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WAITING LINES AND QUEUING THEORY MODELS PROBLEMS
Problem-1:
The Schmedley Discount Department Store has approximately 300 customers shopping in its
store between 9 a.m. and 5 p.m. on Saturdays. In deciding how many cash registers to keep
open each Saturday, Schmedley’s manager considers two factors: customer waiting time (and
the associated waiting cost) and the service costs of employing additional checkout clerks.
Checkout clerks are paid an average of $8 per hour. When only one is on duty, the waiting time
per customer is about 10 minutes (or ⅙ hour); when two clerks are on duty, the average
checkout time is 6 minutes per person; 4 minutes when three clerks are working; and 3 minutes
when four clerks are on duty.
Schmedley’s management has conducted customer satisfaction surveys and has been able to
estimate that the store suffers approximately $10 in lost sales and goodwill for every hour of
customer time spent waiting in checkout lines. Using the information provided, determine the
optimal number of clerks to have on duty each Saturday to minimize the store’s total expected
cost.
Problem-2:
The Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that
occur on an average of λ = 3 per day (approximately Poisson in nature).
The crew can service an average of μ = 8 machines per day, with a repair time distribution that
resembles the exponential distribution.
a. What is the utilization rate of this service system?
b. What is the average downtime for a machine that is broken?
c. How many machines are waiting to be serviced at any given time?
d. What is the probability that more than one machine is in the system? Probability that
more than two are broken and waiting to be repaired or being serviced? More than
three? More than four?
Problem-3:
Automobiles arrive at the drive-through window at a post office at the rate of 4 every 10
minutes. The average service time is 2 minutes. The Poisson distribution is appropriate for the
arrival rate and service times are exponentially distributed.
a) What is the average time a car is in the system?
b) What is the average number of cars in the system?
c) What is the average time cars spend waiting to receive service?
d) What is the average number of cars in line behind the customer receiving
service?
e) What is the probability that there are no cars at the window?
f) What percentage of the time is the postal clerk busy?
g) What is the probability that there are exactly two cars in the system?
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Problem-4:
For the post office in the previous problem, a second drive-through window is being
considered. A single line would be formed and as a car reached the front of the line it would go
to the next available clerk. The clerk at the new window works at the same rate as the current
one.
• What is the average time a car is in the system?
• What is the average number of cars in the system?
• What is the average time cars spend waiting to receive service?
• What is the average number of cars in line behind the customer receiving service?
• What is the probability that there are no cars in the system?
• What percentage of the time are the clerks busy?
• What is the probability that there are exactly two cars in the system?
Case Study
Winter Park Hotel
Donna Shader, manager of the Winter Park Hotel, is considering how to restructure the front
desk to reach an optimum level of staff efficiency and guest service. At present, the hotel has
five clerks on duty, each with a separate waiting line, during the peak check-in time of 3:00 p.m.
to 5:00 p.m. Observation of arrivals during this time show that an average of 90 guests arrive
each hour (although there is no upward limit on the number that could arrive at any given
time). It takes an average of 3 minutes for the front-desk clerk to register each guest.
Donna is considering three plans for improving guest service by reducing the length of time
guests spend waiting in line. The first proposal would designate one employee as a quickservice
clerk for guests registering under corporate accounts, a market segment that fills about
30% of all occupied rooms. Because corporate guests are preregistered, their registration takes
just 2 minutes. With these guests separated from the rest of the clientele, the average time for
registering a typical guest would climb to 3.4 minutes. Under plan 1, noncorporate guests
would choose any of the remaining four lines.
The second plan is to implement a single-line system. All guests could form a single waiting line
to be served by whichever of five clerks became available. This option would require sufficient
lobby space for what could be a substantial queue.
The third proposal involves using an automatic teller machine (ATM) for check-ins. This ATM
would provide approximately the same service rate as a clerk would. Given that initial use of
this technology might be minimal, Shader estimated that 20% of customers, primarily frequent
guests, would be willing to use the machines. (This might be a conservative estimate if the
guests perceive direct benefits from using the ATM, as bank customers do. Citibank reports that
some 95% of its Manhattan customers use its ATMs.) Donna would set up a single queue for
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customers who prefer human check-in clerks. This would be served by the five clerks, although
Donna is hopeful that the machine will allow a reduction to four.
Questions
• Determine the average amount of time that a guest spends checking in. How would this
change under each of the stated options?
• Which option do you recommend?
SIMULATION MODELING PROBLEMS
Problem-1:
Clark Property Management is responsible for the maintenance, rental, and day-to-day
operation of a large apartment complex on the east side of New Orleans. George Clark is
especially concerned about the cost projections for replacing air conditioner compressors. He
would like to simulate the number of compressor failures each year over the next 20 years.
Using data from a similar apartment building he manages in a New Orleans suburb, Clark
establishes a table of relative frequency of failures during a year as shown in the following
table:
NUMBER OF A.C. COMPRESSOR FAILURES PROBABILITY (RELATIVE FREQUENCY)
0 0.06
1 0.13
2 0.25
3 0.28
4 0.20
5 0.07
6 0.01
He decides to simulate the 20-year period by selecting two-digit random numbers from the
random number table.
Conduct the simulation for Clark. Is it common to have three or more consecutive years of
operation with two or fewer compressor failures per year?
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Problem-2:
The number of cars arriving per hour at Lundberg’s Car Wash during the past 200 hours of
operation is observed to be the following:
NUMBER OF CARS ARRIVING FREQUENCY
3 or fewer 0
4 20
5 30
6 50
7 60
8 40
9 or more 0
Total 200
• Set up a probability and cumulative probability distribution for the variable of car
arrivals.
• Establish random number intervals for the variable.
• Simulate 15 hours of car arrivals and compute the average number of arrivals per hour.
Select the random numbers needed from the random number table.
Problem-3:
Compute the expected number of cars arriving in the previous problem using the expected
value formula. Compare this with the results obtained in the simulation.
Problem-4:
Every home football game for the past eight years at Eastern State University has been sold out.
The revenues from ticket sales are significant, but the sale of food, beverages, and souvenirs
has contributed greatly to the overall profitability of the football program. One particular
souvenir is the football program for each game. The number of programs sold at each game is
described by the following probability distribution:
NUMBER (IN 100s) OF PROGRAMS SOLD PROBABILITY
23 0.15
24 0.22
25 0.24
26 0.21
27 0.18
Historically, Eastern has never sold fewer than 2,300 programs or more than 2,700 programs at
one game. Each program costs $0.80 to produce and sells for $2.00. Any programs that are not
sold are donated to a recycling center and do not produce any revenue.
a. Simulate the sales of programs at 10 football games. Use the random number
table.
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b. If the university decided to print 2,500 programs for each game, what would the
average profits be for the 10 games simulated in part (a)?
c. If the university decided to print 2,600 programs for each game, what would the
average profits be for the 10 games simulated in part (a)?
Problem-5:
In the previous problem suppose the sale of football programs described by the probability
distribution only applies to days when the weather is good. When poor weather occurs on the
day of a football game, the crowd that attends the game is only half of capacity. When this
occurs, the sales of programs decreases, and the total sales are given in the following table:
NUMBER (IN 100s) OF PROGRAMS SOLD PROBABILITY
12 0.25
13 0.24
14 0.19
15 0.17
16 0.15
Programs must be printed two days prior to game day. The university is trying to establish a
policy for determining the number of programs to print based on the weather forecast.
a) If the forecast is for a 20% chance of bad weather, simulate the weather for ten
games with this forecast.
b) Simulate the demand for programs at 10 games in which the weather is bad.
c) Beginning with a 20% chance of bad weather and an 80% chance of good weather,
develop a flowchart that would be used to prepare a simulation of the demand for
football programs for 10 games.
d) Suppose there is a 20% chance of bad weather, and the university has decided to
print 2,500 programs. Simulate the total profits that would be achieved for 10
football games.
Problem-6:
Milwaukee’s General Hospital has an emergency room that is divided into six departments: (1)
the initial exam station, to treat minor problems or make diagnoses; (2) an x-ray department;
(3) an operating room; (4) a cast-fitting room; (5) an observation room for recovery and general
observation before final diagnoses or release; and (6) an out-processing department where
clerks check patients out and arrange for payment or insurance forms.
The probabilities that a patient will go from one department to another are presented in the
table below:
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FROM TO PROBABILITY
Initial exam at emergency room entrance X-ray department 0.45
Operating room 0.15
Observation room 0.10
Out-processing clerk 0.30
X-ray department Operating room 0.10
Cast-fitting room 0.25
Observation room 0.35
Out-processing clerk 0.30
Operating room Cast-fitting room 0.25
Observation room 0.70
Out-processing clerk 0.05
Cast-fitting room Observation room 0.55
X-ray department 0.05
Out-processing clerk 0.40
Observation room Operating room 0.15
X-ray department 0.15
Out-processing clerk 0.70
• Simulate the trail followed by 10 emergency room patients. Proceed one patient at a
time from each one’s entry at the initial exam station until he or she leaves through outprocessing.
You should be aware that a patient can enter the same department more
than once.
• Using your simulation data, what are the chances that a patient enters the x-ray
department twice?
Problem-7:
Management of the First Syracuse Bank is concerned about a loss of customers at its main
office downtown. One solution that has been proposed is to add one or more drive-through
teller stations to make it easier for customers in cars to obtain quick service without parking.
Chris Carlson, the bank president, thinks the bank should only risk the cost of installing one
drive-through. He is informed by his staff that the cost (amortized over a 20-year period) of
building a drive-through is $12,000 per year. It also costs $16,000 per year in wages and
benefits to staff each new teller window.
The director of management analysis, Beth Shader, believes that the following two factors
encourage the immediate construction of two drive-through stations, however. According to a
recent article in Banking Research magazine, customers who wait in long lines for drive-through
teller service will cost banks an average of $1 per minute in loss of goodwill. Also, adding a
second drive-through will cost an additional $16,000 in staffing, but amortized construction
costs can be cut to a total of $20,000 per year if two drive-throughs are installed together
instead of one at a time. To complete her analysis, Shader collected one month’s arrival and
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service rates at a competing downtown bank’s drive-through stations. These data are shown as
observation analyses 1 and 2 in the following tables.
• Simulate a 1-hour time period, from 1 to 2 p.m., for a single-teller drive-through.
• Simulate a 1-hour time period, from 1 to 2 p.m., for a two-teller system.
• Conduct a cost analysis of the two options. Assume that the bank is open 7 hours per
day and 200 days per year.
OBSERVATION ANALYSIS 1: INTERARRIVAL TIMES FOR 1,000 OBSERVATIONS
TIME BETWEEN ARRIVALS (MINUTES) NUMBER OF OCCURRENCES
1 200
2 250
3 300
4 150
5 100
OBSERVATION ANALYSIS 2: CUSTOMER SERVICE TIME FOR 1,000 CUSTOMERS
SERVICE TIME (MINUTES) NUMBER OF OCCURRENCES
1 100
2 150
3 350
4 150
5 150
6 100
FORECASTING PROBLEMS
Problem-1:
Sales of industrial vacuum cleaners at R. Lowenthal Supply Co. over the past 13 months are as
follows:
SALES ($1,000s) MONTH
11 January
14 February
16 March
10 April
15 May
17 June
11 July
14 August
17 September
12 October
14 November
16 December
11 January
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• Using a moving average with three periods, determine the demand for vacuum cleaners
for next February.
• Using a weighted moving average with three periods, determine the demand for
vacuum cleaners for February. Use 3, 2, and 1 for the weights of the most recent,
second most recent, and third most recent periods, respectively. For example, if you
were forecasting the demand for February, November would have a weight of 1,
December would have a weight of 2, and January would have a weight of 3.
• Evaluate the accuracy of each of these methods.
• What other factors might R. Lowenthal consider in forecasting sales?
Problem-2:
Passenger miles flown on Northeast Airlines, a commuter firm serving the Boston hub, are as
follows for the past 12 weeks:
WEEK ACTUAL PASSENGER MILES (1,000S)
1 17
2 21
3 19
4 23
5 18
6 16
7 20
8 18
9 22
10 20
11 15
12 22
• Assuming an initial forecast for week 1 of 17,000 miles, use exponential smoothing to
compute miles for weeks 2 through 12. Use α = 0.2.
• What is the MAD for this model?
• Compute the RSFE and tracking signals. Are they within acceptable limits?
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Problem-3:
The following table provides the Dow Jones Industrial Average (DJIA) opening index value on
the first working day of 1991–2010:
YEAR DJIA YEAR 2 DJIA
2010 10,431 2000 11,502
2009 8,772 1999 9,213
2008 13,262 1998 7,908
2007 12,460 1997 6,448
2006 10,718 1996 5,117
2005 10,784 1995 3,834
2004 10,453 1994 3,754
2003 8,342 1993 3,301
2002 10,022 1992 3,169
2001 10,791 1991 2,634
• Develop a trend line and use it to predict the opening DJIA index value for years 2011,
2012, and 2013. Find the MSE for this model.
Case Study
Forecasting Attendance at SWU Football Games
Southwestern University (SWU), a large state college in Stephenville, Texas, 30 miles southwest
of the Dallas/Fort Worth metroplex, enrolls close to 20,000 students. In a typical town–gown
relationship, the school is a dominant force in the small city, with more students during fall and
spring than permanent residents.
A longtime football powerhouse, SWU is a member of the Big Eleven conference and is usually
in the top 20 in college football rankings. To bolster its chances of reaching the elusive and
long-desired number-one ranking, in 2005 SWU hired the legendary Bo Pitterno as its head
coach. Although the number-one ranking remained out of reach, attendance at the five
Saturday home games each year increased. Prior to Pitterno’s arrival, attendance generally
averaged 25,000 to 29,000 per game. Season ticket sales bumped up by 10,000 just with the
announcement of the new coach’s arrival. Stephenville and SWU were ready to move to the big
time!
The immediate issue facing SWU, however, was not NCAA ranking. It was capacity. The existing
SWU stadium, built in 1953, has seating for 54,000 fans. The following table indicates
attendance at each game for the past six years.
One of Pitterno’s demands upon joining SWU had been a stadium expansion, or possibly even a
new stadium. With attendance increasing, SWU administrators began to face the issue head-on.
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Pitterno had wanted dormitories solely for his athletes in the stadium as an additional feature
of any expansion.
SWU’s president, Dr. Marty Starr, decided it was time for his vice president of development to
forecast when the existing stadium would “max out.” He also sought a revenue projection,
assuming an average ticket price of $20 in 2011 and a 5% increase each year in future prices.
Questions
• Develop a forecasting model, justify its selection over other techniques, and project
attendance through 2012.
• What revenues are to be expected in 2011 and 2012?
• Discuss the school’s options.
