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3.2.2 Solution Tevkique for Safe Failure Markov Model. The effective repair rate includes the repair for detected and undetected safe failures. Detected safe failures can be repaired on-line at a much faster rate. Undetected safe failures can only be repaired after the system is taken off-line for periodic testing. The effective repair rate is determined below. The safe failure rate can be broken down as: iS = CSXSD + (1_ CS) 'sU Where: XSD iSU Cs Safe failure rate of a component Safe detected failure rate of a component Safe undetected failure rate of a component Fraction of safe failures detected by diagnostic coverage The generalized Markov model for safe failures is shown below: Where:r 0 /-ot A2pt Failure rate from the intermediate state to the spurious trip state Repair rate when detected due to on-line testing Repair rate for off-line periodic testing This model can be simplified to the following by determining the effective repair rate. Firste MPR Associates, Inc. I M P R 320 King Street Alexandria, VA 22314 Calculation No. Pre red By Checked By 426-001-CBS-01 Where: = Effective repair rate The effective repair rate can be determined by equating the MT'YF for each model. After algebraic manipulation, the MTTF's can be shown to be equal if: 1 / (, + 0) = Cs / (Lot + 0) + (1- CS) / (@,Lpt + 0) Solving for the effective repair rate yields: A, = [(1 - Cs) /Apt.+c + C0t+ AptA03 / [CSAP, + (1- CS) A, + 0] The MTTF can be determined from the Markov model by integrating the probability for the time that the system is in a non-failed states. States 1 through 11 are the non-failed states. Therefore, the MT1FF is: - 11 MTTF= f P(t) ]dt 0 Where: Pi(t) " Probability to be in the ith state at time t A closed form solution to this model exists. From Reference 5, the MTTF is given below. Note that this solution has been verified using alternative techniques outlined in Reference 4
3.2.2 Solution Tevkique for Safe Failure Markov Model. The effective repair rate includes the repair for detected and undetected safe failures. Detected safe failures can be repaired on-line at a much faster rate. Undetected safe failures can only be repaired after the system is taken off-line for periodic testing. The effective repair rate is determined below. The safe failure rate can be broken down as: iS = CSXSD + (1_ CS) 'sU Where: XSD iSU Cs Safe failure rate of a component Safe detected failure rate of a component Safe undetected failure rate of a component Fraction of safe failures detected by diagnostic coverage The generalized Markov model for safe failures is shown below: Where:r 0 /-ot A2pt Failure rate from the intermediate state to the spurious trip state Repair rate when detected due to on-line testing Repair rate for off-line periodic testing This model can be simplified to the following by determining the effective repair rate. Firste MPR Associates, Inc. I M P R 320 King Street Alexandria, VA 22314 Calculation No. Pre red By Checked By 426-001-CBS-01 Where: = Effective repair rate The effective repair rate can be determined by equating the MT'YF for each model. After algebraic manipulation, the MTTF's can be shown to be equal if: 1 / (, + 0) = Cs / (Lot + 0) + (1- CS) / (@,Lpt + 0) Solving for the effective repair rate yields: A, = [(1 - Cs) /Apt.+c + C0t+ AptA03 / [CSAP, + (1- CS) A, + 0] The MTTF can be determined from the Markov model by integrating the probability for the time that the system is in a non-failed states. States 1 through 11 are the non-failed states. Therefore, the MT1FF is: - 11 MTTF= f P(t) ]dt 0 Where: Pi(t) " Probability to be in the ith state at time t A closed form solution to this model exists. From Reference 5, the MTTF is given below. Note that this solution has been verified using alternative techniques outlined in Reference 4
3.2.2 Solution Tevkique for Safe Failure Markov Model. The effective repair rate includes the repair for detected and undetected safe failures. Detected safe failures can be repaired on-line at a much faster rate. Undetected safe failures can only be repaired after the system is taken off-line for periodic testing. The effective repair rate is determined below. The safe failure rate can be broken down as: iS = CSXSD + (1_ CS) 'sU Where: XSD iSU Cs Safe failure rate of a component Safe detected failure rate of a component Safe undetected failure rate of a component Fraction of safe failures detected by diagnostic coverage The generalized Markov model for safe failures is shown below: Where:r 0 /-ot A2pt Failure rate from the intermediate state to the spurious trip state Repair rate when detected due to on-line testing Repair rate for off-line periodic testing This model can be simplified to the following by determining the effective repair rate. Firste MPR Associates, Inc. I M P R 320 King Street Alexandria, VA 22314 Calculation No. Pre red By Checked By 426-001-CBS-01 Where: = Effective repair rate The effective repair rate can be determined by equating the MT'YF for each model. After algebraic manipulation, the MTTF's can be shown to be equal if: 1 / (, + 0) = Cs / (Lot + 0) + (1- CS) / (@,Lpt + 0) Solving for the effective repair rate yields: A, = [(1 - Cs) /Apt.+c + C0t+ AptA03 / [CSAP, + (1- CS) A, + 0] The MTTF can be determined from the Markov model by integrating the probability for the time that the system is in a non-failed states. States 1 through 11 are the non-failed states. Therefore, the MT1FF is: - 11 MTTF= f P(t) ]dt 0 Where: Pi(t) " Probability to be in the ith state at time t A closed form solution to this model exists. From Reference 5, the MTTF is given below. Note that this solution has been verified using alternative techniques outlined in Reference 4
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