Page 313 - A First Course In Stochastic Models

308 ADVANCED RENEWAL THEORY

we have that 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. 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. Since F(0) < 1 the number of renewals up to time t is finite
with probability 1 for any t ≥ 0. The renewal function M(t) is defined by

M(t) = E[N(t)], t ≥ 0.

For n = 1, 2, . . . , define the probability distribution function F n (t) by

F n (t) = P {S n ≤ t}, t ≥ 0.
The function F n (t) is the n-fold convolution of F(t) with itself. Using the important
observation that N(t) ≥ n if and only if S n ≤ t, it was shown in Section 2.1 that

∞

E[N(t)] = F n (t), t ≥ 0. (8.1.1)
n=1
Moreover, it was established in Section 2.1 that M(t) < ∞ for all t ≥ 0. Another
important quantity introduced in Section 2.1 is the excess or residual life at time
t. This random variable is defined by
γ t = S N(t)+1 − t

and denotes the waiting time from time t onwards until the first occurrence of an
event after time t. Using Wald’s equation, it was shown in Section 2.1 that
E(γ t ) = µ 1 {1 + M(t)} − t. (8.1.2)

The following bounds apply to the renewal function:

t t µ 2
− 1 ≤ M(t) ≤ + 2 ,
µ 1 µ 1 µ
1
2
where µ 2 = E(X ). The left inequality is an immediate consequence of (8.1.2) and
1
the fact that γ t ≥ 0. The proof of the other inequality is demanding and lengthy.
The interested reader is referred to Lorden (1970).

8.1.1 The Renewal Equation
A useful characterization of the renewal function is provided by the so-called
renewal equation.