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

5.4 Formal Analysis
latency constraint and to avoid additional overhead for this transformation, we do not
use state migration but use two partition trees (with the two schemes) and the two plans
in parallel. New messages are enqueued into the new partition tree, while we execute
message partitions from the old partition tree with the old plan until this partition tree is
empty. When this termination condition is reached, we exchange plans and execute partitions
from the new partition tree. Third, during cost-based optimization, we estimate the
costs and compute the optimal waiting time with regard to minimizing the total latency
time. Note that there are several side effects between MFO and the overall cost-based
optimization. For example, there are bidirectional influences on cost estimation and plan
rewriting. Therefore, during optimization the MFO technique is applied before the operator
specific rewriting techniques. This is a worst-case scenario (no other optimizations
applied) consideration with regard to the latency time and we benefit from subsequently
applied techniques (e.g., rewriting iterations, which have been introduced by MFO, to parallel
flows). Finally, the computed waiting time ∆tw is used in order to asynchronously
issue instances of the rewritten plan.
To summarize, we discussed how to enable the execution of horizontally partitioned
message batches and how to compute the optimal waiting time with regard to the total
latency time minimization that implies message throughput maximization. Furthermore,
we explained how the overall multi-flow optimization technique is seamlessly integrated
into the general cost-based optimization framework.
5.4 Formal Analysis
In this section, we now formally analyze our solution of horizontal message queue partitioning
and waiting time computation according to the defined multi-flow optimization
problem (P-MFO, see Definition 5.2). This includes (1) the optimality analysis with regard
to latency time minimization as well as the two additional restrictions of (2) the maximum
soft latency constraint for single messages and (3) the serialized external behavior.
5.4.1 Optimality Analysis
First of all, we analyze the optimality with regard to the computed waiting time ∆tw. In
detail, we discuss (1) the monotonically non-increasing execution time function that (2)
asymptotically tends towards a lower bound as shown in Figure 5.13.
Monotonically Non-Increasing Relative and Total Execution Time
Based on the extended cost model, we can give an optimality guarantee for W (P ′ , k ′ ) with
regard to the computed waiting time.
Theorem 5.1 (Optimality of Partitioned Execution). The horizontal message queue partitioning
solves the P-MFO with optimality guarantees of min T L (M ′ ) and monotonically
non-increasing total execution time of
W (P ′ , k ′ )
k ′ · |M ′ | ≤ W (P ′ , k ′ − 1)
k ′ · |M ′ | ≤ W (P ) · M ′ , where k ′ > 1. (5.10)
− 1
Proof. We compute the waiting time ∆tw, where min T L (M ′ ) under the given restrictions.
This determines the batch size k ′ = R · ∆tw and ensures optimal throughput because the
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