Two-type Heterogeneous Multiprocessor
Scheduling: Is there a Phase Transition?
Gurulingesh Raravi, Björn Andersson and Konstantinos Bletsas
CISTER-ISEP Research Centre
Polytechnic Institute of Porto
01/08/2017
1
Introduction
• System Model
– Computing Platform
• Two-type Heterogeneous platform
– A platform with two unrelated processor types
– Task set
• implicit-deadline sporadic tasks
– Assumptions
• Independent Tasks
• No Migrations
• No job parallelism
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Introduction
• Phase Transition
– Transition of a system from one state to another upon
changing some system parameters
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Introduction
• Phase Transition
– Transition of a system from one state to another upon
changing some system parameters
• Phase Transition in Real-Time Scheduling
– Transition of a system from “almost surely
schedulable” state to “almost surely not schedulable”
state upon changing the task set characteristics
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Introduction
• Phase Transition
– Transition of a system from one state to another upon
changing some system parameters
• Phase Transition in Real-Time Scheduling
– Transition of a system from “almost surely
schedulable” state to “almost surely not schedulable”
state upon changing the task set characteristics
• Uni-processor Rate Monotonic Scheduling:
– U*RM: Utilization threshold
» U(τ) ≤ U*RM then τ is almost surely schedulable
» U(τ) > U*RM then τ is almost surely not schedulable
• Identical multiprocessor scheduling
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The Problem
• Does there exist a phase transition behavior for
the two-type heterogeneous multiprocessor
scheduling problem?
– Is there a threshold for a parameter (or combination
of parameters) which classifies the task set from
“almost surely schedulable” state to “almost surely
not schedulable” state
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Some Insights
• Simulations and Observations:
– Simulation setup
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Some Insights
• Simulations and Observations:
– Simulation setup
at most 15 tasks
and 4 processors (2
of each type)
Generate a random
problem instance
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Some Insights
• Simulations and Observations:
– Simulation setup
Generate a random
problem instance
at most 15 tasks
and 4 processors (2
of each type)
Using ILP
formulation
Is there a
feasible
assignment?
Z<=1
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Some Insights
• Simulations and Observations:
– Simulation setup
Generate a random
problem instance
NO
01/08/2017
at most 15 tasks
and 4 processors (2
of each type)
Using ILP
formulation
Is there a
feasible
assignment?
Z<=1
10
Some Insights
• Simulations and Observations:
– Simulation setup
Generate a random
problem instance
NO
Is there a
feasible
assignment?
Z<=1
at most 15 tasks
and 4 processors (2
of each type)
Using ILP
formulation
Using Exhaustive
Enumeration:
ratio = Nsucc/Nvalid
YES
Compute the
“success ratio”
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Some Insights
• Simulations and Observations:
– Simulation setup
Generate a random
problem instance
NO
Is there a
feasible
assignment?
Z<=1
at most 15 tasks
and 4 processors (2
of each type)
Using ILP
formulation
Using Exhaustive
Enumeration:
ratio = Nsucc/Nvalid
YES
Compute the
“success ratio”
• Repeat till 10000 feasible task sets are found
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Some Insights
• Simulations and Observations:
– Observations
• Plotted for 10000 feasible task sets
1
0.9
Average success ratio
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Maximum fraction of capacity used on any processor (Z returned by ILP solver)
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Some Insights
• Simulations and Observations:
Average success ratio
– Observations (for 10000 feasible task sets)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00
Maximum fraction of capacity used on any processor (Z returned
by ILP solver)
– Observations:
• No sharp threshold
• Fluctuations/peaks in the range 0 ≤ Z ≤ 0.4 is probably
due to imbalanced task generation
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The Question
Average success ratio
– Observations (for 10000 feasible task sets)
• Questions:
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00
Maximum fraction of capacity used on any processor (Z returned
by ILP solver)
– Is there a phase transition?
• Yes: What parameters should we observe?
• No: What is its implication?
– considering such a behavior has been observed for:
» uni-processor (RM) and identical multiprocessor scheduling
• Any insights will be useful 
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Few Questions
Average success ratio
– Observations (for 10000 feasible task sets)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Thank You !
• Questions:
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00
Maximum fraction of capacity used on any processor (Z returnd
by ILP solver)
– Is there a phase transition?
• Yes: What parameters should we observe?
• No: What is its implication?
– Since such a behavior has been observed for:
– uni-processor and identical multiprocessor scheduling
• Any insights will be useful 
01/08/2017
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