5.3. Consider a light bulb whose life is a continuous random variable X with probability density function f(x), for x > 0. Assuming that one starts with a fresh bulb and that each failed bulb is immediately replaced by a new one, let M(t) = E[N(t)] be the expected number of renewals up to time t. Consider a block replacement policy (see Section 2.1) that replaces each failed bulb immediately at a cost of c per bulb and replaces all bulbs at the fixed times T, 2T, 3T, .... Let the block replacement cost per bulb be b <

c. Show that the long run total mean cost per bulb per unit time is O(T)= b+cM(T)

T Investigate the choice of a cost minimizing value T* when M(t) _ t + 1 - exp(-at).