practice problems for Milestone QUEUING THEORY
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University of Southern California *
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Course
504A
Subject
Industrial Engineering
Date
Feb 20, 2024
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docx
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6
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ue
queue
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P
RACTICE
PROBLEMS
FOR
Q
UEUING
T
HEORY
Q
UESTION
1
Coldice Hospital admits a certain kind of hernia patients every day except Sunday. The
hospital has 1,000 beds. Assume a constant level of daily admittance and an average stay of
four days.
a)
What is the maximum number of patients the hospital can handle in one week?
b)
The company is considering undertaking a new kind of hernia operation, which
requires an average stay of five days. If the profit margin of the new hernia operation
is $150, should Coldice perform the new kind of hernia operation (assume that the
four-day hernia operation has a profit Q
UESTION
2
Global Financial Corporation (GF) is a subsidiary of the Global Equipment Company (GEC), a
construction equipment manufacturer, set up to handle the financing for those clients who
request it. The steps through which a loan application passes are shown graphically below.
There are four steps, each preceded with an unlimited buffer.
1.
The first step is Evaluation and Analysis. This involves looking at the financial
feasibility of the equipment loan from the buyer’s perspective, as well as a credit
analysis. The Evaluation and Analysis step is performed by three teams, each
responsible for different sales regions. This step is thorough and as a consequence
lengthy, taking on average 6 hours per team.
2.
After the Evaluation and Analysis step has been completed, a loan application file is
passed on to the Interest Rate desk. Here, the appropriate interest rate for the loan
is determined. This is a short step, taking about one hour per application, performed
by one individual.
3.
The next step in the application processing is to determine the exact Loan and
Agreement Terms. This often requires some financial analysis, and takes about 2
hours. It is performed by one individual. 4.
The final step is the process is the Final Issuing, i.e., the actual preparation of the
paperwork for the loan. This takes about 4 hours and is performed by either of two
individuals.
P
AGE
1
Region 1 Evaluatio
n & Analysis
Region 2 Evaluatio
n & Analysis
Region 3 Evaluatio
n & Analysis
Interest Rate
Loan Terms
Final Issuing
On average, GF processes 46 loan applications every month, broken down into 18 for Region
1, 12 for Region 2, and 16 for Region 3. An average month is 20 working days, and an
average working day is 6 hours.
a)
For each stage depicted above, assess the average arrival rate A
, the average service
rate S
, the number of servers m
, the average service capacity m x S, and the average
utilization u. Stage
Arrival Rate A
(applications/day
)
Service Rate S
(applications/day
)
Number of
Servers m
Service Capacity m x S (applications/d
ay)
Utilization
u
Region 1
E&A
0.9
1
1
1
90%
Region 2
E&A
0.6
1
1
1
60%
Region 3
E&A
0.8
1
1
1
80%
Interest
Rate
2.3
6
1
6
38%
Loan
Terms
2.3
3
1
3
77%
Final
Issuing
2.3
1.5
2
3
77%
b)
How would your answer above change with variability in arrivals and service times?
c)
Which stage is the bottleneck? Why? If there is variability in arrivals and service times,
what is the implication of that bottleneck on waiting times?
In the following, assume unit coefficients of variation for the inter-arrival times and for the
service times. Assume also that the arrivals to intermediate stages (which correspond to
the departures from upstream stages) have also unit coefficient of variations.
d)
For each stage, determine the expected number of applications waiting to be processed
L
q
, the expected waiting time W
q
, the expected service time, and the expected
throughput time W of a loan application. Stage
Expected
Queue Length Expected
Waiting Time
(days)
Expected
Service Time
(days)
Expected
Throughput Time
(days)
P
AGE
2
queu
e
queue
queue
queue
L
q
=
u
√
2
(
m
+
1
)
1
−
u
W
q
=
1
A
L
q
1/S
W
=
W
q
+
1
S
Region 1 E&A
8.1
9
1
10
Region 2 E&A
.9
1.5
1
2.5
Region 3 E&A
3.2
4
1
5
Interest Rate
.238
.104
1/6
.27
Loan Terms
2.51
1.09
1/3
1.428
Final Issuing
2.23
.97
2/3
1.63
e)
For each region, determine the total expected throughput time of a loan application.
Averaging across all originating regions, what is the total expected throughput time of a
random loan application?
f)
Overall, how much work in process do you expect?
g)
If all regions were consolidated as shown below, with three teams working in parallel on
the same queue of applications, describe qualitatively what would be the effect on the
average throughput time and explain why.
Q
UESTION
3
The manager of a grocery store in downtown Nashville is interested in providing good
service to her customers. Presently, the store has one express check-out counter that can
serve the customers at an average rate of 35 customers per hour. On average, 30
customers arrive at the express counter per hour. Assume that the coefficient of variation
for both inter-arrival times and service times is 1, i.e., CV
IAT = CV
ST = 1
. a)
Compute the utilization of the express checkout clerk.
P
AGE
3
Team 1 Evaluatio
n & Analysis
Team 2 Evaluatio
n & Analysis
Team 3 Evaluatio
n & Analysis
Interest Rate
Loan Terms
Final Issuing
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b)
Compute the average number of customers in the express line.
c)
Compute the average time spent in the express line
.
d)
Compute the average time spent in the express system
.
e)
Compute the average number of customers in the express system
. f)
Customers are complaining about long waiting lines. Which of the following changes
will not
reduce these complaints? A)
Training the check-out clerk to work faster. B)
Training the check-out clerk to reduce service time variability.
C)
Adding a second check-out counter.
D)
Running a marketing campaign to attract more customers.
Q
UESTION
4
Due to its size, Incredibly Big Discount Store positions service desks at both ends of the store, about a mile apart. John staffs the north entrance, where on average 14 customers arrive per hour. Marsha staffs the south entrance, where on average 18 customers arrive per hour. Both can serve a customer in three minutes on average. a)
How long is the line, and how much time do customers spend waiting at the north entrance? How about the south entrance? [Assume that the coefficient of variation for inter-arrival times and service times for both John and Marsha are 1, i.e., CV
IAT = CV
ST = 1
]
b)
John and Marsha fell in love, and petitioned management to move their service desks
next to each other. In the new layout, customers will wait in a single line. Aside from the humanitarian benefits, what will be the effect on customer waiting?
Q
UESTION
5
Bob owns a hair salon, where he uses a first-come first-served system. He cuts both men’s and women’s hair resulting in significant variability in the time it takes him to complete a haircut. More specifically, the average and the standard deviation of the time it takes to complete a haircut equal 20 minutes and 10 minutes, respectively. He is facing steady demand throughout the day with the average arrival rate to his shop being 2 customers per hour. The standard deviation of inter-arrival times equals 30 minutes. a)
What is the average number of customers waiting to get a haircut?
b)
What is the average time a customer spends in the hair salon?
P
AGE
4
c)
In order to reduce waiting times, Bob is planning to implement an appointment policy. This new policy will eliminate the variability in arrival times without changing the arrival rate because a new customer will punctually arrive in every 30 minutes. Find the average amount of time a customer will spend in the hair salon under the appointment policy. Q
UESTION
6
Kathryn’s Income Tax Services is analyzing its customer service operations during the month prior to the April filing deadline. Based on past data, Kathryn estimates that customers, on average, arrive at a rate of 3 customers/hr. Kathryn can complete a tax return in 12 minutes on average with a standard deviation of 6 minutes. The variability in processing time is due to the complexity of preparing tax returns for different customers (i.e., some tax returns are straightforward and some are complex). [Assume that the coefficient of variation for inter-arrival times equals 1, i.e., CV
IAT = 1
]
a)
How many customers should Kathryn expect to have in the waiting area? b)
If you went to Kathryn for your tax returns, how much time would you allow for your tax returns to be completed?
c)
Kathryn realizes that some of her potential customers walk away because of the long wait. She is considering hiring her friend Taylor to help her prior to the April filing deadline. Taylor can also complete a tax return in 12 minutes on average with a standard deviation of 6 minutes. Kathryn’s analysis indicates that hiring Taylor would
increase the average demand to 6 customers/hr. Assuming that either Kathryn or Taylor can serve an incoming customer, what would be the impact of hiring Taylor on
the average total time customers spend on filing taxes? [Assume that the coefficient of variation for inter-arrival times equals 1, i.e., CV
IAT = 1
]
Q
UESTION
7
A hair salon has five stylists’ chairs and four stylists. The customer arrivals average 6 per hour, and the service times average 30 minutes. The coefficients of variation of the inter-
arrival times and the service times are both equal to 1 (i.e., CV
IAT = CV
ST = 1)
.
a)
Determine the utilization of the four stylists (i.e., system utilization). b)
What is the average number of customers waiting (
L
q
) in the salon? c)
What is the average waiting time (
W
q
)?
d)
Two of the four stylists wish to take their vacations during the same week. What effect will this have on waiting times?
One of the customers in the hair salon turns out to be an operations professor and suggests the following measures to reduce waiting times:
P
AGE
5
Option 1:
Implement a reservation system. The professor claims that the implementation of a reservation system, which will reduce the coefficient of variation of arrivals to zero, will reduce waiting times significantly. e)
What will be the reduction in waiting times if the hair salon implements a reservation system? Option 2
: Hire an additional stylist. The professor claims (and the owner of the salon agrees) that hiring an additional stylist to fill in the fifth chair will also reduce the average waiting times. f)
Quantify the reduction in waiting times due to an addition of another stylist.
g)
Compare the two options in part E. Which option would you recommend implementing?
Q
UESTION
8
Whole Foods Market in Westwood has 6 checkout lanes. On average, a shopper finishes shopping and joins a checkout lane in every 40 seconds. The coefficient of variation of the inter-arrival time is equal to 1. The service times for all the lanes have a mean of 3 minutes and a standard deviation of 3 minutes for each lane. Currently, the store operates such that each checkout lane has its own queue. This results in significant waiting times for the customers. One of the customers (who is an ops professor) tells the store manager about an initiative at a Whole Foods Store in New York City where customers form a single queue and are serviced by the first available checkout lane. If the Westwood store implements the single queue layout, what will be the reduction in waiting times? (Assume that there is a feasible way to implement the single queue solution at the Whole Foods store in Westwood.)
Q
UESTION
9
An auto-rental agency has two employees at its service counter. Customers arrive at an average rate of 24 per hour. Each employee can serve at a rate of 15 customers per hour. The coefficients of variation of the inter-arrival and service times are both equal to 1.
a)
What is the average utilization of the service counter? b)
What is the average waiting time, W
q
?
c)
Find the average number of customers in the auto rental agency, L
?
d)
The auto-rental agency is considering hiring another worker to reduce the waiting times? What is the percentage reduction in waiting times if a third worker with the same service rate (and coefficient of variation) is added?
P
AGE
6
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