STAT3021-Assignment 2
1. There are 6 phones in an insurance company working independently. The calls to
each phone are assumed to follow a Possion process and it is known that, during a
working day, phones 1, 2, 3 and 4 are called with rate 5 per hour each, and phones
5 and 6 with rate 10 per hour each. The working day starts at 9am.
(a) Find the probability that there will be no calls to phone 1 from 2 to 2:30pm.
(b) Find the distribution of the time of the first call to the office.
(c) Find the probability that the first call is to phone 1.
(d) Find the probability that, in the first 10 mins from 10am, only phone 1 receives
2 calls and no other phones receive calls.
(e) Suppose that 20 calls have arrived in one hour. Find the probability that only
2 calls are to phone 1.
(f) Find the average interarrival time between two new calls to the company.
2. In an insurance company, the number of claims from 8am to 5pm is a nonhomoge-
neous Poisson process with intensity function:
λ(t) =

1 if 0 ≤ t < 1
2, if 1 ≤ t < 2
3, if t ≥ 2,
Suppose that the amount of each claim is uniformly distributed on [$200, $800],
the amounts of claims are independent and are also independent of the number of
claims.
(a) Find the expected value and the variance for the total amount received from
all claims in t hours from 8am.
(b) Let P be the number of claims with claim amount less than $500 by 11am.
Find the distribution of P
(c) Let Q be the number of claims with claim amount greater than $500 by time
2pm. Given Q = 4, find the expected value of P .
3. Consider a continuous MC {Xt}t≥0 with the following generator matrix:
Q =
−α α 00 −β β
1/2 1/2 −1
 , α, β > 0.
(a) Find the corresponding jump matrix P and the average holding time for each
state.
(b) Find the stationary distribution of {Xt}t≥0.
(c) Find the limit distribution of {Xt}t≥0.
(d) Find limt→∞ P (t), where P (t) is the transition matrix.
4. A rent-a-car maintenance facility has capabilities for routine maintenance for only
one car at a time. Cars arrive for this routine maintenance according to the Poisson
process at the mean rate of 3 per day, and the service time to perform this main-
tenance has the exponential distribution with the mean of 7/24 days. It costs the
company a fixed $150 a day to operate the facility and the company estimates a
loss $10 per day in profit for each car that is tied up in the shop. The company,
by changing certain procedures and hiring faster mechanics, can decrease the mean
service time to 1/4 day. This also increase their operating costs. Up to what value
can the operating cost increase before it is no longer economically attractive to make
the change?
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