Chapter
14 Practice Questions
14.148
A new shopping mall is considering setting up an information desk manned
by one employee. Based upon
information obtained from similar information desks, it is believed that people
will arrive at the desk at the rate of 15 per hour.
It takes an average of two minutes to answer a question. It is assumed
that arrivals are Poisson and answer times are exponentially distributed.
(a) Find the probability
that the employee is idle.
(b) Find the proportion
of the time that the employee is busy.
(c) Find the average
number of people receiving and waiting to receive some information.
(d) Find the average
number of people waiting in line to get some information.
(e) Find the average time
a person seeking information spends at the desk.
(f) Find the expected
time a person spends just waiting in line to have his question answered.
14.151
Sam the Vet is running a rabies vaccination clinic for dogs at the local
grade school. Sam can
"shoot" a dog every three minutes.
It is estimated that the dogs will arrive independently and randomly
throughout the day at a rate of one dog every six minutes, according to a
Poisson distribution. Also assume
that Sam's shooting times are exponentially distributed.
Find the:
(a) probability that Sam
is idle.
(b) proportion of the
time that Sam is busy.
(c) average number of
dogs being shot or waiting to be shot.
(d) average number of
dogs waiting to be shot.
(e) average time a dog
waits before getting shot.
(f) average amount (mean)
of time a dog spends between waiting in line and getting shot.
14.153
Calls arrive at the hotel switchboard at a rate of two per minute.
The average time to handle each of these is 15 seconds.
There is only one switchboard operator at the current time. The Poisson
and exponential distribution appear to be relevant in this situation.
(a) What is the
probability that the operator is busy?
(b) What is the average
time that a call must wait before reaching the operator?
(c) What is the average
number of calls waiting to be answered?
14.159
The new Providence shopping mall is considering setting up an information
desk manned by one employee. The
layout for this mall is quite complex, leading the mall manager to expect a
higher than normal arrival rate for persons seeking assistance.
It appears that a reasonable expectation is an arrival rate of
approximately 25 patrons per hour. Under
the original plan, the manager expected that it would take approximately four
minutes for the Information Desk employee to help the average person. By utilizing a new map and special guide signs, the manager
believes that the required service time can be reduced to an average of two
minutes per person. Assuming that
he implements the new map and guide signs:
(a) Find the probability
that the Information Desk employee is idle.
(b) Find the proportion
of the time that the Information Desk employee is busy.
(c) Find the average
number of people receiving and waiting to receive some information.
(d) Find the average
number of people waiting in line to get some information.
(e) Find the average time
a person seeking information spends waiting and at the desk.
(f) Find the expected
time a person spends just waiting in line to have his question answered.
14.162
Bank Boston now has a branch at Bryant College.
The branch is always busiest at the beginning of the college year when
freshmen and transfer students arrive on campus and open accounts.
This year, freshmen arrived at the office at a rate of 40 per day (8-hour
day). On average, it takes the Bank
Boston staff person about ten minutes to process each account application.
(a) What is the staff
person's utilization rate?
(b) What is the average
time a student has to wait before getting his application processed?
(c) What is the
probability that there are more than three students waiting in line?
(d) The
office has a total of five chairs for students, four for waiting, and one at the
service desk. Is it likely that any
students will have to stand?
14.163
Sam the Vet is running a rabies vaccination clinic for cats at the local
grade school. Sam can
"shoot" a cat every four minutes.
It is estimated that the cats will arrive independently and randomly
throughout the day at a rate of one cat every five minutes, according to a
Poisson distribution. Also assume
that Sam's shooting times are exponentially distributed.
(a) What is the probability that a cat will have to wait?
(b) On the average, how many cats will be in the waiting
room?
(c) If a cat has to wait more than 20 minutes, it will become
obnoxious. Is this likely to
present a serious problem?
14.161
The new Providence shopping mall has been considering setting up an
information desk manned by one employee. The
layout for this mall is quite complex, leading the mall manager to expect a
higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate
of approximately 25 patrons per hour. Under
the original plan, the manager expected that it would take approximately 4
minutes for the Information Desk employee to help the average person.
He has now come to realize that employing only a single person at the
information desk would lead to a very lengthy line – theoretically, an
infinite line! He has decided,
therefore, to employ two staff members at the information desk.
(a) Find the proportion of the time that the employees
are busy.
(b) Find the average number of people waiting in line
to get some information.
(c) Find the expected time a person spends just waiting
in line to have his question answered.
(d) If the manager has a goal that, for the average
patron, the time spent having one's question answered is less than half the time
spent waiting, has he met that goal?
(e) Assuming that the manager also has the goal that
the combined idle time for the two workers does not exceed 45 minutes in an 8
hour day has he met that goal?
(f) What would the arrival rate have to be for the manager to meet the 45 minute idle time goal?
Answers
for Practice for Chapters 14 Quantitative Methods