Page 80 - A First Course In Stochastic Models

EXERCISES 71

In Chapter 3 the method of embedded Markov chains will be used to derive an
explicit expression for the customer-average probabilities π j .

EXERCISES

2.1 A street lamp is replaced by a new one upon failure and upon scheduled times T, 2T, . . . .
There is always a replacement at the scheduled times regardless of the age of the street
lamp in use. The lifetimes of the street lamps are independent random variables and have a
common Erlang (2, µ) distribution. What is the expected number of street lamps used in a
scheduling interval?
2.2 The municipality of Gotham City has opened a depot for temporarily storing chemical
waste. The amount of waste brought in each week has a gamma distribution with given
shape parameter α and scale parameter λ. The amounts brought in during the successive
weeks are independent of each other.
(a) What is the expected number of weeks until the total amount of waste in the depot
exceeds the critical level L?
(b) Give an asymptotic estimate for the expected value from question (a).
2.3 Limousines depart from the railway station to the airport from the early morning till
late at night. The limousines leave from the railway station with independent interdeparture
times that are uniformly distributed between 10 and 20 minutes. Suppose you plan to arrive
at the railway station at 3 o’clock in the afternoon. What are the estimates for the mean and
the standard deviation of your waiting time at the railway station until a limousine leaves
for the airport?
2.4 Consider the expression (2.1.4) for the renewal function M(t).
(a) Prove that for any k = 0, 1, . . .
∞
F k (t)F(t)

F n (t) ≤
1 − F(t)
n=k+1
for any t with F(t) < 1. (Hint: use P{X 1 +· · ·+X n ≤ t} ≤ P{X 1 +· · ·+X k ≤ t}P{X k+1 ≤
t} · · · P{X n ≤ t}.)
(b) Conclude that M(t) < ∞ for all t ≥ 0.
2.5 Consider a renewal process with Erlang (r, λ) distributed interoccurrence times. Use the
phase method to prove:
(a) For any t > 0,
∞ j
−λt (λt)
P{N(t) > k} = e , k = 0, 1, . . . .
j!
j=(k+1)r
(b) The excess variable γ t is Erlang (j, λ) distributed with probability

∞ kr−j
−λt (λt)
p j (t) = e , j = 1, . . . , r.
(kr − j)!
k=1
2.6 Consider a continuous-time stochastic process {X(t), t ≥ 0} that can assume only the
two states 1 and 2. If the process is currently in state i, it moves to the next state after an
exponentially distributed time with mean 1/λ i for i = 1, 2. The next state is state 1 with
probability p 1 and state 2 with probability p 2 = 1−p 1 irrespective of the past of the process.