Page 39 - A First Course In Stochastic Models

30 THE POISSON PROCESS AND RELATED PROCESSES

(b) Each bus brings 50, 75 or 100 tourists with respective probabilities 1 4 , 1 2 and 1 4 .
Calculate a normal approximation to the probability that more than 1000 bus tourists are in
Edam at 4 o’clock in the afternoon. (Hint: the number of bus tourists is distributed as the
convolution of two compound Poisson distributions.)
1.14 Batches of containers arrive at a stockyard according to a Poisson process with rate
λ. The batch sizes are independent random variables having a common discrete probability
distribution {β j , j = 1, 2, . . . } with finite second moment. The stockyard has ample space to
store any number of containers. The containers are temporarily stored at the stockyard. The
holding times of the containers at the stockyard are independent random variables having a
general probability distribution function B(x) with finite mean µ. Also, the holding times
of containers from the same batch are independent of each other. This model is called
X ∞ j
β
the batch-arrival M /G/∞ queue with individual service. Let β (z) = j=1 j z be the
generating function of the batch size and let {p j } denote the limiting distribution of the
number of the containers present at the stockyard.
∞ j
p
(a) Use Theorem 1.1.5 to prove that P (z) = j=0 j z is given by

∞
P (z) = exp −λ [1 − β ((1 − z)B(x) + z)] dx .
0
(b) Verify that the mean m and the variance ν of the limiting distribution of the number
of containers at the stockyard are given by
∞

2
m = λE(X)µ and ν = λE(X)µ + λE [X(X − 1)] {1 − B (x)} dx,
0
where the random variable X has the batch-size distribution {β j }.
(c) Investigate how good the approximation to {p j } performs when a negative binomial
distribution is fitted to the mean m and the variance ν. Verify that this approximation is
exact when the service times are exponentially distributed and the batch size is geometrically
distributed with mean β > 1.
1.15 Consider Exercise 1.14 assuming this time that containers from the same batch are
kept at the stockyard over the same holding time and are thus simultaneously removed. The
holding times for the various batches have a general distribution function B (x). This model
X
is called the batch-arrival M /G/∞ queue with group service.
(a) Argue that the limiting distribution {p j } of the number of containers present at the
stockyard is insensitive to the form of the holding-time distribution and requires only its
mean µ.
(b) Argue that the limiting distribution {p j } is a compound Poisson distribution with
generating function exp (−λD{1 − β(z)}) with D = µ.
1.16 In a certain region, traffic accidents occur according to a Poisson process. Calculate
the probability that exactly one accident has occurred on each day of some week when it is
given that seven accidents have occurred in that week. Can you explain why this probability
is so small?
1.17 Suppose calls arrive at a computer-controlled exchange according to a Poisson process
at a rate of 25 calls per second. Compute an approximate value for the probability that
during the busy hour there is some period of 3 seconds in which 125 or more calls arrive.
1.18 In any given year claims arrive at an insurance company according to a Poisson process
with an unknown parameter λ, where λ is the outcome of a gamma distribution with shape
parameter α and scale parameter β. Prove that the total number of claims during a given
year has a negative binomial distribution with parameters α and β/(β + 1).
1.19 Claims arrive at an insurance company according to a Poisson process with rate λ. The
claim sizes are independent random variables and have the common discrete distribution
k
a k = −α [k ln(1 − α)] −1 for k = 1, 2, . . . , where α is a constant between 0 and 1. Verify