Page 235 - DCAP601_SIMULATION_AND_MODELING

Unit 12: Design and Evaluation of Simulation Experiments (II)

We also can analyze the OU (a, b) processes by considering only the special case in which a =— Notes
1 and b = 1, which we call canonical OU. We can analyze OU(a, b) in terms of OU(-1,1) because we
can relate the two OU’s by appropriately scaling time and space, just as we did for RBM. For that
purpose, let {Z(t, a, b, X) : t  0} denote OU (a, b) with initial distribution according to the random
variable X. The key relation between the general OU (a, b) and canonical OU(-1,1) is:

d


1
1
Z
{ ( ; , , ) : t  0} {c Z (d t ; 1,1,cX t  0}
t


X
a
b
) :
or, equivalently,
d
b
c

X
X

a
{ ( ; 1,1, ) :t  0} {cZ ( ; , , / ) : t  0}.
t
Z
dt
where
a
| | b  1 1
c  ,d  ,a  andb  .
2
b a 2 cd c d
Then the stationary density of canonical OU is normal with mean 0 and variance 1/2. The mean
of canonical OU starting at x is
t
x
Z
E [ ( ; 1,1, )] xe t  ,t  0.


Paralleling our treatment of RBM, let Z*  {Z*(t, a, b) : t  0} be stationary OU, obtained by
initializing the OU(a,b) with the stationary normal distribution. For stationary canonical OU,
the autocovariance function is
1
t
Z
Z
Cov ( * (0), * ( ))  e t  , t  0.
2
Hence, the asymptotic parameters for canonical OU are
1
2
 ( )   and 

2
Just as with RBM, we can apply Section to determine the effect of scaling. The mean and variance
of the steady-state distribution of OU(a,b) are
b
  0and 2  ,
, a b , a b
2| |
a
and the asymptotic parameters are
b 2 b 3
  x and 2  .
a , ,x 3 , a b 4
b
| | 2a
a
The relative-width criterion makes less sense here because the random variables are not
non-negative.
12.3.5 Stochastic-Process Limits
In this section we discuss stochastic-process limits that make the RBM and OU diffusion processes
serve as useful approximations for queueing models. We start by discussing the impact of
scaling space and time. The scaling is often the key part.
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