Chapter 05 Exercises collection

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Chapter 5 Reliability of simple systems 5.1 Satellite with two transmitters Consider a satellite with two transmitters, one of which is in cold standby. Loss of transmission can occur if either both transmitters have failed or solar disturbances permanently interfere with transmission. lf the rate of failure of the on-line transmitter is λ and the rate of solar disturbances is λcm , find: l. The reliability of transmission. 2. The mean time to transmission failure. 5.2 Simple system Consider a system of two independent components with exponentially distributed failure times. The failure rates are λ1 and λ2, respectively. Determine the probability that component l fails before component 2. 5.3 Parallel system Suppose that in the system shown below, the two components have the same cost, and their reliabilities are R1 = 0.7, R2 = 0.95, respectively. lf it is perrmissible to add two components to the system, would it be preferable to a) replace component l by three components in parallel or b) to replace components l and 2 each by simple parallel systems? 5.5 Common mode failure Suppose that a unit has a design-life reliability of 0.95. Assume an exponential distribution for the failure times and the rare-event approximation for the reliability, R(t) = exp(-λt) ≈ 1- λt. l. Estimate the reliability if two of these units are put in active parallel. 2. Consider now the possibility of common-mode failures (shocks which simultaneously fail both components). In this case, the failure rate λ has two contributions from independent (I) and common-mode (C) failures λ = λI + λC = (1-β)λ + βλ where β=λC/λ Estimate the maximum fraction β of common failures that is acceptable if the parallel units in part l are to retain a system reliability of at least 0.99. 1 2 5.6 Active parallel…