Cost-Based Optimization of Integration Flows - Datenbanken ...

5.4 Formal Analysis
Theorem 5.2. The lower bound of relative and total execution times W (P ′ , k ′ )/k ′ and
W (M ′ , k ′ )/k ′ is given by W + (P ′ ) and W + (P ′ )·|M ′ |, respectively, as the costs that linearly
depend on k ′ .
Proof. Recall that the execution time W (P ′ , k ′ ) is computed by
W (P ′ , k ′ ) = W − (P ) + W + (P ) · k ′ , (5.15)
where, W − (P ′ , k ′ ) is independent of k ′ by definition. If we now use the relative execution
time W (P ′ , k ′ )/k ′ and let k ′ tend to ∞ with
W (P ′ , k ′ )
lim
k ′ →∞
lim
k ′ →∞
k ′ = W + (P ′ ) with W (P ′ , k ′ )
k ′ = W − (P )
k ′ + W + (P ) · k ′
k ′
W (M ′ , k ′ )
k ′ = W + (P ′ ) · |M ′ |
with W (M ′ , k ′ )
k ′ =
( W − (P ′ )
k ′ + W + (P ′ ) · k ′ )
k ′ · |M ′ |,
(5.16)
we see that W (P ′ , k ′ )/k ′ asymptotically tends to W + (P ) and thus, it represents the lower
bound of the relative execution time, while the lower bound of the total execution time is
W + (P ) · |M ′ |. Hence, Theorem 5.2 holds.
These lower bounds are shown in Figure 5.13 and they are the reason why minimizing
the total latency time leads to maximal message throughput because the additional benefit
in terms of lower execution time decreases with increasing waiting time.
5.4.2 Maximum Latency Constraint
Beside the property of optimality with regard to the message throughput, our waiting
time computation ensures that the restriction of the maximum latency constraint holds,
in expectation, at the same time as well.
Theorem 5.3 (Soft Guarantee of Maximum Latency). The waiting time computation
ensures that—for a given fixed message rate R—the latency time of a single message
T L (m i ) with m i ∈ M ′ will, in expectation, and for non-overload situations, not exceed the
maximum latency constraint lc with T L (m i ) ≤ lc.
Proof. In the worst case, 1/sel distinct messages m i arrive simultaneously in the system.
Hence, the highest possible latency time T L (m i ) is given by the total latency time 1/sel ·
∆tw + W (P ′ , k ′ ). Due to our validity condition of ˆT L (M ′ ) ≤ lc with |M ′ | = k ′ /sel, we
need to show that T L (m i ) ≤ ˆT L even for this worst case. Further, our validity condition
ensures that ∆tw ≥ W (P ′ , k ′ ). Thus, we can write T L (m i ) ≤ ˆT L (∆tw · R) as
1
sel · ∆tw + W (P ′ , k ′ ) ≤
⌈ |M ′ |
k ′
1
sel · ∆tw ≤ |M ′ |
k ′ · ∆tw.
⌉
· ∆tw + W (P ′ , k ′ )
(5.17)
If we now substitute |M ′ | by k ′ /sel (in the sense that the cardinality |M ′ | is equal to the
number of partitions 1/sel times the cardinality of a partition k ′ ), we get
1
sel · ∆tw ≤ |M ′ |
k ′ = 1 · ∆tw. (5.18)
sel
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