Page 225 - DCAP601_SIMULATION_AND_MODELING

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

variance  is substituted for the ordinary variance  ; e.g., the required simulation run length Notes
2
2
with a relative-width criterion is
4 2 2  2
z
k

n r ( , )   /2 andn r (10 ,0.05) 16 (10) 2k (4)



2 2
   2
From (1) and (4), we immediately see that the required run length is approximately  2 / 2
times greater when sampling from one run than with independent sampling (assuming that we
could directly observe independent samples from the steady-state distribution, which of course
is typically not possible).
As with independent replications, established simulation methodology and statistical theory
tells how to estimate the unknown quantities ,  and  from data; e.g., see Bratley et al. (1987)
2
and Fishman (2001). Instead, we apply additional information about the model to obtain rough
preliminary estimates for these parameters without data. For  , the representation of the
2
asymptotic variance in terms of the autocovariances is usually too complicated to be of much
help, but fortunately there is another approach, which we will describe in Section

12.3.2 The Asymptotic Parameters for a Function of a Markov Chain

From the previous section, it should be apparent that we can do the intended preliminary
planning if we can estimate the asymptotic bias and the asymptotic variance. We now start to
describe how we can calculate these important parameters. We first consider functions of a
Markov chain. That illustrates available general results. However, fast back-of-the-envelope
calculations usually depend on diffusion approximations, based on stochastic-process limits,
after doing appropriate scaling. Indeed, the scaling is usually the key part, and that is so simple
that back-of-the-envelope calculations are actually possible.
In this section, drawing on Whitt (1992), which itself is primarily a survey of known results
(including Glynn (1984) and Grassman (1987a,b) among others), we observe that (again under
regularity conditions) we can calculate the asymptotic bias and the asymptotic variance whenever
the stochastic process of interest is a function of a (positive-recurrent irreducible) Markov chain,
i.e., when X = f(Y ) for n  1, where f is a real-valued function and {Y : n  1} is a Markov chain
n n n
or when X(t) = f(Y (t)) for t  0, where again f is a real-valued function and {Y (t) : t  0} is a
Markov chain. As noted before, we usually obtain the required Markov structure by
approximating the given stochastic process by a related one with the Markov property.
In fact, as in Whitt (1992), we only discuss the case in which the underlying Markov chain has a
finite state space (by which we mean countably finite, i.e., {0, 1, ...., m}, not [c, d]), but the theory
extends to more general state spaces under regularity conditions. For illustrations, see Glynn
(1994) and Glynn and Meyn (1996). But the finite-state-space condition is very useful. Under the
finite-state-space condition, we can compute the asymptotic parameters numerically, without
relying on special model structure. However, when we do have special structure, we can
sometimes go further to obtain relatively simple closed-form formulas. We also obtain relatively
simple closed-form formulas when we establish diffusion-process approximations via stochastic-
process limits.

Continuous-time Markov Chains

We will discuss the case of a Continuous-time Markov Chain (CTMC); similar results hold for
discrete-time Markov chains. Suppose that the CTMC {Y (t) : t  0} is irreducible with finite state
space {0, 1, ..., m} (which implies that it is positive recurrent). Our sample-mean estimator is

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