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

5.4 Formal Analysis
because E(k ′ ) = ∑ ∞
i=1 k′ i · P (k′ = k i ′ ). In order to show that the latency constraint lc is,
in expectation, not exceeded by skewed arrival rate distributions, we need to show that
the effects of overestimation W (P ′ , k ′ − 1) − W (P ′ , k ′ ) (lower execution time) amortize
the effects of underestimation W (P ′ , k ′ + 1) − W (P ′ , k ′ ) (higher execution time). The
composed effect ∆W (P ′ ) is computed by
∆W (P ′ ) = ( W (P ′ , k ′ − 1) − W (P ′ , k ′ ) ) + ( W (P ′ , k ′ + 1) − W (P ′ , k ′ ) )
= ( W − (P ) + W + (P ) · (k ′ − 1) − (W − (P ) + W + (P ) · k ′ ) )
(5.22)
+ ( W − (P ) + W + (P ) · (k ′ + 1) − (W − (P ) + W + (P ) · k ′ ) ) = 0
Thus, for the worst case, the condition T L (M ′ ) = lc holds in expectation. Hence, Theorem
5.4 holds.
This extended soft guarantee of maximum latency holds for both right-skewed and leftskewed
distributions functions due to the equal average over- and under-estimation. However,
it is important to note that these guarantees of maximum latency are soft constraints
that hold, in expectation, and for non-overload situations, while changing workload characteristics
in combination with a long optimization interval ∆t might lead to temporarily
exceeding the latency constraint.
5.4.3 Serialized External Behavior
According to the requirement of serialized external behavior, we additionally might need
to serialize messages at the outbound side. We analyze once again the given maximum
latency guarantee with regard to arbitrary serialization concepts.
Theorem 5.5 (Serialized Behavior). Theorems 5.3 (Soft Guarantee of Maximum Latency)
and 5.4 (Extended Soft Guarantee of Maximum Latency) also hold for serialized external
behavior.
Proof. We need to prove that the condition T L (m i ) ≤ ˆT L (M ′ ) ≤ lc is true even in the
case, where we have to serialize the external behavior. Therefore, recall the worst case
(Theorem 5.3), where the latency time is given by
T L (m i ) = 1
sel · ∆tw + W (P ′ , k ′ ). (5.23)
In that case, the message m i has not outrun other messages such that no serialization time
is required. For all other messages that exhibit a general latency time of
( ) 1
T L (m i ) =
sel − x · ∆tw + W (P ′ , k ′ ), (5.24)
where x denotes the number of partitions after the partition of m i , this message has
outrun at most x · k ′ messages. Thus, additional serialization time of x · ∆tw + W ∗ (P ′ , k ′ )
is needed. In conclusion, we get
( ) 1
T L (m i ) =
sel − x · ∆tw + W ∗ (P ′ , k ′ ) + x · ∆tw + W (P ′ , k ′ )
= 1
(5.25)
sel · ∆tw + W (P ′ , k ′ ).
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