic interface for displaying simulation results."

"...comprehensive coverage of this powerful technique for analyzing the reliability, maintainability, and safety of complex systems...with necessary background material and a windows-based computer program."

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Editorial Reviews

For practicing systems and reliability engineers engaged in designing redundant systems, provides the necessary background material for probability theory and Markov analysis, and an interactive Windows-based computer program suitable for solving small to medium-sized problems, available on the Internet. Explains how to surmount the mathematical computations that have previously prevented effective reliability analysis. The Computer-Aided Rate Modeling and Simulation software includes a diagram-based environment for model setup, a spreadsheet-like interface for data entry, an expert system link for automatic model construction, and an interactive graphic interface for displaying simulation results. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Meet the Author

About the Authors...
Jan Pukite has been actively involved in military and commercial system design for over 30 years. His experience includes process and flight control system analysis and design, fault-tolerant system design, analysis and simulation of complex electronic systems, software development, and microcomputer applications. He served as the principal investigator on the following SBIR contracts: Logistics Software Implementation (Office of Naval Research); Fail-Safe, Fault-Tolerant Electronics, Phases I and II (Air Force Avionics Laboratory), and Intelligent Built-In Test Module (Naval Air Systems Command). In 1984, he founded DAINA to engage in advanced technology research and development.
Paul Pukite has co-authored 30 refereed papers in various basic and applied research topics dealing with advanced electronics system design and software engineering. His projects have included developing new yield analysis techniques for semiconductor manufacturing, using digital signal processors (DSP) to perform a wide range of computationally intensive statistical analysis tasks that have normally been relegated to supercomputers, and building the Ada expert system and support software that formed the basis of the Redundancy and Reconfiguration Manager (RRM) developed for the Air Force Pave Pillar Integrated Test Bed at Wright-Patterson Air Force Base.

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Read an Excerpt

Modelling for Realiability Analysis

Markov Modelling for Realiability, Maintainability, Safety, and Supportability Analyses of Complex Systems
By Jan Pukite Paul Pukite

John Wiley & Sons

ISBN: 0-7803-3482-5

Chapter One


This book addresses the problems associated with the practical design of complex, yet reliable systems. These systems are used in many applications, including avionics and banking. Designers of such systems have to be methodical and careful to meet all specifications and requirements, many of which have origins in safety or mission-critical applications. Criticality has several meanings in this context; safety-critical systems are those where loss of life must be avoided, mission-critical systems stress mission completion, and business-critical (often also called mission critical) are those that are needed to keep a business operating.

The design of these systems is a lengthy and time-consuming process. Thus, from a designer's standpoint, a major problem is the lack of an established integrated approach that considers overall design concurrently with reliability optimization. Throughout the text, we stress the Markov model approach as a means of providing a unified approach to reliability, performability, and system- and cost-effectiveness evaluation.


The complexity of a fault-tolerant system is one of the major problems facing the system designer. This complexity is due partly to the addition of redundant components and partly to the interaction between the components. Correspondingly, the complexity of the fault-tolerant system directly mirrors the reliability model.

As the number of system components and their failure modes increase, there is an exponential increase in system states, making the resulting reliability model more difficult to analyze. For example, if the system consists of n different components, then the resulting number of system states is [2.sup.n] (without considering the fault sequence), or approximately e X n! states (when considering the fault sequence, where e is the natural logarithm base). Thus, even for a relatively simple system the resulting Markov model may contain an extremely large number of states.

The large number of system states makes it difficult to solve the resulting model, to interpret state probabilities, and to conduct sensitivity analyses. In particular, it is difficult to identify the critical components. The basic issue to address here is how and when to use techniques for both largeness tolerance and largeness avoidance. Largeness tolerance is dependent on automated design tools with adequate computer resources, whereas largeness avoidance requires designer ingenuity and experience.


During the design of fault-tolerant systems, the system analyst and the designer must understand the details of the process: what options are available to meet the objectives, which parameters are approximate, and which parameters are secondary (can be neglected). Software-based system designers, in particular, need methods to handle the overwhelming state-space complexity.

Knowledge of the design process determines the need for specific design tools. Tools (or a toolkit) supported by an interactive and integrated environment should be capable of providing reasonably fast answers to design problems. Support tool designers must also realize that the information and data available during the early design phase will be approximate. Thus, even in the best case, the computed answers to the design problems will also be only approximate and will not require the computing process to achieve ultrahigh accuracy.

In the latter part of this book, we will describe and illustrate the application of the CARMS reliability program. This tool has been developed for fault-tolerant system reliability and effectiveness analyses. To give an idea of the most basic type of analysis that the CARMS program can provide, we give a short example of a problem application.


One of the most common and tangible examples of a fault-tolerant system is the automobile with spare tire. As a practical matter, it makes good common sense to take a spare tire along on a long trip. The reason? To improve the reliability or the probability of reaching the destination without delay.

As an example, consider a jeep with a single spare tire (see Figure 1.1). From thousands of similar vehicles on the road, one can estimate the failure rate of a tire (that is, a flat) per mile traveled by taking the total number of flats encountered and dividing by the miles driven. This number takes into account unexpected road hazards as well as wearout. Since we are using computed average failure rate, an exponential distribution should be used to provide an unbiased estimate for the probability of successfully completing a trip.

After creating the model with CARMS (see Chapter 19), we can evaluate it. Assuming that the failure rate for the tire is one flat per 1000 miles, we obtain the trip reliability for the situations with and without a spare, as shown in Figure 1.2.

As one can see, the probability of becoming stranded is much higher without a spare tire. However, as the trip becomes longer, the curves tend to merge.

Although the example analysis may be intuitively obvious, the methods and tools on which it, as well as this book, are based give a foundation on which to formulate a quantitative analysis.


System effectiveness analysis is part of the overall system design process. It specifically deals with the definition and evaluation of system figure-of-merit (FOM) measures.

In general, a figure of merit is any index that indicates the quality of a system. In the simplest case, it may be a measured physical quantity, such as range or payload. In other cases, it may be a calculated quantity based on measurement, such as the mean down time or the mean time between maintenance actions. Lastly, it may be a predicted quantity based on measurement or simulation. For example, "the probability that a system can meet an operational demand at a random point in time," will require prediction since there will be some uncertainty about the operational environment.

Figures of merit indicate what can be expected from the system. They must be in an operationally oriented form that can be readily understood and used in planning. Where the number of significantly different mission outcomes is small, the probabilities of each of these outcomes can be useful figures of merit. When the number of mission outcomes is large or when a continuous range of outcomes requires consideration, a measure of relative "adequacy" may be assigned to each possible outcome, and the expected "adequacy" may be used as a figure of merit.

System Effectiveness Definition. System effectiveness measures the extent to which a system may be expected to achieve a set of specific mission requirements. System effectiveness is usually defined as a function of availability, dependability, and capability:

Mission. The mission definition is a precise statement of the intended purpose(s) of the system and of the environmental conditions (natural and synthetic) under which it is required to operate.

Availability. Availability is a measure of the system condition at the start of the mission and is a function of the relationships among hardware, personnel, and procedures.

Dependability. Dependability is a measure of the system condition(s) at one or more points during the mission, given the system condition at the start of the mission. It may be stated as the probability(ies) (or other suitable mission-oriented measure) that the system

1. Will enter and/or occupy any one of its significant states during a specified mission and,

2. Will perform the functions associated with those states.

Capability. Capability measures the ability of the system to achieve the mission objectives, given the system condition(s) during the mission, and specifically accounts for the performance of a system.

Thus, system effectivness is a compound measure combining availability, dependability (reliability), and capability (performance). Details of the system effectiveness model derivation and evaluation are discussed in Chapter 15.


The design of a real system involves the following steps:

1. Define the key components of the system. To simplify the design process, the key components of the system must be identified. These should be selected on the basis of their functions and independence.

2. Estimate base component reliability. For each of these components we need to obtain the associated failure rates. These failure rates may be based on MILHDBK (Military Handbook) 217 data or may be obtained through available commercial programs. It is important to remember that these failure rate estimates are only approximate, instead of being highly accurate. Once the failure rates are known, the next step is to perform a first-cut reliability estimation. This will provide a nonredundant, single-thread, system reliability estimate, which will also help to identify those subsystems where redundancy will be required.

3. Select a number of redundancy approaches. In the beginning, only the desired reliability goal will be known. If a nonredundant single-thread system is unable to meet this goal, then redundancy will have to be introduced. The initial redundancy configuration considered should be the simplest that will have the potential of meeting the specified reliability goal.

4. Develop simplified models for these components. At the start of the design process, the designer should work with a simplified redundancy model to gain a better understanding of the potential gains of redundancy and to be able to select an acceptable redundancy level. For example, the designer may consider using different numbers of active components, passive spares, hot standby units, and so on. Each of these configurations will have a different model. A simplified model will allow testing of more options and more time to select the viable one. The initial reliability model should be based on ideal conditions (perfect recovery, no interaction between subsystems, etc.) for simplicity reasons.

5. Select a simple system model and expand it as necessary. Repeat the previous step at the system level. Since the initial model was chosen to be a simple one, the next step will introduce additional factors. These additions will include modifications for recovery, standby failure rates, and the like. Again, these model expansions should be introduced gradually, and their effect on the model noted. The main objective is to work with a minimum complexity model.

6. Identify the critical design areas. The simplest way of identifying the critical areas is to examine the shortest transition paths and determine which specific failure or recovery rates affect these transition paths. The critical components are the least reliable and as such form the weakest link in the proposed design. Then, one can develop more detailed reliability models for the critical subsystems and derive simple parametric equations relating component failure rates to the probability of failure.

7. Determine the key factors affecting failure probability. Once the critical factors have been identified, available options for further improvement should be identified. These options may include the use of higher reliability parts or additional levels of redundancy. For each of these options, reliability and performance models should be developed and evaluated.

8. Determine the level of the required redundancy. Once the desired reliability goal has been reached in the preliminary stage, a more detailed reliability model is developed to determine if all of the goals have been met.

9. Select fault recovery process. Since the fault-tolerant system is vulnerable during the recovery process, a dependable recovery scheme must be implemented. To achieve the desired reliability improvement, the interaction between the components must be controlled to avoid fault propagation to adjacent components or to other subsystems.


Excerpted from Modelling for Realiability Analysis by Jan Pukite Paul Pukite Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Series Introduction.



System Requirements and Design.

Foundations of Probability Theory.

Basic Reliability Concepts.

Basic Reliability Models.

Markov Process Fundamentals.

Hardware Reliability Modeling.

Software Reliability Modeling.

Combined Hardware-Software Reliability Modeling.

Modeling of Large and Complex Systems.

Maintainability Modeling.

Availability Modeling.

Safety Modeling.

Markov Model Evaluation.

Effectiveness Modeling.

Support Analyses.

Application Examples.

Practical Design of Fault-Tolerant Systems.

CARMS User's Guide.

CARMS Model Library.

CARMS Reference.

Definitions and Acronyms.



About the Authors.

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