Page 237 - DCAP601_SIMULATION_AND_MODELING

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

and service processes). For example, in the standard GI/GI/s/  model the heavy-traffic limit Notes
holds with

2
2
a = —s and b = s( c  c ).
a
s
2
2
where c and c are the SCV’s of an interarrival time and a service time, respectively (provided
a
s
that the second moments are finite). Similar limits hold for workload processes, recording the
amount of remaining unfinished work in service time in the system.
We thus can apply the stochastic-process limit with the scaling properties in Section and the
properties of RBM to obtain approximations paralleling the exact results for the M/M/1 queue.
We apply the stochastic-process limit to obtain the approximation
—1
2
{Q (t) : t  0}  {(1 —  ) R(t(1 —  ) ; a, b) : t  0}.

The resulting approximations for the mean and variance of the steady-state distribution of the
queue-length process are thus
b b 2
2
[ E Q  ( )]  and  Var (Q  ( ))  ;



2
a
a
2 (1   ) 4 (1  ) 2
the approximations for the asymptotic parameters are
b 2 2 b 3
  and  .
 3 3  4 4
a
a
4| | (1   ) 2 (1   )
In the GI/GI/s/  case, we just substitute the specific parameters a and b above. The resulting
approximate asymptotic variance is
2
)
(c  c 2 3
2
   2 2  a s .
 ( , , a c 2 , s c ) 2 (1   ) 4

s
s
Notes Note that these formulas agree with the limits of the M/M/1 formulas as  1.
Thus, we see that the M/M/1 formulas are remarkably descriptive more generally. But we also
see the impact of s servers and the GI arrival and service processes. The asymptotic variance is
2
directly proportional to 1/s and to the third power of the overall “variability parameter” (c  c 2 )
a s
1

as well as to the fourth power of (1   ) .
More generally, we see how the parameters s,  , a and b in more general G/G/s/  models (with
non-renewal arrival processes and non-IID service times) will affect the required simulation run
length. Once we have established the corresponding heavy-traffic limit and identified the new
values of a and b for these alternative models, we can apply the results above. For the relative-
width criterion, the key ratios are

 , a b b  2 , a b 2b
 and  .
2
 2 (1   ) 2  2 a  (1   ) 2
a
, a b , a b
Values of the key parameters a and b in alternative models have been determined; e.g., see
Sections of Whitt (1989) and Fendick, Saksena and Whitt (1989).

LOVELY PROFESSIONAL UNIVERSITY 231