A new Approach for Minimizing Buffer Capacities with Throughput Constraint for Embedded System Design

Transcription

1 A new Approach for Minimizing Buffer Capacities with Throughput Constraint for Embedded System Design Mohamed Benazouz, Olivier Marchetti, Alix Munier-Kordon, Pascal Urard LIP6 Université P. et M. Curie Paris Montpellier, le 23 avril 2009

2 Outline 1 Model and Notations 2 3 4

3 Marked Timed Weighted Event Graph (MTWEG) Definition G = (T,P,l, M 0 ) is a Marked Timed Weighted Event Graph (MTWEG) where 1 T = {t 1,, t n } transitions; 2 P = {p 1,, p m } places; 3 l : T N duration function; 4 M 0 : P N initial marking;

4 Marked Timed Weighted Event Graph (MTWEG) t i p u(p) M 0 (p) v(p) t j Figure: A place p = (t i, t j ) of a MTWEG. 1 Each place p P is defined between two transitions t i and t j ; 2 p P u(p) and v(p) are integers called the marking functions.

5 Firing of a transition P + (t i ) = {p = (t i, t j ) P, t j T } if t i is fired at time τ: P (t i ) = {p = (t j, t i ) P, t j T } 1 At time τ, v(p) tokens are removed from every place p P (t i ). 2 At time τ + l(t i ), u(p) tokens are added to every place p P + (t i ). M(τ, p)= The instantaneaous marking of a place p P at time τ 0

6 Schedule and Periodic Schedule Definition Let G be a MTWEG. A schedule is a function s : T N Q + which associates, with any tuple (t i, q) T N, the starting time of the qth firing of t i. Definition A schedule s is periodic if there exists a vector w = (w 1,..., w n ) Q +n such that, for any couple (t i, q) T N, s(t i, q) = s(t i, 1) + (q 1)w i. w i is then the period of the transition t i and λ s (t i ) = 1 w i its throughput.

7 A car radio application (Wiggers et al.) MP3 Reader Cell Phone t 7 t 1 t 9 t 5 Out To Speaker Microphone t 10 t 3 Signal to eliminate Figure: Block diagram of a car-radio application

8 Modelling using a MTWEG 1 Transitions corresponds to treatments; 2 Places corresponds to buffered transfers. But...the size of the buffers should be limited!

9 Bounded capacity Definition A place p = (t i, t j ) has a bounded capacity F(p) > 0 if the number of tokens stored in p can not exceed F(p): τ 0, M(τ, p) F(p) Definition A MTWEG G = (T,P, M 0,l, F) is said to be a bounded capacity graph if the capacity of every place p P is bounded by F(p).

10 Bounded capacity (Marchetti, Munier 2009) t i u(p) p M 0 (p) M(p, τ) F(p) v(p) t j t i M 0 (p 1 ) = M 0 (p) u(p) p 1 p 2 v(p) M 0 (p 2 ) = F(p) M 0 (p) Figure: A place p with limited capacity F(p) and the couple of places (p 1, p 2 ) c without capacities that models place p. t j Definition G is a symmetric MTWEG if every place p = (t i, t j ) is associated with a backward place p = (t j, t i ) modelling the limited capacity.

11 Modelling of a car radio using a symmetric MTWEG p t 7 7 t 8 t 9 p p p 7 p 8 p p 80 t 6 6 p p t 1 t p 1 80 p 5 p 5 1 t p p 4 p p 2 2 p 10 p 80 3 t t 4 t p p 10 3 Figure: A MTWEG G modelling a car-radio application

12 Problem Formulation Let G be a symmetric (non marked) WTEG. p P, θ(p) is the size of a data stored in place p. The problem consists in computing an initial marking M 0 of G such that: 1 The weighted sum of initial marking p P θ(p)m 0(p) is minimum; 2 There exists a periodic schedule s such that λ s (t i ), t i T.

13 Unitary MTWEG (Karp, Miller 1966) Definition The weight (or gain) of every path µ of a MTWEG G by W(µ) = p P µ u(p) v(p). Definition A MTWEG G is unitary if every circuit c of G verifies W(c) = 1. Since our symmetric MTWEG must be live, we only consider unitary MTWEG.

14 Normalized MTWEG (Marchetti, Munier 2009) Definition A MTWEG is normalized if all adjacent marking functions of every transition t i T are equal to a single value denoted by Z i, i.e. p P + (t i ), u(p) = Z i and p P (t i ), v(p) = Z i. Note that the number of tokens in every circuit of a normalized MTWEG does not vary.

15 Theorem Every unitary MTWEG may be transformed into an equivalent normalized MTWEG. 1 For any value α Q +, the markings functions and the initial markings of any place p P may be replaced simultaneously by respectively αu(p), αv(p) and αm 0 (p) without any influence on the schedules; 2 Positive integer values α(p), p P such that, t i T, there exists an integer Z i with, p P + (t i ), α(p)u(p) = Z i and p P (t i ), α(p)v(p) = Z i may be computed in polynomial time.

16 Normalization of the MTWEG of the car Radio Z 1 = α(p 1 ) = α(p 6 ) = α(p 8 ) = α(p 9 ) Z 2 = 441α(p 1 ) = 80α(p 2 ) Z 3 = 80α(p 2 ) = 80α(p 3 ) = 80α(p 10 ) Z 4 = α(p 3 ) = α(p 4 ) Z 5 = α(p 4 ) = α(p 5 ) Z 6 = 80α(p 5 ) = 441α(p 6 ) Z 7 = 1152α(p 7 ) Z 8 = 480α(p 7 ) = 441α(p 8 ) Z 9 = α(p 9 ) Z 10 = α(p 10 )

17 Normalization of the MTWEG of the car Radio p 7 t 7 t 8 t 9 p p p 7 p 8 p t p 6 6 p p t 1 t p p 5 p 5 p p 2 2 p t p p t t 4 t p p p Figure: The normalized MTWEG associated to G.

18 Precedence constraints induced by a place (Munier 1993) Let G = (T,P,l, M 0 ) a MGTEG. For any (t i,ν i ) T N,(t i,ν i ) is the ν i th firing of t i. Theorem A place p = (t i, t j ) will induce a precedence constraint between (t i,ν i ) and (t j,ν j ) iff u(p) M 0 (p) > u(p)ν i v(p)ν j max(u(p) v(p), 0) M 0 (p)

19 Characterization of a periodic schedule (Benabid et al. 2008) Theorem Let G be a normalized MTWEG. For any feasible periodic schedule s of G, there exists K Q + called the normalized period of s such that, for any couple of transitions (t i, t j ) T 2, w i Z i = w j Z j = K. Moreover, s is feasible iff, for any place p = (t i, t j ) P, s(t j, 1) s(t i, 1) l(t i ) + K(Z j M 0 (p) gcd i,j ). where gcd i,j = gcd(z i, Z j ).

20 Computation of the maximum processing times for the car radio example The output (t 9 ) must have a frequency equal to 44.1kHz. l(t 9 ) = w 9 = Thus, K = w 9 Z 9 = sec; t i T {t 9 }, w i = Z i K and l(t i ) w i. Table: Upper bound w i of the processing times, t i T in milliseconds t 1, t 9 t 2, t 6, t 8 t 3 t 4, t 5, t 10 t 7 l

21 Computation of a periodic Schedule input A normalized MTWEG G; output Minimum normalized period K.

22 Computation of a periodic schedule t 2 p p 4 p t 3 2 t 1 2 p t p 2 15 Figure: A MTWEG with l(t 1 ) = 10, l(t 2 ) = 12, l(t 3 ) = 6 and l(t 4 ) = 5.

23 12 6 K σ K σ t K σ 12 t K σ t 1 t K σ 10 2 K σ 5 11 K σ 5 5 K σ Figure: Valued graph G = (X, A) associated with the normalized Marked WEG pictured by Figure 6.

24 Computation of a periodic schedule with a minimum period min K subject to s(t 4, 1) s(t 2, 1) 12 s(t 1, 1) s(t 4, 1) 5 11K s(t 2, 1) s(t 1, 1) K K K t i T, s(t i, 1) 0

25 Computation of a periodic schedule with a minimum period min K subject to p = (t i, t j ) P, s(t j, 1) s(t i, 1) l(t i )+ K (Z j M 0 (p) gcd i,j ) t i T, s(t i, 1) 0 Polynomially solved using Linear Programming or a variant of Bellman-Ford algorithm.

26 Computation of the minimum size of the buffers under for a given K G is a normalized symmetric MTWEG. System Π(K): ( ) min p P θ(p)m 0(p) subject to p = (t i, t j ) P, s(t j, 1) s(t i, 1) l(t i )+ K (Z j M 0 (p) gcd i,j ) p = (t i, t j ) P, M 0 (p) = k i,j gcd i,j p = (t i, t j ) P k i,j N t i T, s(t i, 1) 0

27 Study of a buffer p 2 Z 1 Z 2 t 1 t 2 Z 1 p 1 Z 2 Figure: A unitary WEG with two places. F(p 1, p 2 ) = M 0 (p 1 ) + M 0 (p 2 ) { K opt l(t1 ) = max, l(t } 2) l(t 1 ) + l(t 2 ), Z 1 Z 2 F(p 1, p 2 ) + 2gcd 1,2 (Z 1 + Z 2 )

28 K opt (p 7, p 7 ) gcd 7,8 17gcd 7,8 18gcd 7,8 19gcd 7,8 31gcd 7,8 32gcd 7,8 F(p 7 p 7 ) Figure: K opt according to F(p 7, p 7 ) = M 0(p 7 ) + M 0 (p 7 ). F min (p 7, p 7 ) = 16gcd 7,8, K min (p 7, p 7 ) = ms, F max (p 7, p 7 ) = 32gcd 7,8 and K max (p 7, p 7 ) = ms.

29 An optimal O(m log( max i {1,...,n} {Z i})) algorithm for a symmetric MWTEG without circuits of more than two transitions { } Let K max l(ti ) t i T Z i and (p, p ) c a couple of places corresponding to a buffer. F K (p, p ) = minimum capacity of buffer p to achieve a normalized period K. 1 If K K max (p, p ), set M 0 (p) = F K (p, p ) and M 0 (p ) = 0; 2 Else, K > K max (p, p ). Set M 0 (p) = F min (p, p ) and M 0 (p ) = 0. This solution is clearly minimum for every buffer. So, it minimizes the overall weighted capacity of G.

30 Example for the car radio Let G limited to the transitions T = {t 1, t 5, t 6, t 7, t 8, t 9 } corresponding to the mixing of the sounds coming from the MP3 reader and the cell phone to the output. The corresponding undirected graph defined as G = (T, {{t 7, t 8 }, {t 5, t 6 }, {t 6, t 1 }, {t 8, t 1 }, {t 1, t 9 }}) is clearly a tree (see Figure 10). t 9 t 5 t 6 t 1 t 8 t 7 Figure: The undirected graph G is a tree.

31 Example for the car radio Table: Optimal initial markings of the subgraph G for different processing time of t 8. l(t 8 ) = θ(p 6 )F(p 6, p 6 ) θ(p 7 )F(p 7, p 7 ) θ(p 8 )F(p 8, p 8 ) θ(p 9 )F(p 9, p 9 ) Sum

32 An Approximation Algorithm for the General Case Let K be a fixed value. Let us consider the Linear Program Π (K) obtained from Π(K) by replacing the condition k i,j N by k i,j Q +. 1 Compute an optimum solution M 0 (p) Q, p P of Π (K). This step can be done in polynomial time; 2 M 0 (p) = M0 (p) is then a feasible solution of Π(K).

33 An Approximation Algorithm for the General Case Theorem The competitive ratio of our algorithm is 2. Worst case may easily be achieved for a symmetric Timed Non Weighted Event graph (i.e., Z i = 1, t i T ).

34 Application to the car radio Table: Optimal buffers capacities for the MTWEG pictured by Figure 4 Buffers θ(p i )a K (p i, p i ) θ(p i)f App K (p i, p i ) (p 1, p 1 ) c (p 2, p 2 ) c (p 3, p 3 ) c (p 4, p 4 ) c 2 2 (p 5, p 5 ) c (p 6, p 6 ) c (p 7, p 7 ) c (p 8, p 8 ) c (p 9, p 9 ) c 2 2 (p 10, p 10 ) c Sum

35 Conclusion and Perspectives 1 First analytical and polynomial approach to compute the minimum size of the buffers; 2 Implementation in an industrial context; 3 Extension to cyclo-static Synchronous DataFlow Graphs.

36 Olivier Marchetti, Alix Munier Kordon A sufficient condition for the liveness of weighted event graphs. EJOR, vol 197, no 2, pp , Olivier Marchetti, Alix Munier Kordon Cyclic scheduling for the synthesis of embedded systems. Introduction to Scheduling, Y.Robert et F.Vivien Eds, à paraître. Mohamed Benazouz, Olivier Marchetti, Alix Munier Kordon, Pascal Urard A new Approach for Minimzing Buffer capacities with Throughput Constraint for Embedded System Design. submitted.