below which is slightly more general than Palm’s, and a somewhat different approach is adopted. The viewpoint is more particularly taken here that the problem is in the nature of a probabilistic limit theorem, and that accordingly the addition of independent variables and the central limit theorem is a useful prototype. In fact, the proof given below for the existence of an exponential limit distribution is patterned after one by Kolmogorov and Gnedenko [3] for the central limit theorem. The presentation is organized as follows. Section 2 introduces some relevant concepts of renewal theory and of reliability. Section 3 contains the statement and proof of the theorem which asserts in effect that a complex piece of equipment, after an extended period of operation, will tend to exhibit a failure pattern with an exponential distribution for the time between failures. In 4, it is shown that a similar line of reasoning leads to conditions which insure that the time up to the first failure is also nearly exponentially distributed. Section 5 contains another parallel to the addition of random variables, namely, an asymptotic expansion of the Edgeworth-type for some of the distributions involved. Section 6, finally, contains comments on the results obtained, related to practical applications and to extensions of the theory.