Page 341 - A First Course In Stochastic Models

336 ADVANCED RENEWAL THEORY

(b) By repeated substitution, verify that
∞
(n+1)∗ (n+1)∗
H on (t, x) = {F n∗ (t − x) − F (t − x)}F on (x), 0 ≤ x < t,
off off
n=0
∞
(n+1)∗
n∗
n∗
H off (t, x) = {F off (t − x) − F off (t − x)}F on (x), 0 ≤ x < t,
n=0
where F n∗ (x) denotes the n-fold convolution of a probability distribution of F(x) with
itself for n ≥ 1 and F 0∗ (x) = 1 for all x ≥ 0.
8.8 Consider Exercise 8.7 again. Define for any fixed t 0 > 0,

(t 0 , x) = lim P{the cumulative on-time during the time
t→∞
interval [t, t + t 0 ] is no more than x}
for 0 ≤ x < t 0 . Use results from Exercise 8.7 to argue that
(t 0 , x) is given by

∞
µ on (n+1)∗ e
n∗
n∗
{F off (t 0 − x) − F off (t 0 − x)}F on ∗ F on (x)
µ on + µ off
n=0
∞
µ off e e (n+1)∗ (n+1)∗
n∗
+ {F off ∗ F off (t 0 − x) − F off ∗ F off (t 0 − x)}F on (x)
µ on + µ off
n=0
µ off e
(t
+ {1 − F off 0 − x)}, 0 ≤ x < t 0 ,
µ on + µ off
e
where F (x) denotes the equilibrium excess distribution function of a probability distribution
function F(x) and A ∗ B(x) denotes the convolution of two distribution functions A(x) and
B(x).
8.9 Consider an age-replacement model in which preventive replacements are only possible
at special times. Opportunities for preventive replacements occur according to a Poisson
process with rate λ. The item is replaced by a new one upon failure or upon a preventive
replacement opportunity occurring when the age of the item is T or more, whichever occurs
first. The lifetime of the item has a probability density f (x). The cost of replacing the
item upon failure is c 0 and the cost of a preventive replacement is c 1 with 0 < c 1 < c 0 .
Determine the long-run average cost per time unit. This problem is motivated by Dekker
and Smeitink (1994).
8.10 A production machine gradually deteriorates in time. The machine has N possible work-
ing conditions 1, . . . , N which describe increasing degrees of deterioration. Here working
condition 1 represents a new system and working condition N represents a failed system.
If the system reaches the working condition i, it stays in this condition during an expo-
nentially distributed time with mean 1/µ for each i with 1 ≤ i < N. A change of the
working condition cannot be observed except for a failure which is detected immediately.
The machine is replaced by a new one upon failure or upon having worked during a time T ,
whichever occurs first. Each planned replacement involves a fixed cost of J 1 > 0, whereas
a replacement because of a failure involves a fixed cost of J 2 > 0. The replacement time
is negligible in both cases. Also, the system incurs an operating cost of a i > 0 for each
time unit the system is operating in working condition i. Use Lemma 1.1.4 to verify that