7.5.3 Consider a lightbulb 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/ D E[N.t/] be the expected number of renewals up to time t. Consider a block replacement policy (see Section 7.2.1) 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

b+cM(T) (T) = T Investigate the choice of a cost minimizing value 7* when M(t)=1+1-exp(-at).