Page 43 - A First Course In Stochastic Models

34 RENEWAL-REWARD PROCESSES

2.1 RENEWAL THEORY

As a generalization of the Poisson process, renewal theory concerns the study of
stochastic processes counting the number of events that take place as a function
of time. Here the interoccurrence times between successive events are indepen-
dent and identically distributed random variables. For instance, the events could
be the arrival of customers to a waiting line or the successive replacements of
light bulbs. Although renewal theory originated from the analysis of replacement
problems for components such as light bulbs, the theory has many applications to
quite a wide range of practical probability problems. In inventory, queueing and
reliability problems, the analysis is often based on an appropriate identification of
embedded renewal processes for the specific problem considered. For example, in
a queueing process the embedded events could be the arrival of customers who
find the system empty, or in an inventory process the embedded events could be
the replenishment of stock when the inventory position drops to the reorder point
or below it.
Formally, let X 1 , X 2 , . . . be a sequence 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 n denotes the interoccurrence time between the (n − 1)th
and nth event in some specific probability problem. Define

n

S 0 = 0 and S n = X i , n = 1, 2, . . . .
i=1
Then S n is the epoch at which the nth event occurs. For each t ≥ 0, let
N(t) = the largest integer n ≥ 0 for which S n ≤ t.

Then the random variable N(t) represents the number of events up to time t.

Definition 2.1.1 The counting process {N(t), t ≥ 0} is called the renewal process
generated by the interoccurrence times X 1 , X 2 , . . . .

It is said that a renewal occurs at time t if S n = t for some n. For each t ≥ 0, the
number of renewals up to time t is finite with probability 1. This is an immediate
consequence of the strong law of large numbers stating that S n /n → E(X 1 ) with
probability 1 as n → ∞ and thus S n ≤ t only for finitely many n. The Poisson
process is a special case of a renewal process. Here we give some other examples
of a renewal process.