Page 312 - A First Course In Stochastic Models

CHAPTER 8

Advanced Renewal Theory

8.0 INTRODUCTION
A renewal process is a counting process that generalizes the Poisson process. In
the Poisson process the interoccurrence times between the events are independent
random variables with an exponential distribution, whereas in a renewal process the
interoccurrence times have a general distribution. A first introduction to renewal
theory has been already given in Section 2.1. In that section several limit theorems
were given without proof. These limit theorems will be proved in Section 8.2 after
having discussed the renewal function in more detail in Section 8.1. A key tool
in proving the limit theorems is the so-called key renewal theorem. Section 8.3
deals with the alternating renewal model and gives an application of this model to
a reliability problem. In queueing and insurance problems it is often important to
have asymptotic estimates for the waiting-time probability and the ruin probability.
In Section 8.4 such estimates are derived by using renewal-theoretic methods. This
derivation illustrates the simplicity of analysis to be achieved by a general renewal-
theoretic approach to hard individual problems.

8.1 THE RENEWAL FUNCTION
Let us first repeat some definitions and results that were given earlier in Section 2.1.
The starting point is a sequence X 1 , X 2 , . . . of non-negative independent random
variables having a common probability distribution function

F(x) = P {X k ≤ x}, x ≥ 0

for k = 1, 2, . . . . Letting µ 1 = E(X k ), it is assumed that 0 < µ 1 < ∞. The
random variable X k denotes the interoccurrence time between the (k − 1)th and
kth events in some specific probability problem; see Section 2.1 for examples.
Letting
n

S 0 = 0 and S n = X i , n = 1, 2, . . . ,
i=1

A First Course in Stochastic Models H.C. Tijms
c 2003 John Wiley & Sons, Ltd. ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)