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Assignment on Queuing Models for Coffee kiosk, Car wash and Storage Facility

Problem 1

Problem 1

Quicklane Coffee is a drive-through coffee kiosk.  During rush hour, the owner estimates that customers arrive at a rate of 12 per hour.  It takes an average of 3 minutes to serve each customer, and one customer at a time is served.  Assume Poisson arrivals and exponential service times.  Use Excel to solve this problem. Also insert a text box that shows the formulas you use to calculate the requested value for each part of the problem below.  For example, if the problem asked you to calculate the profit for part a, you would insert a text box and type

Part a. Profit = Revenue – Expenses

a.What is the average service rate?  What is the average arrival rate?   What is the average number of customers in line?

b.What is the average time a customer waits in line before being served?

c.What is the average time a customer is in the service system?

d.What is the utilization rate of the coffee kiosk?

e.What is the probability there are more than three customers in the system?

Problem 2

Harry’s Car Wash washes one car at a time.  On Saturdays, one car arrives every eight minutes on average.  The average service time is six minutes.  Arrivals follow a Poisson distribution, and service times are exponentially distributed. Use Excel to solve this problem. Also insert a text box that shows the formulas you use to calculate the requested value for each part of the problem below.  For example, if the problem asked you to calculate the profit for part a, you would insert a text box and type

Part a. Profit = Revenue – Expenses

a.What is the average arrival rate?  What is the average service rate?

b.What is the average time a car is in the system?

c.What is the average number of cars in the system?

d.What is the average time cars spend waiting to be washed?

e.What is the average number of cars in line behind the one being washed?

f.What is the probability the car wash is idle?

g.What percentage of time is the car wash busy?

Problem 3

Larry’s Car Wash is an automated car wash where customers arrive at a rate of 3 cars every 13 minutes.  The time to wash a car is always exactly 3 minutes.  Use Excel to solve this problem. Also insert a text box that shows the formulas you use to calculate the requested value for each part of the problem below.  For example, if the problem asked you to calculate the profit for part a, you would insert a text box and type

## Problem 2

Part a. Profit = Revenue – Expenses

a.What is the average arrival rate?  What is the average service rate?

b.What is the average time a car is in the system?

c.What is the average number of cars in the system?

d.What is the average time cars spend waiting to be washed?

e.What is the average number of cars in line behind the one being washed?

f.What is the probability the car wash is idle?

g.What percentage of time is the car wash busy?

Problem 4

An operator processes jobs on a first-come first-serve basis.  The jobs have Poisson arrival rates, with an average of 6 minutes between arrivals.  The objective is to process these jobs so there are no more than 3 jobs in the system on average.  At what rate does the operator have to process jobs, on average, to meet this objective?

a.Insert a text box in Excel.  Derive a formula to show the mathematical expression of the service rate necessary to meet this objective.

b.In Excel, use the formula derived in Part a to calculate the service rate that will meet the objective.

Problem 5

Soybean farmers in a cooperative in the Midwest deliver their harvest within a three-week period to a central storage facility.  The farmers own the storage facility cooperatively and therefore would like to keep costs as low as possible.  Because the harvest season is relatively short, a line of trucks frequently backs up, resulting in costs of waiting and unloading.  These costs are estimated to be \$21/hour.  The storage facility is open 24 hours per day, 7 days per week during the harvest season.  The average time taken to unload a truck is 2.5 minutes, and trucks arrive every three minutes on average.  Unloading times follow an exponential distribution, and arrivals follow a Poisson distribution.  Use Excel to solve this problem. Also insert a text box that shows the formulas you use to calculate the requested value for each part of the problem below.  For example, if the problem asked you to calculate the profit for part a, you would insert a text box and typ

Part a. Profit = Revenue – Expenses

a.How many trucks on average are in the system?

b.How long, on average, does a truck spend in the system?

c.What is the daily cost to the farmers of having trucks in the unloading process?

d.Enlarging the storage facility and adding workers to staff it would cost \$30,000.  However, the total costs of waiting and unloading would be reduced by 40%.  Should the cooperative expand the storage facility?  Provide evidence to support your recommendation.

## Problem 3

Problem 1

Trucks arrive at Smith’s Storage and must be unloaded by a team of workers.  The probability distributions of the number of trucks arriving every day and the number unloaded every day are shown in the tables below.

 Truck arrivals Number Probability 0 0.12 1 0.18 2 0.14 3 0.26 4 0.19 5 0.11
 Unloading rates Number Probability 1 0.03 2 0.05 3 0.4 4 0.4 5 0.12

Simulate 15 days of operation for Smith’s Storage using the random number table in Table 4 in the textbook.  For arrivals, start at the top of the rightmost column and select the first 15 numbers in the column.  For unloading, start at the first number on the left in the bottom row and select the first 15 numbers in the row.  Use these random numbers and the probability distributions shown above to simulate the number of trucks arriving and the number of trucks possibly unloaded each day for 15 days. You should use Excel for all calculations, but you will generate the random numbers manually using Table 4 as specified above.  You will use these random numbers to determine the simulated arrivals and unloadings.

Trucks are unloaded on a first-come, first-served basis.  Any trucks that are not unloaded must wait until the following day and are unloaded in addition to any new trucks arriving the following day.  Of course, if there are fewer trucks waiting to be unloaded than the number that could be unloaded, only the number of trucks waiting are actually unloaded.

a.Based on this simulation, what is the average number of trucks arriving each day?

b.Based on the simulation, what is the average number of trucks delayed?

c.Based on the simulation, what is the average number of trucks that are actually unloaded?

d.What is the expected number of trucks arriving each day?

e.What is the expected number of trucks that could be unloaded each day?

Problem 2

For each home basketball game at Northwestern State University, game programs are offered for sale.  The number of programs demanded for each game is described by the following probability distribution.

Programs are sold for \$4.00 each, and each program costs \$1.25 to be printed.

 Probability Programs demanded 0.1 1600 0.21 1700 0.28 1800 0.22 1900 0.19 2000

a.Simulate demand for game programs for 15 home games.  Use Excel and the random number function RAND() to generate random numbers.  Use the VLOOKUP function to simulate demand from the probability distribution.

b.Assume that 1,700 programs are printed for each game.  Any unsold programs are recycled and yield no revenue.  Calculate the revenue and profit for each home game.  What is the average profit per game?

c.Assume that 1,700 programs are printed for each game.  Estimate the probability that more programs will be demanded than have been printed.

d.Assume that 1,900 programs are printed for each game. Calculate the revenue and profit for each home game.  What is the average profit per game?

e.Assume that 1,900 programs are printed each game.  Estimate the probability that more programs will be demanded than have been printed.

Problem 3

Jennifer Brown runs a hot dog stand in a busy part of downtown.  The demand for hot dogs follows a uniform distribution with a minimum of 100 and a maximum of 200, which means that any quantity between 100 and 200 (inclusive) is equally likely during the day. Her supplier is not particularly reliable, and number of hot dogs supplied every day also follows a uniform distribution with a minimum of 100 and maximum of 200 hot dogs supplied every day.  The price of a hot dog is \$3.00, and each hot dog costs \$1.00 to buy from the supplier.  The fixed costs are \$100 per day.  Any unsold hot dogs must be discarded at the end of the day.  Remember that Jennifer pays for all of the hot dogs provided by the supplier each day regardless of the demand. Also remember that she can only sell as many hot dogs as she buys from the supplier regardless of the demand.

a.Simulate 1 month (30 days) of operation for this hot dog stand.  For each day, you should include the following quantities:  Number supplied, number demanded, number sold, number short (excess of demand over supply), surplus (number of unsold hot dogs), revenue, variable costs, gross profit (revenue – variable costs), total profit (gross profit – fixed costs).

b.For 1 month, what are the totals for each of the quantities calculated in part a?

c.For 1 month, estimate the probability that Jennifer will run out of hot dogs on a particular day.