IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2012, Professor Whitt OPTIONAL EXTRA INFORMATION ABOUT RENEWAL THEORY The Renewal Function, The Renewal Equation and Renewal Theorems

These notes cover material beyond the topics covered in IEOR 3106. We refer to the books by [Asmussen (2003)] and [Ross (1996)] (a more advanced book than the one we use). First, renewal theory is about renewal processes. A key quantity is the renewal function. The renewal function is important because it is a key component of the solution of the renewal equation. In applications, we are often interested in the asymptotic form of the renewal function and the solution of the renewal equation. Several theorems describe this asymptotic behavior: (i) the elementary renewal theorem, (ii) Blackwell’s renewal theorem, and (iii) the key renewal theorem. These notes explain. The first three sections do cover material that we did discuss, at least to some extent. 1. a renewal process Let {Xn : n ≥ 1} be a sequence of IID nonnegative real-valued random variables with cdf F having finite mean EX. Avoid trivialities by assuming that P (X > 0) > 0. Let Sn ≡ X1 + · · ·+ Xn, n ≥ 1 , (1) with S0 ≡ 0 and N(t) ≡ max {n ≥ 0 : Sn ≤ t}, t ≥ 0 . (2) Then N ≡ {N(t) : t ≥ 0} is a renewal (counting) process, for which Xn is the length of the interval between the (n− 1)st point and the nth point, for n ≥ 1. (We do not put a point at 0 unless X1 = 0.) In addition, for the asymptotic results we will want to assume that the distribution of X1 is non-lattice. A distribution is a lattice distribution if it concentrates all mass (has support) on the sequence {cn : n ≥ 0} for some constant c > 0. The most common lattice distributions have mass concentrated on the integers (the case c = 1). For lattice distributions, there are corresponding limits, along the lattice of support. 2. the renewal function Then the renewal function is m(t) ≡ E[N(t)], t ≥ 0 .