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

5 Multi-Flow Optimization
Execution Time
W + (P)·|M’|
Total Execution Time W(M’,k’)
W + (P)
lower bound
Relative Execution Time W(P’,k’) / k’
Waiting Time
∆tw
Figure 5.13: Monotonically Non-Increasing Execution Time with Lower Bound
total execution time is decreased as long as we benefit from it with regard to the total
latency time. The execution time W (P ′ , k ′ ) is computed by
W (P ′ , k ′ ) = W − (P ′ ) + W + (P ′ ) · k ′ , (5.11)
where W + (P ′ ) denote the costs (execution time) of operators that do not benefit from
partitioning, while W − (P ′ ) denotes the costs of operators that benefit from partitioning
(independent of k ′ ). As a result, in the worst case, the execution time W (P ′ , k ′ ) increases
linearly with increasing k ′ . Thus, the relative execution time W (P ′ , k ′ )/k ′ is a monotonically
non-increasing function with
∀k ′ , k ′′ ∈ [1, |M ′ |] : k ′ < k ′′ ⇒ W (P ′ , k ′ )
k ′ ≥ W (P ′ , k ′′ )
k ′′ . (5.12)
If we now fix a certain |M ′ |, this also implies that the total execution time W (M ′ , k ′ ) is a
monotonically non-increasing function with
Then, it follows directly that
W (M ′ , k ′ ) = W (P ′ , k ′ )
k ′ · |M ′ |. (5.13)
W (P ′ , k ′ )
k ′ ≤ W (P ′ , k ′ − 1)
k ′ ≤ W (P )
− 1
W (P ′ , k ′ )
k ′ · |M ′ | ≤ W (P ′ , k ′ − 1)
k ′ · |M ′ | ≤ W (P ) · |M ′ |.
− 1
(5.14)
Hence, Theorem 5.1 holds.
This result of monotonic non-increasing relative and total execution time functions is
illustrated in Figure 5.13 as (strictly) monotonically decreasing function. However, due
to (1) integer batch sizes k ′ and (2) a constant function in the case, where no operator
benefits from partitioning, the functions are, in general, monotonic non-increasing.
Lower Bound of Relative and Total Execution Time
In analogy to Amdahl’s law, where the fraction of a task (execution time) that cannot be
executed in parallel determines the upper bound for the reachable speedup, we compute
the lower bound of the relative and total execution times. The existence of this lower bound
was already discussed in Section 5.3. Now, we investigate this lower bound analytically.
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