Page 37 - A First Course In Stochastic Models

28 THE POISSON PROCESS AND RELATED PROCESSES

e −r 1 (z)t − e −r 2 (z)t e −r 1 (z)t − e −r 2 (z)t
∗ ∗
P (z, t) = ω 1 and P (z, t) = ω 2 ,
21
12
r 2 (z) − r 1 (z) r 2 (z) − r 1 (z)
where
1
r 1,2 (z) = (λ 1 (1 − z) + ω 1 + λ 2 (1 − z) + ω 2 )
2
1 2
± {λ 1 (1 − z) + ω 1 + λ 2 (1 − z) + ω 2 }
2
1/2

− 4{(λ 1 (1 − z) + ω 1 )(λ 2 (1 − z) + ω 2 ) − ω 1 ω 2 } .
It is a matter of straightforward but tedious algebra to derive these expressions. The
probabilities P ij (k, t) can be readily computed from these expressions by applying
the discrete FFT method.

EXERCISES
1.1 A businessman parks his car illegally in the streets of Amsterdam twice a day for a
period of exactly one hour. Parking surveillances occur according to a Poisson process with
an average of λ passes per hour. What is the probability of the businessman getting a fine
on a given day?
1.2 At a shuttle station, passengers arrive according to a Poisson process with rate λ. A
shuttle departs as soon as seven passengers have arrived. There is an ample number of
shuttles at the station.
(a) What is the conditional distribution of the time a customer has to wait until departure
when upon arrival the customer finds j other customers waiting for j = 0, 1, . . . , 6?
(b) What is the probability that the nth customer will not have to wait? (Hint: distinguish
between the case that n is a multiple of 7 and the case that n is not a multiple of 7.)
(c) What is the long-run fraction of customers who, upon arrival, find j other customers
waiting for j = 0, 1, . . . 6?
(d) What is the long-run fraction of customers who wait more than x time units until
departure?
1.3 Answer (a), (b) and (c) in Exercise 1.2 assuming that the interarrival times of the
customers have an Erlang (2, λ) distribution.
1.4 You leave work at random times between 5 pm and 6 pm to take the bus home. Bus
numbers 1 and 3 bring you home. You take the first bus that arrives. Bus number 1 arrives
exactly every 10 minutes, whereas bus number 3 arrives according to a Poisson process
with the same average frequency as bus number 1. What is the probability that you take bus
number 1 home on a given day? Can you explain why this probability is larger than 1/2?
1.5 You wish to cross a one-way traffic road on which cars drive at a constant speed and
pass according to a Poisson process with rate λ. You can only cross the road when no car
has come round the corner for c time units. What is the probability of the number of passing
cars before you can cross the road when you arrive at a random moment? What property of
the Poisson process do you use?
1.6 Consider a Poisson arrival process with rate λ. For each fixed t > 0, define the random
variable δ t as the time elapsed since the last arrival before or at time t (assume that an
arrival occurs at epoch 0).
(a) Show that the random variable δ t has a truncated exponential distribution: P{δ t =
−λt −λx
t} = e and P{δ t > x} = e for 0 ≤ x < t.