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By: Abby Almonte
Crystal Chea
Anthony Yoohanna
IE 417: Operations Research
HOT DOG KING RESTAURANT
Extra CreditChapter 20 section 5, #2
Overview
• Problem Statement• Assumptions• M/M/1/GD/C/∞• Manual Solutions• WinQSB Solutions• Extra Questions• Sensitivity Analysis• Report to Manager
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?
Assumptions
1. The information in the problem is accurate
2. The queuing problem is a M/M/1/GD/C/∞ problem
M/M/S/GD/C/∞
ExponentialDistribution
Number of Servers
GeneralQueue
Discipline
Limit Capacity
CustomerPopulation
Type of System
Arrival Process
Service Process
M/M/1/GD/4/∞
ExponentialDistribution
Number of Servers
GeneralQueue
Discipline
Limit Capacity
CustomerPopulation
Type of System
FOR EXAMPLE…
Manual Solutions1. Make pre-calculations
- Identify Arrival Rate (λ)- Calculate Effective Arrival Rate (λe)
- Identify Service Rate (μ) - Calculate rho (ρ) = λ/μ
Pre-Calculations • 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 =
• Then, we calculate service rate by using the given average service time which is 4 minutes per car.
μ = 60 minute = 15 cars per hour 4 minutes per car
• Using λ & μ values, we find ρ
λ = 40 cars per hour = 2.667μ 15 cars per hour
λ ( 1 – π4) = 40 (1 – 0.62967) = 14.9 cars per hour
Part A
Lq= L – Ls
a) What is the average number of cars waiting for the drive- in window? (not including the car at the window)
We use this equation which represents the number of customers in the queue
with respect to an M/M/1/GD/∞ system
L = ρ [ 1 – (C + 1) ρC + CρC+1] (1 – ρC+1) ( 1 – ρ)L = 2.667 [ 1 – (4 + 1) 2.6674 + 4(2.667 4 + 1)] = ( 1 – 2.6674 + 1) ( 1 – 2.667)
Part A continued. • Begin by finding L
3.437
• Then Ls
Lq= L – Ls
Equation
Ls = 1 – π0
Π0 = _( 1 – ρ )_ ( 1 – ρ C + 1) = _( 1 – 2.667)_ = ( 1 – 2.667 4 + 1 )
0.012 Ls = 1 – 0.012 = 0.988
Part A continued. • Finally Lq
Lq = L – Ls = 3.437 – 0.0988 = 2.449
Solution: About 2.5 cars
Part B
π4= (ρ4)(π0)
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 = 2.6674 (0.012452) =0.62967
Refer to Part A
Part B continued. • Then
λ ( 1 – π4) = 40 (1 – 0.62967) = 14.8132
Solution: About 14.8 cars per hour
Part Cc) 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?
We use this equation which represents time a customer spends in the system.
W = ___L___ λ(1 – πc)
C = 4
W = ___3.43709___ = 40 ( 1- 0.629675)
π4 Refer to Part B
Solution: About 13.9 minutes
0.23203 hours
Using WinQSB
• The red box indicates the cost values we decided to apply in WinQSB to calculate hourly cost.
Output: WinQSB
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.
Manual Solutions vs. WinQSBQuestion Manual WinQSB
a)
What is the average number of cars waiting for the drive-in
window?
2.449 cars 2.4498 cars
b)
On the average, how many cars will be served
per hour?
14.8132 cars 14.8132 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?
0.2320 minutes 0.2320 hours
Questions
1. What happens if the average service time changes?
2. What happens if the arrival rate changes?
3. What happens if the number of servers changes?
4. What can be done to reduce the hourly cost?
5. What is the best option for reducing the hourly cost?
Initiate SENSITIVITY ANALYSIS
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.
1 2 3 4 5 $-
$100.00
$200.00
$300.00
$400.00
$500.00
$600.00
$529.69
$314.70
$184.02
$130.24 $116.44
Number of Servers
Cost
Servers 1 - 5
Sensitivity Analysis: Number of Servers
• More than 5 servers results in a gain of cost.
Servers 1 - 7
1 2 3 4 5 6 7 $-
$100.00
$200.00
$300.00
$400.00
$500.00
$600.00
$529.69
$314.70
$184.02
$130.24 $116.44 $117.78 $123.94
Number of Servers
Cost
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.
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.
Report to Manager
• The total hourly cost is $529.69• The total hourly cost from balking is $453.36
Original Data
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)
Questions?