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40.520 Stochastic Models 2015
Homework 4
due on Thursday, 23rd April, 2015 (in class)
Answer any five of the following questions.
1. At time 0, cars are positioned along an infinite highway according to HPP(α). Assume that
the initial position of the nth car is Xn . Each car chooses a velocity independent of all other
cars and proceeds to travel at the fixed velocity. Call the velocities {Vn }, and assume these
are iid random variables with state space (−∞, ∞). A negative velocity means the car
travels to the left and a positive velocity means it travels to the right. Assume collisions are
impossible; cars can pass through each other rather than collide. Assume E|V1 | < ∞.
(a) Let N0 (·) be the Poisson process of initial positions and let Nt (·) denote the point
process describing positions at time t. What sort of process is Nt (·)?
(b) Let Tn be the
P time the nth car passes through 0, so that Tn satisfies Xn + Tn Vn = 0.
Show that
εTn is Poisson and find its mean measure.
2. Incidents of drug usage occur in the bathroom of Harry’s restaurant. The times of these
incidents constitute a Poisson process of rate two per hour. Harry is concerned about this
because of the possible effect in business and because he worries about how effective the
government’s war on drugs will be. Consequently, Harry visits the bathroom at points of
time which constitutes a Poisson process of rate one per hour. Assume this Poisson process
is independent of the Poisson process of drug incidents. Assume also that if a drug incident
has occurred, Harry detects it.
(a) On his first visit, what is the probability that Harry finds evidence of drug use; i.e.,
what is the probability that a drug incident occurs before Harry checks the bathroom
for the first time?
(b) What is the expected time until Harry detects evidence of drug use?
P
3. Consider the Poisson process N = k jk on (0, ∞] with points {jk }k≥1 and mean measure
µ(dx) = αx−α−1 dx for x > 0. Let Y1 = supk≥1 jk be the largest point and Y2 be the second
largest point. Compute the joint distribution of Y1 , Y2 .
4. M/M/∞ queue. Arrivals are Poisson with rate α, service times are exponential with
parameter b, and there are infinite number of servers. Let Q(t) be the number in the system
at time t. Give the generator matrix and then show by solving η 0 A = 0 that the stationary
and limiting distribution is Poisson.
5. Consider a birth and death process with birth rates {λn } and death rates {µn }. Starting at
state i, find the probability that the first k events are all births.
6. Consider a birth and death process X(t) with birth rates λn = λ and death rates µn = nµ;
let Xn be the embedded chain, and show that it has a stationary distribution
πn =
1
n
(1 + )ρn e−ρ
2(n!)
ρ
where ρ = µλ . Explain why this differs from the stationary distribution of X(t).
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