1. Problem 15.6 A control device contains two parallel circuit boards. Both cir cuit boards are switched on. The device operates properly as long as at least one of the circuit boards functions. Each circuit board is subject to random failure. The failure rate increases with the age of the circuit board. The cir cuit boards are identical and their lifetimes are independent. Let ri denote the probability of a circuit board failing during the next week if the circuit board has functioned for the past i weeks. Past records of circuit boards give the failure function r0 = 0,r1 = 0.05,r2 = 0.07,r3 = 0.12,r4 = 0.25,r5 = 0.50, and r6 = 1. Any failed circuit board is replaced at the beginning of the fol lowing week. Also, any six-week-old circuit board is replaced. Formulate an appropriate Markov chain for the failure analysis of the device and specify the one-step transition probabilities.