Memory and Time-Efficient Schedulability
Analysis of Task Sets with Stochastic
Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
Department of Computer and Information Science
Linköpings universitet
1
Outline
 Introduction
 Task model and problem formulation
 Analysis method
 Experimental results
 Conclusions and future work
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
2
Introduction
Functionality as an annotated task graph
Partitioning
Allocation
Mapping
The schedulability
analysis gives the
design fitness estimate
Scheduling
Mapped and scheduled tasks on the allocated processors
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
3
Motivation
 “Classical” schedulability analysis works on the WCET
model
 Established analysis methods
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
4
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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
5
Sources of Variability
 Application characteristics (data dependent loops and
branches)
 Architectural factors (pipeline hazards, cache misses)
 External factors (network load)
 Insufficient knowledge
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
6
Related Work
 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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
7
Outline
 Introduction
 Task model and problem formulation
 Analysis method
 Experimental results
 Conclusions and future work
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
8
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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
9
Task Model
360
120
A
2
15
9
B
C
4
6
G
H
3
5
J
D
E
60
12
I
F
9
15
24
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
10
Outline
 Introduction
 Task model and problem formulation
 Analysis method
 Experimental results
 Conclusions and future work
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
11
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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
12
PMIs
0
3
5
A, 0, {B}
B, t0, {}
B, t1, {}
B, tk, {A}
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
B, tk+1, {A}
13
PMIs
A, 0, {B}
B, t0, {}
0
B, [0, 3),
B, t{}
1, {}
3
5
B, B,
tk, [3,
{A}5), {A} B, tk+1, {A}
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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
14
Stochastic Process
0
3
5
0
3
0
3
5
A, [0, 3), {B}
0
3
0
B, [0, 3), {}
0
3
5
3
B, [3, 5), {A}
8
-, [0, 3), {}
A, [3, 5), {}
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
A, [5, 6), {B}
15
Analysis
[0, 3)
[3, 5)
[5, 6)
[6, 9)
[9, 10)
[10, 12)
[12, 15)
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
16
Outline
 Introduction
 Task model and problem formulation
 Analysis method
 Experimental results
 Conclusions and future work
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
17
Experimental Results
Number of process states
Influence of number of tasks on the process size
155000
110000
65000
20000
10
11
12
13
14
15
16
17
18
19
Tasks
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
18
Experimental Results
Influence of dependency degree on the process size
Number of process states
1000000
100000
10000
1000
0
1
2
3
4
5
6
7
8
9
Dependency degree
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
19
Experimental Results
Influence of the period LCM on the process size
Number of process states
1800000
1200000
600000
0
1000
2500
4000
5500
Least common multiple
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
20
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
Memory- and Time-Efficient Schedulability Analysis of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
21