Download Advance Waiting Line Theory and Simulation Modeling

Advance Waiting Line
Theory and Simulation
Modeling
Supplement Objectives
Be able to:
 Describe different types of waiting line systems.
 Use statistics-based formulas to estimate waiting
line lengths and waiting times for three different
types of waiting line systems.
 Explain the purpose, advantages and
disadvantages, and steps of simulation modeling.
 Develop a simple Monte Carlo simulation using
Microsoft Excel.
 Develop and analyze a system using SimQuick.
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 2
Alternative Waiting Lines
• Single-Channel, Single-Phase
– Ticket window at theater,
• Multiple-Channel, Single-Phase
– Tellers at the bank, windows at post office
• Single-Channel, Multiple-Phase
– Line at the Laundromat, DMV
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 3
Alternative Waiting Lines
Multiple-Channel, Single-Phase
Single-Channel, Single-Phase
Single-Channel, Multiple-Phase
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 4
Assumptions
• Arrivals
– At random (Poisson, exponential distributions)
– Fixed (appointments, service intervals)
• Service times
– Variable (exponential, normal distributions)
– Fixed (constant service time)
• Other
– Size of arrival population, priority rules,
balking, reneging
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 5
Poisson Distribution
Probability of n arrivals in T time periods
( T )  T
Pn 
e
n!
n
where  = arrival rate
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 6
Waiting Line Formulas
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 7
P0 = Probability of 0 Units in
Multiple-Channel System
P0 
1
M 1 1    n  1    M  M 

       
 n 0 n!    M!    M   
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 8
Single-Channel, Single-Phase
Manual Car Wash Example
• Arrival rate  = 7.5 cars per hour
• Service rate  = an average of 10 cars per hour
• Utilization  = / = 75%
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 9
Single-Channel, Single-Phase
Automated Car Wash Example
• Arrival rate  = 7.5 cars per hour
• Service rate  = a constant rate of 10 cars per hour
• Utilization  = / = 75%
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 10
Comparisons
Manual wash,
single server
Automated wash, Manual wash,
single server
two servers
Cars
waiting
2.25
1.125
0.1227
Cars in
system
3
1.875
1.517
Time
waiting
18 minutes
9 minutes
1 minute
Time in
System
24 minutes
15 minutes
7 minutes
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 11
Simulation Modeling
Advantages
• Off-line evaluation of
new processes or
process changes
• Time compression
• “What-if” analysis
• Provides variance
estimates in addition
to averages
Disadvantages
• Does not provide
optimal solution
• More realistic  the
more costly and more
difficult to interpret
• Still just a simulation
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 12
Monte Carlo Simulation
• Maps random numbers to cumulative
probability distributions of variables
• Probability distributions can be either
discrete (coin flip, roll of a die) or
continuous (exponential service time or
time between arrivals)
• Random numbers 0 to 99 supplied by
computer functions such as =
INT(100*RAND()) in Excel.
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 13
Monte Carlo Simulation
Examples
• Coin toss: Random numbers 0 to 49 for
‘heads’, 50 to 99 for ‘tails’
• Dice throw: Use Excel function
= RANDBETWEEN(1,6) for throws
• Service time: Use Excel function
= –(avg service time)*ln(RAND()) for
exponential service time
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 14
Building a Simulation Model
Four basic steps
1) Develop a picture of system to be modeled (process
mapping)
2) Identify objects, elements, and probability distributions
that define the system


Objects = items moving through system
Elements = pieces of the system
3) Determine experiment conditions (constraints) and
desired outputs
4) Build and test model, capture the output data
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 15
Simulation Example
(Using single-channel, single-phase waiting line)
1) Process map
2) Time between arrivals (exponential distribution),
service time (exponential distribution), objects =
cars, elements = line and wash station
3) Maximum length for line, time spent in the system
4) Run model for a total of 100 cars entering the car
wash, average the results for waiting time, cars in
line, etc.
© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply
Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036
Supplement 8S, Slide 16
‘SimQuick’ Simulation
An Excel-based application for simulating
processes that allows use of constraints
(see text example 8S.5)