Page 233 - DCAP601_SIMULATION_AND_MODELING

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

where Notes

2
y
m ( ) 
y
s
 2 ( ) ( )
y
is the speed density,
y
x
 1 1 s  [2 ( )/ 2 ( )]dx

x
s ( )  e
y
is the scale density and
y
M ( )  1 s  m ( )dx ,s  y  s 2 .
y
x
1
provided that the integrals are finite.
Let p(t, x, y) be the transition kernel. Then, paralleling the fundamental matrix of a CTMC, we
can define the fundamental function of a diffusion process, Z  Z(x, y), by

y
x
p
y
x
y
Z ( , )  0  [ ( , , )   ( )] .
dt
t
As before, let  be the average of f with respect to the stationary probability density , i.e.,
2 s
  1 s   ( ) ( )dx .
x
f
x
Then the integral representations for the asymptotic bias ( )  starting with initial probability
density  and the asymptotic variance  are:
2
2 s 1  y y 
 ( ) 2 1 s   2 ( ) ( )   1 s  ( ( )   ) ( )dx 1 s  ( ( )   ( ))dz dy




x
z
z
x
f
 
y 
y
and
2 s 1 y 2
2
x
x

  4 1 s   2 ( ) ( )    1 s  ( ( )   ) ( )dx    dy .
f
y
y 
We now discuss two examples of diffusion processes, which are especially important because
they arise as limit processes for queueing models.
Example: RBM
Suppose that the diffusion process is reflected Brownian motion (RBM) on the interval [0,  ) with
2
drift function (x) = a and diffusion function  (x) = b, where a < 0 < b, which we refer to by RBM(a;
b); see Harrison (1985), Whitt (2002) and references therein for more background. RBM is the
continuous analog of the queue-length process for the M/M/1 queue (as we will explain in the next
section). It is a relatively simple stochastic process with only the two parameters a and b.
In fact, we can analyze the RBM(a, b) processes by considering only the special case in which
a = —1 and b = 1, which we call canonical RBM because there are no free parameters. We can analyze
RBM(a, b) in terms of RBM(-1,1) because we can relate the two RBM’s by appropriately scaling
time and space. For that purpose, let {R(t; a, b, X) : t  0} denote RBM (a, b) with initial distribution
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