Page 230 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes
It is interesting to see that the asymptotic bias ( )  and the asymptotic variance  can be
2
arbitrarily large in a very small BD model with bounded rates. Suppose that m = 2, so that the BD
x
process has only 3 states: 0, 1 and 2. Consider the symmetric model in which     and
2
0
    1 , where 0 < x  1. Then the stationary distribution is:
1
1
1 x
    and  .
1
2
0

2 x 2 x

Let f = i for all i, so that we are estimating the mean . Then the mean is  = 1 and the asymptotic
i
variance is
4 2
2
   for small . x
x (2 x ) x

2
This model has a high asymptotic variance  for small x because the model is bistable, tending
to remain in the states 0 and 2 a long time before leaving. To see this, note that the mean first
passage time from state 0 or state 2 to state 1 is 1/x.
2
Note that the large asymptotic variance  cannot be detected from the variance of the steady-
state distribution,  . As x  0,  , the variance of , increases to 1. Thus, the ratio  /  is of
2
2
2
2
order O(1/x). The steady-state distribution has moderate variance, but the process has quite
strong dependence (but not so strong that the asymptotic variance becomes infinite).
The asymptotic bias starting in state 0 (or state 2) is also large for small x. The asymptotic bias
starting in state 0 is
 (x  1) 2  1
 (0)   for small . x
( x x  2) 2 4x
As a function of the key model parameter x, the bias is much more important here than it was for
the previousM/M/1 queue example. Here, both the asymptotic bias and the asymptotic variance
are of order O(1/x), so that as a function of x, for very small x, the width of the confidence interval
is (1/O x ), while the bias is of order O(1/x). Thus the bias tends to be much more important in
this example. In particular, the run length required to make the bias suitably small is of the same
order as the run length required to make the width of a confidence interval suitably small. For
this example, using simulation to estimate the mean  when the parameter x is very small
would be difficult at best.
This model is clearly pathological. For very small x, a relatively long simulation run of this
model starting in state 0 could yield a sample path that is identically zero. We might never
experience even a single transition! This example demonstrates that it can be very helpful to
know something about model structure when conducting a simulation.

Example: The M/M/  queue.
A queueing system with many servers tends to behave quite differently from a single-server
queue. A queueing system with many servers can often be well approximated by an infinite-
server queue. Thus we consider the number of busy servers at time t, also denoted by Q(t), in an
M/M/  queue. As before, let the mean individual service time be 1, but now let the arrival rate
be  (since the previous notion of traffic intensity is no longer appropriate). Now the arrival
rate  can be arbitrarily large.

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