Simulation

Where real stuff starts

ToC

1. What, transience, stationarity2. How, discrete event, recurrence

3. Accuracy of output4. Monte Carlo

5. Random Number Generators6. How to sample

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1 What is a simulation ?

An experiment in computerImportant differences

Simulated vs real timeSerialization of events

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Stationarity

A simulation can be terminating or not

For a terminating simulation you should make sure it is stationary

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Information server, scenario 1

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Information server, scenario 2

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Stationarity

In scenario 1:Transient phase, followed by “typical” (=stationary) phase You want to measure things only in the stationary phase

Otherwise: non reproduceable, non typical

In scenario 2:There is no stationary regime“walk to infinity”

you should not do a non terminating simulation with this scenario

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Definition of Stationarity

A property of a stochastic model Xt

Let Xt be a stochastic model that represents the state of the simulator at time t. It is stationary iff

for any s>0

(Xt1, Xt2, …, Xtk) has same distribution as (Xt1+s, Xt2+s, …, Xtk+s)

i.e. simulation does not get “old”

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Classical Cases

Markov modelsState Xt is sufficient to draw the future of the simulation

Common case for all simulations

For a Markov model, over a discrete state spaceIf you run the simulation long enough it will either walk to infinity (unstable) or converge to stationary

Ex: queue with >1: unstablequeue with <1: becomes stationary after transient

If the state space is strongly connected (any state can be reached from any state) then there is 0 or 1 stationary regime

Ex: queue

Else, there may be several distinct stationary regimes Ex: system with failure modes

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Stationarity and Transience

Knowing whether a model is stationary is sometimes a hard problemWe will see important models where this is solvedEx: Solved for single queues, not for networksEx: time series models

Reasoning about your system may give you indicationsDo you expect growth ? Do you expect seasonality ?

Once you believe your model is stationary, you should handle transientsRemove (how ? Look at your output and guess)Sometimes it is possible to avoid transients at all (perfect simulation)

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Non-Stationary (Time Dependent Inputs)

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Typical Reasons For Non Stationarity

Obvious dependency on timeSeasonality, growthCan be ignored at small time scale (minute)

Non Stability: ExplosionQueue with utilization factor >1

Non Stability: Freezing Simulation System becomes slower with time (aging)

Typically because there are rare events of large impact (« Kings »)The longer the simulation, the larger the largest king

Ex: time between regeneration points has infinite mean

We’ll come back to this in the chapter « Importance of the View Point »

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2 Simulation Techniques

Discrete event simulationsRecurrences

Stochastic recurrences

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Discrete event simulationUses an Event Scheduler

Example:Information system modelled as a single server queueThree event classes

arrivalserviceDeparture

One event scheduler(global system clock)

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0 t1 t2 t3

queuelength

1

2

t4

arrival service departure

Scheduler: Timeline

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departuret=t2

servicet=t2

arrivalt=t4

servicet=t2

departuret=t3

arrivalt=t4

departuret=t3

arrivalt=t4

servicet=0

arrivalt=t1

arrivalt=0

arrivalt=t1

departuret=t2

servicet=0

departuret=t2

arrivalt=t4

arrivalt=t1

arrivalt=0

initialization

0 t1 t2 t3

queuelength

1

2

t4

Stat

e of

sch

edul

erE

ven

t bei

ng

exec

ute

d

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

Statistical Counters

Assume we want to output: mean queue length and mean response time. How do we do this ?

Note the difference betweenEvent based statisticTime based statistic

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A Classical Organization of Simulation Code

Events contain specific codeA main loop advances the state of the schedulerExample: in the code of a departure event

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Main program

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Stochastic Recurrence

An alternative to discrete event simulationfaster but requires more work on the modelnot always applicable

Defined by iteration:

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Example: random waypoint mobility model

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Queuing System implemented as Stochastic Recurrence

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3 Accuracy of Simulation Output

A stochastic simulation produces a random output, we need confidence intervalsMethod of choice: independent replicationsRemove transients

For non terminating simulations

Be careful to have truly random seedsEx: use computer time as seed

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Results of 30 Independent Replications

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Do They Look Normal ?

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Bootstrap Replicates

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Confidence Intervals

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Mean, normal approx

Median

Mean, bootstrap

5 Random Number Generator

A stochastic simulation does not use truly random numbers but pseudo-random numbers

Produces a random number » U(0,1)Example (obsolete but commonly used, eg in ns2: )

Output appears to be random (see next slides)

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The Linear Congruential Generator of ns2

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uniform qq plot

autocorrelation

Lag Diagram, 1000 points

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Period of RNG

RNG is in fact periodicPeriod should be much larger than maximum number of uses

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Two Parallel Streams with too simple a RNG

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Impact of RNG

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Take Home Message

Be careful to have a RNG that has a period orders of magnitude larger than what you will ever use in the simulationSerialize the use of the RNG rather than parallel streams

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6 Sampling From A Distribution

Pb is: Given a distribution F(), and a RNG, produce some sample X that is drawn from this distribution

A common task in simulationMatlab does it for us most of the time, but not alwaysTwo generic methods

CDF inversionRejection sampling

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CDF Inversion

Applies to real or integer valued RV

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Example: integer valued RV

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Rejection SamplingApplies more generally, also to joint n-dimensional distributionsExample 1: conditional distribution on this areaStep1 :

Can you sample a point uniformly in the bounding rectangle ?

Step 2:How can you go from there to a uniform sample inside the non convex area ?

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1000 samples

Rejection Sampling for Conditional Distribution

This is the main idea

How can you apply this to the example in the previous slide ?

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Rejection Sampling for General Distributions

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A Sample from a Weird Distribution

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Another Sample from a Weird Distribution

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6.3 Ad-Hoc Methods

Optimized methods exist for some common distributionsOptimization = reduce computing time

If implemented in your tool, use them !Example: simulating a normal distribution

Inversion method is not simple (no closed form for F-1)Rejection method is possibleBut a more efficient method exists, for drawing jointly 2 independent normal RV

There are also ad-hoc methods for n-dimensional normal distributions (gaussian vectors)

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A Sample from a 2d- Normal Vector

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4 Monte Carlo Simulation

A simple method to compute integrals of all kindsIdea: interpret the integral as Assume you can simulate as many independent samples of X as you want

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Take Home Message

Most hard problems relative to computing a probability or an integral can be solved with Monte Carlo

Braindumb but why notRun time may be large -> importance sampling techniques (see later in this lecture)

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Conclusion

Simulating well requires knowing the concepts ofTransienceConfidence intervalsSampling methods

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