1. Where real stuff starts
28/05/2019
Simulation
Where real stuff starts

2. ToC What, transience, stationarity How, discrete event, recurrence
28/05/2019
What, transience, stationarity
How, discrete event, recurrence
Accuracy of output
Monte Carlo
Random Number Generators
How to sample 1

3. 1 What is a simulation ? An experiment in computer
28/05/2019
An experiment in computer
Important differences
Simulated vs real time
Serialization of events 2

4. Stationarity A simulation can be terminating or not
28/05/2019
A simulation can be terminating or not
For a terminating simulation you should make sure it is stationary 3

5. Information server, scenario 1
28/05/2019
Information server, scenario 1 4

6. Information server, scenario 2
28/05/2019
Information server, scenario 2 5

7. Stationarity In scenario 1: In scenario 2:
28/05/2019
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 6

8. Definition of Stationarity
28/05/2019
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” 7

9. Classical Cases Markov models
28/05/2019
Markov models
State Xt is sufficient to draw the future of the simulation
Common case for all simulations
For a Markov model, over a discrete state space
If you run the simulation long enough it will either walk to infinity (unstable) or converge to stationary
Ex: queue with  >1: unstable
queue 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 8

10. Stationarity and Transience
28/05/2019
Knowing whether a model is stationary is sometimes a hard problem
We will see important models where this is solved
Ex: Solved for single queues, not for networks
Ex: time series models
Reasoning about your system may give you indications
Do you expect growth ?
Do you expect seasonality ?
Once you believe your model is stationary, you should handle transients
Remove (how ? Look at your output and guess)
Sometimes it is possible to avoid transients at all (perfect simulation) 9

11. Non-Stationary (Time Dependent Inputs)
28/05/2019
Non-Stationary (Time Dependent Inputs) 10

12. Typical Reasons For Non Stationarity
Obvious dependency on time
Seasonality, growth
Can be ignored at small time scale (minute)
Non Stability: Explosion
Queue 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 » 11

13. Quiz
28/05/2019 12

14. 2 Simulation Techniques
28/05/2019
Discrete event simulations
Recurrences
Stochastic recurrences 13

15. Discrete event simulation Uses an Event Scheduler
28/05/2019 t1 t2 t3
queue
length 1 2 t4
Example:
Information system modelled as a single server queue
Three event classes
arrival
service
Departure
One event scheduler (global system clock)
arrival
service
departure 14

16. Scheduler: Timeline t1 t2 t3 queue length 1 2 t4 initialization Step 1t1 t2 t3
queue
length 1 2 t4
Scheduler: Timeline
28/05/2019
initialization
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
service
t=0
arrival
t=t1
arrival
t=t1
departure
t=t2
service
t=0
departure
t=t2
arrival
t=t4
t=t1
departure
t=t2
service
arrival
t=t4
service
t=t2
departure
t=t3
arrival
t=t4
departure
t=t3
arrival
t=t4
Event being executed
arrival
t=0
State of scheduler 15

17. Statistical Counters
28/05/2019
Assume we want to output: mean queue length and mean response time. How do we do this ?
Note the difference between
Event based statistic
Time based statistic 16

18. A Classical Organization of Simulation Code
28/05/2019
Events contain specific code
A main loop advances the state of the scheduler
Example: in the code of a departure event 17

19. 28/05/2019
Main program 18

20. 28/05/2019 19

21. Stochastic Recurrence
28/05/2019
An alternative to discrete event simulation
faster but requires more work on the model
not always applicable
Defined by iteration: 20

22. Example: random waypoint mobility model
28/05/2019 21

23. 28/05/2019 22

24. 28/05/2019 23

25. 24

26. Queuing System implemented as Stochastic Recurrence
28/05/2019 25

27. 28/05/2019 26

28. 3 Accuracy of Simulation Output
28/05/2019
A stochastic simulation produces a random output, we need confidence intervals
Method of choice: independent replications
Remove transients
For non terminating simulations
Be careful to have truly random seeds
Ex: use computer time as seed 27

29. Results of 30 Independent Replications
28/05/2019 28

30. 28/05/2019
Do They Look Normal ? 29

31. 28/05/2019
Bootstrap Replicates 30

32. Confidence Intervals Mean, bootstrap Median Mean, normal approx 31
28/05/2019
Mean, bootstrap
Median
Mean, normal approx 31

33. 5 Random Number Generator
28/05/2019
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) 32

34. The Linear Congruential Generator of ns2
28/05/2019
uniform qq plot
autocorrelation 33

35. Lag Diagram, 1000 points
28/05/2019 34

36. Period of RNG RNG is in fact periodic
28/05/2019
RNG is in fact periodic
Period should be much larger than maximum number of uses 35

37. Two Parallel Streams with too simple a RNG
28/05/2019 36

38. 28/05/2019
Impact of RNG 37

39. Take Home Message
28/05/2019
Be careful to have a RNG that has a period orders of magnitude larger than what you will ever use in the simulation
Serialize the use of the RNG rather than parallel streams 38

40. 6 Sampling From A Distribution
28/05/2019
Pb is:
Given a distribution F(), and a RNG, produce some sample X that is drawn from this distribution
A common task in simulation
Matlab does it for us most of the time, but not always
Two generic methods
CDF inversion
Rejection sampling 39

41. CDF Inversion
28/05/2019
Applies to real or integer valued RV 40

42. 28/05/2019 41

43. Example: integer valued RV
28/05/2019 42

44. 28/05/2019 43

45. 28/05/2019 44

46. Rejection Sampling
28/05/2019
Applies more generally, also to joint n-dimensional distributions
Example 1: conditional distribution on this area
Step1 :
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 ?
1000 samples 45

47. Rejection Sampling for Conditional Distribution
28/05/2019
This is the main idea
How can you apply this to the example in the previous slide ? 46

48. Rejection Sampling for General Distributions
28/05/2019 47

49. 28/05/2019 48

50. A Sample from a Weird Distribution
28/05/2019 49

51. 28/05/2019 50

52. Another Sample from a Weird Distribution
28/05/2019 51

53. 6.3 Ad-Hoc Methods
28/05/2019
Optimized methods exist for some common distributions
Optimization = 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 possible
But 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) 52

54. A Sample from a 2d- Normal Vector
28/05/2019 53

55. 4 Monte Carlo Simulation
28/05/2019
A simple method to compute integrals of all kinds
Idea: interpret the integral as
Assume you can simulate as many independent samples of X as you want 54

56. 28/05/2019 55

57. 28/05/2019 56

58. 28/05/2019 57

59. 28/05/2019 58

60. Take Home Message
28/05/2019
Most hard problems relative to computing a probability or an integral can be solved with Monte Carlo
Braindumb but why not
Run time may be large -> importance sampling techniques (see later in this lecture) 59

61. Questions
28/05/2019 60

62. Questions
Q4.2
A node receives messages from 2 sensors S1 and S2. Each of the sources sends messages independently of each other. The sequence of time intervals between messages sent by source S1 is iid, with a Gaussian distribution with mean m1 and variance s12 and similarly for S2. The node works as follows. It waits for the next message from S1. When it has received one message from S1, it waits for the next message from S2, and sends a message of its own.
How do you implement a simulator for this system ?
Q4.1
A system contains a population of tasks that arrive as a Poisson process. Each task spends a time that is random, iid, independent of the arrival process, and with Pareto distribution. How do you implement a simulation of this system ? 61

63. Conclusion Simulating well requires knowing the concepts of Transience
28/05/2019
Simulating well requires knowing the concepts of
Transience
Confidence intervals
Sampling methods 62