Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event,...
-
date post
19-Dec-2015 -
Category
Documents
-
view
217 -
download
1
Transcript of Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event,...
![Page 1: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/1.jpg)
Simulation
Where real stuff starts
![Page 2: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/2.jpg)
ToC
1. What, transience, stationarity2. How, discrete event, recurrence
3. Accuracy of output4. Monte Carlo
5. Random Number Generators6. How to sample
2
![Page 3: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/3.jpg)
1 What is a simulation ?
An experiment in computerImportant differences
Simulated vs real timeSerialization of events
3
![Page 4: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/4.jpg)
Stationarity
A simulation can be terminating or not
For a terminating simulation you should make sure it is stationary
4
![Page 5: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/5.jpg)
Information server, scenario 1
5
![Page 6: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/6.jpg)
Information server, scenario 2
6
![Page 7: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/7.jpg)
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
7
![Page 8: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/8.jpg)
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”
8
![Page 9: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/9.jpg)
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
9
![Page 10: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/10.jpg)
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)
10
![Page 11: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/11.jpg)
Non-Stationary (Time Dependent Inputs)
11
![Page 12: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/12.jpg)
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 »
12
![Page 13: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/13.jpg)
2 Simulation Techniques
Discrete event simulationsRecurrences
Stochastic recurrences
13
![Page 14: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/14.jpg)
Discrete event simulationUses an Event Scheduler
Example:Information system modelled as a single server queueThree event classes
arrivalserviceDeparture
One event scheduler(global system clock)
14
0 t1 t2 t3
queuelength
1
2
t4
arrival service departure
![Page 15: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/15.jpg)
Scheduler: Timeline
15
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
![Page 16: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/16.jpg)
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
16
![Page 17: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/17.jpg)
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
17
![Page 18: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/18.jpg)
Main program
18
![Page 19: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/19.jpg)
Stochastic Recurrence
An alternative to discrete event simulationfaster but requires more work on the modelnot always applicable
Defined by iteration:
19
![Page 20: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/20.jpg)
Example: random waypoint mobility model
20
![Page 21: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/21.jpg)
21
![Page 22: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/22.jpg)
22
![Page 23: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/23.jpg)
Queuing System implemented as Stochastic Recurrence
23
![Page 24: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/24.jpg)
24
![Page 25: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/25.jpg)
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
25
![Page 26: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/26.jpg)
Results of 30 Independent Replications
26
![Page 27: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/27.jpg)
Do They Look Normal ?
27
![Page 28: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/28.jpg)
Bootstrap Replicates
28
![Page 29: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/29.jpg)
Confidence Intervals
29
Mean, normal approx
Median
Mean, bootstrap
![Page 30: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/30.jpg)
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)
30
![Page 31: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/31.jpg)
The Linear Congruential Generator of ns2
31
uniform qq plot
autocorrelation
![Page 32: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/32.jpg)
Lag Diagram, 1000 points
32
![Page 33: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/33.jpg)
Period of RNG
RNG is in fact periodicPeriod should be much larger than maximum number of uses
33
![Page 34: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/34.jpg)
Two Parallel Streams with too simple a RNG
34
![Page 35: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/35.jpg)
Impact of RNG
35
![Page 36: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/36.jpg)
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
36
![Page 37: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/37.jpg)
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
37
![Page 38: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/38.jpg)
CDF Inversion
Applies to real or integer valued RV
38
![Page 39: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/39.jpg)
39
![Page 40: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/40.jpg)
Example: integer valued RV
40
![Page 41: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/41.jpg)
41
![Page 42: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/42.jpg)
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 ?
42
1000 samples
![Page 43: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/43.jpg)
Rejection Sampling for Conditional Distribution
This is the main idea
How can you apply this to the example in the previous slide ?
43
![Page 44: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/44.jpg)
Rejection Sampling for General Distributions
44
![Page 45: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/45.jpg)
45
![Page 46: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/46.jpg)
A Sample from a Weird Distribution
46
![Page 47: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/47.jpg)
47
![Page 48: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/48.jpg)
Another Sample from a Weird Distribution
48
![Page 49: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/49.jpg)
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)
49
![Page 50: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/50.jpg)
A Sample from a 2d- Normal Vector
50
![Page 51: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/51.jpg)
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
51
![Page 52: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/52.jpg)
52
![Page 53: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/53.jpg)
53
![Page 54: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/54.jpg)
54
![Page 55: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/55.jpg)
55
![Page 56: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/56.jpg)
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)
56
![Page 57: Simulation Where real stuff starts. ToC 1.What, transience, stationarity 2.How, discrete event, recurrence 3.Accuracy of output 4.Monte Carlo 5.Random.](https://reader034.fdocuments.net/reader034/viewer/2022051516/56649d2f5503460f94a076d1/html5/thumbnails/57.jpg)
Conclusion
Simulating well requires knowing the concepts ofTransienceConfidence intervalsSampling methods
57