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?