Practical Schedulability Analysis for Generalized Sporadic Tasks in Distributed Real-Time Systems Yuanfang Zhang 1, Donald K. Krecker 2, Christopher Gill.

Practical Schedulability Analysis for Generalized Sporadic Tasks in Distributed Real-Time SystemsYuanfang Zhang1, Donald K. Krecker2,

Christopher Gill1, Chenyang Lu1, Gautam H. Thaker2

1.Washington University, St. Louis, MO, USA

2.Lockheed Martin Advanced Technology Laboratories

Outline

Motivation Generalized Sporadic Task Model Schedulability Test for independent generalized

sporadic tasks Schedulability Test for end-to-end generalized

sporadic tasks Generalized release guard synchronization protocol

Simulations Conclusions

Motivation

Offline schedulability analysis Periodic task: period Traditional sporadic task: minimum inter-arrival

time Generalized sporadic Tasks

Higher instantaneous arrival rate Lower average arrival rate

Problem Overestimate the task’s time demand Pessimistic schedulability analysis

Generalized sporadic task model Traditional sporadic task Ti

(1, Wi) limit, window

Generalized sporadic task Ti

Introduce a greater limit Allow multiple pairs of limits and windows {(Zi,k, Wi,k) 1≤k≤K(i)}

Existing leaky bucket filter model ρ: token input rate σ: Bucket size

Leaky bucket vs. generalized sporadic task Generalized sporadic

task K(i) = 3 {(1,2), (3, 10), (5, 18)}

Leaky bucket model ρ =0.5 σ =1

Leaky bucket model greatly overestimates the workload

Time-Demand Analysis

Maximum number of arrivals

Maximum execution time demand

Earliest arrival time

The longest response time for Ti occurs during a level-i busy period if the arrivals of all tasks with equal or higher priority satisfy the maximum number of arrivals in that period

, ,

0 0( )

min ( ) |1 ( ) 0ii i k i k

if tMNA t

MNA t w z k K i if t

( ) ( )*i i iTD t MNA t

,1

,1, ,

0

( ) 0 0

max ( ) |1 ( )

i i

ii i k i k

if n

EAT n if n z

if n zEAT n z w k K i

Schedulability Test for independent generalized sporadic tasks One example

T1: a periodic task

T2: a generalized sporadic task

Arrival time constraints priority Exec. time Deadline

T1 {(1,40)} 1 10 40

T2 {(1, 10), (2, 30), (3, 50)} 2 8 50

Schedulability Test for T2 Compute an upper bound on the duration of a level-i busy period

D2 = min { t>0 | t = MNA1(t)*10 + MNA2(t)*8} = 26 Compute an upper bound on the number of instances Ti in a level-i

busy period of duration Di M2 = MNA(D2) = 2

For m=1 to M2 Compute an upper bound on the completion time of the mth job of

Ti in a level-i busy period C2(1) = min{ t>0 | t = MNA1(t)*10+8} = 18, C2(2) = 26

Compute an upper bound on the response time of the mth job of Ti in the busy period V2(1) = C2(1) – EATi(1) = 18, V2(2) = 16

The maximum is the WCRT for Ti W2 = 18

Compare WCRT with Di W2 < D2, task 2 is schedulable.

Schedulability Test for end-to-end generalized sporadic tasks Generalized release guard gi,j for each non-initial

subtask Ti,j (j>1) At the initial time, set gi,j=0 When m-1th job of Ti,j is released at time ri,j(m-1), update

gi,j=ri,j(m-1)+(ri,1(m)-ri,1(m-1)) Update gi,j to the current time if the current time is a processor

idle point on the processor where Ti,j executes An upper bound Wi to the end-to-end response time

of any generalized sporadic task Ti in a fixed-priority system synchronized according to the generalized release guard protocol is the sum of the upper bounds of WCRTs for all its subtasks

Simple Example

1 periodic task Task 1

period 40, highest priority 2 sporadic tasks

Task 2 Time constrains {(1, 10), (2, 30), (3, 50)}, 2nd highest priority

Task 3 Time constrains{(1, 30), (2, 80)}, lowest priority

TiTi,j Exec time Phase Pi

T1 T1,1 10 0 P1

T2 T2,1 8 0 P1

T2,2 5 0 P2

T3 T3,1 15 18 P2

Schedule with RG

0 10 40 50

10 18 26 30 38

18 23 28 33 43 48

23 28 33 43 48

50 58 60 68

68 73

63

T1,1

T2,1

T2,2

T3,1

Release time

Sync signal

RG

63

P1

P2

Idle point

Idle point

End-to-end WCRTs

WCRT in the above schedule T1 is 10

T2 is 23

T3 is 25

WCRT bounded by our generalized sporadic task analysis T1 is 10

T2 is 23

T3 is 25

Since they are matching, WCRT bounds calculated by our analysis are true WCRTs.

Simulations 4 end-to-end periodic tasks

Allocated to 3 processors (execution time)

T1 T2 T3 T4

Period 312 90 162 203

Deadline 284 90 162 203

Priority 4 1 2 3

T1,1 T1,2 T1,3 T2,1 T2,2 T2,3 T3,1 T3,2 T3,3 T4,1 T4,1

P1 21 75 30 42

P2 23 30 58

P3 24 13 18 20

Sporadic Task Periodic task T3 (for example T3,1 on P1)

Make T3 jittery (1-x/100)*162 Arrival constrains {(1, 113), (2, 324)}

Arrival constrains {(1, 65), (2, 324)}

0 113

30 30

324

30

X=30

X=60

0 65

30 30

324

30

0 162

30 30

When T3 is allowed 30 percent jitter

0

200

400

600

800

1000

T1 T2 T3 T4

WC

RT

Bo

un

d

PT/RGST/RG

PT/RG Minimum inter-arrival

time analysis ST/RG

Generalized sporadic task analysis

ST/RG outperforms PT/RG on WCRT bound for T1

WCRT bounds for T1 and T3 in PT/RG are infinite

When T3 is allowed 60 percent jitter

0

200

400

600

800

1000

T1 T2 T3 T4

WC

RT

Bo

un

dPT/RGST/RG

When T3 jitter increases

0

200

400

600

800

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Jitter Percentage

WC

RT

Bo

un

d f

or

T1

PT/RG

ST/RG

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

Jitter Percentage

Mis

s R

atio

PT/RG

ST/RG

When the jitter percentage reaches 37.5, the WCRT bound for T1 is infinite when PT/RG is used.

The WCRT bounds for T1 are under 600 when ST/RG is used The miss ratios that are calculated by PT/RG reach 75% The miss ratios that are calculated by ST/RG are always 0

Representative Example

Military shipboard computing 15 end-to-end periodic tasks on 50

processors T10 becomes a generalized sporadic task

Highest priority Share processors with other 12 tasks Original period 200

The WCRT bounds for 6 tasks in PT/RG are infinite

Arrival time constraints {(1, 50), (2, 400)

0

2000

4000

6000

8000

10000

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

WC

RT

Bo

un

d

PT/RG ST/RG

Comparisons under different scenarios

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

Jitter Percentage

Mis

s R

atio

PT/RG

ST/RG

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7

Number of Simultaneous Release

Mis

s R

atio

PT/RG

ST/RG

0

0.2

0.4

0.6

0.8

1

2 3 4 5 6 7 8

Number of Time Constraints

Mis

s R

atio

PT/RG

ST/RG

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of Sporadic Tasks

Mis

s R

atio

PT/RG

ST/RG

• T10 jitter increases • Limit in the first window increases

• Number of time constraints for T10 increases • Number of sporadic tasks increases

Conclusions

Generalized sporadic task model characterizes arrival time more precisely

Our schedulability analysis tightens the bounds on worst case response time

Our schedulability analysis more effectively guarantee schedulability when arrival time jitter increases