Page 202 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes 11.3.3 Control Variates

The method of control variates is based on the following idea. Suppose that we want to estimate
some unknown performance measure E(X) by doing K independent simulation runs, the i one
th
yielding the output random variable X with E(X ) = E(X). An unbiased estimator for E(X) is
i i
K
given by X   i 1 X i  /K. However, assume that at the same time we are able to simulate a

related output variable Y , with E(Y ) = E(Y), where E(Y ) is known. If we denote by
i i
K
Y   i 1 Y i  /K, then, for any constant c, the quantity X   c Y E Y   is also an unbiased

estimator of E(X). Furthermore, from the formula
2


Var(X c(Y E(Y))  Var(X) c Var(Y) 2cCov(X,Y).



it is easy to deduce that Var X c Y E Y     is minimized if we take c =c*, where
Cov(X,Y)
c*   .
Var(Y)
Unfortunately, the quantities Cov(X,Y) and Var(Y) are usually not known beforehand and must
be estimated from the simulated data. The quantity Y is called a control variate for the estimator
X . To see why the method works, suppose that X and Y are positively correlated. If a simulation
results in a large value of Y (i.e. Y larger than its mean E(Y )) then probably also X is larger than
its mean E(X) and so we correct for this by lowering the value of the estimator X . A similar
argument holds when X and Y are negatively correlated.

!

Caution In the simulation of the production line of section 2, a natural control variate
would be the long-term average production rate for the line with zero buffers.

11.3.4 Conditioning

The method of conditioning is based on the following two formulas. If X and Y are two arbitrary
random variables, then E(X) = E(E(X|Y )) and Var(X) = E(Var(X|Y )) +Var(E(X|Y ))  Var(E(X|Y)).
Hence, we conclude that the random variable E(X|Y ) has the same mean as and a smaller
variance than the random variable X.
How can we use these results to reduce the variance in a simulation? Let E(X) be the performance
measure that we want to estimate. If Y is a random variable such that E(X|Y = y) is known, then
the above formulas tell us that we can better simulate Y and use E(X|Y ) than that we directly
simulate X.
The method is illustrated using the example of an M/Mµ/1/N queueing model in which
customers who find upon arrival N other customers in the system are lost. The performance
measure that we want to simulate is E(X), the expected number of lost customers at some fixed
time t. A direct simulation would consist of K simulation runs until time t. Denoting by Xi the

number of lost customers in run i,then X = (  K i=1 X )/K is an unbiased estimator of E(X).
i
However, we can reduce the variance of the estimator in the following way. Let Y be the total
i

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