1. Sanjoy Baruah The University of North Carolina at Chapel Hill
Cost-efficient synthesis of real-time systems upon heterogeneous multiprocessor platforms
Sanjoy Baruah
The University of North Carolina at Chapel Hill
Supported in part by the U. S. National Science Foundation

2. Multiprocessor real-time scheduling
Multiprocessor systems are important
more computing capacity, at less cost
arise naturally in many situations
Long-term goal: Extend real-time scheduling theory to multiprocessors
In this work: schedule periodic tasks on heterogeneous multiprocessors
Outline of talk:
background
problem description
results
background – problem – results

3. Real-time process model
Jobs: basic units of work
characterized by three parameters
arrival/ release time
execution requirement
deadline
are preemptable
Recurring tasks
generate the jobs
are independent
background – problem – results

4. The periodic (Liu & Layland) task model
Task T = (e,p)
execution requirement
period
(utilization  e/p)
(2) 5 10 15 20
Example: T = (2,5)
time
Jobs:
first job arrives at any time; consecutive arrivals at least p time units apart
each job has execution requirement e
each job’s deadline is p time units after arrival
background – problem – results

5. Migration issues on multiprocessors
Partitioned scheduling
Each task may only execute on a specific processor
Global scheduling
Any task’s job may execute on any processor
background – problem – results

6. Migration issues on multiprocessors
Partitioned scheduling:
1. Determine a mapping of tasks to processors
2. Perform run-time scheduling
The Earliest Deadline First (EDF) scheduling algorithm
- provably optimal (utilization bound = 1.0) on uniprocessors
Partitioned with EDF 
Assign tasks to the processors, such that no processor’s capacity is exceeded
Schedule each processor using EDF
background – problem – results

7. Multiprocessor models
identical multiprocessors:
each processor has the same computing capacity
uniform multiprocessors:
different processors have different computing capacities
heterogeneous multiprocessors:
each (task, processor) pair may have a different computing capacity
background – problem – results

8. Multiprocessor models
Fraction of computing capacity
background – problem – results

9. Multiprocessor models
identical multiprocessors:
each processor has the same computing capacity P1 P2 P3
Task T1
Task T2
background – problem – results

10. Multiprocessor models
uniform multiprocessors:
different processors have different computing capacities P1 P2 P3
Task T1
Task T2 y
y/2
y/3 x
x/2
x/3
speed = 1
speed = 2
speed = 3
background – problem – results

11. Multiprocessor models
heterogeneous multiprocessors:
each (task, processor) pair may have a different computing capacity P1 P2 P3
Task T1
Task T2 y y
1.5 y x
x/2
x/3
background – problem – results

12. Why study heterogeneous multiprocessors?
Natural generalization of earlier models
Systems synthesized using specialized COTS processors
x/2
x/3 x
CPU
DSP chip
Graphics co-processor
Graphics-intensive task:
Number-crunching task: y
1.5 y
background – problem – results

13. Problem statement Determine Given
system specification, as a collection of periodic real-time tasks
a library of available processing units (and corresponding costs)
Determine
a (multiprocessor) implementation, comprised of the available kinds of processing units, of minimum cost
COST-OPTIMAL
SYSTEM SYNTHESIZER
Implementation of A
Available processors
Periodic task system A
background – problem – results

14. A = Problem example System specification: as a utilization matrix
+ a cost vector
Cost/ processor:
0.4
0.3
0.5
0.35
0.8
0.25 
0.6 T1 T2 T3 T4 T5
P P P3
T2 needs 0.6 of P3’s capacity
A =
T4 cannot execute on P2
Available processors
Periodic task system A
COST-OPTIMAL
SYSTEM SYNTHESIZER
Implementation of A
background – problem – results

15. A = Problem example System specification: as a utilization matrix
+ a cost vector
0.4
0.3
0.5
0.35
0.8
0.25 
0.6 T1 T2 T3 T4 T5
P P P3
A = P1 T1 T2 T3 T4 T5
A possible implementation:
uses P1 procs only
cost = = 30
background – problem – results

16. A = Problem example System specification: as a utilization matrix
+ a cost vector
0.4
0.3
0.5
0.35
0.8
0.25 
0.6 T1 T2 T3 T4 T5
P P P3
A = P1 P2 T2 P3 T3 T1
A better implementation: T5
uses one processor of each type
cost = = 16 T4
background – problem – results

17. A = Problem example System specification: as a utilization matrix
+ a cost vector
0.4
0.3
0.5
0.35
0.8
0.25 
0.6 T1 T2 T3 T4 T5
P P P3
A = P1 P2 T2 T1
An even better implementation: T3 T5
uses one P1 and two P2’s
cost = = 14 T4 P2
background – problem – results

18. Results-I: intractability
Given: An (n  m) matrix representing a task system, and an m-vector of processor costs, determine a mapping of the n tasks onto processors to minimize the total processor cost.
Result: The minimum cost heterogeneous multiprocessor synthesis problem is NP-hard in the strong sense
Proof: Transform from bin-packing
Consequence: Unlikely to be able to solve efficiently
even for relatively small systems
background – problem – results

19. Results-II: Approximation algorithms
The minimum cost heterogeneous real-time system implementation problem is NP-hard in the strong sense
OBJECTIVE: Obtain approximate solutions to this problem:
Obtain low-cost implementations…
…with cost a bounded amount greater than the minimum cost
RESULTS: Polynomial time algorithms for obtaining system implementations of cost at most
costOPT + c, for global scheduling algorithms
2  costOPT + c, for partitioned scheduling algorithms
where c is a constant independent of the system being designed
background – problem – results

20. Summary Fact: Low-cost implementations are important
Fact: Heterogeneous multiprocessor implementations are common
But, the heterogeneity is usually not exploited
… because heterogeneous systems are not well understood
NSF CCR : Real-time Scheduling on Heterogeneous Multiprocessors
Results: Approximation algorithms for minimum-cost synthesis of periodic task systems upon heterogeneous multiprocessor platforms that are
asymptotically optimal, under the global scheduling paradigm
asymptotically 2-approximate, under the partitioned scheduling paradigm