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

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

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

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

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

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

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

Multiprocessor models Fraction of computing capacity background – problem – results

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

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

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

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

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

A = Problem example System specification: as a utilization matrix + a cost vector Cost/ processor: 10 2 4 0.4 0.3 0.5 0.35 0.8 0.25  0.6 T1 T2 T3 T4 T5 P1 P2 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

A = Problem example System specification: as a utilization matrix + a cost vector 10 2 4 0.4 0.3 0.5 0.35 0.8 0.25  0.6 T1 T2 T3 T4 T5 P1 P2 P3 A = P1 T1 T2 T3 T4 T5 A possible implementation: uses P1 procs only cost = 10 + 10 + 10 = 30 background – problem – results

A = Problem example System specification: as a utilization matrix + a cost vector 10 2 4 0.4 0.3 0.5 0.35 0.8 0.25  0.6 T1 T2 T3 T4 T5 P1 P2 P3 A = P1 P2 T2 P3 T3 T1 A better implementation: T5 uses one processor of each type cost = 10 + 2 + 4 = 16 T4 background – problem – results

A = Problem example System specification: as a utilization matrix + a cost vector 10 2 4 0.4 0.3 0.5 0.35 0.8 0.25  0.6 T1 T2 T3 T4 T5 P1 P2 P3 A = P1 P2 T2 T1 An even better implementation: T3 T5 uses one P1 and two P2’s cost = 10 + 2 + 2 = 14 T4 P2 background – problem – results

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

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

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-0309825: 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