Page 234 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes according to the random variable X. The key relation between the general RBM and canonical
RBM is:

d
1


1
a
X
b

{ ( ; , , ) : t  0} {c R (d t ; 1,1,cX t  0}
t
R

) :
or, equivalently,
d
R
{ ( ; 1,1, ) :t  0} {cR ( ; , , / ) : t  0}.
dt
a
X

t

X
c
b
where
| | b  1 1
a
c  ,d  ,a  andb  ,
2
b a 2 cd c d
where  d means equal in distribution (here as stochastic processes).
Hence it suffices to focus on canonical RBM. It has stationary density (x) = 2e —2x , x  0. If we
initialize RBM with its stationary distribution, then we obtain a stationary process. Let
R*  {R*(t; a, b) : t  0} denote stationary RBM, initialized by the stationary distribution.
If f(x) = x for canonical RBM, then we would be estimating the steady-state mean  = 1/2. In this
case, the asymptotic bias is  = 1/4 (Theorem 1.3 of Abate and Whitt (1987a)) and the asymptotic
2
variance (for R*) is  = 1/2 (Abate and Whitt (1988b)).
To describe the general RBM with parameters a and b, we apply the scaling relations in Subsection
6.1. As a consequence of those scaling properties, the mean and variance of the steady-state
distribution of RBM(a, b) are:
b b 2
 , a b  and 2 , a b   2 , a b  ,
a
2| | 4a 2
and the asymptotic parameters are:
b 2 2 b 3
  and  .
, a b 3 , a b 4
a
4| | 2a
For the relative-width criterion, the key ratios are:
 b  2 2b
, a b , a b
 and  .
 , a b 2a 2  2 , a b a 2
Thus we see that the relative asymptotic bias is about the same as the relative asymptotic
variance. Since the bias of the sample mean X is of order O(1/t), while the square root of the
t
variance of the sample mean X is of order O(1/ t ), the bias tends to be negligible for large t.
t
Example: OU
Suppose that the diffusion process is the Ornstein-Uhlenbeck (OU) diffusion process on the
interval (—  ,  ) with drift function (x) = ax and diffusion function  (x) = b, where a < 0 < b,
2
which we refer to as OU(a; b). It is the continuous analog of the queue-length process in the
M/M/  queue when we center appropriately.

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