Chapter 10 Exercises collection

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Chapter 10 Markov chains 10.1 Nuclear steam supply system A nuclear steam supply system has two turbo-generator units; unit l operates and unit 2 is in standby whenever both are good. The units have a constant MTTF of λi-1=l and 2, during active operation while during standby unit 2 has a MTTF of (λ2*)-1. The repair of a unit is assumed to begin instantaneously after it fails, but its duration is random so that the instantaneous repair rates will be μ1 and μ2 , respectively. The repairs can be done on only one unit at a time and any unit under repair will remain so until the task is completed. l. Draw the system diagram. 2. Write the Markov equations. 10.2 Alarm system An alarm system is subject to both unrevealed (u) and revealed (r) faults each of which have time to occurrence which are exponentially distributed with mean values of 200 h and 100 h respectively. lf a revealed fault occurs, then the complete system is restored to the time-zero condition by a repair process which has exponentially distributed times to completion with a mean value of 10 h. lf an unrevealed fault occurs, then it remains in existence until a revealed fau1t occurs when it is repaired along with the revealed fault. l. What is the asymptotic unavai1ability of the alarm system? 2. What is the mean number of system failures in a total time of 1,000 h? 10.3 Two unit standby system Consider a two-unit standby system. The failure rate of the operating unit is λ1 and that of the standby unit is λ2 . Both components are operative at time 0. l. Find the reliability of the system. 2. lf, in addition, the system is exposed to a hazard, characterized by a Poisson process with parameter λ12, that is detrimental to both components, develop an expression for the reliability of the system. 3. Find the MTTF…