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SUMMARY This is an expository article on the theoretical and some computational aspects of renewal theory. §1 describes basic theory, including Blackwell's theorem; renewal density theorem; cumulants and asymptotic normality of the number of renewals in (0, t); and the integral equations of renewal theory. §2 is concerned with asymptotic behaviour of processes having an embedded renewal process: regenerative stochastic processes; semi‐Markov processes; cumulative processes. §3 discusses infinite sums and products connected with a renewal process which arise out of the study of electronic particle counters, and an integral equation connected with the infinite products. §4 describes some generalizations of renewal theory that have been proposed by different writers, and also “infinitesimal” renewal processes. Illustrative examples are drawn from biology and from the theories of queues, dams, and electronic counters; a table summarizing some of the papers on the last subject is given.

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The hypothetical model that we shall be considering in this paper is referred to as the single-server queue, and the details of this model are given in a recent paper by Lindley(5). The present treatment involves exactly the same assumptions as Lindley has given already, and we refer to his paper for a rigorous statement of them. Briefly, we shall be assuming general independent service times and general independent input or arrival times. Theoretical studies of the single-server queue are capable of wide applications, many of which are described in a paper by Kendall (4) and in the discussion to that paper.

SynopsisA sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete processwhere Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under whichThese new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.