Fast, but Approximate,
Workflow-Runtime Estimation
Using the Bell-Curve Calculus
Alan Bundy
Joint work with Lin Yang,
Conrad Hughes and Dave Berry
University of Edinburgh
31 May 2016
1
Overview
The Bell Curve Calculus (BCC)
 Application to quality of service.

– Estimating runtimes of e-Science
workflows.

Evaluation of accuracy and
efficiency.
– Compared to piecewise estimation
(Agrajag)
– Faster, but less accurate.
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2
Problem

Scientists need to estimate quality of
service properties of workflows.
– Eg runtime, accuracy, reliability.

Not just a number, but range and
likelihood,
– e.g. probability density function

However, propagating PDFs within
large workflows is computationally
expensive.
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3
Key Idea

BCC extends arithmetic operations to
normal distributions (aka bell curves).
– Analogous to Interval Arithmetic.

A bell curve can be fully described with
two parameters.
– Mean  and Standard Deviation .

Assume output is also a bell curve
– Calculate its  &  from input ones.
– How bad an approximation is this?
– How much does it speed up calculations?
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4
Bell Curve
BC(,) =  x.
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2/22
-(x-)
e
/(2p)0.5
5
Service Combinators

Sequential:
Runtime
S1
S1


Parallel All:
Parallel First:
S2
+
S
Max
S
Min
S
Cond
S2
S1
S2

Disjunctional:
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S1
Succeed
Fail
S2
6
Formal Definition of Problem


Input Curves: BC(1,1) and BC(2,2).
Perfect Curve: Fc(BC(1,1),BC(2,2)).
– where c is Seq, PA, PF or D.
– Not a bell curve in general.
– Use Agrajag to estimate by piecewise approx.

Best Bell Curve: BC((Fc (…)),(Fc (…)))
– bell curve with  and  of perfect curve.

BCC Estimate: BC(Mc(1,1,2,2),c(1,1,2,2))
– for each value of c.
– Is always a bell curve.
– Approximating Best Bell Curve, but without piecewise
estimation.
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7
Sum of Two Bell Curves
BBC Estimate = Perfect Curve
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8
Methodology






Generate perfect curves and, hence, best
bell curves.
Inspect best bell curves and guess Mc and
 c.
Plot errors of Mc and c and curve fit.
Use error functions to improve Mc and c .
Repeat until accuracy acceptable.
Resulting definitions of Mc and c are very
messy.
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9
Max of Two Bell Curves
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Worst case when input curves have
similar means.
10
Accuracy of BCC on Workflow
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BCC estimate close to best bell
curve, but not perfect curve.
11
BCC Efficiency on Workflow Family
Both linear, but with very different slopes.
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12
Conclusion

BCC shows both range and likelihood of
QoS properties.
– Tested for workflow runtimes.

Extends arithmetic to bell curves.
– Only approximate.

Less accurate but much more efficient than
piecewise estimation.
– Good for quick, rough estimate.

Extend to other QoS properties.
 Incorporate into workflow construction tool.
31 May 2016
13