VARIANCE REDUCTION

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

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

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.

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

spreadsheet...

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

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

OPTIMAL CONTROL

IMPORTANT CALCULATIONS Fusing many results in statistics

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