Page 339 - A First Course In Stochastic Models

334 ADVANCED RENEWAL THEORY

EXERCISES
8.1 Use Laplace transform theory to verify the following results:
(a) The renewal function associated with the interoccurrence-time density f (x) =
pλ 1 e −λ 1 x + (1 − p)λ 2 e −λ 2 x is
x 1 2 −(pλ 1 +(1−p)λ 2 )x
M(x) = + (c − 1)[1 − e ], x ≥ 0,
X
E(X) 2
where the random variable X denotes the interoccurrence time.
(b) The renewal function associated with the interoccurrence-time density f (x) =
2
pλe −λx + (1 − p)λ xe −λx is
x 1 2 −λ(2−p)x
M(x) = + (c − 1)[1 − e ], x ≥ 0.
X
E(X) 2
2
8.2 For a renewal process let M 2 (t) = E[N (t)] be the second moment of the number of
renewals up to time t. Verify that M 2 (t) satisfies the renewal equation
t
M 2 (t) = 2M(t) − F(t) + M 2 (t − x)f (x) dx, t ≥ 0,
0
where f (x) is the probability density of the interoccurrence times. Next verify that

2 2
2 t 2µ 2 3 3µ 2 2µ 3 3µ 2
lim E[N (t)] − + − t = − − + 1,
t→∞ µ 2 µ 3 µ 1 2µ 4 3µ 3 2µ 2
1 1 1 1 1
where µ k denotes the kth moment of the density f (x). Also, prove that

t t 3 3 3µ 2
2
2
lim E[N (y)] dy − + µ 2 − t + 2 − 2µ 3 − 3µ 2 + 1 t
t→∞ 0 3µ 3 1 µ 3 1 2µ 1 2µ 4 1 3µ 3 1 2µ 2 1
3 2
µ 4 µ 2 µ 3 µ 2 µ 3 3µ 2
= 3 − 4 + 5 + 2 − 3 .
6µ µ µ 2µ 4µ
1 1 1 1 1
8.3 Consider a renewal process generated by the interoccurrence times X 1 , X 2 , . . . with
mean µ 1 and second moment µ 2 . Let L 1 be the length of the interoccurrence time covering
epoch t. Derive a renewal equation for E(L t ). Verify the following results:
(a) E(L t ) = 2µ 1 − µ 1 e −t/µ 1 for all t when the X i are exponentially distributed.
(b) lim t→∞ E(L t ) = µ 2 /µ 1 when the X i are continuously distributed.
Also derive a renewal equation for P{L t > x}. Prove that the limiting distribution of L t
has the density xf (x)/µ 1 when the X i have a probability density f (x). Can you give a
heuristic explanation of why E(L t ) ≥ µ 1 ?
8.4 Consider an alternating renewal process in which the on-times and the off-times are
generally distributed. The on-times are assumed to have a probability density. Let P on (t)
be the probability that the process is in the on-state at time t given that an on-time starts at
epoch 0.
(a) Prove that

t
P on (t) = 1 − F on (t) + [1 − F on (t − x)]m(x) dx, t ≥ 0,
0