1. VARIANCE REDUCTION

2. CALCULATIONS ON VARIANCES: SOME BASICS
Let X and Y be random variables
COV=0 if X and Y are independent.

3. COMMON RANDOM NUMBERS Built for distinguishing among two systems
di = yi – xi
Variance reduced by COV(X, Y)
Streaming induces MORE Covariance

4. STREAMING Zi=aZi-1 mod m
Segregate the random number generation task into streams connected to phenomena
Zi=aZi-1 mod m
seed1
seed2
Inter-arrival
times
Service
times
1. Change features of the service.
2. Use exact same arrival stream for
comparing each service setting.

5. ANTITHETIC VARIATES Use Uniforms U1, U2, ... to generate a sample
Use Uniforms 1-U1, 1-U2, ... to generate a second sample
Combine the samples
Extreme values get canceled out
Depends on...
effective streaming
straightforward F-1(U) method of variate generation

6. spreadsheet...

7. CONTROL VARIATES X is your output variable Y is a random variable
You seek the Expected Value of X
Y is a random variable
Y is one of the variables that we are generating
We know the Expected Value of Y
Example
X is the total waiting time of a customer
Y is the inter-arrival time before he entered service

8. ...more CONTROL VARIATES
Xc is a random variable with less Variance and the same Expected Value
pick b to minimize VAR(Xc)

9. OPTIMAL CONTROL

10. IMPORTANT CALCULATIONS Fusing many results in statistics

11. ALSO KNOWN AS... We are regressing X vs. Y
b* is the parameter that a regression package would calculate
r = SQRT[COV(X,Y)2/VAR(X)VAR(Y)] is the correlation coefficient of X and Y
r =1 or -1 implies
Y completely explains X and
VAR(Xc)=0