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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 ScienceLinköping University, Sweden

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping 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 TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Introduction

Mapped and scheduled tasks on the Mapped and scheduled tasks on the allocated processorsallocated processors

The schedulability analysis gives the design fitness estimate

Scheduling

Allocation

P1 P2

Partitioning

Mapping

Functionality as an annotated task graphFunctionality as an annotated task graph

FitFit??NoNo

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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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 TimesSorin 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 TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Applications (2)

Early design phases, early estimations for future design guidance

Alternative Models: Average Interval Stochastic

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Application characteristics (data dependent loops and branches)

Architectural factors (pipeline hazards, cache misses) External factors (network load) Insufficient knowledge

Sources of Variability

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin 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 Deadlines less than or equal to the periods Designer controlled rejection (discarding)

execution timeexecution time

prob

ab

prob

ab

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin 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% 3%

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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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 TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Approach Outline (2)

Approximation

Coxian distribs

Task graphs

GSPN

CTMC

Results

Modelling

CTMC constr.

Analysis

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Application Modelling (1)

Approximation

Coxian distribs

Task graphs

GSPN

CTMC

Results

Modelling

CTMC constr.

Analysis

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Application Modelling (2)

B C

D

A

E

F

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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C

Application Modelling (3)

A F D B E

firing delay

prob

ab Firing delay equals execution time

A

B C

D

E

F

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Approximation (1)

Approximation

Coxian distribs

Task graphs

GSPN

CTMC

Results

Modelling

CTMC constr.

Analysis

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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Approximation (2)

j

ji

jj

i

ir

ii ss

sG

)1()(1

11

11 22 33

2)21)1

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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CTMC Construction (1)

Approximation

Coxian distribs

Task graphs

GSPN

CTMC

Results

Modelling

CTMC constr.

Analysis

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden

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CTMC Construction (2)

X

X, Y X, Y X

Approximation of the SMPSMP

Approximation of X

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin 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

jjenjiji

enij

enjij

jenj

DIBI

IAM

)()(

)(

,,

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 TimesSorin 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 TimesSorin 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 TimesSorin 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 TimesSorin 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 5Relative error 8.7% 4.1% 1.04% 0.4%

Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution TimesSorin 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