Template:Failure rate and failure intensity - Revision history

Richard House: Redirected page to RGA Overview#Failure Rate and Failure Intensity

2012-08-24T00:11:22Z

Redirected page to RGA Overview#Failure Rate and Failure Intensity

← Older revision		Revision as of 00:11, 24 August 2012
Line 1:		Line 1:
	~~==Failure Rate and Failure Intensity==~~		#REDIRECT [[RGA_Overview#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>0</math> . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution <math>F(x)</math> with failure rate <math>r(x)</math> . Each succeeding failure is governed by the intensity function <math>u(t)</math> of the process. Let <math>t</math> be the age of the system and <math>\Delta t</math> is very small. The probability that a system of age <math>t</math> fails between <math>t</math> and <math>t+\Delta t</math> is given by the intensity function <math>u(t)\Delta t</math> . Notice that this probability is not conditioned on not having any system failures up to time <math>t</math> , as is the case for a failure rate. The failure intensity <math>u(t)</math> for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, <math>u(t)=r(t)</math> , where <math>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>r(x)=\lambda \beta {{x}^{\beta -1}}</math>~~


	~~Under minimal repair, the system intensity function is:~~


	~~::<math>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 [[~~RGA Chapter 13\|here~~]].

Athanasios Gerokostopoulos at 18:39, 6 June 2012

2012-06-06T18:39:27Z

← Older revision		Revision as of 18:39, 6 June 2012
Line 12:		Line 12:


	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.		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 [[RGA Chapter 13\|here]].

Athanasios Gerokostopoulos: /* Failure Rate and Failure Intensity */

2012-06-06T18:36:01Z

Failure Rate and Failure Intensity

← Older revision		Revision as of 18:36, 6 June 2012
Line 1:		Line 1:
	==Failure Rate and Failure Intensity==		==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>0</math> . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution <math>F(x)</math> with failure rate <math>r(x)</math> . Each succeeding failure is governed by the intensity function <math>u(t)</math> of the process. Let <math>t</math> be the age of the system and <math>\Delta t</math> is very small. The probability that a system of age <math>t</math> fails between <math>t</math> and <math>t+\Delta t</math> is given by the intensity function <math>u(t)\Delta t</math> . Notice that this probability is not conditioned on not having any system failures up to time <math>t</math> , as is the case for a failure rate. The failure intensity <math>u(t)</math> for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, <math>u(t)=r(t)</math> , where <math>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:		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>0</math> . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution <math>F(x)</math> with failure rate <math>r(x)</math> . Each succeeding failure is governed by the intensity function <math>u(t)</math> of the process. Let <math>t</math> be the age of the system and <math>\Delta t</math> is very small. The probability that a system of age <math>t</math> fails between <math>t</math> and <math>t+\Delta t</math> is given by the intensity function <math>u(t)\Delta t</math> . Notice that this probability is not conditioned on not having any system failures up to time <math>t</math> , as is the case for a failure rate. The failure intensity <math>u(t)</math> for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, <math>u(t)=r(t)</math> , where <math>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>r(x)=\lambda \beta {{x}^{\beta -1}}</math>		::<math>r(x)=\lambda \beta {{x}^{\beta -1}}</math>


	Under minimal repair, the system intensity function is:		Under minimal repair, the system intensity function is:


	::<math>u(t)=\lambda \beta {{t}^{\beta -1}}</math>		::<math>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.		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.

Nicolette Young: Created page with '==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 g…'

2012-01-05T17:46:19Z

Created page with '==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 g…'

New page

==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>0</math> . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution <math>F(x)</math> with failure rate <math>r(x)</math> . Each succeeding failure is governed by the intensity function <math>u(t)</math> of the process. Let <math>t</math> be the age of the system and <math>\Delta t</math> is very small. The probability that a system of age <math>t</math> fails between <math>t</math> and <math>t+\Delta t</math> is given by the intensity function <math>u(t)\Delta t</math> . Notice that this probability is not conditioned on not having any system failures up to time <math>t</math> , as is the case for a failure rate. The failure intensity <math>u(t)</math> for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, <math>u(t)=r(t)</math> , where <math>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>r(x)=\lambda \beta {{x}^{\beta -1}}</math>

Under minimal repair, the system intensity function is:

::<math>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.