where f(x) is the probability density function. In reliability analysis we are interested in the expected value of the time of failure(MTTF),so TTF=tf(t)dt where f(r) is the failure density function, and the integral runs from 0 to oo because the failure density function is undefined for times less than 0. We know, however, that the failure density function is f()=Q dt so, the mttF can be written as dQ() MTTF=t Using integration by parts and the fact that dQ(o/dt=-dR(n)/dt we can show that dQ(t) MTTF=t i. dt=[- tR()+R(d -jr()dr Consequently, the MTTF is defined in terms of the reliability function as MTTF=R(t)dt which is valid for any reliability function that satisfies R(oo)=0 The mean time to repair(MTTR) is simply the average time required to repair a system. The MTTR xtremely difficult to estimate and is often determined experimentally by injecting a set of faults, one at a tim into a system and measuring the time required to repair the system in each case. The MTTR is normally specified terms of a repair rate, H, which is the average number of repairs that occur per time period. The units of the repair rate are normally number of repairs per hour. The MTTR and the rate, u, are related by MTTR 1 μ It is very important to understand the difference between the MTTF and the mean time between failur (MTBF). Unfortunately, these two terms are often used interchangeably. While the numerical difference is small in many cases, the conceptual difference is very important. The MTTF is the average time until the first failure of a system, while the mtBF is the average time between failures of a system. If we assume that all repairs to make the system perfect once again just as it was when it was new, the relationship between the MTTF MTBF can be determined easily. Once successfully placed into operation, a system will operate, on the a time corresponding to the mTTF before encountering the first failure. The system will then require e 2000 by CRC Press LLC© 2000 by CRC Press LLC where f(x) is the probability density function. In reliability analysis we are interested in the expected value of the time of failure (MTTF), so where f(t) is the failure density function, and the integral runs from 0 to • because the failure density function is undefined for times less than 0. We know, however, that the failure density function is so, the MTTF can be written as Using integration by parts and the fact that dQ(t)/dt = –dR(t)/dt we can show that Consequently, the MTTF is defined in terms of the reliability function as which is valid for any reliability function that satisfies R(•) = 0. The mean time to repair (MTTR) is simply the average time required to repair a system. The MTTR is extremely difficult to estimate and is often determined experimentally by injecting a set of faults, one at a time, into a system and measuring the time required to repair the system in each case. The MTTR is normally specified in terms of a repair rate, m, which is the average number of repairs that occur per time period. The units of the repair rate are normally number of repairs per hour. The MTTR and the rate, m, are related by It is very important to understand the difference between the MTTF and the mean time between failure (MTBF). Unfortunately, these two terms are often used interchangeably. While the numerical difference is small in many cases, the conceptual difference is very important. The MTTF is the average time until the first failure of a system, while the MTBF is the average time between failures of a system. If we assume that all repairs to a system make the system perfect once again just as it was when it was new, the relationship between the MTTF and the MTBF can be determined easily. Once successfully placed into operation, a system will operate, on the average, a time corresponding to the MTTF before encountering the first failure. The system will then require MTTF = • Ú t f t dt ( ) 0 f t dQ t dt ( ) ( ) = MTTF = • Ú t dQ t dt dt ( ) 0 MTTF = = =+ = • • • • Ú Ú ÚÚ t dQ t dt dt t dR t dt dt tR t R t dt R t dt ( ) – ( ) [ ] – () () () 0 0 0 0 U MTTF = • Ú R t dt ( ) 0 MTTR = 1 m