Simulation Lecture 1

39 Pages·2012·0.65 MB·English

2WB05 Simulation
Lecture 1: Introduction and
Monte Carlo simulation
Marko Boon
http://www.win.tue.nl/courses/2WB05
November12,2012
Organisation
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• studeerwijzer available (with notes, slides, programs, assignments)
• examination consists of 4 or 5 take home assignments. Assignments are given on Mondays. Some assign-
ments take two weeks to complete, while others have to be handed in one week later.
• assignments done in groups of 2
• register your group (on the homepage of 2WB05) and register yourself in OASE.
Schedule
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Week Dates Monday Thursday
1 12-18 Nov Course 1
2 19-25 Nov Course 2 Course 3
3 26 Nov - 2 Dec Course 4
4 3-9 Dec Course 5
5 10-16 Dec Course 6 Extra course ???
6 17-23 Dec Course 7
7 7-13 Jan Course 8
Warning: this is a preliminary schedule. Check the course website for the definite schedule.
Topics
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• Monte Carlo simulation
• Modelling of discrete-event systems
• Programming
• Random number generation
• Output analysis
Discrete-event systems
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State changes at (random) discrete points in time
Examples:
• Manufacturing systems
Completion times of jobs at machines, machine break-downs
• Inventory systems
Arrival times of customer demand, replenishments
• Communication systems
Arrival times of messages at communication links
Discrete-event systems
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Programming tools:
• General purpose languages (C, C++, Java, Mathematica, ...);
• Simulation language χ developed by the Systems Engineering group
• Simulation system Arena
Monte Carlo simulation
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The law of large numbers
Z , Z ,... Z are i.i.d. random variables with mean z := E[Z ] and finite variance.
1 2 n 1
ε >
The probability of the sample mean being close to z is large. In fact, for every 0,
(cid:18)(cid:12)Z + Z + ... + Z (cid:12) (cid:19)
lim P (cid:12)(cid:12) 1 2 n − z(cid:12)(cid:12) < ε = 1.
n→∞ (cid:12) n (cid:12)
Estimator for z:
Z + Z + ··· + Z
1 2 n
Z ≡ Z :=
n
n
P( > )
Estimator for Z t :
i
E[1{Z >t}]
i
Simple probabilistic problems
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Simulation is a perfect tool to develop and sharpen your intuition
for probabilistic models (see also Tijms’ book Spelen met kansen)
In no time probabilistic properties can be illustrated by a simulation
experiment and the results can be shown in graphs or tables!
Problems
• coin tossing
• (nearly) birth day problem
• lottery
Simple probabilistic problems
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Coin tossing
Two players A and B throw a fair coin N times.
If Head, then A gets 1 point; otherwise B.
• What happens to the absolute difference in points as N increases?
• What is the probability that one of the players is leading between 50% and 55% of the time? Or more than
95% of the time?
• In case of 20 trials, say, what is the probability of 5 Heads in a row?
Simple probabilistic problems
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Birthday problem
Consider a group of N randomly chosen persons.
What is the probability that at least 2 persons have the same birthday?
Nearly birthday problem
What is the probability that at least 2 persons have their birthday within r days of each other?