Introduction into Simulation

Introduction into SimulationBasic Simulation Modeling

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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 characteristics040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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System, Models and Simulation


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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)


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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


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different ways to study a system


system

experiment with actual system

experiment with model of the

system

mathematical model

physical model

analytical solution

Simulation

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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

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Discrete Event Simulation


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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)


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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


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Simulation of Single-Server Queuing System (M/M/1)


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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 idle040669 || 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)


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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)


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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 expected proportion of time there are i customers in queueT(n) time necessary to observe n delays in queueTi total time during the simulation the queue is of length 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


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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

queue040669 || 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 timesA1 = 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• service times

S1 = 2.0S2 = 0.7 S3 = 0.2 S4 = 1.1 S5 = 3.7

S6 = 0.6

initializiation (t = 0)system starts emtpy (no customers yet) and idle (server not busy)


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0



A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

Arrivals

Departureinitialize system at t = 0

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1 ea4= 3.8 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ea4= 3.8 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8

ed4= 4.9

ea5= 4.0

Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8

ed4= 4.9

ea5= 4.0

ea6= 5.6 Arrivals

Departure

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A1 = 0.4 A2 = 1.2 A3 = 0.5 A4 = 1.7 A5 = 0.2A6 = 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

t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8

ed4= 4.9

ea5= 4.0

ed5= 8.6

ea6= 5.6 Arrivals

Departure

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t

Events

0 1 5 9

t

tB(t)

Q(t)

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8

ed4= 4.9

ea5= 4.0

ed5= 8.6

ea6= 5.6

ea7= 5.8

ea8= 7.2 Arrivals

Departure

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average waiting time d(n)


t

Events

0 1 5 9

ea1 = 0.4

ed1= 2.4

ea2= 1.6 ea

3= 2.1

ed2= 3.1

ed3= 3.3

ea4= 3.8

ed4= 4.9

ea5= 4.0

ed5= 8.6

ea6= 5.6

ea7= 5.8

ea8= 7.2 Arrivals

Departure

x x x

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average number in queue q(n)


Q(t)

0.0

1.6

2.1

2.4

3.1

4.0

5.6

4.9

5.8

7.2

8.6

040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I 33

average utilization u(n)B(t)

0.0

0.4

3.3

3.8

8.6

fraction of time server

is busy


Necessary Steps for Simulation


formulate problem and plan the study

collect data anddefine model

assumptions still valid

construct a computer program & verify

test runs

model valid

design experiments

make production runs

analyze output data

present results

yes

no

no

yes


Advantages and Disadvantages of Simulation

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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


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disadvantages

each run of a stochastic modelproduces 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


Documents

Introduction into Simulation