ComputerAided_Design_Engineering_amp_Manufactur.pdf
ComputerAided_Design_Engineering_amp_Manufactur.pdf
ComputerAided_Design_Engineering_amp_Manufactur.pdf
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( 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