Page 232 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes the M/G/  model, there is a natural way to do this: The steady-state distribution of the number
of busy servers is again Poisson with mean  , just as for the M/M/  model. In addition, in
steady-state, conditional upon the number of busy servers, the remaining service times of those
customers in service are distributed as IID random variables with the stationary-excess (or
equilibrium residual-life) Cumulative Distribution Function (CDF) G associated with the service-
e
time CDF G, i.e.,
t
1
G ( ) m  [1 G ( )]du , (1)

u

t
e 0 
where m is the mean of G (here m = 1); e.g., see Takács (1962).
It is natural to apply this insight to more general many-server queueing models. Even in more
general G/G/s models, it is natural to initialize the simulation by putting s customers in the
system at time 0 and letting their remaining service times be distributed as s IID random
variables with cdf G . For large s, that should be much better than starting the system empty.
e
For many-server queues, we may be interested in different congestion measures. By Little’s law
(L =  W), we know that the expected steady-state number of busy servers in the G/G/s/  model
is exactly  (provided that  < s). Thus, in simulation experiments, we usually are more
+
+
interested in estimating quantities such as E[(Q(  ) — s) ], where (x)  max{0, x}, or P(Q(  ) >
s + k). Note that we can calculate the asymptotic bias and the asymptotic variance for these
quantities in the M/M/s model by applying the BD recursion with appropriate functions f. With
large s, it often helps to start the recursion at s and move away in both directions. The initial
value at s can be taken to be 1; afterwards the correct value is obtained by choosing the appropriate
normalizing constant.

12.3.4 Diffusion Processes

Diffusion processes are continuous analogues of BD processes; e.g., see Karlin and Taylor (1981)
and Browne and Whitt (1995). In this chapter we discuss diffusion processes because we are
interested in them as approximations of other processes that we might naturally simulate using
discrete-event simulation. We want to use the diffusion processes to approximate the asymptotic
bias and the asymptotic variance of sample means in the original process.
Diffusion processes tend to be complicated to simulate directly because they have continuous,
continuously fluctuating, sample paths. Nevertheless, there also is great interest in simulating
diffusion processes and stochastic differential equations, e.g., for finance applications, and special
techniques have been developed; see Kloeden, Platen and Schurz (1994) and Kloeden and Platen
(1995). Hence the analysis in this section may have broader application.

For diffusion processes, there are integral representations of the asymptotic parameters,
paralleling the sums exhibited for BD processes. Corresponding to the finite-state-space
assumption for the BD processes, assume that the state space of the diffusion is the finite interval
[s , s ] and let the diffusion be reflecting at the boundary points s and s , but under regularity
1 2 1 2
conditions the integral representations will be valid over unbounded intervals. Let {Y (t) : t  0}
be the diffusion process and let X(t) = f(Y (t)) for a real-valued function f. The diffusion process is
2
characterized by its drift function (x) and its diffusion function  (x).
Let  be the stationary probability density. The stationary probability density can be represented
as
y
m ( )
y
 ( )  ,s  y  s 2 .
1
M ( )
s
2
226 LOVELY PROFESSIONAL UNIVERSITY