1. By: Abby Almonte Crystal Chea Anthony Yoohanna
IE 417: Operations Research
HOT DOG KING RESTAURANT
Extra Credit
Chapter 20 section 5, #2
By: Abby Almonte
Crystal Chea
Anthony Yoohanna

2. Overview Problem Statement Assumptions M/M/1/GD/C/∞ Manual Solutions
WinQSB Solutions
Extra Questions
Sensitivity Analysis
Report to Manager

3. Problem Statement
An average of 40 cars per hour which are exponentially distributed want 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?

4. Assumptions The information in the problem is accurate
The queuing problem is a M/M/1/GD/C/∞ problem

5. Type of System M/M/S/GD/C/∞
Arrival
Process
Service
Process
General
Queue
Discipline
Customer
Population
Exponential
Distribution
Number of
Servers
Limit
Capacity

6. FOR EXAMPLE… Type of System M/M/1/GD/4/∞ Exponential Distribution
General
Queue
Discipline
Customer
Population
Number of
Servers
Limit
Capacity
FOR EXAMPLE…

7. Manual Solutions Make pre-calculations Identify Arrival Rate (λ)
Calculate Effective Arrival Rate (λe)
Identify Service Rate (μ)
Calculate rho (ρ) = λ/μ

8. Pre-Calculations λ = 40 cars per hour λe = λ ( 1 – π4)
Given that there are an average of 40 cars per hour to use the drive-in
λ = 40 cars per hour
BUT there can only be 4 cars in the drive-in, therefore the effective
arrival rate is:
λe =
λ ( 1 – π4)
= 40 (1 – )
= 14.9 cars per hour
Then, we calculate service rate by using the given average service
time which is 4 minutes per car.
μ = minute = 15 cars per hour
4 minutes per car
Using λ & μ values, we find ρ
λ = cars per hour = 2.667
μ cars per hour

9. with respect to an M/M/1/GD/∞ system
Part A
a) What is the average number of cars waiting for the drive- in window? (not including the car at the window)
Lq= L – Ls
We use this equation which represents the number of customers in the queue
with respect to an M/M/1/GD/∞ system

10. Part A continued. Lq= L – Ls L = ρ [ 1 – (C + 1) ρC + CρC+1]
Equation
Lq= L – Ls
Begin by finding L
L = ρ [ 1 – (C + 1) ρC + CρC+1]
(1 – ρC+1) ( 1 – ρ)
L = [ 1 – (4 + 1) ( )] =
( 1 – ) ( 1 – 2.667)
3.437
Then Ls
Ls = 1 – π0
Π0 = _( 1 – ρ )_
( 1 – ρ C + 1)  
= _( 1 – 2.667)_ =
( 1 – )
0.012
Ls = 1 – 0.012
= 0.988

11. Part A continued. Lq = L – Ls = 3.437 – 0.0988 = 2.449 Finally Lq
Solution: About 2.5 cars

12. Part B π4= (ρ4)(π0) Π4 = ρ4 π0 Π4 = 2.6674 (0.012452) = 0.62967
b) On the average, how many cars will be served per hour?
First we find the probability of having 4 cars in the drive-in, which
is noted as π4
π4= (ρ4)(π0)
Π4 = ρ4 π0
Π4 = ( ) =
Refer to Part A

13. Solution: About 14.8 cars per hour
Part B continued.
Then
λ ( 1 – π4)
= 40 (1 – ) =
Solution: About 14.8 cars per hour

14. Solution: About 13.9 minutes
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?
C = 4 π4
W = ___L___ λ(1 – πc)
Refer to Part B
We use this equation which represents time a customer spends in the system.
W = ___ ___ = ( )
hours
Solution: About 13.9 minutes

15. Using WinQSB
The red box indicates the cost values we decided to apply in WinQSB to
calculate hourly cost.

16. Output: WinQSB

17. Probability of Customers
This table represents the probability of having the number of customers in the line. Since there is a capacity of 4 customers and an arrival rate of 40 customers per hour, the table shows that having 4 customers in line has a higher probability compared to having zero.

18. Manual Solutions vs. WinQSB
Question
Manual
WinQSB a)
What is the average number of cars waiting for the drive-in window?
2.449 cars
cars b)
On the average, how many cars will be served per hour?
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?
minutes
hours

19. Questions SENSITIVITY ANALYSIS
What happens if the average service time changes?
What happens if the arrival rate changes?
What happens if the number of servers changes?
What can be done to reduce the hourly cost?
What is the best option for reducing the hourly cost?
Initiate
SENSITIVITY ANALYSIS

20. Sensitivity Analysis: Number of Servers
Average balked customers drops to nearly zero
Total hourly cost of the system drops to $116.44
From 1 server to 2 the cost drops the most.

21. Sensitivity Analysis: Number of Servers
More than 5 servers results in a gain of cost.

22. Sensitivity Analysis: Service Rate
From increasing your service rate from 15 cars to 25 cars per hour, the total cost is still higher than increasing the number of servers.

23. Sensitivity Analysis: Arrival Rate
From increasing your service rate from 15 cars to 25 cars per hour, the total cost is still higher due to the amount of customers balking.

24. Report to Manager Original Data The total hourly cost is $529.69
The total hourly cost from balking is $453.36
Sensitivity Analysis
Add up to 4 servers to reduce total hourly cost significantly
Increased service rate has small effect on total hourly cost
If arrival rate decreases, so will total hourly cost (not ideal)

25. Questions?