Back to Basics: Homogeneous Representations of Multi-Rate - PowerPoint PPT Presentation

Back to Basics: Homogeneous Representations of Multi-Rate Synchronous Dataflow Graphs Robert de Groote, Philip H¨ olzenspies, Jan Kuper, and Hajo Broersma Computer Architectures for Embedded Systems Group Dept. of Electrical Engineering, Mathematics and Computer Science University of Twente, Enschede, The Netherlands http://caes.ewi.utwente.nl MEMOCODE 2013

Multi-Rate Synchronous Dataflow Graphs (1/3) d 1 d 2 MP3 SRC DAC 1152 480 1 441 v MP3 v SRC v DAC 480 1 1152 441 d 1 d 2 1 1 1 1 1 1 1 1 1 ◮ Capture task graphs ◮ Potential parallelism and interactions explicit ◮ Well suited for modelling DSP applications ◮ Annotations for analysis Robert de Groote (University of Twente) Back to Basics 10/18/2013 2 / 22

Multi-Rate Synchronous Dataflow Graphs (2/3) execution time production rate consumption rate 1152 480 441 1 v 1 , 2 v 2 , 1 v 3 , 5 1152 480 441 1 d 1 d 2 1 1 1 1 1 1 1 1 1 tokens ◮ Rates, auto-concurrency ◮ Consistency, Iteration, Periodicity ◮ Homogeneous, Cyclo-Static, Scenario-Aware, ... Robert de Groote (University of Twente) Back to Basics 10/18/2013 3 / 22

Multi-Rate Synchronous Dataflow Graphs (3/3) execution time production rate consumption rate 1152 480 441 1 v 1 , 2 v 2 , 1 v 3 , 5 1152 480 441 1 d 1 d 2 1 1 1 1 1 1 1 1 1 tokens Throughput Analysis ◮ (Average) number of graph iterations per time unit ◮ Find critical cycle Buffer Analysis ◮ Determine buffer capacities required for minimal throughput ◮ Make all cycles equally critical Robert de Groote (University of Twente) Back to Basics 10/18/2013 4 / 22

MRSDF Graphs - Exact Analysis (1/3) Throughput Analysis ◮ Algorithms available for Homogeneous SDF Graphs (marked graphs) ◮ Transform MRSDF graph into HSDF graph ◮ Transformation described in [1], [2], ... [1] Lee, Edward A., and David G. Messerschmitt. ”Synchronous data flow.” Proceedings of the IEEE 75.9 (1987): 1235-1245. [2] Sriram, Sundararajan, and Shuvra S. Bhattacharyya. Embedded multiprocessors: Scheduling and synchronization. CRC press, 2009. Robert de Groote (University of Twente) Back to Basics 10/18/2013 5 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 MRSDF to HSDF Transformation ◮ Represent individual firings in an iteration Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 MRSDF to HSDF Transformation ◮ Represent individual firings in an iteration ◮ Represent each token by a single edge Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 MRSDF to HSDF Transformation ◮ Represent individual firings in an iteration ◮ Represent each token by a single edge Analysis: compute critical cycle (MCR) Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 b 1 a 1 c 1 b 2 a 2 c 2 b 3 a 3 c 3 b 4 a 4 c 4 b 5 a 5 b 6 Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 b 1 a 1 c 1 b 2 a 2 c 2 b 3 a 3 c 3 b 4 a 4 c 4 b 5 a 5 b 6 Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 b 1 a 1 c 1 b 2 a 2 c 2 b 3 a 3 c 3 b 4 a 4 c 4 b 5 a 5 b 6 Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3) 1 6 5 2 3 a, 1 b, 3 c, 9 1 6 5 2 3 1 15 6 b 1 a 1 c 1 b 2 a 2 c 2 b 3 a 3 c 3 b 4 a 4 c 4 b 5 a 5 b 6 Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (3/3) HSDF-based approach abandoned due to high complexity ◮ State-Space Exploration used instead [1] [1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R. Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD , 2006. Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Exact Analysis (3/3) HSDF-based approach abandoned due to high complexity ◮ State-Space Exploration used instead [1] Exact Analysis: costly, but useful? [1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R. Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD , 2006. Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Exact Analysis (3/3) HSDF-based approach abandoned due to high complexity ◮ State-Space Exploration used instead [1] Exact Analysis: costly, but useful? ◮ Only need guarantees [1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R. Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD , 2006. Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Conservative Analysis (1/2) cumulative token transfer α c = r c · t + ϕ ˆ ϕ π ρ α p = r c · π ˇ ϕ ( t − ρ ) + 1 − ϕ − ϕ r c · π + ρ time r c Construct linear bounds: ◮ Upper bound on token consumption times: ˆ α c ◮ Lower bound on token production times: ˇ α p Robert de Groote (University of Twente) Back to Basics 10/18/2013 8 / 22

MRSDF Graphs - Conservative Analysis (1/2) cumulative token transfer α c = r c · t + ϕ ˆ ϕ π ρ delay α p = r c · π ˇ ϕ ( t − ρ ) + 1 − ϕ − ϕ r c · π + ρ time r c Construct linear bounds: ◮ Upper bound on token consumption times: ˆ α c ◮ Lower bound on token production times: ˇ α p Robert de Groote (University of Twente) Back to Basics 10/18/2013 8 / 22

SDF Graphs - Conservative Analysis (2/2) ϕ d π ρ Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference on Embedded software (EMSOFT). ACM, 2012. Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2) ϕ d π ρ ϕ − ϕ − d ρ + π r r r · π r r r ϕ Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference on Embedded software (EMSOFT). ACM, 2012. Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2) ϕ d π ρ ϕ − ϕ − d ρ + π r r r · π r r r ϕ Translate each actor and channel into an edge ( i , j ): ◮ γ : Transfer rate ratio ◮ ǫ : Rate-independent delay ◮ δ : Rate-dependent delay Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference on Embedded software (EMSOFT). ACM, 2012. Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2) ϕ d π ρ ϕ − ϕ − d ρ + π r r r · π r r r ϕ Translate each actor and channel into an edge ( i , j ): ◮ γ : Transfer rate ratio ◮ ǫ : Rate-independent delay ◮ δ : Rate-dependent delay ◮ s : Firing start time ◮ Compute maximum rate, r Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference on Embedded software (EMSOFT). ACM, 2012. Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2) ϕ d π ρ ϕ − ϕ − d ρ + π r r r · π r r r ϕ Translate each actor and channel into an edge ( i , j ): ◮ γ : Transfer rate ratio ◮ ǫ : Rate-independent delay maximize r ◮ δ : Rate-dependent delay s.t. s ( j ) ≥ s ( i ) + ǫ ( i , j ) + δ ( i , j ) r ( i ) ◮ s : Firing start time r ( j ) = γ ( i , j ) · r ( i ) ◮ Compute maximum rate, r Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference on Embedded software (EMSOFT). ACM, 2012. Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

Back to Basics Existing exact analysis of MRSDF graphs ◮ Data-driven transformation into HSDF ◮ Redundancy in resulting HSDF Existing approximate analysis ◮ No upper bound on rate - no sense of error ◮ Opaque solution from an LP Robert de Groote (University of Twente) Back to Basics 10/18/2013 10 / 22

Back to Basics Existing exact analysis of MRSDF graphs ◮ Data-driven transformation into HSDF ◮ Redundancy in resulting HSDF Existing approximate analysis ◮ No upper bound on rate - no sense of error ◮ Opaque solution from an LP No common ground! Robert de Groote (University of Twente) Back to Basics 10/18/2013 10 / 22