RGA Overview

Overview

Overview

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.

Overview

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.

Overview

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.

Overview

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.

Overview

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.

Fielded Systems

When a complex system with new technology is fielded and subjected to a customer use environment, there is often considerable interest in assessing its reliability and other related performance metrics, such as availability. This interest in evaluating the system reliability based on actual customer usage failure data may be motivated by a number of factors. For example, the reliability that is generally measured during development is typically related to the system's inherent reliability capability. This inherent capability may differ from actual use experience because of different operating conditions or environment, different maintenance policies, different levels of experience of maintenance personnel, etc. Although operational tests are conducted for many systems during development, it is generally recognized that in many cases these tests may not yield complete data representative of an actual use environment. Moreover, the testing during development is typically limited by the usual cost and schedule constraints, which prevent obtaining a system's reliability profile over an extended portion of its life. Other interests in measuring the reliability of a fielded system may center on, for example, logistics and maintenance policies, quality and manufacturing issues, burn-in, wearout, mission reliability or warranties.

Most complex systems are repaired, not replaced, when they fail. For example, a complex communication system or a truck would be repaired upon failure, not thrown away and replaced by a new system. A number of books and papers in literature have stressed that the usual non-repairable reliability analysis methodologies, such as the Weibull distribution, are not appropriate for repairable system reliability analyses and have suggested the use of nonhomogeneous Poisson process models instead. The homogeneous process is equivalent to the widely used Poisson distribution and exponential times between system failures can be modeled appropriately when the system's failure intensity is not affected by the system's age. However, to realistically consider burn-in, wearout, useful life, maintenance policies, warranties, mission reliability, etc., the analyst will often require an approach that recognizes that the failure intensity of these systems may not be constant over the operating life of interest but may change with system age. A useful, and generally practical, extension of the homogeneous Poisson process, is the nonhomogeneous Poisson process, which allows for the system failure intensity to change with system age. Typically, the reliability analysis of a repairable system under customer use will involve data generated by multiple systems. Crow [17] proposed the Weibull process or power law nonhomogeneous Poisson process for this type of analysis and developed appropriate statistical procedures for maximum likelihood estimation, goodness-of-fit and confidence bounds.

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the power law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the power law model governs each succeeding system failure. Additional information on the power law model can also be found in Chapter 13.