Transcript Decision Technology - Villanova University

Decision Technology
Modeling, Software and Applications
Matthew J. Liberatore
Robert L. Nydick
John Wiley & Sons, Inc.
Chapter 13
Introduction to Process Simulation
Using Extend
INTRODUCTION TO QUEUING
Waiting in line is part of everyday life. Most people
spend 5 years of their lives waiting in lines and
6 months sitting at traffic stops.
Americans spend 37 billion hours a year in lines.
People also wait to have their calls answered or
files downloaded from the internet.
Jobs and materials wait to be processed and ships
and railcars wait to be unloaded.
INTRODUCTION TO QUEUING
In the past, people spent time
to save money.
Today, people spend money
to save time.
INTRODUCTION TO QUEUING
There is a fundamental tradeoff between the cost of
providing service and the amount of waiting
time.
Adding servers is one approach for reducing
waiting time but it increases costs.
Process redesign is another approach.
Simulation models can explore the performance of
queuing systems when the number of servers or
the process design itself changes.
The desire to improve queuing systems has
impacted your life in a variety of ways:
optical
scanners
express
checkout
lines
ATM machines
single vs.
multiple
waiting lines
right turn on
red (AFTER
STOPPING!)
left turning lanes
paying tolls in
one direction
only
credit cards
at service
stations, and
mirrors
near
elevator
doors
Mobil Speed Pass.
INTRODUCTION TO QUEUING
Formal analysis of queuing
systems began in 1908 with
the work of the Danish
telephone engineer A. K. Erlang.
Mathematical queuing models have been
developed for many queuing situations.
Since these models cannot handle all of the
subtleties found in many practical situations,
simulation is often preferred over
mathematical analysis.
EXTEND OVERVIEW
Extend is a visual simulation software package that can
be used to develop dynamic models of real-life
processes.
Extend’s web site is: http://www.imaginethatinc.com.
Upgrades can be downloaded at this site.
When Extend is launched a worksheet with sample
models automatically opens. We suggest that you turn
off this feature by using the Edit, and Preferences
commands. In the dialog box of the Default Model
box (Model tab), delete demo.mox and click OK.
EXTEND OVERVIEW
We begin with a brief overview of the structure of
Extend blocks.
Blocks specify an action or process.
Information comes into the block and is processed
by the program that is in the block.
The block then sends information to the next block.
Blocks are linked together to form a model.
EXTEND OVERVIEW
Blocks are stored in libraries in Extend’s LIBS
folder.
If an Extend model is opened, the required libraries
are automatically opened.
When a new model is created, you must open the
required libraries using the Library, Open Library
commands.
EXTEND OVERVIEW
We recommend that you configure Extend so that it
automatically opens commonly used libraries as
follows:
First select the Edit and then the Preferences
commands.
Then in the dialog box of the “Libraries” tab enter
the following libraries in the “Preload libraries”
location: Discrete Event.lix, Generic.lix, Mfg.lix,
Optimization.lix, Plotter.lix, Quick Blocks.lix,
Statistics.lix.
EXTEND OVERVIEW
Extend models are composed of connected blocks.
There are four types of connectors: item input, item
output, value input, and value output.
Item input and output are used to move the items
from one block to another.
Value input and output are used to pass parameters
between blocks.
EXTEND OVERVIEW
The Activity, Multiple block shown below illustrates
all four types of connectors. This block will be
discussed in more detail later.
F
U
Item input
Item output
D C
C L W
Value input
Value output
Filled-in connectors (F, U, L, and W) are value
output locations.
EXTEND OVERVIEW
Double click on any block to open the dialog box.
A sample dialog box for the Activity, Multiple block
is shown on the next slide.
Notice that tabs are used to organize the elements of
the dialog box.
Clicking on the “Activity” tab allows the user to set
the Delay (service time) and Maximum number
in activity (number of servers).
EXTEND OVERVIEW
EXTEND OVERVIEW
We use a bank to illustrate the key points of Extend.
Assumptions:
 customers arrive randomly,
 first in first out (FIFO) queue,
 single server or teller, and
 service time is random.
EXTEND OVERVIEW
Extend’s blocks can represent the four key
activities:
 a customer arrives,
 enters the queue,
 receives service, and then
 exits.
We now consider each of the activities to build the
banking model.
EXTEND OVERVIEW: ARRIVALS
1. Customer arrives: Generator (DE.LIX,
Generators):
V 1 2 33
The interarrival time of items in the system, such as
customers, is often random and can be described
by a probability distribution.
Often, interarrival time data are collected and fitted to
a continuous probability distribution.
EXTEND OVERVIEW: ARRIVALS
A variety of continuous distributions are built into
the Generator block.
If interarrival time data are collected, a package
called Stat::Fit can be used to select a distribution
based on goodness of fit tests.
We now demonstrate the use of Stat::Fit on a data
set that is contained in a file called
RANTIME.XLS.
This package is not included with your CD, but is
available on our network.
EXTEND OVERVIEW: ARRIVALS
Copy the 100 data points from RANTIME.XLS to
the clipboard.
Open Stat::Fit, select File, New, Edit, Paste, and
click on the “Auto::Fit” button.
Select “continuous distribution” and “fixed lower
bound options,” and set the “Lowest Allowed
Value” to 0.
The results are shown on the next slide.
EXTEND OVERVIEW: ARRIVALS
EXTEND OVERVIEW: ARRIVALS
We see that the best fit is an exponential distribution
with a minimum value of 0 and a mean
interarrival time of 2.86.
The exponential and the Erlang have the same rank
because an Erlang whose second parameter is 1 is
identical to an exponential.
We also use this example to validate the accuracy of
Stat::Fit.
EXTEND OVERVIEW: ARRIVALS
That is, the 100 data points were generated from an
exponential distribution with a mean of 3.
The difference between 2.86 and 3.00 is due to the
small sample size.
For mathematical simplicity we will use an
exponential distribution with a mean interarrival
time of 3 for the remainder of our banking
example.
EXTEND OVERVIEW: ARRIVALS
Back to Extend, double click on the Generator
block and select the “Items” tab.
Click on the box to the right of Distribution:, select
the exponential, and enter 3 for the Mean.
We can use the “Plot” button to show a graph of an
exponential distribution with a mean of 3.
This plot is based on 20,000 randomly generated
interarrrival times.
EXTEND OVERVIEW: ARRIVALS
The exponential is skewed to the right, so it is more
likely that the interarrival times will be less than
3 minutes than greater than 3 minutes. There is a
small chance of very long interarrival times.
EXTEND OVERVIEW: ARRIVALS
It is important to understand the relationship
between the mean interarrival time (1/) and the
mean arrival rate ().
If the mean interarrival time is 3 minutes, then we
expect 20 arrivals per hour.
For example, if (1/) = 3 minutes / arrival, then
 = 1/3 arrival / minute, or
 = [1/3 arrival/minute]* 60 minutes/hour
 = 20 arrivals/hour.
EXTEND OVERVIEW: QUEUE
2. Customer enters queue: Queue, FIFO (DE.LIX,
Queues):
F
L
W
Link the item output connector of the Generator
block to the item input connector of the Queue,
FIFO block.
EXTEND OVERVIEW: QUEUE
Often, when people are involved, a FIFO queue
discipline is used, as assumed in this block.
Extend sets the default for the maximum queue
length at 1000 (found in the “Queue” tab).
At this time, the only purpose of this factor is to
prevent excessive queue buildup.
EXTEND OVERVIEW: QUEUE
Two important classes of statistics are reported here
(“Results” tab): line (queue) length and waiting
time.
Specifically, the average and maximum values of
these statistics are reported.
EXTEND OVERVIEW: QUEUE
We define LQ as the average length of queue and
WQ as the average waiting time in the queue.
(Extend simply uses L and W to refer to our LQ and
WQ.)
Queue utilization is the proportion of time that
customers are in the queue.
We will discuss these results after running the
simulation.
EXTEND OVERVIEW: SERVICE
3. Customer receives service: Activity, Multiple
(DE.LIX, Activities):
F
U
D C
C L W
Connect the Queue, FIFO block to the Activity,
Multiple block.
Enter the number of servers in the dialog box
marked: Maximum number in activity (“Activity”
tab). For our bank, this number should be 1.
EXTEND OVERVIEW: SERVICE
If the service time is constant, such as in an
automated car wash, simply enter the fixed value
into the “Delay” location.
Similar to our discussion about customer arrivals,
service times are often random and represented by
a probability distribution.
EXTEND OVERVIEW: SERVICE
Random service times cannot be set within the
Activity, Multiple dialog box.
An additional block called Input Random Number
(GENERIC.LIX, Inputs/Outputs) must be used.
Rand
1 2
3
EXTEND OVERVIEW: SERVICE
The distribution parameters are set through the
“Distributions” tab.
Click in the Distribution: box to select the desired
probability distribution and the parameters in the
indicated boxes.
A random service time is generated for each
customer as it enters the Activity, Multiple block.
This value overrides any Delay value that may have
been entered in the dialog box by the user.
EXTEND OVERVIEW: SERVICE
The value output connector of the Input Random
Number block must be linked to the D connector
(that is, Delay) of the Activity, Multiple block.
EXTEND OVERVIEW: SERVICE
The process of choosing a distribution and its
parameters is the same as that described for the
Generator block.
Once again, the service time data can be fitted to a
probability distribution using Stat::Fit.
In our banking simulation we assume that service
time data can be fitted to an Erlang distribution.
The Erlang is often used to describe the variability
of service times.
EXTEND OVERVIEW: SERVICE
Two parameters are needed: the mean service time
(Mean) and an integer shape parameter called k.
The Erlang can approximate a variety of
distributions depending upon parameter values.
A k value of 1 causes the distribution to become an
exponential, while larger integer values tend
toward a normal distribution.
We assume that service time follows an Erlang with
a mean of 10 and a k value of 3.
EXTEND OVERVIEW: SERVICE
Using an argument similar to that made for the
arrival process, if the mean service time is 10
minutes (1/), then we can service 6 customers
per hour ().
This shows that the mean service rate () is the
reciprocal of the mean service time (1/ ).
EXTEND OVERVIEW: SERVICE
Similar to our discussion about the queue, we can
also measure the average number of people in
service (LS) and the average waiting time (WS)
for our service facility (“Results” tab).
(Again, Extend uses L and W to refer to our LS and
WS.)
We also measure utilization which is the percent of
time that the single server is busy.
EXTEND OVERVIEW: EXIT
4. Customer exits the system: Exit (DE.LIX,
Routing):
Exit
#
Connect the Activity, Multiple block to the Exit
block.
This block simply removes the customer from the
system and reports the number of customers that
exited during the simulation run (“Exit” tab).
EXTEND OVERVIEW: KEEPING TIME
Executive: (DE.LIX, none):
count
Place Executive to the left of all other blocks.
It creates a clock and manages simulation time.
You have no reason to change anything in this block.
PLOTTING THE RESULTS
Plotter, Discrete Event: (PLOTTER.LIX, none):
This block offers a flexible plotter that provides
both a table and a graph of the numerical values.
PLOTTING THE RESULTS
We recommend plotting L and W from the Queue,
FIFO block, although you may need to plot other
outputs.
Connect “L” from the Queue, FIFO block to the top
(blue) connector of the Plotter, Discrete Event
block.
Connect “W” to the second (red) connector.
PLOTTING THE RESULTS
Three plotter tools are often used: “Trace
properties”, “Open Dialog”, and “Auto Scale XY.”
Open Dialog
Trace properties
Auto Scale XY
PLOTTING THE RESULTS
The “Trace properties” command brings up a
window that allows the user to specify which Yaxis will be used for each plot (Y1/Y2 option).
Y1/Y2
PLOTTING THE RESULTS
If two plots are to appear on the graph, simply
click the Y1/Y2 option for either one of the
plots and it will switch from the default Y1
axis (left) to the Y2 axis (right).
Y1/Y2
PLOTTING THE RESULTS
The “Open Dialog” tool controls when the graph is
displayed and automatically scales the Y-axis.
Open Dialog
PLOTTING THE RESULTS
Choose the “Display” tab and click “Show plot at
end of simulation,” and select “Autoscale at end
of simulation.”
PLOTTING THE RESULTS
You can manually scale by clicking on the “Auto
Scale XY” tool after the simulation has run.
Auto Scale XY
THE COMPLETE MODEL
The complete model is shown below and appears in
a file called BANK1T.MOX.
RUNNING THE MODEL
We are now ready to run the simulation.
Select Run, Simulation Setup (“Discrete event” tab).
Since our unit of time is minutes, specify 480 in the
End simulation at time location to simulate an 8hour day.
RUNNING THE MODEL
You want to perform multiple runs to improve the
accuracy of such factors as LQ and WQ.
To simulate 365 days, enter 365 in the Number of
runs box (“Discrete event” tab).
Details about how many runs to make will be
discussed later.
For now simulate 1 day.
RUNNING THE MODEL
There are several ways to run the model.
Use the “Run Now” command (“Discrete event”
tab) within the Simulation Setup dialog box.
If the Simulation Setup options have been set
previously, the Run and Run simulation
commands can be used or simply click on the
green right arrow on the toolbar to run the model.
PRINTING THE MODEL
After running the model we recommend printing the
entire model along with the “Results” tab of
selected dialog boxes, including Queue, FIFO
and Activity, Multiple).
EXTEND OVERVIEW: ANIMATION
Extend has the capability for animation by selecting
“Run” and “Show Animation.”
To have the same icon move throughout the model,
open the Generator dialog box, click on the
“Animate” tab, and select the desired icon by
clicking on the box next to “Items will appear
as:”
To have different icons appear, open the dialog box
of the desired block, click on the “Animate” tab,
select the desired icon as above, and click on the
button “Change all pictures to:”
INTERPRETING THE RESULTS
What does the plotter tell us about this bank?
The line and waiting times keep growing.
What is the average waiting time in the queue
(WQ)?
The results tab of Queue, FIFO shows that WQ is
168.91 minutes.
What is the average queue length (LQ)?
LQ is 51.33 customers.
INTERPRETING THE RESULTS
How many people arrived during the simulation?
148 entered the queue which is close to the
expected number of 8*= 8*20 = 160.
How many people exited the queue?
45 people exited which means that 103 people are
still in line at the end of the simulation.
What is the utilization of the queue?
99.4%, so the queue is busy nearly all the time.
INTERPRETING THE RESULTS
What is the average service time (WS)?
The results tab of Activity, Multiple shows that
WS is 10.65 minutes which is close to 1/ of 10.
What is the utilization of the teller?
The utilization of the server is 1 which means that
the server is always busy with customers.
How many people exited the bank?
44 departed from service. This number is also
shown in the “Exit” tab of the Exit block.
INTERPRETING THE RESULTS
What is the average total waiting time and the average
number of customers in the bank?
WT = WQ + WS = 168.91 + 10.65 = 179.56 minutes
LT = LQ + LS = 51.33 + 1.00 = 52.33 customers
MODIFYING THE BANK
No matter how you look at it,
the bank is out of control!
How many tellers are needed?
QUEUING RELATIONSHIPS: , , N
There is an important relationship between , , and
the number of servers.
The minimum number of servers (N) to prevent an
ever expanding queue is the smallest integer
greater than  / .
In our example,  is 20 customers/hour and  is 6
customers/hour.
So, at least 20/6=3.33 => 4 servers are needed.
QUEUING RELATIONSHIPS: , , N
A simple approach to compute N is to divide (1/)by
(1/) and raising the result to the next integer.
Do you understand why this works?
/= (1/) / (1/)
Since 1/is 3 minutes/customer and 1/ is 10
minutes/customer.
10/3 = 3.33 => a minimum of 4 servers are needed .
QUEUING RELATIONSHIPS: , , N
This approach saves a step in the calculation
process since usually interarrival time and service
time data are known.
Obviously, as the number of servers increases
beyond 4, WQ and LQ will decrease.
This point leads to a fundamental trade-off between
the cost of service and the cost of waiting.
Be patient, we will study this relationship soon!
MULTI SERVER BANK
What happens to performance if 4 tellers are available?
Assume a single queue feeds all 4 tellers.
Change the Maximum number in activity: to 4 in the
Activity, Multiple block.
The results are in a file called BANK4T.MOX.
We see that WQ (5.99 minutes) and LQ (2.16
customers) decrease dramatically.
MULTI SERVER BANK
The utilization of the tellers only drops to 88%.
When there are multiple servers, the utilization is the
average percentage of server capacity that is used
over the simulation.
This result can be obtained by dividing LS (3.52) by
the number of servers (4).
This increase of performance comes at a cost of
scheduling 3 more tellers each hour.
MULTI SERVER BANK
What happens to performance if 5 tellers are
available?
The results are in a file called BANK5T.MOX.
WQ (0.42) and LQ (0.13) drop to very low levels.
The utilization of the tellers drops to 62%.
This increase of performance comes at a cost of
scheduling 1 additional teller each hour.
MULTI SERVER BANK
Does the cost of scheduling the fifth teller justify
the savings in waiting time?
It depends on how you value the customers’
waiting time.
Since the cost of waiting time is hard to measure,
a better approach may be to measure the
maximum amount of time that a customer waits
in line.
Can you think of a service business that uses
maximum waiting time in their advertisements?
QUEUING RELATIONSHIPS: , LQ, WQ
Two important outputs of most simulation models
are LQ and WQ.
Interestingly, there is an important relationship
between them.
Believe it or not (Ripley),
it is called the fundamental
theorem of queuing,
or Little’s Law (after
John D. C. Little of M.I.T.).
QUEUING RELATIONSHIPS: , LQ, WQ
We fondly refer to this as the E=mc2 of queuing.
It is simply this,
LQ =(WQ)
This relationship is independent of the number of
servers and the probability distributions
describing the arrival and service processes.
QUEUING RELATIONSHIPS: , LQ, WQ
In running a simulation model, the results will
closely approximate this formula if the
simulation has been run for a “long enough”
(steady state) period of time.
We will not prove this result for two reasons - you
don’t want to see it and we don’t want to do it!
However, we will provide an intuitive illustration as
to why this relationship holds.
QUEUING RELATIONSHIPS: , LQ, WQ
Consider the following example.
Suppose  = 2 customers per minute.
Assume that there is no random variation in the
arrival rate, so that 2 customers arrive every
minute.
Further assume that each person that enters the
queue waits exactly the same amount of time in
the queue, let’s say, WQ = 3 minutes.
QUEUING RELATIONSHIPS: , LQ, WQ
Consider what happens to the queue during the 10th
minute.
The two people that arrived during minute 10 enter
the queue and remain in the queue for minutes 11
and 12.
The two people that arrived during minute 9 are still
in the queue, as are the people who arrived
during minute 8.
QUEUING RELATIONSHIPS: , LQ, WQ
Therefore, there are a total of 6 people in the queue
during minute 10.
This shows that LQ = (WQ), that is, 6 = 2(3),
assuming no randomness.
The following chart illustrates what happens to LQ
over time.
QUEUING RELATIONSHIPS: , LQ, WQ
This argument holds for every minute except for the
first two since “steady state” does not occur until
minute 3.
Queue length by minute
2 arrivals
during minute
. . .
8
9
10
11
12
13
14
15
. . .
Total number in
queue
. . .
. . .
10
11
2
2
2
2
2
2
6
6
12
2
2
2
6
13
2
2
2
6
14
2
2
2
6
15
. . .
2
2
2
6
. . .
QUEUING RELATIONSHIPS: , LQ, WQ
This illustration shows that using averages, Little’s
theorem holds.
A more complex proof is needed to show that the
theorem works under random arrivals and service
times.
The result also holds for service and the complete
system, that is:
LS = (WS), and
LT = (WT).
QUEUING RELATIONSHIPS: , LQ, WQ
In BANK4T.MOX, LQ=2.16, WQ=5.99 minutes,
and =(1/3) customer/minute.
By Little’s Law, LQ = (1/3)*5.99 = 2.00, which is
close to our 2.16.
Once again, the difference is due to the length of the
simulation run.
STATISTICS AND
CONFIDENCE INTERVALS
Multiple runs of a simulation model can improve the
estimates of queuing statistics, such as LQ and
WQ.
Confidence intervals can clarify the interpretation of
simulation output data.
Point estimates can be misleading if confidence
intervals are very wide relative to the size of the
point estimate.
STATISTICS AND
CONFIDENCE INTERVALS
Decision makers must consider how values within
the confidence interval affect their decisions.
All of these objectives can be accomplished by
adding Queue Stats (STATS.LIX, none) to any
Extend model.
Queue
Stats
Update
STATISTICS AND
CONFIDENCE INTERVALS
Queue Stats is not connected to any other block.
Statistics from any queuing block will appear as
output.
Select Append new updates and Update at end of
simulation options.
STATISTICS AND
CONFIDENCE INTERVALS
Suppose we want to make 25 runs of BANK4T.MOX
and compute 95% confidence intervals for LQ and
WQ.
Set the Number of runs in the Run, Simulation Setup
box to 25.
This model is in a file called BANK4TSTATS.MOX.
STATISTICS AND
CONFIDENCE INTERVALS
After running the simulation, double click on Queue
Stats to see the results of each individual run.
Click on “Confidence Interval” to compute the
confidence interval for each statistic.
For example, if a 95% confidence interval for LQ is
1.91 +/- 0.87, there is a 95% probability that the
true population value for LQ is between 1.04 and
2.78.
STATISTICS AND
CONFIDENCE INTERVALS
Increasing the number of runs will reduce the width
of the confidence interval.
Also, increasing the confidence from 95% to 99%
will increase the width of the interval.
For example, a 99% confidence interval for LQ is
1.91 +/- 1.18, there is a 99% probability that the
true population value for LQ is between 0.73 and
3.09.
COLD STARTS
In all of our queueing examples we started with
empty queues and empty service facilities.
Clear Statistics (STATS.LIX, none) should be used to
start computing simulation statistics after the
simulation has been operating for a period of time.
Clear
COLD STARTS
Clear Statistics is not connected to any other block.
To start computing statistics after 30 minutes, set
“Clear statistics at time:” location to 30.
Queuing statistics should now increase since the first
few customers did not have to wait as long in the
queue.
If you simulate long enough, this should not make
much of a difference.
HIERARCHICAL MODELS
Extend can group blocks hierarchically so that a
simplified view of a model can be obtained. This
is useful when making a presentation.
Highlight a set of blocks and select Model and Make
Selection Hierarchical.
Name the block and click on Make H-Block.
The highlighted blocks are replaced with an empty
square connected to the other blocks in the model.
Open the H-block to show the Extend blocks.
HIERARCHICAL MODELS
Highlight the H-block, select Model, and Open
Hierarchical Block Structure to modify the Hblock.
Expand the window in the upper left hand corner
and replace the white box with another shape or
graphic (use Edit, Paste Object) and add text and
color as desired.
BANK4THIERARCHICAL.mox is a hierarchical
representation of BANK4T.mox that has a
hierarchical block for each of the major activities:
arrival, wait in line, receive service, and exit.
HIERARCHICAL MODELS
The use of hierarchical blocks further improves the
clarity and understanding of the model.
Seeing a customer at the door of the bank indicates
that this block generates arrivals.
Images from clipart or photographs can be used.
CLONING AND THE NOTEBOOK
Extend has a notebook available where model
parameters, text, and graphics can be placed.
The notebook is valuable for summarizing and
presenting model inputs and outputs.
Extend has a cloning feature for capturing items that
can be placed into the notebook.
CLONING AND THE NOTEBOOK
To open the notebook and create a clone:

Open the notebook.

Open a block’s dialog box.

Click on the cloning tool.

Drag the desired item to the notebook.
To add text, click on the main cursor.
CLONING AND THE NOTEBOOK
To move text or resize a text box in the notebook the
main cursor must be active.
To move or resize a cloned item the cloning tool must
be active.
A notebook example is shown in
BANK4TNOTE.MOX.
EXCEL SPREADSHEET INTERFACE
Extend can send output to a spreadsheet.
We can modify BANK4TSTATS.mox and send the 25
WQ values to a spreadsheet with the following steps:
 Connect the WQ output connector from Queue Fifo
to the input connector of Mean & Variance to
compute WQ for all runs.
 Connect the “M” connector of Mean & Variance to
the input connector of Data Send.
 Specify the name and location of the Excel file in the
Workbook file name box of Data Send.
EXCEL SPREADSHEET INTERFACE

Within Data Send, the Send data to: Row and
Column values are both 1.
 Check “If neither row nor column connector are used
increment row every” and change setting from step to
run.
The complete model is shown in
BANK4TEXCEL.mox.
REPORT GENERATING FEATURE
Extend can also send output to a text file instead of a
spreadsheet with the following steps.
 Select the Run and Generate Report commands.
 Highlight the block whose information is to be in the
report.
 Select the Run and Add Selected to Report
commands.
 Select the Show Reporting Blocks command.
 If reporting statistics, select the Run, Report Type,
and Statistics commands.
The results of the run will be sent to a file whose default
name is report.txt.
SUMMARY

In this chapter, we:

Provided an introduction to process
simulation using Extend

Used a bank setting to explain the basics of
process simulation

Additional features covered include: Little’s
Law, statistics and confidence intervals,
hierarchical models, and model input and
output
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