Geoscience Reference
In-Depth Information
while when is small, the waiting times in both systems are quite small and the
large relative difference may not be of practical significance. Thus, as a rough
approximation, M=G=1 system can be used in place of M=G= when the expected
waiting times are of primary interest.
How
ev
er, when the primary measure of interest is the expected total time in the
system W , one has to be more care
fu
l. When the system is highly utilized, i.e., is
close to 1, the main determinant of W is the waiting time and the previous argument
applies. However, when the system utilization is lower, the expected service time
will play a large role. Since it is 1=
0
for M=G= and 1=
D
=
0
for M=G=1,
the differe
nc
e is quite large and approximation is no longer appropriate. Thus, with
respect to W , the approximation can only be justified in the heavy utilization case.
Turning our attention back to the M=G=1 system, we would like to
rewrite (
17.12
) in terms of decision variables in our model. This is not difficult
to do, and with a little algebraic manipulation we obtain the following expression
for the expected waiting time at an open facility i
2
I:
i
C
1
D
.1
C
2
/
i
2
i
.
i
i
/
C
1
W
i
D
W
q
(17.15)
i
i
with
i
given by (
17.6
). We assume that W
i
D
0 if there is no facility at i.
Several comments are in order. First, we treat
2
as an intrinsic model parameter,
rather than a decision variable, i.e., we assume that the coefficient of variation of
service times is fixed in advance. While this is certainly the case when a specific
distribution of service times is assumed (e.g., for M=M=1 queues
2
D
1), there is,
in principle, no reason why this should not be a decision parameter in the system.
For example, if the decision on how much capacity to install in facility i also deals
with
what kind
of capacity to install, then the coefficient of variation could well
be affected, as well as
i
: service systems with higher level of automation may have
lower , while more manual processes may have higher (of course the resulting
values may be different at different facilities, so
i
notation would have to be used).
Another case where may be a decision variable is when customers at different
nodes have different service time variabilities, in which case the allocation decisions
x
ij
may well influence the total demand
i
and the variability of service times
i
as well as
i
. Nevertheless, we are not aware of any SLCIS model that treats this
parameter as a decision variable; in fact the value of the coefficient of variation is
assumed to be identical at all facilities, which is reflected in our usage of without
a subscript.
Second, observe that W
i
(and W
q
i
) is decreasing in
i
, increasing in
i
and
convex with respect to both
i
and
i
whenever system stability conditions (
17.7
)
hold. These properties are exploited in many SLCIS models that follow.
Let
WC
.
w
/ represent the “waiting cost”, i.e. the cost incurred by customers
waiting
w
units of time (henceforth we assume that waits include service times,
i.e. use measure W defined earlier; an equivalent treatment can be developed by
focusing on waiting times in queue only, i.e. W
q
). As with the travel costs, we
