Analysis and Optimisation of Real-Time
Systems with Stochastic Behaviour
Sorin Manolache
ESLAB, IDA
Analysis and Optimisation of
Real-Time Systems with Stochastic Behaviour
Outline
 Introduction
,
,
 Analysis and optimisation
 Stochastic task execution times
 Applications implemented on NoC with unreliable
network links
 Conclusions
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Embedded Systems
 Computing systems encapsulated in the device they
control
 Dedicated function, perform pre-defined tasks
 Resource constrained:
 Time
 Power consumption
 Memory
 Size, cost
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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System Design Flow
Informal
specification
Functional
simulation
Formal
verification
Modelling
Architecture
selection
System model
System
architecture
Mapping
Estimation
Scheduling
ok
ok
Mapped and
scheduled model
ok
Simulation
Formal
verification
Analysis
Lower levels of design
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Specification
Informal
specification
Functional  ”Baseband processing
for a multi-standard
simulation
base station in a mobile
Formal
communication
verification
Modelling
Architecture
selection
System model
network”
System
architecture
Mapping
Estimation
Scheduling
Mapped and
ok
scheduled model
ok
ok
Lower levels of design
Simulation
Formal
verification
Analysis  Analogy: “One-floor,
three-bedroom, twobathroom house”
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Modelling
Informal
specification
Functional
simulation
Formal
verification
Modelling
Architecture
selection
System model
System
architecture
Mapping
Estimation
Scheduling
Mapped and
ok
scheduled model
ok
ok
Lower levels of design
 Functionality is
partitioned in
communicating
processes. Expressed
as a set of task graphs,
or process algebra, or
SystemC, VHDL etc.
specifications.
Simulation
Formal
verification
Analysis  Analogy: Blue-prints of
the house, expressed in
e.g. AutoCAD
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Architecture Selection
Informal
specification
Functional  Selecting the set of
resources (processors,
simulation
buses)
Formal
verification
Modelling
Architecture
selection
System model
System
architecture
Mapping
Estimation
Scheduling
Mapped and
ok
scheduled model
ok
ok
Lower levels of design
Simulation
 Analogy: selecting the
Formal
professions needed to
verification
build a house
Analysis
(bricklayer, carpenter,
unspecialised workers,
civil engineer)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Functionality Mapping
Informal
specification
Functional  Assigning tasks to
processors and
simulation
messages to buses or
Formal
links
verification
Modelling
Architecture
selection
System model
System
architecture
Mapping
Estimation
Scheduling
Mapped and
ok
scheduled model
ok
ok
Lower levels of design
Simulation
Formal
verification  Analogy: assigning jobs
to persons (foundation
Analysis
dug by unspecialised
workers, roof structure
built by carpenters)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Task Scheduling
Informal
specification
Functional  Specifying when the
tasks are executed
simulation
Formal
verification
Modelling
Architecture
selection
System model
System
architecture
Mapping
Estimation
Scheduling
Mapped and
ok
scheduled model
ok
ok
Lower levels of design
Simulation  Analogy: specifying that
the construction of the
Formal
walls starts after the
verification
foundation has been
Analysis
laid, and that the roof is
to be built after the
walls have been
erected
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Outline
 Introduction
 Systems
 Real-time
 Stochastic behaviour
 Analysis and optimisation
 Stochastic task execution times
 Applications implemented on NoC with unreliable
network links
 Conclusions
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Real-Time
 Correctness defined not only in terms of functionality but
also in terms of timeliness
 E.g. Deadlines, jitters, periods
 Analogy: House must be ready on the 30th of June
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Hard vs. Soft Real-Time
Hard real-time systems
Soft real-time systems
 Missing a deadline is not
tolerated
 Occasionally missing a deadline
is not desired but accepted
 Their analysis provides yes/no
answers
 Their analysis provides
feasibility degrees
 Construction of the worst-case
scenario
 Construction of a multitude of
scenarios with associated
likelihood
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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probability density
Hard vs. Soft Real-Time
Fast and expensive
WCET
deadline
response time
Slow but inexpensive
deadline
WCET
response time
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Outline
 Introduction
 Systems
 Real-time
 Stochastic behaviour
 Analysis and optimisation
 Stochastic task execution times
 Applications implemented on NoC with unreliable
network links
 Conclusions
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Execution Time Variability
Execution time depends on:
 Input data
 MPEG-encoded motion-intensive video clips require
more processing than videos surveying the night sky
 Features and dynamic state of the architecture
 Cache memories, branch predictions, pipeline bubbles
 Environment
 Component unreliability
 Network congestion
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Execution Time Model
probability density
 Task execution time is more realistically described by an
execution time probability distribution
computation time
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Outline
 Introduction
 Systems
 Real-time
 Stochastic behaviour
 Analysis and optimisation
 Stochastic task execution times
 Applications implemented on NoC with unreliable
network links
 Conclusions
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Formulation of the Analysis Problem
Probability
2s
 Task graphs
 Task periods
 Task execution time
2s
4s
probability densities
 Deadlines
6s
 Processors, buses,
interconnects
10s
 Mapping and scheduling
policy
 Late task policy
10s
Time
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Formulation of the Analysis Problem
Determine the expected deadline miss ratio for
each task and task graph
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Formulation of the Optimisation Problem
Probability
2s
 Task graphs
 Task periods
 Task execution time
2s
4s
probability densities
 Deadlines and miss
thresholds
6s
 Processors, buses,
miss 4%
interconnects
10s
 Fixed priority scheduling
 Late task policy
10s
Time
miss 2%
miss 10%
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Formulation of the Optimisation Problem
Find a task mapping and priority assignment such
that the deviation of the expected deadline miss
ratios from the imposed thresholds is minimised
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Limitations of Previous Work
 Exponentially distributed execution time probability
distributions
 Monoprocessor systems
 No data dependencies among the tasks
 Specific assumptions such as the existence of servers,
admission control mechanisms, processor load close to
one, FIFO scheduling policy
 Complexity makes that often only unrealistically small
systems are amenable to analysis
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Contributions to the Analysis Problem
 Generalised task execution time probability distributions
functions
 Any widely used scheduling policy
 Generous assumptions regarding
 The applications (periods, deadlines, dependencies,
no special treatment of soft tasks etc.)
 The hardware platform (multiprocessor systems, no
restrictions on the interconnections between
processors)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Contributions to the Analysis Problem
1. Approach giving exact deadline miss ratios
 Efficient for monoprocessor systems
 While applicable to multiprocessor systems, its
application is limited by complexity issues
2. Approach giving approximations of the deadline miss
ratios
 Applicable to multiprocessor systems
 Too slow to be put inside a design optimisation loop
3. Fast approximate approach
 Fast enough to be put inside optimisation loops
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Exact Approach
 The deadline miss ratios are obtained by constructing and
analysing the stochastic process underlying the application
 Two approaches to manage complexity
 Clustering of individual completion times into time
intervals
 Concurrent construction and analysis of the stochastic
process
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Clustering of Completion Times
p=d=3 A
p=d=5 B
Earliest deadline first, non-preemptive execution
A, 0, {B}
B, [0, 3),
B,{}t1, {}
B, t0, {}
0
t0
t1
3 tk tk+1 5
[3, 5), {A}
B, tkB,
, {A}
6
9
10
B, tk+1, {A}
12
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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26
Concurrent Construction and Analysis
[0, 3)
[3, 5)
[5, 6)
[6, 9)
[9, 10)
[10, 12)
[12, 15)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Size(Number of Tasks)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Sliding Window Size(Number of Tasks)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Contributions to the Analysis Problem
1. Approach giving exact deadline miss ratios
 Efficient for monoprocessor systems
 While applicable to multiprocessor systems, its
application is limited by complexity issues
2. Approach giving approximations of the deadline miss
ratios
 Applicable to multiprocessor systems
 Too slow to be put inside an design optimisation loop
3. Fast approximate approach
 Fast enough to be put inside optimisation loops
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Approximate Approach
 Relies upon the analysis of a continuous-time Markov
chain that approximates the stochastic process underlying
the application
 Two approaches to manage complexity:
 The generalised execution time probability distribution
functions are approximated by Coxian distributions, i.e.
weighted sums of convoluted exponentials
 Structural properties of the Markov chain are exploited
for low-memory analysis
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Probability density
Coxian Approximation
Real density
P( s )
Q( s )
Computation time
Find  and  such that
r
j
i i 1
 G ( s)    i
(1   j )

s  i j 1
s  j
i 1
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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CTMC Construction
X, Y
X, Y
GSMP
X
CTMC approximating the GSMP
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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CTMC Construction
 The global generator of the Markov chain becomes then
M  (  Aj )  I 
jen
  (  I j )  Bi  (  I j )  D j 
ien
j i ,
jen
j i ,
jen
 M is expressed in terms of small matrices and can be
generated on the fly – memory savings
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Analysis Time(Number of Tasks)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Analysis Time(Number of Processors)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Accuracy
 Accuracy vs analysis complexity compared to the exact
approach for monoprocessors
Stages
Relative error
2
3
4
8.7% 4.1% 1.04%
5
0.4%
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Contributions to the Analysis Problem
1. Approach giving exact deadline miss ratios
 Efficient for monoprocessor systems
 While applicable to multiprocessor systems, its
application is limited by complexity issues
2. Approach giving approximations of the deadline miss
ratios
 Applicable to multiprocessor systems
 Too slow to be put inside an design optimisation loop
3. Fast approximate approach
 Fast enough to be put inside optimisation loops
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Fast Approximate Analysis
 Task completion times are expressed as functions of task
start times and execution times
 The probability distributions of completion times are
approximated by considering some weakly dependent
random variables as independent
 Polynomial analysis time
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Fast Approximation
Exponential, =1/7s-1
A
Fixed, 0.5s
Fixed, 4s
C
Exponential, =1/2s-1
B
E Fixed, 6s
Highest priority
D Fixed, 5s
Lowest priority
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Fast Approximation
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Formulation of the Optimisation Problem
Find a task mapping and priority assignment such
that the deviation of the expected deadline miss
ratios from the imposed thresholds is minimised
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
42
Deadline Miss Ratio Minimisation
System model
System platform
Mapping
Priority assignment
ok
Fast approximate
analysis
Mapped and
scheduled model
ok
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Deadline Miss Ratio Minimisation
Stochastic task execution times
Average task execution times
 Use a heuristic for task mapping
and priority assignment
 Use the same heuristic for task
mapping and priority
assignment
 Drive the heuristic by an
analysis that considers the
average task execution times
and minimises task response
time
 Apply the same deadline miss
ratio analysis on the design
obtained by the heuristic
 Drive the heuristic by the fast
analysis that considers
stochastic task execution times
and minimises deadline miss
ratios
 Apply a deadline miss ratio
analysis on the design obtained
by the heuristic
The deadline miss ratio deviation from the imposed thresholds is one
order of magnitude larger in the case of average task execution times
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
44
Outline
 Introduction
 Systems
 Real-time
 Stochastic behaviour
 Analysis and optimisation
 Stochastic task execution times
 Applications implemented on NoC with unreliable
network links
 Conclusions
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Transient Faults
 Technological advances  increased miniaturisation of
integrated circuits  higher packing density  more
functionality  decreased cost and power consumption per
transistor
 But…
 Increased total power consumption
 Appearance of submicron effects, leakage power, crosstalk, interference
 Increased vulnerability to transient faults
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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NoC
 Networks-on-chip
 Links may fail
temporarily with known
probability
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Problem Formulation
 Communication synthesis, i.e. mapping of messages to
network links and setting their release times such that:
 Reliable communication (message arrival probability
above an imposed threshold)
 Satisfaction of timeliness constraints
 Satisfaction of communication buffer constraints or
buffer space minimisation
 Minimisation of communication energy consumption
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Solution Outline
1. Reliable communication
 Combination of temporally and spatially redundant
communication while keeping energy overhead at
minimum
2. Satisfaction of timeliness constraints
 Based on a worst-case timing analysis
3. Satisfaction of buffer space constraints
 Based on a worst-case buffer space analysis
4. Strategy to explore the set of possible message mappings
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Communication Supports
MAP=0.941<0.99
ECE~5.851
MAP=0.999>0.99
ECE~11.997
MAP=0.998>0.99
ECE~11.935
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Communication Supports
 Communication synthesis problem means selecting one
communication support for each message
 Search problem in the space of communication supports
 Too large a search space search too slow
 Too small a search space low quality result
Build the search space with only “good” communication
support candidates
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Communication Supports
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Latency(Number of Tasks)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Latency(MAP)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Solution Outline
1. Reliable communication
 Combination of temporally and spatially redundant
communication while keeping energy overhead at
minimum
2. Satisfaction of timeliness constraints
 Based on a worst-case timing analysis
3. Satisfaction of buffer space constraints
 Based on a worst-case buffer space analysis
4. Strategy to explore the set of possible message mappings
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Buffer Space Minimisation Problems
 Communication synthesis with extended objective:
 Application specific NoCs: the total amount of buffering
space used by the application at the switches should be
minimised
 Predefined
buffer space: synthesize the communication
such that no buffer overflow occurs during the execution
of the application
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Buffer Space Minimisation
System model
System platform
Buffer space
analysis
Communication
mapping and
Packet release
time
ok
Mapped and
Packet release
scheduled
model
time
ok
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Buffer Space Minimisation
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
58
Buffer Space Constraints
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Conclusions
 Schedulability analysis of real-time systems with
stochastic task execution times
1. Exact method, efficient for monoprocessors
2. Approximate method, efficient for multiprocessors
3. Fast approximate method, may be used for
optimisation
 Deadline miss ratio minimisation by means of task
mapping and priority assignment
 Communication synthesis for NoC with unreliable links,
under reliability, timeliness, buffer space, and energy
constraints
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Related Work
 L. Abeni and G. Buttazzo, “Integrating Multimedia Applications in Hard RealTime Systems”, 1998
 A. Burns, G. Bernat, I. Broster, “A Probabilistic Framework for Schedulability
Analysis”, 2003
 J. L. Diaz et al., “Stochastic Analysis of Periodic Real-Time Systems”, 2002
 M. K. Gardner, “Probabilistic Analysis and Scheduling of Critical Soft Real-Time
Systems”, 1999
 X.S. Hu, T. Zhou, E. H.-M. Sha, “Estimating Probabilistic Timing Performance
for Real-Time Embedded Systems”, 2001
 J. Kim and K. G. Shin, “Execution Time Analysis of Communicating Tasks in
Distributed Systems”, 1996
 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
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
61
Influence of Dependency Degree
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
62
Approximate Approach
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
63
Fast Approximate Analysis
A
Z
Y
X
Z
Y
X
Z
Y
X
P(AX t) = P(FY  t) P(FZ  t)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
64
Average vs. Probabilistic Execution Times
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Latency Reduction(Communication Load)
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Communication Supports
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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Buffer Space Analysis
0
15
27.5
35
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
40 42.5
68
Coxian Approximation
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
69
Coxian Approximation
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
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
70
CTMC Construction
X, Y
X, Y
X
GSMP
CTMC approximating the GSMP
X
Approximation of X
S. Manolache, “Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour”
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