Page 38 - A First Course In Stochastic Models

EXERCISES 29

(b) Prove that the random variables γ t (= waiting time from time t until the next arrival)
and δ t are independent of each other by verifying P{γ t > u, δ t > v} = P{γ t > u}P{δ t > v}
for all u ≥ 0 and 0 ≤ v < t.
1.7 Suppose that fast and slow cars enter a one-way highway according to independent
Poisson processes with respective rates λ 1 and λ 2 . The length of the highway is L. A
fast car travels at a constant speed of s 1 and a slow car at a constant speed of s 2 with
s 2 < s 1 . When a fast car encounters a slower one, it cannot pass it and the car has to
reduce its speed to s 2 . Show that the long-run average travel time per fast car equals
L/s 2 − (1/λ 2 )[1 − exp (−λ 2 (L/s 2 − L/s 1 ))]. (Hint: tag a fast car and express its travel
time in terms of the time elapsed since the last slow car entered the highway.)
1.8 Let {N(t)} be a Poisson process with interarrival times X 1 , X 2 , . . . . Prove for any
t, s > 0 that for all n, k = 0, 1, . . .
P{N(t + s) − N(t) ≤ k, N(t) = n} = P{N(s) ≤ k}P{N(t) = n}.
In other words, the process has stationary and independent increments. (Hint: evaluate the
probability P{X 1 + · · · + X n ≤ t < X 1 + · · · + X n+1 , X 1 + · · · + X n+k+1 > t + s}.)
1.9 An information centre provides services in a bilingual environment. Requests for service
arrive by telephone. Major language service requests and minor language service requests
arrive according to independent Poisson processes with respective rates of λ 1 and λ 2 requests
per hour. The service time of each request is exponentially distributed with a mean of 1/µ 1
minutes for a major language request and a mean of 1/µ 2 minutes for a minor language
request.
(a) What is the probability that in the next hour a total of n service requests will arrive?
(b) What is the probability density of the service time of an arbitrarily chosen service
request?
1.10 Short-term parkers and long-term parkers arrive at a parking lot according to indepen-
dent Poisson processes with respective rates λ 1 and λ 2 . The parking times of the customers
are independent of each other. The parking time of a short-term parker has a uniform dis-
tribution on [a 1 , b 1 ] and that of a long-term parker has a uniform distribution on [a 2 , b 2 ].
The parking lot has ample capacity.
(a) What is the mean parking time of an arriving car?
(b) What is the probability distribution of the number of occupied parking spots at any
time t > b 2 ?
1.11 Oil tankers with world’s largest harbour Rotterdam as destination leave from harbours
in the Middle East according to a Poisson process with an average of two tankers per day.
The sailing time to Rotterdam has a gamma distribution with an expected value of 10 days
and a standard deviation of 4 days. What is the probability distribution of the number of oil
tankers that are under way from the Middle East to Rotterdam at an arbitrary point in time?
1.12 Customers with items to repair arrive at a repair facility according to a Poisson process
with rate λ. The repair time of an item has a uniform distribution on [a, b]. There are ample
repair facilities so that each defective item immediately enters repair. The exact repair time
can be determined upon arrival of the item. If the repair time of an item takes longer than
τ time units with τ a given number between a and b, then the customer gets a loaner for
the defective item until the item returns from repair. A sufficiently large supply of loaners
is available. What is the average number of loaners which are out?
1.13 On a summer day, buses with tourists arrive in the picturesque village of Edam accord-
ing to a Poisson process with an average of five buses per hour. The village of Edam is
world famous for its cheese. Each bus stays either one hour or two hours in Edam with
equal probabilities.
(a) What is the probability distribution of the number of tourist buses in Edam at 4 o’clock
in the afternoon?