Page 340 - A First Course In Stochastic Models

EXERCISES 335

where F on (t) is the probability distribution function of the on-time and m(x) is the renewal
density for the renewal process in which the interoccurrence time is distributed as the sum
of an on-time and an off-time. Express the Laplace transform of P on (t) in terms of the
Laplace transforms of the on-time density and the off-time density. Give an expression for
t
the Laplace transform of E(U(t)) = 0 P on (u) du, where the random variable U(t) denotes
the cumulative on-time during [0, t].
(b) Use the result of (a) to verify that
[t/D] k
(t − kD)
−(t−kD)/µ
P on (t) = e , t ≥ 0,
k
µ k!
k=0
when the off-time is a constant D and the on-time has an exponential distribution with mean
1/µ.
8.5 Consider the alternating renewal process in which both the on-times and the off-times
have a general probability distribution. Assuming that an on-time starts at epoch 0, denote
by the random variable U(t) the cumulative amount of time the system is in the on-state
during [0, t].
(a) Use Theorem 2.2.5 to verify that U(t) is asymptotically normally distributed with
2
3
2
2
mean µ on t/(µ on + µ off ) and variance (µ σ 2 + µ σ )t/(µ on + µ off ) , where µ on (µ off )
on off off on
2
2
and σ (σ ) denote the mean and the variance of the on-time (off-time).
on off
(b) Derive a renewal equation for E(U(t)). Assuming that the on-time distribution and
the off-time distribution are not both arithmetic, prove that
2 2
µ on σ − µ off σ
µ on off on µ on µ off
lim E(U(t)) − t = + .
t→∞ µ on + µ off 2(µ on + µ off ) 2 2(µ on + µ off )
8.6 Consider the alternating renewal process in which both the on-times and the off-times
have a general probability distribution. Let µ on and µ off denote the respective means of an
on-time and an off-time. Denote by G on (x, t) the joint probability that the system is on at
time t and that the residual on-time at time t is no more than x. Derive a renewal equation
for G on (x, t). Assuming that the distribution functions of the on-time and off-time are not
both arithmetic, prove that
µ on 1 x
lim G on (x, t) = × [1 − F on (y)] dy, x ≥ 0,
t→∞ µ on + µ off µ on 0
where F on (x) denotes the probability distribution function of the on-time.
8.7 Consider the alternating renewal process. Let F on (t) and F off (t) denote the probability
distribution functions of the on-time and the off-time. Assume that these distribution func-
tions have respective densities f on (t) and f off (t). For any fixed t > 0, define H on (t, x)
(H off (t, x)) as the probability that the cumulative on-time during [0, t] is no more than x
given that an on-time (off-time) starts at epoch 0.
(a) Argue the integral equations
x
H on (t, x) = H off (t − u, x − u)f on (u) du, 0 ≤ x < t
0
t−x
H off (t, x) = 1 − F off (t − x) + H on (t − u, x)f off (u) du, 0 ≤ x < t.
0