1. Reliability of Systems Series systems: CI C2 B Consider the system with components C1 and C2 in line (series) as above. In order for the system to operate both C1 and C2 must function correctly. Then R(t) = P(T > t) = P(T1 > tand T2 > t) , where Ti is the time to failure of Ci. If component failures are independent, R(t) = P(T1 > t) .P(T2 > t) = R1(t) .R2(t) (6) By examining the expression we can conclude that the reliability of a series system is less than or equal to the reliability of the weakest component. Problem 8. (15 points) What is R(t) if T1 and 72 are exponential with hazard rates 21 and 2.2, and C1 and C2 are independent? Is equation (6) extendable to n independent series components? What happens to system reliability as n increases? Since this is a modeling and simulation course, you are going to run simulations of this system at t = 100. I highly recommend Excel for these simulations. You will have to think about how to go about this. This website will help you get started: http://stattrek.com/experiments/simulation.aspx Hint: you'll also need the COUNTIF excel function to count the number of "working" systems at t = 100. Consider the series system with 21 = .001, 12 = .005 and t= 100. Calculate R(100) analytically and then perform 1000 simulations of this system. The simulations must examine each element individually and develop the system reliability from that. Calculate the proportion of these simulations in which the system operates (does not fail).Repeat the simulation process five times. Calculate the average number of proportions over all simulations. Parallel systems: C2 Consider the system with components connected in parallel as above. System operation requires that C1 or C2 or both are functioning correctly. Then, R(t) = P(T > t) = 1-P(T t)]-[1 - P(T2 > t)] = 1 - [1 - R1(1)]-[1 - R2(1)] = R1(t) + R2(t) - R1(t) .R2(t) (7) By examining the expression (7) we can conclude that the reliability of a parallel system is better than a single component system. Problem 9. (15 points) What is R(t) if 71 and 72 are exponential with hazard rates 2.1 and 12 and C1 and C2 are independent? Is expression (7) extendable to n independent parallel components? What happens to system reliability as n increases?Consider the series system with 21 = .001, 12 = .005 and t= 100. Calculate R(100) analytically and then perform 1000 simulations of this system. The simulations must examine each element individually and develop the system reliability from that. Calculate the proportion of these simulations in which the system operates (does not fail). Repeat the simulation process five times. Calculate the average number of proportions over all simulations. Compare your results for this problem with the result from Problem 8. Series-Parallel systems: C2 Cl R C3 Consider the series-parallel system above. In order for the system to operate C1 and either C2 or C3 must function. Thus, R(t) = P(T > t) = P(T1 > t and [(T2 > t) or (T3 > t)]) Assuming independent failures R(t) = R1(t) .[R2(1) + R3(t) - R2(1) .R3(t)] To analyze the reliability of more complex series-parallel systems, we attempt to decompose them into simpler series or parallel blocks. Problem 10. (20 points) What is R(t) if T1, 72 and 73 are exponential with hazard rates 21, 12 and 23 and C1, C2 and C3 are independent? Consider the series system with 21 = .001, 12 = .005, 23 = .002 and t = 100. Calculate R(100) analytically and then perform 1000 simulations of this system. The simulations must examine each element individually and develop the system reliability from that. Calculate the proportion of these simulations in which the system operates (does not fail). Repeat the simulation process five times. Calculate the average number of proportions over all simulations. Compare your results for this problem with the result from Problems 8 and 9