Page 322 - A First Course In Stochastic Models

ASYMPTOTIC EXPANSIONS 317

where

∞
µ 2
a(t) = 2 {1 − F(x)} dx
2µ 1 t
1 ∞ ∞
+ t {1 − F(x)} dx − x{1 − F(x)} dx .
µ 1 t t
The function a(t) is the sum of two monotone functions. Each of the two terms is
integrable. Using formula (A.8) in Appendix A, we find after some algebra

2
µ 2 µ 3
∞
a(t) dt = 2 − .
0 4µ 1 6µ 1
Next, by applying the key renewal theorem to (8.2.7), we obtain (8.2.5).
The asymptotic expansions in Theorem 8.2.3 are very useful. They are accurate
for practical purposes already for moderate values of t. Asymptotic expansions for
the second moment of N(t) are discussed in Exercise 8.3. An immediate conse-
quence of the relations (8.1.2) and (8.1.7) and Theorem 8.2.3 is the following result
for the excess life γ t .

Corollary 8.2.4 Suppose F(x) is non-arithmetic. Then
µ 2 2 µ 3
lim E(γ t ) = and lim E(γ ) = .
t
t→∞ 2µ 1 t→∞ 3µ 1
Next we discuss the limiting distribution of the excess life γ t for t → ∞.

Theorem 8.2.5 Suppose F(x) is non-arithmetic. Then
1 x
lim P {γ t ≤ x} = {1 − F(y)} dy, x ≥ 0. (8.2.8)
t→∞ µ 1 0
Proof For fixed u ≥ 0, define Z(t) = P {γ t > u}, t ≥ 0. By conditioning on the
time of the first renewal, we derive a renewal equation for Z(t). Since after each
renewal the renewal process probabilistically starts over, it follows that

P {γ t−x > u} if x ≤ t,
P {γ t > u | X 1 = x} = 0 if t < x ≤ t + u,
1 if x > t + u.

By the law of total probability,
∞

P {γ t > u} = P {γ t > u | X 1 = x}f (x) dx.
0
This yields the renewal equation
t
Z(t) = 1 − F(t + u) + Z(t − x)f (x) dx, t ≥ 0.
0