1 Memory and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times...

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Memory and Time-Efficient Schedulability Analysis of Task Sets with Stochastic

Execution Times

Sorin Manolache, Petru Eles, Zebo PengDepartment of Computer and Information Science

Linköpings universitet

Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution TimesSorin Manolache, Petru Eles, Zebo Peng

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Outline

Introduction

Task model and problem formulation

Analysis method

Experimental results

Conclusions and future work


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Introduction

Partitioning Allocation

Mapping

Scheduling

Functionality as an annotated task graphFunctionality as an annotated task graph

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

The schedulability analysis gives the design fitness estimate


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Motivation

“Classical” schedulability analysis works on the WCET model

Established analysis methods


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Applications

Soft real-time applications (missing a deadline is acceptable)

WCET becomes pessimistic

Leads to processor under-utilization

Early design phases, early estimations for future design guidance

Alternative Models:

Average

Interval

Stochastic


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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


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L. Abeni and G. Butazzo, “Integrating Multimedia Applications in Hard Real-Time Systems”, 1998

A. Atlas and A. Bestavros, “Stochastic Rate Monotonic Scheduling”, 1998

A. Kalavade, P. Moghe, “A Tool for Performance Estimation for Networked Embedded Systems”, 1998

J. Lehoczky, “Real Time Queueing Systems”, 1996

T. Tia et al., “Probabilistic Performance Guarantee for Real-Time Tasks with Varying Computation Times”, 1995

T. Zhou et al., “A Probabilistic Performance Metric for Real-Time System Design”, 1999

Related Work


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Outline

Introduction


Analysis method




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Problem Formulation

Input

Set of task graphs

Set of execution time probability distribution functions (continuous)

Scheduling policy

Output

Ratio of missed deadlines per task or per task graph

Limitations

Discarding, non-pre-emption


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Task Model

A

CB

D

E F

G H

I

J

2

64

12

60

120

24

53

15

15

9

9

360


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Outline

Introduction


Analysis method




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Analysis Method

Relies on the analysis of the underlying stochastic process

A state of the process should capture enough information to be able to generate the next states and to compute the corresponding transition probabilities


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PMIs

B, tk, {A} B, tk+1, {A}

0 53

B, t0, {} B, t1, {}

A, 0, {B}


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PMIs

B, tk, {A} B, tk+1, {A}B, t0, {} B, t1, {}

A, 0, {B}

B, [0, 3), {} B, [3, 5), {A}

0 53 6 9 10 12 15

A PMI is delimited by the arrival times and deadlines

The sorting of the tasks according to their priorities is unique inside of a PMI


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Stochastic Process

A, [0, 3), {B}

B, [0, 3), {}

-, [0, 3), {}

B, [3, 5), {A}

A, [3, 5), {} A, [5, 6), {B}

300 53

30 30

0 53

30 5 8


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Analysis

[0, 3)

[3, 5)

[5, 6)

[6, 9)

[9, 10)

[10, 12)

[12, 15)


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Outline

Introduction


Analysis method




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Experimental Results

Influence of number of tasks on the process size

Tasks

10 11 12 13 14 15 16 17 18 19

Num

ber

of

pro

cess

sta

tes

20000

155000

65000

110000


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Influence of dependency degree on the process size

Dependency degree

0 1 2 3 4 5 6 7 8 9

Num

ber

of

pro

cess

sta

tes

1000

1000000

10000

100000


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Influence of the period LCM on the process size

Least common multiple

2500 4000 5500

Num

ber

of

pro

cess

sta

tes

0

1800000

600000

1200000

1000


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Conclusions

Schedulability analysis of set of tasks with stochastic execution times

Construction and analysis of the process at the same time sliding window size between 16 to 172 times smaller than the total number of process states

Future work: extension for multiprocessor case

1 Memory and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times...

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