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Introduction 7 t 0 t 1 t 2 t 3 t 4 t S 1 0 FIGURE 1.5 System operation and repair. Suppose the system is in normal operation at t = 0, it fails at t 1 , and the nor- mal system operation is recovered at t 2 by some software modification, reset, or hardware replacement. Similar failure and repair events happen at t 3 and t 4 . The duration of normal system operation ( T n ), for intervals such as t 1 t 0 and t 3 t 2 , is generally assumed to be a random number that is exponentially distributed. This is known as the exponential failure law . Hence, the probability that a system will operate normally until time t , referred to as reliability , is given by: P T n > t = e t where is the failure rate . Because a system is composed of a number of com-
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Unformatted text preview: ponents, the overall failure rate for the system is the sum of the individual failure rates ( ³ i ) for each of the k components: ³ = k ± i = ³ i The mean time between failures (MTBF) is given by: MTBF = ± ² e − ³t dt = 1 ³ Similarly, the repair time (R) is also assumed to obey an exponential distribution and is given by: P±R > t² = e − ´t where µ is the repair rate . Hence, the mean time to repair (MTTR) is given by: MTTR = 1 ´ The fraction of time that a system is operating normally (failure-free) is the system availability and is given by: System availability = MTBF MTBF + MTTR...
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