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APPENDIX
A Analytic Model (AM)
A.1 Basic Notation
In this section, we use the following notations to for-
malize AM:
S a set of service ids in the inputSer-
vices model input.
C S ⊆ S a set of composite service ids.
AS ⊆ S a set of atomic service ids.
SAS ⊆ S a set of atomic service ids with
state.
inFIds(s) a set of infow ids for service s ∈ S .
outFIds(s) a set of outfow ids for service s ∈
S.
sub(s) a set of subService ids for service
s ∈ C S .
onFlag(s, p) is the onFlag(p) for a service with
id s ∈ AS for period p ∈ P, which
gives ”1” if the service is running
(ON) and ”0” otherwise. (see sec-
tion 3.2.3).
invested(s, p) is the invested(p) for a service with
id s ∈ AS for period p ∈ P, which
gives ”1” if investment occurs and
”0” otherwise. (see section 3.2.3).
investAmt(s, p) is the amount required to invest in
a service with id s ∈ AS in period
p ∈ P.
investedAmt(s) is the invested amount for a ser-
vice with id s ∈ AS .
lb(s, p, f ) is a lower bound lb of a service
with id s ∈ S for period p ∈ P and
flow id f ∈ inFIDs(s).
ub(s, p, f ) is an upper bound ub of a service
with id s ∈ S for period p ∈ P and
flow id f ∈ inFIDs(s).
inQtyPP(s, p, f ) is an inFlow quantity per period
(qtyPP) for a service with id s ∈ S ,
for period p ∈ P and flow id f ∈
inFIDs(s).
outQtyPP(s, p, f ) is an outFlow quantity per period
(qtyPP) for a service with id s ∈ S ,
for period p ∈ P and flow id f ∈
outFIDs(s).
totalQty(s, f ) is the total quantity (totalQty) for
inFlow with flow id f ∈ inFIDs(s)
or outFlow with flow id f ∈
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