Page 44 - A First Course In Stochastic Models

RENEWAL THEORY 35

Example 2.1.1 A replacement problem
Suppose we have an infinite supply of electric bulbs, where the burning times of
the bulbs are independent and identically distributed random variables. If the bulb
in use fails, it is immediately replaced by a new bulb. Let X i be the burning time of
the ith bulb, i = 1, 2, . . . . Then N(t) is the total number of bulbs to be replaced
up to time t.

Example 2.1.2 An inventory problem
Consider a periodic-review inventory system for which the demands for a single
product in the successive weeks t = 1, 2, . . . are independent random variables
having a common continuous distribution. Let X i be the demand in the ith week,
i = 1, 2, . . . . Then 1+N(u) is the number of weeks until depletion of the current
stock u.

2.1.1 The Renewal Function
An important role in renewal theory is played by the renewal function M(t) which
is defined by

M(t) = E[N(t)], t ≥ 0. (2.1.1)
For n = 1, 2, . . . , define the probability distribution function

F n (t) = P {S n ≤ t}, t ≥ 0.

Note that F 1 (t) = F(t). A basic relation is
N(t) ≥ n if and only if S n ≤ t. (2.1.2)

This relation implies that

P {N(t) ≥ n} = F n (t), n = 1, 2, . . . . (2.1.3)

Lemma 2.1.1 For any t ≥ 0,
∞

M(t) = F n (t). (2.1.4)
n=1
Proof Since for any non-negative integer-valued random variable N,

∞ ∞

E(N) = P {N > k} = P {N ≥ n},
k=0 n=1
the relation (2.1.4) is an immediate consequence of (2.1.3).