Geoscience Reference
In-Depth Information
certain (undesirable) events, we will refer to the former as the “Mean SC” and the
latter as the “Probabilistic SC”. While the Mean SC is easily expressed in terms
of the decision variables by substituting (
17.15
)into(
17.17
), the Probabilistic SC
requires an expression for the steady-state distribution of the waiting time, which
is not generally available. One option is to make additional assumptions about
the distribution of service times (e.g., assuming M=M=1 or M=E
k
=1 queues at
the facilities) since steady-state distributions of waiting times have been derived
for many common systems. Another option is to use an approximation. The
one we follow here is based on Baron et al. (
2008
). Assume that the service
constraints (
17.18
) are specified and let
V.T;Ǜ
T
/
D
ln
.Ǜ
T
/
T
I
observe that since
ln
.Ǜ
T
/<0, this is a positive constant that is decreasing in Ǜ
T
and
in T . Then (under certain mild technical assumptions), constraint (
17.18
) is satisfied
whenever
G
S
.
V.T;Ǜ
T
/
i
/.
i
1/
V.T;Ǜ
T
/;
(17.19)
where G
S
.
/ is the MGF of service times defined earlier. Recall that G
S
./ is
an increasing function for >0, implying that the left-hand side of (
17.19
)
is decreasing in
i
. This is quite intuitive: when T or Ǜ
T
are decreased, the
probabilistic SC becomes tighter, requiring more capacity at the facility. In fact,
as V.T;Ǜ
T
/ becomes larger, satisfying (
17.19
) requires more capacity
i
.
This leads to a general view of service constraints: for any arrival rate
i
at
facility i
2
I one can define a minimum capacity level
N
.
i
/ such that SC holds if
and only if
i
N
.
i
/;
(17.20)
where
N
.
i
/ is computed (perhaps numerically) from (
17.17
), (
17.18
), or (
17.19
).
Of course, an equivalent view is to specify a function
N
./, which is just an inverse
of
N
./, so that SC holds whenever
i
N
.
i
/;
(17.21)
i.e., for a given capacity level
i
there is a maximal arrival rate
N
.
i
/ for which
an adequate service level can be provided by facility i. This view extends to other
definitions of SCs (e.g., instead of using waiting time one could use L or another
service level measure)—the only thing that changes is the way functions
N
./ and
N
./ are computed.
We note that system stability conditions imply that
N
./ > (equivalently
N
./ < ) and the difference
N
./
may be interpreted as the amount of
the “capacity cushion” (capacity in excess of the minimal possible level) needed
