for Review Exam II
May 5, 2008
What We'll Do ...
Today
Exam Review
Tomorrow
Nana's office hour: 1-2:30, 3:30-4:20 Mavis's office hour: 2:30-4:00
Wednesday
Exam Day
Review for Exam
General Advices Warm-up Periods and Run Length Random Number and Variate Generation Non-homogeneous Poisson Process Compound Poisson Process
General Advices
Review all homework problems after Exam I
Especially those without Arena/Excel You are not allowed to use laptop
Bring you calculators!!
Do not spend too much time computing
Partial Credit will be given if you indicate the correct approach No need to simplify complex mathematical expressions
Do the easiest questions first
Warm Up and Run Length
Most models start empty and idle
Empty: No entities are present at time 0 Idle: All resources are idle at time 0 In a terminating simulation this is OK if realistic In a steady-state simulation, though, this can bias the output for a while after startup
Bias can go either way Usually downward (results are biased low) in queueing-type models that eventually get congested Depending on model, parameters, and run length, the bias can be very severe
Warm Up and Run Length (cont'd.)
Remedies for initialization bias
Better starting state, more typical of steady state
Throw some entities around the model Can be inconvenient to do this in the model How do you know how many to throw and where?
This is what you're trying to estimate in the first place!
Make the run so long that bias is overwhelmed
Might work if initial bias is weak or dissipates quickly Most practical idea: preliminary runs, plots Simply "eyeball" them
Warm-up and run length times?
Warm Up and Run Length (cont'd.)
Batching in a Single Run
If model warms up very slowly, truncated replications can be costly
Have to "pay" warm-up on each replication Only have to "pay" warm-up once
Alternative: Just one R E A L L Y long run
Batching in a Single Run (cont'd.)
Break each output record from the run into a few large batches
Tally (discrete-time) outputs: Observation-based Time-Persistent (continuous-time): Time-based
Take averages over batches as "basic" statistics for estimation: Batch means
Tally outputs: Simple arithmetic averages Time-Persistent: Continuous-time averages Key: batch size must be big enough for low correlation between successive batches (details in text) Still might want to truncate (once, time-based)
Treat batch means as IID
Example--Homework 7 Question 5
Suppose a warm-up time is 400 hours
a)
a)
a)
Assume you need 1000 replications of 100 hours each. How many hours of operation will you need to simulate if you need to include the warm-up period in each replication? Ans: 100+400=500 What would be an alternative way of generating the needed replications, if including the warm-up period in each replication takes too much computation time? Ans: An alternative way is to use batch-means methods in a single long run with a single Warm-up Period at its beginning, each batch being 100 hours How many fewer hours of operation would you need to simulate if you generate the replications that way (in part b)? Ans: ( 100 + 400 ) * 1000 - ( 400 + 100 * 1000 ) = 399,600 hours
Random-Number Generators (RNGs)
Algorithm to generate independent, identically distributed draws from the continuous UNIF (0, 1) f(x) distribution
These are called random numbers in simulation
1 0 1 x
Basis for generating observations from all other distributions and random processes
Transform random numbers in a way that depends on the desired distribution or process (later in this chapter)
It's essential to have a good RNG There are a lot of bad RNGs -- this is very tricky
Methods, coding are both tricky
Linear Congruential Generators (LCGs) The most common of several different methods
But not the one in Arena (though it's related) ... later
Generate a sequence of integers Z1, Z2, Z3, ... via the recursion Zi = (a Zi1 + c) (mod m) a, c, and m are carefully chosen constants Specify a seed Z0 to start off "mod m" means take the remainder of dividing by m as the next Zi All the Zi's are between 0 and m 1 Return the ith "random number" as Ui = Zi / m
Example of a "Toy" LCG
Parameters m = 63, a = 22, c = 4, Z0 = 19: Zi = (22 Zi1 + 4) (mod 63), seed with Z0 = 19
i 0 1 2 3 4 : 61 62 63 64 65 66 : 22 Zi1+4 422 972 598 686 : 158 708 334 422 972 598 : Zi 19 44 27 31 56 : 32 15 19 44 27 31 : Ui 0.6984 0.4286 0.4921 0.8889 : 0.5079 0.2381 0.3016 0.6984 0.4286 0.4921 :
Cycling -- will repeat forever Cycle length m (could be << m depending on parameters) Pick m BIG But that might not be enough for good statistical properties
Issues with LCGs
Cycle length: < m
Typically, m = 2.1 billion (= 231 1) or more
Which used to be a lot ... more later
Other parameters chosen so that cycle length = m or m 1 Uniformity, independence There are many tests of RNGs
Statistical properties
tests
Empirical Theoretical tests -- "lattice" structure (next slide ...)
Speed, storage -- both are usually fine Must be carefully, cleverly coded -- BIG integers Reproducibility -- streams (long internal subsequences) with fixed seeds
Issues with LCGs (cont'd.)
"Regularity" of LCGs (and other kinds of RNGs): For the earlier "toy" LCG ...
Plot of Ui vs. i Plot of Ui+1 vs. Ui
"Random Numbers Fall Mainly in the Planes" -- George Marsaglia
"Design" RNGs: dense lattice in high dimensions Other kinds of RNGs -- longer memory in recursion, combination of several RNGs
Generating Random Variates
Have: Desired input distribution for model (fitted or specified in some way), and RNG (UNIF (0, 1)) Want: Transform UNIF (0, 1) random numbers into "draws" from the desired input distribution Method: Mathematical transformations of random numbers to "deform" them to the desired distribution
Specific transform depends on desired distribution Details in online Help about methods for all distributions
Do discrete, continuous distributions separately
Inverse Transform Techniques
1. Find the cdf of X, F(x) 2. Set F(x) = U ~ U(0,1) on the range of x 3. Solve x = F (U ) if you can
-1
Generating from Discrete Distributions
Example: probability mass function
2 0 3
Divide [0, 1]
Into subintervals of length 0.1, 0.5, 0.4 Generate U ~ UNIF (0, 1) See which subinterval it's in Return X = corresponding value
Discrete Generation: Another View
Plot cumulative distribution function; generate U and plot on vertical axis; read "across and down"
Inverting the CDF Equivalent to earlier method
Generating from Continuous Distributions Example: EXPO (5) distribution
Density (PDF) Distribution (CDF)
General algorithm (can be rigorously justified): 1. Generate a random number U ~ UNIF(0, 1) 2. Set U = F(X) and solve for X = F1(U)
Solving analytically for X may or may not be simple (or possible) Sometimes use numerical approximation to "solve"
Generating from Continuous Distributions (cont'd.) Solution for EXPO (5) case:
U = F(X) = 1 eX/5 eX/5 = 1 U X/5 = ln (1 U) X = 5 ln (1 U) Picture (inverting the CDF, as in discrete case):
Intuition (garden hose): More U's will hit F(x) where it's steep This is where the density f(x) is tallest, and we want a denser distribution of X's
Set
Example: Homework 8, Question1, 4
Use LCG to generate some random numbers
Hint: You must be able to calculate them without Excel!! Sensitivity issues
The impact of different parameters a, m, c, Z0
Use some random numbers to generate random variates
Review of Exponential Distribution
Density
e- x , x 0 f ( x) = , >0 0, x < 0
1 - e- x , x 0 F ( x) = 0, x < 0
1 E[ X ] = 1 Var [ X ] = 2
CDF
Mean
so...
Variance
Poisson Process Definition A counting process {N(t), t 0} is a Poisson process with rate , > 0, if N(0) = 0 The process has independent increments...