Transcript Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng {sorma, petel, zebpe}@ida.liu.se Department of Computer and Information.

Schedulability Analysis of Multiprocessor
Real-Time Applications with
Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
{sorma, petel, zebpe}@ida.liu.se
Department of Computer and Information Science
Linköping University, Sweden
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Outline
 Introduction
 Problem formulation
 Application modelling
 Approximation method
 Markov chain generator construction
 Analysis
 Experimental results
 Conclusions
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Introduction
Functionality as an annotated task graph
Partitioning
Allocation
Mapping
Scheduling
Mapped and scheduled tasks on the
allocated processors
P1
P2
The schedulability
analysis gives the
design fitness
estimate
No
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
Fit?
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Motivation
 “Classical” schedulability analysis works on the worst
case execution time (WCET) model
 Established analysis methods
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Applications (1)
 Soft real-time applications (missing a deadline could
be acceptable)
 WCET becomes pessimistic
 Leads to processor under-utilization
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Applications (2)
 Early design
 Alternative
phases, early
Models:
estimations for future
design guidance
 Average
 Interval
 Stochastic
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Sources of Variability
 Application characteristics (data dependent loops and
branches)
 Architectural factors (pipeline hazards, cache misses)
 External factors (network load)
 Insufficient knowledge
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Problem Formulation (1)
Input:
 Set of task graphs, periodic tasks, deadlines equal
periods, statically mapped
 Set of execution times probability density functions
(continuous)
 Scheduling policy
probab
probab
 Deadlines less than or equal to the periods
 Designer controlled rejection (discarding)
execution time
execution time
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Problem Formulation (2)
Output:
 Ratio of missed deadlines per task graph
Limitations:
 Non-preemption
15%
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
3%
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Approach Outline (1)
The application with stochastic task execution times can
be regarded as a system with random character
 The solution can be obtained by constructing and
analysing the underlying stochastic process
 Very difficult to solve in the case of arbitrary
task execution time PDFs (ETPDFs)
Approximate the ETPDFs by Coxian distributions
A much larger Markov chain is obtained, but it is easier
to solve
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Approach Outline (2)
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Application Modelling (1)
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Application Modelling (2)
A
E
B
C
F
D
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Application Modelling (3)
A
B
C
D
F
C
F
probab
A
E
D
B
E
Firing delay equals
execution time
firing delay
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Approximation (1)
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Approximation (2)
11
(11)1
22
33
(12)2
j
i i 1
G(s)   i
(1   j )

s  i j 1
s  j
i 1
r
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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CTMC Construction (1)
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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CTMC Construction (2)
X, Y
X, Y
X
SMP
Approximation of the SMP
X
Approximation of X
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Construction of the CTMC
 The global generator of the Markov chain becomes
then
M  (  Aj )  I 
jen
  (  I j )  Bi  (  I j )  D j 
j i ,
ien jen
j i ,
jen
 M is expressed in terms of small matrices and can be
generated on the fly – memory savings
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Analysis Time vs. Number of Tasks
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Analysis Time vs. Number of Procs
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Growth with Number of Stages
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Accuracy
Accuracy vs analysis complexity compared to
an exact approach presented in previous work
Stages
2
3
4
5
Relative error 8.7% 4.1% 1.04% 0.4%
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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Conclusions
 Approximation approach to performance analysis of
multiprocessor real-time applications with stochastic
execution times
 Larger scale applications can be analysed due to an
efficient scheme to store the underlying stochastic
process
 Provides the possibility to trade-off analysis speed and
memory demand with analysis accuracy
Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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