Page 315 - A First Course In Stochastic Models

310 ADVANCED RENEWAL THEORY

t
Z(t) = a(t) + a(t − x)m(x) dx, t ≥ 0, (8.1.6)
0
where the renewal density m(x) denotes the derivative of M(x).

Proof We give only a sketch of the proof. The proof is similar to the proof of
the second part of Theorem 8.1.1. Substituting the equation (8.1.5) repeatedly into
itself yields
n t t

Z(t) = a(t) + a(t − x)f k (x) dx + Z(t − x)f n+1 (x) dx.
0 0
k=1
Next, by letting n → ∞, the desired result readily follows. It is left to the reader
to verify that the various mathematical operations are allowed.

The integral equation (8.1.5) is called the renewal equation. This important
equation arises in many applied probability problems. As an application of The-
orem 8.1.2, we derive an expression for the second moment of the excess life at
time t.
2
Lemma 8.1.3 Assuming that µ 2 = E(X ) is finite,
1
t
2 2
E(γ ) = µ 2 [1 + M(t)] − 2µ 1 t + M(x) dx + t , t ≥ 0. (8.1.7)
t
0
Proof Fix t ≥ 0. Given that the epoch of the first renewal is x, the random
variable γ t is distributed as γ t−x when x ≤ t and γ t equals x − t otherwise. Thus
∞

2 2
E(γ ) = E(γ | X 1 = x)f (x) dx
t
t
0
t ∞
2
= E(γ 2 )f (x) dx + (x − t) f (x) dx.
t−x
0 t
2 ∞ 2
Hence, by letting Z(t) = E(γ ) and a(t) = t (x − t) f (x) dx, we obtain a
t
renewal equation of the form (8.1.5). Next it is a question of tedious algebra to
derive (8.1.7) from (8.1.6). The details of the derivation are omitted.
8.1.2 Computation of the Renewal Function
The following tools are available for the numerical computation of the renewal
function:
(a) the series representation,
(b) numerical Laplace inversion,

(c) discretization of the renewal equation.