Relaxation du parallélisme synchrone pour les systèmes asynchrones communicant par buffers : réseaux de Kahn n-synchrones

Transcription

1 Master Parisien de Recherche en Informatique Relaxation du parallélisme synchrone pour les systèmes asynchrones communicant par buffers : réseaux de Kahn n-synchrones Louis Mandel LRI, Université Paris-Sud INRIA Paris-Rocquencourt 7 novembre 22

2 Picture-in-Picture example picture in picture encrust p downscaler small p2 encrust ot big merge o Incrustation of a Standard Definition (SD) image in a High Definition (HD) one downscaler: reduction of an HD image (92 8 pixels) to an SD image (72 48 pixels) : removal of a part of an HD image merge: incrustation of an SD image in an HD image The constraints resolution algorithm has to deal with clocks of 2 millions of elements. 2

3 Motivation Real time constraints the frequency of the display cannot be changed Efficiency constraints handling of High Definition images necessitate billions of operations per second Safety constraints guaranty of deterministic and deadlock free execution 3

4 Kahn Process Networks [Gilles Kahn, 974] i x y o P Q R Network of processes concurrent execution communication through buffers of sufficient size If processes are deterministic then the network is deterministic 4

5 Programming Kahn Process Networks Problem: computation of sufficient buffer sizes risk of data loss, of deadlock sometimes, need of infinite buffers Goal: do not accept infinite buffers automatic sizing of buffers Related work: Synchronous Data Flow and variants [Lee et al.] [Buck] scheduling [Carlier, Chretienne] [Baccelli, Cohen, Quadrat] Network Calculus [Cruz], Real-time Calculus [Thiele et al.] 5

6 Dataflow Synchronous Model i x y o P Q R Programming Kahn networks without buffers: programming languages: Lustre, Signal, Lucid Synchrone instantaneous consumption of produced data strong guaranties: bounded memory, absence of deadlocks But: communication without buffers sometimes too restrictive (e.g. multimedia applications) 6

7 n-synchronous Model i x y o P Q R Programming Kahn networks with bounded memory Automatic methods at compile time to: reject networks needing infinite memory compute activation paces of computations nodes compute sufficient buffers sizes More flexibility with the same guaranties 7

8 Overview. Lucy-n: a n-synchronous Extension of Lustre 2. Adaptability Relation 3. Ultimately Periodic Clocks 4. Abstract Clocks 5. Latency Insensitive Designs 8

9 Lucy-n: a n-synchronous Extension of Lustre

10 Synchronous Model x not so bad () () x x2 + o stream values x x x

11 n-synchronous Model x good () () x x2 + o stream values x x buffer(x) x buffer(x) + x

12 The Lucy-n Language let node good x = o where 2 rec x = x () 3 and x2 = x () 4 and o = buffer(x) + x2 x good () () x x2 + o good :: forall a. a -> a on () Buffer line 4, characters -2: size = 2

13 Programming a stuttering node Synchronous version stutter () let node stutter x = o where rec o = merge () x x_bis and x_bis = (( fby o) ot ()) stutter :: forall a. a on () -> a x fby () ot x bis merge o stream values x ()... fby o x_bis = ( fby o) ot () o = merge () x x_bis

14 Programming a stuttering node Synchronous version stutter () let node stutter x = o where rec o = merge () x x_bis and x_bis = (( fby o) ot ()) stutter :: forall a. a on () -> a x fby () ot x bis merge o n-synchronous version let node new_stutter x = o where new stutter () rec o = merge () x (buffer(x)) new_stutter :: forall a. a on () -> a Buffer line 2, characters 24-33: size = x merge x ()... buffer(x) merge () x (buffer(x))

15 Flows and Clocks

16 Flows and Clocks x w x w = clock(x)... 6

17 Sampling x w w2 x w2 w on w 2 x w2... x w w... w 2... w on w 2... clock(x w2) = clock(x) on w 2 Definition:.w on w 2 def =.(w on w 2 ).w on.w 2 def =.(w on w 2 ).w on.w 2 def =.(w on w 2 ) 7

18 Composition x w y w + z w x y z = x + y clock(x) = clock(y) = clock(z) 8

19 Buffering x buffer x w w 2 Communication through a bounded buffer: the input s clock must be adaptable to the output s clock w <: w 2 9

20 Adaptability Relation

21 Cumulative Function Number of ones Instants w O w O w : cumulative function of the word w 2

22 Adaptability Relation Number of ones buffer size adaptability Writings in the buffer Readings in the buffer Instants O w O w2 size(w,w 2 ) = max i N (O w (i) O w2 (i)) w <: w 2 def n N, i, O w (i) O w2 (i) n 22

23 Adaptability Relation Number of ones Writings in the buffer Readings in the buffer Instants O w O w2 buffer size adaptability size(w,w 2 ) = max i N (O w (i) O w2 (i)) w <: w 2 def n N, i, O w (i) O w2 (i) n synchronizability w w 2 def b,b 2 Z, i, b O w (i) O w2 (i) b 2 precedence w w 2 def i, O w (i) O w2 (i) 22

24 Ultimately Periodic Clocks definition: p = u(v) def p = u.w with w = v.w example: () =... notation: prefix: ().u = periodic part: ().v = Index of the j ième of w notation: I w (j) example: I w (4) = 6 23

25 on operator Example: p ( ) p 2 ( ) p on p 2 ( ) Properties: size and number of : Let p and p 2 such that p.u = p 2.u and p.v = p 2.v. Then: (p on p 2 ).u = p.u (p on p 2 ).u = p 2.u (p on p 2 ).v = p.v (p on p 2 ).v = p 2.v index of the j ième of w on w 2 : j, I w on w 2 (j) = I w (I w2 (j)) 24

26 Adaptability x buffer x w w 2 Communication through a bounded buffer: synchronizability test: p p 2 p.v p.v = p 2.v p 2.v example: () () precedence test: let h = max( p.u, p 2.u )+ lcm ( p.v, p 2.v ), p p 2 j, j h, I p (j) I p2 (j) example: () () adaptability test: p <: p 2 p p 2 p p 2 25

27 Clock Typing

28 Clock Calculus H e : ct C H e 2 : ct C 2 H e +e 2 : ct C C 2 H e : ct C H buffer(e) : ct {ct <: ct } C Adaptability constraints are collected during the clocking of a node. They are solved at the end of node clocking. 27

29 Typing plus plus x y + z () () t t r + o 4 let node plus_plus (x,y) = o where 5 rec z = x + y 6 and t = z () 7 and t = buffer(t) 8 and r = y () 9 and o = t + r val plus_plus : (int * int) -> int val plus_plus :: forall a. ( a * a) -> a on () Buffer line 7, characters -2: size = 28

30 Typing plus plus x α y α + α z () () t t r + o 4 let node plus_plus (x,y) = o where 5 rec z = x + y 6 and t = z () 7 and t = buffer(t) 8 and r = y () 9 and o = t + r val plus_plus : (int * int) -> int val plus_plus :: forall a. ( a * a) -> a on () Buffer line 7, characters -2: size = 28

31 Typing plus plus x α y α + α z () () t t α on () r α on () + o 4 let node plus_plus (x,y) = o where 5 rec z = x + y 6 and t = z () 7 and t = buffer(t) 8 and r = y () 9 and o = t + r val plus_plus : (int * int) -> int val plus_plus :: forall a. ( a * a) -> a on () Buffer line 7, characters -2: size = 28

32 Typing plus plus x α y α + α z () () t t α on () α on () r α on () + o α on () for all α such that {α on () <: α on ()}, (α α) α on () Subtyping constraint solving simple case: α on w <: α on w 2 w <: w 2 29

33 Typing plus plus x α y α + α z () () t t α on () α on () r α on () + o α on () for all α such that { () <: ()}, (α α) α on () Subtyping constraint solving simple case: α on w <: α on w 2 w <: w 2 29

34 Typing plus plus x α y α + α z () () t t α on () α on () r α on () + o α on () for all α, (α α) α on () Subtyping constraint solving simple case: α on w <: α on w 2 w <: w 2 29

35 Typing: More Difficult Case i i2 for all α,α such that () () aux aux2 α on () <: α α 2 on () <: α + g o, α α 2 α Goal of the constraints resolution algorithm : find instantiations for α, α 2 and α such that the constraints are always verified Idea of the constraints resolution algorithm: translate adaptability constraints into constraints on the size of the words and the indexes of the s 3

36 Example of Lucy-n program i i2 f () () () + aux2 + aux () () + o let node f (i, i2) = o where rec aux = buffer (i ()) + aux2 and aux2 = buffer (i2 ()) + i2 () and o = buffer (aux ()) + buffer (aux ()) 3

37 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o f :: α α 2 α 3 with the following constraints: α on () <: α 2 on () α 2 on () <: α 2 on () α 2 on () on () <: α 3 on () α 2 on () on () <: α 3 on () Question: find types α, α 2 and α 3 such that the constraints are always satisfied. 32

38 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o f :: α α 2 α 3 with the following constraints: α on () <: α 2 on () α 2 on () <: α 2 on () α 2 on () on () <: α 3 on () α 2 on () on () <: α 3 on () We can simplify constraints that depends on the same type variable Property: α on ce <: α on ce 2 ce <: ce 2

39 α on () <: α 2 on () () <: () α 2 on () on () <: α 3 on () α 2 on () on () <: α 3 on () We can check that the adaptability constraint is satisfied. α on () <: α 2 on () α 2 on () on () <: α 3 on () α 2 on () on () <: α 3 on () We can express this system in function of a unique type variable by instantiation of α, α 2, α 3 θ = {α α on c ; α 2 α on c 2 ; α 3 α on c 3 ;}

40 α on c on () <: α on c 2 on () α on c 2 on () on () <: α on c 3 on () α on c 2 on () on () <: α on c 3 on () We can simplify the constraints c on () <: c 2 on () c 2 on () on () <: c 3 on () c 2 on () on () <: c 3 on () Question: find ultimately periodic binary words c, c 2 and c 3 such that the constraints are always satisfied. 35

41 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o f :: α on c α on c 2 α on c 3 with the following constraints: c on () <: c 2 on () c 2 on () on () <: c 3 on () c 2 on () on () <: c 3 on () We can compute on.

42 c on () <: c 2 on () c 2 on () <: c 3 on () c 2 on () <: c 3 on () We can adjust the system such that all the samplers of a same variable have the same size. c on () <: c 2 on () c 2 on () <: c 3 on () c 2 on () <: c 3 on () We choose the number of of each c n such that it is equal to the size of its samplers. c.u = 2 c.v = c 2.u = c 2.v = 4 c 3.u = c 3.v =

43 We can split adaptability constraints into synchronizability and precedence constraints. c on () c 2 on () c 2 on () c 3 on () c 2 on () c 3 on () c on () c 2 on () c 2 on () c 3 on () c 2 on () c 3 on () (Sync) (Prec) We can apply the synchronizability and precedence tests. Thanks to the choice of the number of of the c n, we can simplify the formulas.

44 We can apply the synchronizability test. (c on ()).v (c on ()).v (c 2 on ()).v (c 2 on ()).v = (c 2 on ()).v (c 2 on ()).v = (c 3 on ()).v (c 3 on ()).v (Prec) (c 2 on ()).v (c 2 on ()).v = (c 3 on ()).v (c 3 on ()).v Thanks to the choice of the number of of the c n, we can simplify the formulas. ().v c.v ().v c 2.v = ().v c 2.v = ().v c 3.v (Prec) ().v c 2.v = ().v c 3.v

45 We can rewrite the system. ().v c.v = (()).v c 2.v ().v c 2.v = ().v c 3.v ().v c 2.v = ().v c 3.v (Prec) We can compute the number of of the samplers. 2 c.v = c 2.v c 2.v = c 3.v c 2.v = c 3.v c on () c 2 on () c 2 on () c 3 on () c 2 on () c 3 on () Thanks to the choice of the number of of the c n, we can apply the precedence test.

46 (Sync) j, j 3, I c on () (j) I c 2 on() (j) j, j, I c2 on() (j) I c 3 on() (j) j, j, I c2 on() (j) I c 3 on() (j) We can apply the on formula. (Sync) j, j 3, I c (I () (j)) I c 2 (I () (j)) j, j, I c2 (I () (j)) I c 3 (I () (j)) j, j, I c2 (I () (j)) I c 3 (I () (j)) We can compute the index of the s in the periodic words.

47 2 c.v = c 2.v c 2.v = c 3.v c 2.v = c 3.v I c () I c2 (2) I c (3) I c2 (4) I c (4) I c2 (6) I c2 (2) I c3 () I c2 (4) I c3 () Question: find the sizes and the index of such that the constraints are always satisfied and they define well formed ultimately periodic words. 42

48 Well formed ultimately periodic binary words ultimately periodic word ( ) infinite word... index Well formation constraints: increasing indexes: j, I w (j) < I w (j +) sufficient indexes: periodicity: j, I w (j) j j > p.u, I p (j + p.v ) = I p (j)+ p.v sufficient size: p.v +I p ( p.u + p.v ) I p ( p.u +) 43

49 Typing f :: α on c α on c 2 α on c 3 with the following constraints: 2 c.v = c 2.v - (v_size_c - ) c 2.v = c 3.v c 2.v = c 3.v I_c() 2 I_c(3) v_size_c I_c(4) source I_c2() - (v_size_c2 - ) 2 I_c2(4) 2 I_c2(6) I_c2(2) v_size_c2 - (v_size_c3 - ) I_c3() We can solve the Integer Linear Programming problem using GLPK. 44

50 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o f :: α on c α on c 2 α on c 3 with the following constraints: c.v = 2 c 2.v = 4 c 3.v = 4 I c () = I c (3) = 3 I c (4) = 5 I c2 () = I c2 (2) = 2 I c2 (4) = 4 I c2 (6) = 6 I c3 () = 4 We can build the following solution: c = () c 2 = () = () c 3 = () = ( 3 ) 45

51 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o let node f (i, i2) = o where rec aux = buffer (i ()) + aux2 and aux2 = buffer (i2 ()) + i2 () and o = buffer (aux ()) + buffer (aux ()) val f :: forall a. ( a on () * a) -> a on ^3(^3) Buffer line 2, characters 3-35: size = Buffer line 3, characters 3-36: size = Buffer line 4, characters -33: size = Buffer line 4, characters 36-59: size = 46

52 Completeness: Choice of the number of s All the steps of the constraints resolution algorithm are based on equivalence relations except the choice of number of s The choice of number of s is directed by the property: Let p and p 2 such that p.u = p 2.u and p.v = p 2.v. Then: (p on p 2 ).u = p.u (p on p 2 ).u = p 2.u (p on p 2 ).v = p.v (p on p 2 ).v = p 2.v 47

53 Completeness: choice of the number of s Example: p ( ) p 2 ( ) p on p 2 ( ) Property: size and number of : Let p and p 2 such that p.u = p 2.u +k p 2.v and p.v = k p 2.v. Then: (p on p 2 ).u = p.u (p on p 2 ).u = p 2.u +k p 2.v (p on p 2 ).v = p.v (p on p 2 ).v = k p 2.v 48

54 Throughput v.s. buffering () g i i2 () aux aux2 + o let node g (i, i2) = o where rec aux = i () and aux2 = i2 () and o = buffer aux + buffer aux2 49

55 Throughput v.s. buffering () g i i2 () aux aux2 + o let node g (i, i2) = o where rec aux = i () and aux2 = i2 () and o = buffer aux + buffer aux2 With bufferization: val g :: forall a. ( a * a) -> a on () Buffer line 4, characters -2: size = Buffer line 4, characters 24-35: size = Without bufferization: val g :: forall a. ( a on (^4 ) * a on ()) -> a on () Buffer line 4, characters -2: size = Buffer line 4, characters 24-35: size = 5

56 Objective function It is possible to choose the objective function the linear constraints are solves ASAP: minimize the index of s rate: minimize the sizes of the words buffer: minimize the precedence constraints 5

57 GSM speech encoding and decoding

58 Example of a GSM speech encoder and decoder cyclic encoding ( 5 ) i redundancy bit bit bit2 merge o ( 5 ) merge o2 ( 52 ) merge o3 53

59 Example of a GSM speech encoder and decoder gsm encoding ( ) ( ) cyclic encoding merge convolutional encoding split speech merge 54

60 Global unification H e : ct C H e 2 : ct C 2 H e +e 2 : ct C C 2 Collection of equality constraints H e : ct C H e 2 : ct 2 C 2 H e +e 2 : ct 3 {ct = ct 2 = ct 3 } C C 2 55

61 Example of a GSM speech encoder and decoder gsm decoding convolutional decoding split Ia Ib cyclic decoding i split I II join speech o 56

62 Example of a GSM speech encoder and decoder cyclic decoding x ( 5 3 ) ot ( 5 3 ) data cyclic encoding redondancy reencoded data ( 5 3 ) ot code ok frame ok = and by 3 stutter5 data ok 5 current frame if o 5 () true 5 merge last frame 57

63 Abstract Clocks

64 Abstract Clocks: abs(w) = b,b (r) Number of ones concr( b,b (r)) = a =, 4 5 ( ) Instants O w w w[i] = O w (i) (i) w[i] = O w (i) (i) : r i+b : r i+b 59

65 Abstract Clocks: early and late Nombre de uns a 9 = 2 5, 5 ( 9 3 ) O earlya9 5 O latea Instants i, O earlya (i) = max(,min(i, r i+b )) O latea (i) = max(,min(i, r i+b )) most of the proofs rely on these definitions 6

66 Abstract Clocks: normal form Nombre de uns 5 4 a = 4, 4 ( 7 ) 2 ( a = 2,3 2 2) Instants a = b,b (r), a k = (n, ) k l l l with gcd(n,l) = such that concr(()a ) = concr(()a). We have: n l = red(r), k = b l and k = b l. 6

67 Abstract Clocks: prefix Nombre de uns a 7 = 6 3, 3 ( ) 2 3 Nombre de uns a8 = ( 3, 6 3 3) O w O w Instants Instants concr( b,b (r)) = w w[i] = O w (i) (i) w[i] = O w (i) (i) 62

68 Abstract Clocks: automaton Nombre de uns a 2 = 9 5, 3 5 ( 3 5 ) O w , 5, 6,2 7, Instants, 2, 3, 4, Acceptors and generators : States (i,j) : coordinates in the graph if j + (i) Transitions : if j (i) Finite number of equivalent states. 63

69 Clocks Generator (Lucid Synchrone) type rat = { num: int; den: int; } let norm { num = n; den = l; } i j = if i > l && j >= n then (i - l, j - n) else (i, j) let node generate choice b b r = clk where rec i, j = (,) fby norm r (i+) (if clk then j + else j) and one = rat (j + ) <=.. r *.. (rat i) +.. b and zero = rat j >=.. r *.. (rat i) +.. b and clk = choice one zero let node early b b r = generate (fun x y -> x) b b r let node late b b r = generate (fun x y -> not y) b b r 64

70 Adaptability Relation on Abstract Clocks Number of ones a =, 4 5 ( ) a 2 = 9 5, 3 5 ( 3 5 ) Instants O w O w2 synchronizability b,b (r ) b 2,b 2 (r 2 ) def r = r 2 precedence b,b (r) b 2,b 2 (r) def b 2 b < buffer size size (a,a 2 ) = b b 2 65

71 Abstract Operators Clock expressions: ce ::= w not w ce on ce Abstraction of a clock expression: abs(not w) = not abs(w) abs(c on c 2 ) = abs(c ) on abs(c 2 ) Operators correctness property: not w concr(not abs(w)) c on c 2 concr(abs(c ) on abs(c 2 )) 66

72 Abstract Operators: not Nombre de uns a8 = 3, 6 3 ( 3) O w8 6 5 O w a 7 = 6 3, 3 ( ) Instants not operator definition: not b,b (r) = b, b ( r) 67

73 Abstract Operators: on 9,4,4 7,3 8,3 5,2 6,2 7,2, 2, 3, 3, 4, 5, 2 3, ( ) 2 3 on 5 6, ( ) 5 6 on operator definition: b, b ( r ) on b 2, b 2 ( r 2 ) = b r 2 +b 2, b r 2 +b 2 (r r 2 ) with b, b 2 68

74 Typing f () i α α on () () α 2 on () () i2 α 2 α 2 on () α 2 on () + α 2 on () aux2 α 2 on () + aux α 2 on () () () α 2 on () on () α 3 α 2 on () on () α 3 + α 3 o f :: α on c α on c 2 α on c 3 with the following constraints: c on () <: c 2 on () c 2 on () on () <: c 3 on () c 2 on () on () <: c 3 on () We solve the constraints in the abstract domain.

75 abs(c ) on abs(()) <: abs(c 2 ) on abs(()) abs(c 2 ) on abs(() on ()) <: abs(c 2 ) on abs(() on ()) <: abs(c 3 ) on abs(()) abs(c 3 ) on abs(()) We compute the abstractions. b,b (r ) on, () <: b 2,b 2 (r 2 ) on 2, ( ) 2 b 2,b 2 (r 2 ) on 4, 2 ( ) 4 <: b 3,b 3 (r 3 ) on, () b 2,b 2 (r 2 ) on 3 4, ( ) 4 <: b 3,b 3 (r 3 ) on, () We apply the definition of on. b,b + (r ) <: b 2 2 2,b ( ) r 2 2 b 2 4 4,b ( ) 2 r2 4 <: b 3 +,b 3 + (r 3 ) b ,b ( ) r 2 4 <: b 3 +,b 3 + (r 3 )

76 We can split adaptability constraints into synchronizability and precedence constraints and apply the tests. r = r 2 2 r 2 4 = r 3 r 2 4 = r 3 b b < b 3 + b < b 3 + b < (Sync) (Prec) To solve synchronizability constraints we put all the constraints into the for r i = k r j with k r = 2 r 2 r 3 = 4 r 2 (Prec) We choose r = for the variables that do not appear on the left.

77 r 2 = b b < r = 2 r 3 = 4 b 3 + b < b 3 + b < Question: find the values of the b s and b s such that the constraints are always satisfied and they define well formed abstractions. 72

78 Abstract Clocks: non empty Nombre de uns 4 3 a = 5, 2 5 ( 3 5 ) Instants a = k (n ) l, k, l l k l k l l concr(a) 73

79 Abstract Operators: on definition domain Nombre de uns , ( 6 3 6) (, 6 3 6) O w on w () on 3 () = () , ( Instants 3 ) = 2 ) = , 2 +2 ( , 6 ( ) 3 6, 6 ( ) 3 6 b,b (r ) on b 2,b 2 (r 2 ) is define b and b 2 74

80 Typing f :: α on c α on c 2 α on c 3 with the following constraints: c concr( b,b (r )) c 2 concr( b 2,b 2 (r 2 )) c 3 concr( b 3,b 3 (r 3 )) and r 2 = b r = 2 r 3 = 4 (Prec) (NonEmpty) b 2 b 3 We can solve the constraints using GLPK. 75

81 Application to Latency Insensitive Designs

82 Latency Insensitive Design [Carloni et al. 2] Method used to design synchronous circuits that tolerate data transfer latency design synchronous IPs and interconnect them at each instant, each IP is activated at each activation, an IP consumes a token on each input and produces a token on each output data transfer between each IP takes one instant add relay stations on the wires and shell wrappers around IPs relay-station = split a wire into two pieces shell wrapper = buffers on inputs + a controller to activate the IP Question: do IPs have to be activated by their controller? 77

83 Scheduling Latency Insensitive Design Existing answers: elastic circuits dynamic schedule [Carloni et al. 2, Carmona et al. 29]: every wire is transformed into a channel carrying data and control bits the wrappers dynamically decide activation of IPs by analysing control bits and applying an ASAP strategy a back pressure protocol must be used to avoid buffer overflows static schedule [Casu et al. 24, Boucaron et al. 27, Carmona et al. 29]: computation of an explicit schedule avoids additionnal control pathes and runtime overhead of dynamic schedule maximizes rate (by computing sufficient buffer sizes) minimizes buffer sizes (by choosing other strategies than ASAP) 78

84 Modeling Latency Insensitive Designs with Lucy-n Wire x w delay delay x () on w Relay station x w relay relay x w Shell wrapper x w y w 2 IP z w with w <: w and w 2 <: w 79

85 Example: composition of ip A and ip B i delay ip AB init B delay () merge relay delay ip A () out A delay out B ip B merge init A Schedule computed by the compiler val ip_ab :: forall a. a on () -> a on () 8

86 Example: composition of ip A and ip B i delay ip AB init B delay () merge relay delay ip A () out A delay out B ip B merge init A Schedule computed by the compiler val ip_ab :: forall a. a on () -> a on () Better throughput obtained with the help of the user (option -nbones 2): val ip_ab :: forall a. a on () -> a on () 8

87 Composition of Statically Scheduled IPs ip AAB i delay ip A delay ip AB out AB () delay delay relay merge init Schedule computed by the compiler val ip_aab :: forall a. a on () -> a on () The Lucy-n compiler can schedule IPs that do not necessarily consume a token on each input and produce a token on each output at each activation. 82

88 MPEG-2 video encoder [Carloni et al. 22, Casu et al. 24] Frame mem + DCT Quantizer Regulator Inverse quantizer Buffer Preprocessing Motion comp. + + IDCT Frame mem Variable length code ENC Motion est. input output 83

89 Conclusion 84