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

5 Multi-Flow Optimization
Thus, for the worst case, T L (m i ) = lc, while for all other cases, T L (m i ) ≤ lc is true. Hence,
Theorem 5.3 holds.
Probability P(R’)
Probability P(k’)
E(R’)
E(k’) with E(k’) = Σ (k i’ · P(k’=k i’))
(a) Message Arrival Rate R ′
Message
Rate R’
(b) Batch Size k ′
Batch Size
k’
Figure 5.14: Waiting Time Computation With Skewed Message Arrival Rate
So far, we have assumed a fixed message rate R. Now, we formally analyze the influence
of the assumption of a stochastic arrival rate R ′ that exhibits a potentially skewed probability
distribution D such as the Poisson distribution. Figure 5.14(a) illustrates the probability
of a specific arrival rate (for a continuous arrival rate distribution function), while
Figure 5.14(b) illustrates the resulting batch sizes k ′ . Here, D can be a discrete or continuous
function, where the expected value E(R ′ ) is computed as the probability-weighted
integral of continuous values or as the probability-weighted sum of discrete values, respectively.
For example, the discrete Poisson distribution is relevant because Xiao et al. argued
that the arrival process of workflow instances is typically Poisson-distributed [XCY06].
The main difference is that we now include uncertainty—in the form of an arbitrary message
rate—into the min ˆT L computation because until now, we have used k ′ = R · ∆tw as
batch size estimation. For D = poisson, the fixed arrival rate R is substituted with an
uncertain arrival rate R ′ such that
k ′ = R ′ · ∆tw with P R (R ′ = r) = Rr
r! · e−R , (5.19)
with an expected value of E(R ′ ) = R. Due to the introduced uncertainty, we need to
extend Theorem 5.3 to skewed probability distributions functions.
Theorem 5.4 (Extended Soft Guarantee of Maximum Latency). The waiting time computation
ensures that—for a given uncertain message rate R ′ with skewed probability distribution
function D—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. Recall the worst case of T L (M ′ ) = lc with
⌈ |M
T L (M ′ ′ ⌉
|
) = · ∆tw + W (P ′ , k ′ ). (5.20)
k ′
There, the over- and underestimation of batch size k ′ does affect the execution time
W (P ′ , k ′ ). Hence, the worst case is given, where ∆tw = W (P ′ , k ′ ). With regard to
the uncertain arrival rate, the average discrete over- and under-estimation of k ′ is, in
expectation, equal with
∑
k ′ i ≤E(k′ )
(
k
′
i · P (k ′ = k ′ i) ) = ∑
k ′ i ≥E(k′ )
(
k
′
i · P (k ′ = k ′ i) ) (5.21)
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