### Intro into Simulation

```Introduction into Simulation
Basic Simulation Modeling
The Nature of Simulation
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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definitions
•simulation
–imitate operations of real-world facilities or processes
•system
–facility or process of interest
–assumptions needed (mathematical, logical)
•model
–set of assumptions
–used to gain understanding how corresponding system works
–simple enough? → solve analytically to obtain exact information
–mostly too complex → evaluate model numerically using simulation
and estimate desired true characteristics
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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System, Models and
Simulation
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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system
•set of entities (people, machine, etc.) that (inter)act
–example [bank]: tellers, customers, loan officers
•state of system
–collection of variables to describe system at particular time
–example [bank]: number of busy tellers, number of customers in the
bank, arrival time of each customer
•entities
–characterized by data values (attributes)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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types of system
•discrete system
–state variables change instantaneously at separated points in time
–example: bank
•number of customers changes: new customer arrives, service
finished
•continuous system
–state variables change continuously with respect to time
–example: airplane moving through air
•position, velocity can change continuously
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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different ways to study a system
analytical
solution
system
experiment with
model of the
system
mathematical
model
Simulation
experiment with
actual system
physical model
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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classification of simulation models
•static vs. dynamic simulation models
–static model: time plays no role
–dynamic model: represents model as it evolves over time
•deterministic vs. stochastic simulation models
–deterministic: no probabilistic (i.e. random) components
–stochastic: random components, output itself is random (estimate of
true models characteristics)
•continuous vs. discrete simulation models
–continuous: state variables change instantaneously
–discrete: changes only happen at discrete point in time
DSM discrete event simulation models
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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Discrete Event Simulation
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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discrete event simulation
•system evolves over time
•state variables change at separate points in time only
–whenever an event occurs
•example [bank]: (single server, estimate average waiting time in queue)
–state variables:
•status of server (idle or busy)
•number of customers in queue (or in system)
•time of arrival of each customer (for calculation of waiting time)
–events: customer arrives, service complete (customer leaves)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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simulation clock
•time-advance mechanism
–keep track of current value of simulated time
–no explicit unit of measurement → same unit as input parameters (be
consistent!!)
•two approaches
–next-event time advance
•simulation clock initialized at time 0
•times of future events are determined
•clock is advanced to the next future event (nothing happens/changes
between)
–fixed-increment time advance
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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Simulation of Single-Server
Queuing System (M/M/1)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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M/M/1
•interarrival times (service times)
–A1, A2, A3, …. (S1, S2, S3, ….)
–iid (independent and identically distributed) random variables
•arriving customer (served FCFS/FIFO)
–who finds the server idle: is served immediately
–who finds the server busy: joins the end of a single queue
•upon completion of service
–queue: first customer in queue will be serviced
–no queue: server is idle again
•start of simulation: empty and idle
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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M/M/1
•performance measures
–expected average delay in queue d(n)
–expected average number of customers in queue q(n)
–expected utilization of server u(n)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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delay d(n)
•estimator of systems performance from customers point of view
•on a given run: observed average delay
–depends on random service and arrival times
–is random itself
– estimator for d(n)
Di
customer delays on a very long (infinite) run
delay of a customer can also be equal to zero (D1 = 0)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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number in queue q(n)
•customers in queue
•customers in system (not being served)
•again: observation is just an estimator of true expected value
pi
T(n)
Ti
expected proportion of time there are i customers in queue
time necessary to observe n delays in queue
total time during the simulation the queue is of length i
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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utilization u(n)
•measures how busy server is
•expected utilization = expected proportion of time server is busy (not
idle)
•busy function
•estimator
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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performance measures
•discrete time statistic
–average delay
(defined relative to discrete random variables Di)
•continuous-time statistic
–average number in queue
–utilization
(defined on continuous random variables Q(t) and B(t))
•other statistics than just averages
–minimum, maximum, proportion of time there’re at least 5 customer in
queue
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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simulation “by hand”
necessary random variables
(generated from their corresponding probability distribution)
•interarrival times
A1 = 0.4
A2 = 1.2
A3 = 0.5
A4 = 1.7
A6 = 1.6
A7 = 0.2
A8 = 1.4
A9 = 1.9
S2 = 0.7
S3 = 0.2
S4 = 1.1
A5 = 0.2
•service times
S1 = 2.0
S5 = 3.7
S6 = 0.6
initializiation (t = 0)
system starts emtpy (no customers yet) and idle (server not busy)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4
initialize system at t = 0
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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t
Arrivals
Departure
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6
9
t
Arrivals
Departure
ed1= 2.4
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
9
t
Arrivals
Departure
ed1= 2.4
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
ea4= 3.8
9
t
Arrivals
Departure
ed1= 2.4
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
ea4= 3.8
9
t
Arrivals
Departure
ed1= 2.4 ed2= 3.1
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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t
Arrivals
Departure
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
5
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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t
Arrivals
Departure
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
ea1
ea5=
= 0.4 ea2= 1.6 ea = 2.1
3
4.0
5
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
9
t
Arrivals
Departure
ed4= 4.9
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
ea1
ea5=
= 0.4 ea2= 1.6 ea = 2.1
3
4.0
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
9
5
ea6= 5.6
t
Arrivals
Departure
ed4= 4.9
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2
A6 = 1.6 A7 = 0.2 A8 = 1.4 A9 = 1.9
S1 = 2.0 S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7 S6 = 0.6
simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
ea1
ea5=
= 0.4 ea2= 1.6 ea = 2.1
3
4.0
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
9
5
ea6= 5.6
t
Arrivals
Departure
ed4= 4.9
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
ed5= 8.6
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simulation “by hand”
Q(t)
t
B(t)
t
Events
0
1
ea1
ea5=
= 0.4 ea2= 1.6 ea = 2.1
3
4.0
5
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
ea7=
ea6= 5.6
9
t
5.8
ea8= 7.2
Arrivals
Departure
ed4= 4.9
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
ed5= 8.6
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average waiting time d(n)
Events
0
x
x
1
ea1 = 0.4 ea2= 1.6 ea = 2.1
3
ea5= 4.0
ea4= 3.8
ed3= 3.3
ed1= 2.4 ed2= 3.1
5
ea7= 5.8
ea6= 5.6
x
9
ea8= 7.2
t
Arrivals
Departure
ed4=
4.9
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
ed5=
8.6
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Q(t)
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
8.6
7.2
5.6
5.8
4.9
4.0
3.1
2.4
2.1
1.6
0.0
average number in queue q(n)
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average utilization u(n)
8.6
3.8
3.3
0.4
0.0
B(t)
fraction of
time server
is busy
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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Necessary Steps for Simulation
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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formulate problem and
plan the study
model valid
collect data and
define model
no
yes
no
assumptions
still valid
yes
design experiments
make production runs
construct a computer
program & verify
analyze output data
test runs
present results
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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Advantages and Disadvantages
of Simulation
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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advantages
complex models cannot be solved analytically
→ only simulation possible
allows to estimate the performance of an existing system under some
projected set of operating conditions
alternative proposed system designs (operating policies) can be
compared easily
better control over experimental conditions
study system over long time frame
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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disadvantages
each run of a stochastic model
produces only estimates of true measures
→ several independent runs (or one very long one) needed
expensive and time consuming
need to make sure the model is valid
040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I
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```