some time, MTTR, to repair the system and place it back into operation once again. The system will then be perfect once again and will operate for a time corresponding to the mttF before encountering its next failure. The time between the two failures is the sum of the mttf and the mttr and is the mtbf. thus the difference between the MTtF and the MTBF is the mTTR. Specifically, the mTBF is given by MTBF MTTF mttR In most practical applications the MTTR is a small fraction of the MTTE, so the approximation that the MTBF and MTTF are equal is often quite good. Conceptually, however, it is crucial to understand the difference between the mtbf and the mtte s An extremely important parameter in the design and analysis of fault-tolerant systems is fault coverage.The lult coverage available in a system can have a tremendous impact on the reliability, safety, and other attributes of the system. Fault coverage is mathematically defined as the conditional probability that, given the existence of a fault, the system recovers Bouricius et al., 1969]. The fundamental problem with fault coverage is that it is extremely difficult to calculate. Probably the most common approach to estimating fault coverage is to develop a list all of the faults that can occur in a system and to form, from that list, a list of faults from which the system can recover. The fault coverage factor is then calculated appropriately. Reliability is perhaps one of the most important attributes of systems. The reliability of a system is generally derived in terms of the reliabilities of the individual components of the system. The two models of systems that are most common in practice are the series and the parallel. In a series system, each element of the system is required to operate correctly for the system to operate correctly In a parallel system, on the other hand, only one of several elements must be operational for the system to perform its functions correctly. The series system is best thought of as a system that contains no redundancy; that is, each element of the system is needed to make the system function correctly. In general, a system may contain N elements, and in a series system each of the N elements is required for the system to function correctly. The reliability of the series system can be calculated as the probability that none of the elements will fail. anothe this is that the reliability of the series system is the probability that all of the elements are working properly The reliability of a series system is given by Reries (t)=R(OR(r).Rt series ()= ∏Rt) An interesting relationship exists in a series system if each individual component satisfies the exponential failure law. Suppose that we have a series system made up of N components, and each component, i, has a constant failure rate of n, Also assume that each component satisfies the exponential failure law. The reliability of the series system is given by The distinguishing feature of the basic parallel system is that only one of N identical elements is required for the system to function. The reliability of the parallel system can be written as Rs()=10- parallel(t)=10-Q()=10-(10-R() e 2000 by CRC Press LLC© 2000 by CRC Press LLC some time, MTTR, to repair the system and place it back into operation once again. The system will then be perfect once again and will operate for a time corresponding to the MTTF before encountering its next failure. The time between the two failures is the sum of the MTTF and the MTTR and is the MTBF. Thus, the difference between the MTTF and the MTBF is the MTTR. Specifically, the MTBF is given by MTBF = MTTF + MTTR In most practical applications the MTTR is a small fraction of the MTTF, so the approximation that the MTBF and MTTF are equal is often quite good. Conceptually, however, it is crucial to understand the difference between the MTBF and the MTTF. An extremely important parameter in the design and analysis of fault-tolerant systems is fault coverage. The fault coverage available in a system can have a tremendous impact on the reliability, safety, and other attributes of the system. Fault coverage is mathematically defined as the conditional probability that, given the existence of a fault, the system recovers [Bouricius et al., 1969]. The fundamental problem with fault coverage is that it is extremely difficult to calculate. Probably the most common approach to estimating fault coverage is to develop a list all of the faults that can occur in a system and to form, from that list, a list of faults from which the system can recover. The fault coverage factor is then calculated appropriately. Reliability is perhaps one of the most important attributes of systems. The reliability of a system is generally derived in terms of the reliabilities of the individual components of the system. The two models of systems that are most common in practice are the series and the parallel. In a series system, each element of the system is required to operate correctly for the system to operate correctly. In a parallel system, on the other hand, only one of several elements must be operational for the system to perform its functions correctly. The series system is best thought of as a system that contains no redundancy; that is, each element of the system is needed to make the system function correctly. In general, a system may contain N elements, and in a series system each of the N elements is required for the system to function correctly. The reliability of the series system can be calculated as the probability that none of the elements will fail. Another way to look at this is that the reliability of the series system is the probability that all of the elements are working properly. The reliability of a series system is given by Rseries (t) = R1(t)R2(t) . . . RN(t) or An interesting relationship exists in a series system if each individual component satisfies the exponential failure law. Suppose that we have a series system made up of N components, and each component, i, has a constant failure rate of li .Also assume that each component satisfies the exponential failure law. The reliability of the series system is given by The distinguishing feature of the basic parallel system is that only one of N identical elements is required for the system to function. The reliability of the parallel system can be written as R t Rt i i N series() () = = ’1 R t ee e tt t N series( ) . . . –– – = ll l 1 2 Rte it i N series( ) – = = Â l 1 R t Q t Qt Rt i i i N i N parallel parallel ( ) . – ( ) . – ( ) . – ( . – ( )) = == == 10 10 10 10 ’’ 11