10.01.2013 Views

ComputerAided_Design_Engineering_amp_Manufactur.pdf

ComputerAided_Design_Engineering_amp_Manufactur.pdf

ComputerAided_Design_Engineering_amp_Manufactur.pdf

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

( R12) Lemma 7: Let n1 → n2 → n3, then R13 � where gcd is the greatest common denominator<br />

between and .<br />

u � gcd<br />

( R23) l -----------------------------<br />

� gcd<br />

( ) l<br />

( ) u<br />

R 12<br />

The following observation regarding the relationship between two nodes x and y is useful to find the<br />

least ratios between nodes that are not sequential to each other.<br />

Observation 2:<br />

1. If x → y, then �P1<br />

from x to y, which does not pass through any home place.<br />

2. If x � y or x | y, then the P1 from x to y contains the home place.<br />

3. If x � y (x | y),<br />

then �P1 and P2 containing x and y,<br />

respectively, which do not share common<br />

PSPs, but share the same end transition (place).<br />

Ex<strong>amp</strong>le: In Figure 8.12(a), t1→t3;<br />

P1 [ t1<br />

p2<br />

t2<br />

p3<br />

t3]<br />

does not include the home place p1.<br />

p3<br />

| p7;<br />

the<br />

P1 from p3<br />

to p7<br />

[ p3<br />

t3<br />

p4<br />

t4<br />

p5<br />

t5<br />

p6<br />

t6<br />

p1<br />

t1<br />

p2<br />

t7<br />

p7]<br />

includes the home place p1.<br />

P1 [ p3<br />

t3<br />

p4]<br />

and<br />

P2 [ p7<br />

t8<br />

p4]<br />

join at p4.<br />

In Figure 8.16(b), t3<br />

� t5;<br />

the P1 from t3<br />

to t5<br />

[ t3<br />

p3<br />

t4<br />

p4<br />

t1<br />

p1<br />

t2<br />

p5<br />

t5]<br />

includes<br />

the home place .<br />

Definition:<br />

A PP-path with a generation point, (which has more than one output transition), is<br />

said to have partial flow if its tg, the output transition of pg on the path, must fire more than once to<br />

have no tokens blocked inside the path.<br />

Partial flow may lose some tokens in a PP-path if these tokens are insufficient to enable some transition<br />

in the path and thus may cause a deadlock.<br />

Definition:<br />

m<br />

The input ratio of a PP-path is the ratio of ( ) to the arc weight between and .<br />

u<br />

Ex<strong>amp</strong>le: Figure 8.17 illustrates an ex<strong>amp</strong>le of partial flow that causes a deadlock as shown in Figure<br />

8.17(b). The input ratio of path ( p2t 2p3t 3p4) is 2/1 and that of path ( p2t 5p5t 6p4) is 4/1.<br />

Because the transitions in Xj1 g � LEX ( t are mutually exclusive, the number of transitions that<br />

1<br />

j1, tg1) are fired during one iteration may differ from that during another iteration. The weighted firing of<br />

transitions in is defined as<br />

�( tgs)Rg1g. s<br />

The following theorem implies that the ratio of weighted firings of Xg1 j to that of X is ,<br />

1<br />

j1g [ R<br />

1<br />

g1j ]�<br />

1<br />

i.e., the prime ratio of the least firing between tg1 and tj1, if only tg1 in Xg1 j and t in fire during<br />

1<br />

j1 Xj1 g1 an iteration, respectively.<br />

Theorem 7:<br />

p 1<br />

X g1 j 1<br />

R 23<br />

�<br />

tgs�X g1j 1<br />

If tg1�t j1 in a synthesized strongly connected net,<br />

�<br />

t gs �X g1 j 1<br />

�<br />

tgs�X j1g 1<br />

p g<br />

f<br />

� ( tgs)Rg1gs �<br />

� ( tjs)Rj1js f<br />

��<br />

such that M0[� � M0, then<br />

Proof:<br />

Assuming that the paths containing and , respectively, intersect at , when ts<br />

fires w times,<br />

f<br />

f<br />

then each member tgs and tjs, respectively of Xg1 j and X , may fire and times,<br />

1<br />

j1g ( v<br />

1<br />

gs)<br />

( Rgs s)<br />

( vjs) ( Rjs s)<br />

respectively, where v�s ( �w)[<br />

� � g,<br />

j]<br />

is the number of firings of ts applied to the path containing �s. We have<br />

f<br />

R gj<br />

--------------------------------------- f<br />

[ Rg1 j ]�.<br />

1<br />

t g1<br />

f<br />

f f<br />

��( t�s)R�1� �v s<br />

�s(<br />

R�s s)<br />

( R�1 � ) ⎛�⎞ f<br />

� �<br />

s ⎝ v ( R�1<br />

�s⎠<br />

s)<br />

, � �<br />

g, j.<br />

� s<br />

� s<br />

t j1<br />

t s<br />

p g<br />

f<br />

t g

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!