Page 229 - DCAP601_SIMULATION_AND_MODELING

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

Notes
 p
2
  and  ;
1   (1   ) 2
e.g., see Cohen (1982).

To estimate the required simulation run length from a single long run, we use the asymptotic
bias and the asymptotic variance. Let the system start out empty, so that the initial state is 0. As
an argument of ( )  , let 0 also denote the initial probability vector  that puts mass 1 on the
state 0. Then

 2 (1   )

2
 (0)  and   .
(1   ) 3 (1   ) 4
These formulas can be derived from the general BD formulas or directly; see Abate and Whitt
(1987b, 1988 a,b).
Ignoring the initial transient (assuming that the queue-length process we observe is a
stationary process), the required run length with a relative-width criterion, specified in general
in (4), here is
8(1   )z 2 /2 32(1  )(10) 2k
k


t r ( , )   andt r (10 ,0.05) .

2
)
 (1    2  (1  ) 2
For 10% statistical precision ( = 0.1) with 95% confidence intervals ( = 0.05), when the traffic
intensity is  = 0.9, the required run length is about 675,000 (mean service times, which
corresponds to an expected number of arrivals equal to 0.9675,000 = 608,000); when the traffic
intensity is  = 0.99, the required run length is 64,300,000 (mean service times, which corresponds
to an expected number of arrivals equal to 0.964,300, 000 = 57,900,000). To summarize, for high
traffic intensities, the required run length is of order 10 or more mean service times. We can
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anticipate great computational difficulty as the traffic intensity  increases toward the critical
value for stability.
Compared to independent sampling of the steady-state queue length (which is typically not
directly an option), the required run length is greater by a factor of
 2 2(1   )
 .
 2  (1   ) 2

which equals 422 when  = 0.9 and 40,200 when  = 0.99. Clearly, the dependence can make a
great difference.

Now let us consider the bias. The relative bias is
 (0)  (0) 1
t   ,
 t (1  ) 2t
so that, for  = 0.9 the relative bias starting empty is about 100/t, where t is the run length. For
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a run length of 675,000, the relative bias is 1.510 or 0.015%, which is indeed negligible
compared to the specified 10% relative width of the confidence interval. Hence the bias is in the
noise; it can be ignored for high traffic intensities. The situation is even more extreme for higher
traffic intensities such as  = 0.99.

Example. A small example with large asymptotic parameters

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