Page 81 - A First Course In Stochastic Models

72 RENEWAL-REWARD PROCESSES

(a) Use the renewal-reward model to find the long-run fraction of time the process {X(t)}
is in state i for i = 1, 2. Does lim t→∞ P{X(t) = i} exist for i = 1, 2? If so, what is
the limit?
(b) Consider a renewal process in which the interoccurrence times have an H 2 distribution
with density p 1 λ 1 e −λ 1 t + p 2 λ 2 e −λ 2 t . Argue that
p 1 λ 2 −λ 1 x p 2 λ 1 −λ 2 x
lim P{γ t > x} = e + e , x ≥ 0.
t→∞ p 1 λ 2 + p 2 λ 1 p 1 λ 2 + p 2 λ 1

2.7 Consider a renewal process with Erlang (r, λ) distributed interoccurrence times. Let the
probability p j (t) be defined as in part (b) of Exercise 2.5. Use the renewal-reward model
to argue that lim t→∞ p j (t) = 1/r for j = 1, . . . , r and conclude that
r j−1 k
1 −λx (λx)
lim P{γ t > x} = e , x ≥ 0.
t→∞ r k!
j=1 k=0
Generalize these results when the interoccurrence time is distributed as an Erlang (j, λ)
random variable with probability β j for j = 1, . . . , r.
2.8 Consider the E r /D/∞ queueing system with infinitely many servers. Customers arrive
according to a renewal process in which the interoccurence times have an Erlang (r,λ)
distribution and the service time of each customer is a constant D. Each newly arriving
customer gets immediately assigned a free server. Let p n (t) denote the probability that n
servers will be busy at time t. Use an appropriate conditioning argument to verify that
r j−1 k
1 −µD (µD)
lim p 0 (t) = e
t→∞ r k!
j=1 k=0
r r−1 r−j+1+(n−1)r+k
1 −µD (µD)
lim p n (t) = e , n ≥ 1.
t→∞ r (r − j + 1 + (n − 1)r + k)!
j=1 k=0
(Hint: the only customers present at time t are those customers who have arrived in
(t − D, t].)
2.9 The lifetime of a street lamp has a given probability distribution function F(x) with
probability density f (x). The street lamp is replaced by a new one upon failure or upon
reaching the critical age T , whichever occurs first. A cost of c f > 0 is incurred for each
failure replacement and a cost of c p > 0 for each preventive replacement, where c p < c f .
The lifetimes of the street lamps are independent of each other.
(a) Define a regenerative process and specify its regeneration epochs.
(b) Show that the long-run average cost per time unit under the age-replacement rule
T
equals g(T ) = [c p + (c f − c p )F(T )]/ 0 {1 − F(x)} dx.
(c) Verify that the optimal value of T satisfies g(T ) = (c f − c p )r(T ), where r(x) is the
failure rate function of the lifetime.
2.10 Consider the M/G/∞ queue from Section 1.1.3 again. Let the random variable L be
the length of a busy period. A busy period begins when an arrival finds the system empty
and ends when there are no longer any customers in the system. Use the result (2.2.1) to
λµ
argue that E(L) = (e − 1)/λ.
2.11 Consider an electronic system having n identical components that operate independently
of each other. If a component breaks down, it goes immediately into repair. There are ample