Extra Credit Solution & Report
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(Solution & Report) (Extra Credit Assignment) Abby Almonte, Crystal Chea, Anthony Yoohanna2/6/2012
Table of ContentsProblem Statement3Original Problem Statement Assumptions4Notations5Manual Formulation of Problem6Part A7Part B9Part C10Original Problem Statement: WinQSB Solution11Data input for WinQSB “Queuing Analysis”11Cost Assumptions12Data output for WinQSB “Queuing Analysis”13Part A: WinQSB15Part B: WinQSB16Part C: WinQSB17Comparison between Manual Solution versus WinQSB18WinQSB Summary Analysis19Sensitivity Analysis:20Number of Servers:20Service Rate:26Queue Capacity29Report to Manager32
Problem Statement
An average of 40 cars per hour, which are exponentially distributed, are tempted to use the drive-in at the Hot Dog King Restaurant. If more than 4 cars are in the line, including the car at the window, a car will not enter the line. It takes an average of 4 minutes to serve a car.
a) What is the average number of cars waiting for the drive-in window? (not including the car at the window)
b) On the average, how many cars will be served per hour?
c) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?
Original Problem Statement Assumptions
Within this problem, the drive through line can only accommodate 3 cars, plus the car at the service window. Customers arriving when the drive in line is full leave rather than wait for the line to clear because the queue has reached its full capacity. This is known as balking. Balking is the action of which a customer leaves the system due to the lack of space and unsatisfying needs of the customer. We assume that this type of system is an M/M/1/GD/C/infinity. “M” meaning exponential, “GD” meaning general queue discipline, “C” is the capacity, and infinity is the population size. The number 1 in this system represents the number of servers.
Notations
Averages for a steady state queuing system
= Arrival rate approaching the system
e = Arrival rate (effective) entering the system
= Maximum (possible) service rate
e = Practical (effective) service rate
L = Number of customers present in the system
Lq = Number of customers waiting in the line
Ls = Number of customers in service
W =Time a customer spends in the system
Wq = Time a customer spends in the line
Ws = Time a customer spends in service
Manual Formulation of Problem
To begin, we identify what the arrival rate, is which is given in the problem statement:
Arrival Rate (λ) = 40 cars per hour
Then to find the service rate, we use the average service time per customer which is 4 minutes, and we calculate how many customers can be served within an hour time. Service Rate (μ) = 15 cars per hour
Next, we must find the traffic intensity - rho (ρ)
Traffic Intensity is a measure of how occupied a server is which vital in determining the amount of people or equipment needed to develop a steady system.
Note that if rho (ρ) is greater than 1 then the system will blow up.To find rho (ρ) we follow these calculations:
From these values we can now solve parts a, b, and c.
Part A
a) What is the average number of cars waiting for the drive-in window? (not including the car at the window)
To determine the average number of cars waiting for the drive-in, we use the formula:
1) We first solve for L:
Next we solve for Ls:
Note that π0 is the probability that there is no one in the system.
To find π0 we use the following equation
2) Now we solve for Lq
The average number of cars waiting for the drive – in window is 2.5 cars.
Part B
b) On the average, how many cars will be served per hour?
We use the equation:
So we must calculate π4 to find the probability that 4 cars are in the system since the capacity of the drive-in are 4 cars.
We calculate as follows.
Now we input the values for:
The average number of cars being served per hour is 14.8 cars.
Part C
c) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?
To determine the amount of time a customer just joining the line we must use the equation as follows:
The average amount of time it will take for a customer who just entered the line is 13.9 minutes
Original Problem Statement: WinQSB SolutionData input for WinQSB “Queuing Analysis”
Figure 1.1
When working with WinQSB, our team input all of the data from the question, which covered the number of servers, service rate per hour, customer arrival rate per hour, queue capacity, and the customer population. The values that require cost were fabricated by our team since all cost values are subjective.
Cost Assumptions
The cost of busy server cost per hour and idle server cost represents how much a server is paid per hour. We assumed that the costs were marked at $10 because we believe that a worker at a fast food restaurant with a drive-in usually earns $10 an hour on the high end. The idle server cost and busy server cost are marked at the same price because servers are paid by the hour, and not by how much work they have actually performed. The customer waiting cost per hour represents how much money the business loses when a customer waits for an hour, and we placed it at $18. We assume that if a customer has to wait that long, their food should be free to compensate for the delay, which we assume to be an average of $10 per car. We also assume that if a customer waits for over an hour, they will be upset and not visit the restaurant again for at least two weeks, or if they do visit, they do not purchase as much food. The cost of a customer being balked represents the cost that Hot Dog King suffers from losing a customer due to a customer not entering the system. This price was placed at $20 because we used the average cost of a meal and the cost of losing a customer which is assumed to sum up to 20.
Data output for WinQSB “Queuing Analysis”
Figure 1.2
Figure 1.2 shows a table of the performance measures and the results from entering in the information from Figure 1.1 into WinQSB. It shows information such as the system utilization, the average number of customers in the queue, the average time a customer spends in a queue, and a probability of a system being busy. This table also shows how much the company would be paying for each factor,
and the total price of the whole system. Line number 20 shows that the total cost of customers being balked per hour is $503.74, which is the majority of the hourly cost of the system.
Figure 1.3
Figure 1.3 shows the estimated probability of a each number of customer in the system, and the cumulative probability of each. There is a 0.02 probability that there are no cars in the system, 0.03 probability that there is one car in the system, 0.09 probability that there are 2 cars in the system, 0.24 probability that there are 3 cars, and 0.63 probability that there are 4 cars in the system. All of the probabiltites add up to 1.
Part A: WinQSB
a) What is the average number of cars waiting for the drive-in window? (not including the car at the window)
Figure 1.4
Figure 1.4 shows the average number of cars waiting in the drive in window, or Lq, enclosed in a red rectangle, which equals 2.4498
There is an average of 2.45 Cars waiting in the drive-in window
Part B: WinQSB
On the average, how many cars will be served per hour?
Figure 1.5
Figure 1.5 shows the average overall system effective service rate per hour, enclosed in a red rectangle.
Average amount of cars served per hour = 14.8 cars
Part C: WinQSB
b) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?
Figure 1.6
Figure 1.6 shows the average time a customer spends in the system, enclosed in a red rectangle.
Time it takes for a customer to receive their food = 0.2320 hoursconverted to minutes = 13.9 minutes
It takes an average of 13.9 minutes until food is received
Comparison between Manual Solution versus WinQSB
Figure 1.16
Question
Manual
WinQSB
a)
What is the average number of cars waiting for the drive-in window?
2.45 cars
2.45 cars
b)
On the average, how many cars will be served per hour?
14.81 cars
14.81 cars
c)
If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?
13.9 minutes
13.9 minutes
In Figure 1.16 is a chart that compares the results from manually solving the problem statement versus using WinQSB. Using WinQSB is a great way to double check your own work; however it is also very important to input the correct values and abbreviations. It is only beneficial to use WinQSB if it is used correctly, and garbage in is garbage out. For instance, if you were to change customer population to a number 4 instead of an “M” the results would have skewed more than 5% from the real values.
WinQSB Summary Analysis
The WinQSB solutions matched our manually calculated solutions. The various server and customer costs we chose for the WinQSB input had no effect on the solution to the problem other than showing hourly costs for the system. To go further, we created extra questions that relate to the problem at hand.
1. What happens if the number of servers changes?
2. What happens if the average service rate changes?
3. What happens if the queue capacity is increased?
4. What can be done to reduce the hourly cost?
5. What is the best option for reducing the hourly cost?
Our team decided to perform sensitivity analysis on the arrival rate, service time, and queue capacity to see what effect changes in those variables on the solution and its related costs. With the sensitivity analysis we would be able to answer the questions and make recommendations to the management that would help reduce the total hourly cost.
Sensitivity Analysis:Number of Servers:
For our first sensitivity analysis, we chose to look at the number of servers. In the original problem statement, we currently only have one server, but we wanted to see how the system would run differently if we had up to 5 servers.
Figure 1.7
(Figure 1.7 shows the values that we chose for the sensitivity analysis for number of servers. We ran the program so that we would be able to see the difference from 1 to 5 workers, in increments of 1. )
Figure 1.8
Figure 1.8 shows the results of having 1 to 5 numbers of servers. The first item that our team would like to point out is that the effective arrival rate of customers continues to increase as more servers are added to the system. The effective arrival rate represents the average number of cars that actually enter the system. This causes the average amount of customers balked to decrease dramatically from an average of 25.2 cars to 0.5 cars, which also causes our balked customer cost to decrease as well. As stated previously in the WinQSB data output, most of the total cost of the system comes from the balked customer cost, and by increasing the amount of servers, the total cost of the system decreases from $557.83 to $61.42.
The second item that we would like to point out is that the system utilization decreases with each added server. The reason for this being is because with more servers, each server does less amount of work. The system utilization is directly proportional to the waiting time and the number of customers in line.
In this problem, you can see that the utilization of the system drop down from about 99% to 53%, number of customers in line, or Lq, went from an average of 2.45 cars to 0.79 cars, and the average waiting time in line, or Wq, decreased from .16 of an hour, or .003 of an hour, which proves the relationship show above.
The third item that we would like to point out is that even though the busy server cost and idle server cost increase with each added server, the total cost of the system continues to decrease because of the saved money in the waiting customer costs and balked customer costs.
Figure 1.9
Figure 1.9 is a graph that shows the number of servers versus the total cost of the system. It shows how the total cost of the system continues to decrease as the number of servers increase. From 1 to 2 servers, there is a steep line, representing a large decrease in the total cost of the system per hour, and from 4 to 5 servers, line is more horizonal. This shows us that as the number of servers increase, the amount of money saved in the total cost of the system decreases more with each server added. Our team decided to extend our sensitivity analysis to see what would happen with the total cost of the system if more servers were added.
Figure 1.10
Figire 1.10 shows the results of having up to 7 servers in total. We extended our sensitivity analysis to see what would happen to the trend if more than 5 servers were in a system. Here you can see that the effective arrival rate and system utilization continue to decrease as number of servers increase. At a certain point though, the utilization of the system drops to a level that is unacceptable. Due to the lack of work of each server, the idle server cost increases, and causes the total cost to increase as well. This shows us that having 5 servers would cause the total cost of the system to be at its lowest at a value of $61.42.
Figure 1.11
Figure 1.11 visually shows us that although the increasing number of servers decreases the total cost of the system, if too many servers are added, the cost of the system starts to increase again. Going back to question 1, what happens if the number of servers changes? The answer is that the total cost of the system decreases until too many servers are added, and at that point, the total cost of the system begins to rise again.
Service Rate:
For our second sensitivity analysis, we chose to look at the service rate. In the original problem statement, we currently only had a service rate of 15 customers per hour, but we wanted to see how the system would run differently if we had a rate up to 25 customers an hour. We only changed the service rate, and left all of the other factors in place of the original problem statement.
Figure 1.12
Figure 1.12 shows the set up for the service rate. We wanted to see how the service rate would change all of the other factors of this problem. We started the sensitivity analysis from 15 customers an hour because that is the service rate from the original problem statement, and ended the study on 25 customers an hour to see how it would be if the service rate was faster. The way that we can increase the service rate is to provide the servers with extra training, purchase new machines that cook the food at a faster rate, and redesign the system and layout so that servers can save time on walking. We studied the system in service rate changes of 1.
Figure 1.13
1. Figure 1.13 shows the results for changing the service rate in the problem. The first item at hand that we would like to point out is that effective arrival rate increases as the service rate increases. Because of the increase in customers entering the system, the average number of balked customers decrease, and the balked customer cost decreases as well. The second item that our team would like to point out is that the system utilization slightly decreases, which is correlated to the number of cars in the system and the waiting time in line slightly decreasing. The waiting customer cost also decreases because fewer
customers are waiting in line. Overall, an increase in the service rate causes a decrease in the total cost of the system. To answer question number 2, what happens if the average service rate changes? If the average service rate changes, the effective arrival rate increases, causing the balked customer cost to decrease, and the system utilization slightly decreases, causing the waiting time in line, and number of cars in line to decrease as well. By increasing the service rate, all parameters shown through WinQSB are improved.
Queue Capacity
For this sensitivity analysis, our team looked at how the system would change if the queue capacity were increased. The way that we would increase our queue capacity would be to purchase more land, and do construction so that more cars are able to enter into the system.
Figure 1.14
Figure 1.14 shows the set up for how our team set up the sensitivity analysis. The original problem statement said that the queue capacity of the system is 3, but our team is trying to see if purchasing more land and doing construction for up to ten cars to extend the queue capacity will be worth it.
Figure 1.15
Figure 1.15 shows our solution for changing the queue capacity in the system. This solution shows that the effective arrival rate stays at the same value for the most part. The effective arrival rate is connected to the average balked, and the balked customer cost, which also stay close to their original values. The system utilization increases to a point where the system is unsteady at a queue capacity of 9 and 10. The reason for this is because more customers are entering into the line, but there is only one server and the server is working at the same rate. The cars are piling up in the queue waiting to be served, which can be seen in the waiting customer cost. The cost increases from $44 to $169 because customers are waiting for a longer time in line. Increasing the queue capacity, while keeping all other parameters the same is not a good idea. The only way that increasing the capacity would be worth it is if the number of servers and/or service rate were also increased.
To answer question number 3, what happens if the queue capacity is increased? If the queue capacity is increased, the waiting time in line and the number of cars in line will be increased because essentially, nothing else is changing. Increasing the queue capacity too much would cause the system to go into an unsteady state. The total cost of the system would continue to increase with each added queue capacity space.
Question number 4 asks, what can be done to reduce the hourly cost? The only parameter that we have control over is the amount of servers. By increasing the servers to 5, the total cost of the system decreases to $116. By increasing the amount of servers, the utilization of the system decreases, which causes the waiting time in line and average number of cars in line to also decrease. Also, by increasing the servers, more cars are able to enter the system, saving money on the balked cost. We can also reduce the hourly cost of the system by having a quicker service rate. We can increase the service rate by training servers, changing the layout of the restaurant, and purchasing new machines that cook the food at a faster rate.
Question number 5 asks, what is the best option for reducing the hourly cost? The best option for reducing the hourly cost is to have 5 servers working. The total cost of the system can drop down to $116 by just changing one parameter. Although labor costs will increase, money will be saved in the balked customer cost.
Report to Manager
Dear Manager,
The Hot Dog King drive through currently costs $529.69 per hour of operation. The biggest contributor to the hourly cost is the cost of customers leaving when the drive through line is full. $453.36 per hour is lost due to customers refusing to wait for the line to clear. The one server is almost never idle which indicates that the system is inadequate to handle the current volume of drive through customers. Further analysis showed that increasing number of servers and the service rate can significantly help reduce the hourly cost. Sensitivity analysis was performed on the average service time as well as the number of servers. Even with a 40% reduction in average service time, the hourly cost was only reduced to $368.82. Increasing the number of servers had a much greater effect as shown in the table below:
Servers
Total Cost per Hour
1
$ 529.69
2
$ 314.70
3
$ 184.02
4
$ 130.24
5
$ 116.44
Adding one more server reduces total hourly cost more than decreasing average service time by 40%. In order to reduce the hourly cost as much as possible, it is recommended that the company increase the number of servers from one to five. With five servers the total hourly cost will drop to $116.44 with only $8.35 cost per hour due to lost customers.
Number of Servers Vs. Total Cost
557.82999999999947301.16000000000008143.6978.3161.42
Number of Servers
Total Cost
Number of Servers Vs. Total Cost
557.82999999999947301.16000000000008143.6978.3161.4263.12000000000001270.72
Number of Servers
Total Cost
32
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