CHAPTER 9
QUEUING MODELS
TRUE/FALSE QUESTIONS
1. One method for reducing the randomness of customer arrivals at
a retail business is to allow for customer appointments.
2. Random arrival processes must be modeled using continuous
3. Airline passenger arrivals at U.S. Customs counters, at a
port-of-entry airport, would not likely be modeled as Poisson due to
4. The size of the population of potential customers may have an
impact on the validity of the Poisson arrival pattern assumption.
5. “Jockeying” occurs when a customer in the waiting line gets
upset with the time the process is taking and leaves the system.
6. The priority rule chosen for determining the next customer to
be served affects both waiting time variance and average customer
7. A Poisson distribution with mean λ is equivalent to an
8. μ, in the service segment of a queuing process, is the average
9. The steady state service measure formulas for the M/G/k/k
10. Although service times may be relatively constant at
sequential work stations in an assembly line, the line still may be
11. The optimal situation (in terms of minimizing the average
number of customers in the queue) in an M/M/1 queuing system is when
12. The probability distribution for the arrival process can be
estimated if we know either the time between customer arrivals or
13. If the service times are not memoryless, then the Erlangian
distribution might be a better basis than the exponential
14. The average waiting time in the system is less than the sum of
the average waiting time in the queue plus the average service time
because some customers do not have to wait in the queue.
15. For an M/M/k queue to reach steady state, the service rate for
each server μ must be greater than the arrival rate λ.
MULTIPLE CHOICE QUESTIONS
1. Necessary assumptions underlying a Poisson arrival process do
not include:
a. orderliness.
b. homogeneity.
c. independence.
d. stationarity.
2. Which of the following would best be characterized as a
Poisson arrival process?
a. Football game attendees,
b. Tax returns received by a regional IRS office.
c. Incoming phone calls to a business switchboard.
d. Ladies visiting a hair salon.
3. In order to achieve steady state performance in a queuing
system, the sum of the effective service rates of all servers must:
4. “Balking” is:
a. refusing service.
b. leaving the queue.
c. refusing to enter the queue.
d. refusing to leave after service.
5. Where would one most likely face a tandem queue?
a. A local post office.
b. A supermarket.
c. A cafeteria.
d. A bank.
6. If a service facility changes from a first-come, first-served
basis to a random basis in selecting the next customer to be served,
one should expect average customer waiting time to:
a. decrease slightly.
b. remain constant.
c. increase slightly.
d. increase dramatically.
7. In the exponential distribution of service times, the mean
customer service time equals the:
a. variance of customer service times.
b. standard deviation of customer service times.
c. median customer service time.
d. most likely customer service time.
8. In a situation where the distribution of service times is
assumed exponential, suppose the probability that service time is
under five minutes is 0.40. If a given customer has already had five
minutes of service, the probability that he/she will have service
totally completed in less than ten minutes, is:
a. 0.40.
b. greater than 0.40.
c. less than 0.40.
d. indeterminate.
9. A Markovian queuing process has a(n) __________ arrival
pattern, and a(n)__________ service pattern.
a. Poisson; Poisson.
b. Poisson; exponential.
c. exponential; Poisson.
d. exponential; exponential.
10. The exponential distribution is:
a. generally discrete.
b. never symmetrical.
c. usually symmetrical.
d. not related to the Poisson distribution.
11. Suppose that an office has one secretary who can put up to two
callers on hold while speaking to a third caller. (If two callers
are on hold, additional callers will get a busy signal and will not
call back.) If the arrival rate of calls follows a Poisson
distribution with a mean rate of 20 per hour and the average length
of a telephone conversation is 2 minutes, the average number of
callers who will be on hold is approximately:
a. 2.0.
b. 1.3333
c. 1.0154.
d. .4308.
12. Suppose a penny arcade worker is in charge of keeping ten
machines in operation. The failure rate of each machine follows an
exponential distribution with a mean time of ten hours and the time
required to repair each machine follows an exponential distribution
with a mean time of thirty minutes. Over the long run, approximately
what percentage of machines, on average, will be operating?
a. 76.5%
b. 92.4%
c. 53.8%
d. 46.2%
13. Ray’s Barber Shop has 3 barber’s chairs, and 8 seats for
waiting customers. The greatest number of people ever waiting for a
haircut, in Ray’s experience, is 6. For analytical purposes, then,
Ray’s queue length is:
a. infinite.
b. 6.
c. 8.
d. 11.
14. You go to your local hospital for your complete annual
physical exam. From a queuing standpoint, you are facing:
a. single server; single queue.
b. multiple servers; single queue.
c. multiple servers; multiple queues.
d. tandem queues.
15. In queuing analysis, the exponential distribution is a special
case of which other distribution?
a. Erlangian.
b. Poisson.
c. normal.
d. Markovian.
16. Which of the following is a basic component of a queuing
system?
a. Poisson distribution.
b. Waiting in a queue.
c. Priority rules.
d. Exponential distribution.
17. Homogeneity means:
a. all customers arrive according to the same pattern and
receive the same service.
b. past service time does not affect future service time.
c. processes reach steady state.
d. all servers are available as long as the queue
functions.
18. You are studying service times at Derman’s Department Store,
which is open 7 days a week from 10:00 AM to 9:00 PM. Why might you
ignore data from 10:00 to 10:30 AM?
a. Insufficient sample size.
b. Start up bias.
c. The doors do not open at exactly 10:00 AM.
d. The data do not fit the hypothesis.
19. Which of the following is not a common steady state
performance measure?
a. The probability that a customer does not have to wait in
the queue.
b. The average number of customers who do not have to wait
in the queue.
c. The probability all servers are idle.
d. The average time a customer spends between entering and
leaving the system.
20. The Pollaczek-Khintchine formula comes from the study of:
a. M/G/1 queues.
b. simulation.
c. M/M/1 queues.
d. Markov chains.
SHORT ANSWER QUESTIONS
1. If λ is the non–constant average arrival rate, and μ (or kμ
with multiple servers) is the average service rate, why does a
queuing system not approach maximum efficiency when these two values
are approximately the same?
2. In an M/M/1 queuing system, λ = 6, and the system is idle 40%
of the time. What is the average service rate, μ?
3. A local dry cleaning establishment is open from 7 a.m. to 7
p.m., weekdays. The average arrival rate of customers is 15 per
hour, both from 7 to 10 a.m., and from 4 to 7 p.m. In between, it is
9 per hour. In all time periods, the Poisson assumption for the
arrival process seems to be valid. Can this be treated as a queuing
system, with λ = 12 ([15 + 9]/2)?
4. What does the Excel formula EXPONDIST(A10,B3,FALSE) return?
5. Using Little’s formula, if we know the mean arrival rate, mean
service rate, and the average time a customer spends in the queue
(Wq), what else can we calculate?
6. Explain Kendall’s notation: G/M/3/10/20.
7. If the standard deviation of service time in an M/G/1 queue is
zero, what type of system does it become? What if σ = 1/μ?
8. What is the formula for the effective arrival rate of an
M/M/k/F queue?
9. What is the Kendall notation for a queue with
maximum queue length equal to the number of servers;
multiple servers working at the same, non-exponential rate;
and customer interarrival time exponential.
10. For many queue types, Pw, the probability a customer must wait
for service, = ρ, the server utilization rate. For what types of
queues is this not true?
FORMULATION/SOLUTION/ANALYSIS QUESTIONS
1. Based upon the experience gathered at the bank’s other
locations, the following is thought to be an accurate relative
frequency distribution of customer service times (to the nearest
minute) at the bank’s outside walk-up window, at the branch in your
neighborhood:
Minutes (y) f(y)
1 .15
2 .25 Mean: 3.00
3 .25
4 .20 Variance: 1.90
5 .10
6 .05
7 or more .00
Pedestrian arrivals at the walk-up window are assumed to be
Poisson-distributed, with an average of 0.25 arrivals each minute
(i.e., an average of one customer every four minutes):
Arrivals/Min (x) f(x)
0 .7788
1 .1947
2 .0243 Mean: 0.25
3 .0020
4 .0002 Variance: 0.25
5 or more .0000
What is the average number of customers waiting in line, Lq?
2. Harry and Larry have opened up an automated carwash near the
edge of town. There is a single service lane, and cars line up in a
single line to enter the carwash. No special services are offered,
and the time required for each vehicle to go through the facility is
exactly three minutes. Throughout the day, customers arrive
independently and largely at random at an average rate of twelve
per hour. Because of the location, even when the queue is full,
potential customers may line up in the adjacent side street.
A. What percentage of time is the carwash idle?
C. What will be the average elapsed time between when a vehicle
enters and leaves the system?
Suppose Harry and Larry offer a range of services: double washing,
waxing, interior cleaning, etc., and the mean service time is now
exponentially distributed with a mean of four minutes.
D. What percentage of time is the carwash idle?
E. What is the average number of waiting vehicles?
F. What will be the average elapsed time between when a vehicle
3. Ajax, Inc. specializes in the maintenance and repair of all
types of electronic products. Tools and test equipment needed by its
many skilled technicians on various diverse jobs must be drawn from
and returned to a centrally-located tool room. Technicians arrive at
the tool room at the rate of 20 per hour according to a Poisson
process and earn $16 per hour per person. The company can hire tool
room clerks at $10 per hour. It is estimated that it takes an
average of 4 minutes for a clerk to serve a technician and service
time follows an exponential distribution. Determine the optimal
number of tool room clerks that Ajax should hire.
4. The new community of Lemon Heights is planning to set up a
paramedic station. It is estimated that calls will come into this
station according to a Poisson distribution, and the station
receives an average of twenty calls a day. The time an ambulance is
out responding to a call follows an exponential distribution with a
mean time of one hour and thirty minutes. If no ambulance is
available, an ambulance from a nearby town will be dispatched, but
this will significantly increase the response time. Due to the
potentially tragic consequences associated with not having an
ambulance readily available when a call comes in, the city council
has mandated that the probability of this happening should be no
more than .005. Determine how many ambulances the paramedic station
should purchase.
5. The Red Rock School District has six buses in its fleet.
Maintenance of the fleet is handled by Joe Clem, a mechanic who
works for the district. The time between bus failures follows an
exponential distribution with a mean of twenty days. During the time
a bus is out of commission, the district must lease another bus at a
cost of $80 per day.
Joe has put in for retirement and the district is considering hiring
either Tom Meyers or Andy Johnson. Based on a skills assessment
test, it is estimated that Tom can repair a bus in an average of 10
hours, while Andy will take an average of 8 hours. Tom wants a
salary equivalent to $160 per working day, while Andy wants a salary
equivalent to $170 per working day. A working day lasts 8 hours, and
there are 200 working days per year. Which employee should the
district hire? Give your reasons.
6. The computer help desk at Averill University receives an
average of 40 calls per hour and calls come in according to a
Poisson distribution. The average time a technician takes to
diagnose a problem is three minutes and twenty seconds, and the
service time follows an exponential distribution. For 60% of the
callers the diagnosis is satisfactory, but for 40% of the callers,
the technician must transfer the call to a specialist. The time a
caller speaks to a specialist follows an exponential distribution
with a mean of five minutes. For ninety five percent of callers
transferred to a specialist, the problem is solved, but five percent
of callers will need an on-site visit from a service technician.
The average time a technician takes to fix a computer on-site is
forty minutes with a standard deviation of ten minutes.
The help desk operation has three technicians answering the incoming
calls, two specialists handling calls, and one on-site technician.
If a caller needs to have on-site service, determine the average
time it will take to have the service completed form the time the
initial call is made.
7. Customers arrive at the First Fidelity Bank branch on Friday
afternoon according to a Poisson process at a mean rate of 45 per
hour. The average time a teller takes to serve a customer is two
and a half minutes and service time follows an exponential
distribution. At present, the bank has two tellers on staff to
serve customers during this time.
A. Determine the average time a customer will wait in line before
being served.
B. Determine the probability an arriving customer will have to wait
in line.
C. Determine the average number of customers in the system.
D. Suppose that the bank would like the average time a customer
spends waiting in line to be three minutes or less. What is the
fewest number of tellers they should employ to meet this goal?
8. For the post-Christmas gift returns, Paver’s store added a
second clerk on December 26. Paver’s expects 10 customers an hour,
and the mean service time is 10 minutes. Assuming an M/M/2 queue
and a single waiting line, compute the average time a customer
spends in the queue. What if there were two lines with jockeying
allowed?
9. Given these parameters: λ = 25 per hour, μ = 30 per hour, and
Wq = .3 hours, calculate the average number of customers in the
system, average number of customers in the queue, and the average
time a customer spends in the system.
10. For an M/G/1 queue, λ = 12 per hour, μ = 24 per hour, and the
standard deviation of the service time σ = .05. Calculate the
average number of customers in the system, average time a customer
spends in the system, and the probability that there are exactly two
customers in the system.