ComputerAided_Design_Engineering_amp_Manufactur.pdf

Although the example in Figure 8.13 synchronizes only two mutually exclusive transitions, the same
idea can apply to a set of more than two mutually exclusive transitions. As a result of synchronization,
they are sequentialized; i.e., their mutual synchronic distance changes from � to one. For the net shown
in Figure 8.14, this set contains three mutually exclusive transitions.
Summarizing the above discussion, we have the following synchronization rule: If a TT-path ( �1) is
generated from tg to tj , tg tj and tj is a control transition, then generate a series of TT-paths ( �2, �3,…) from t�g to t�j such that all above �s are in a cycle and each t�j is also a control transition with
the same decision place for tg and tj .
Note that tg to t�g need not be a control transition. On the other hand, if t�j is not a control transition,
the token in the decision place may flow to the input place of this control transition, not matching to the
token in the cycle and hence resulting in a deadlock.
This rule must be checked concurrently along with other TT rules. For instance, after the above cycle
(with a token) generation, the following TT-path generation may become a backward generation; then
Rule TT.2 must be applied to insert a token in this TT-path.
The correctness of this rule can be proved by showing that it satisfies the two guidelines. We omit the
proof here because it is not the main purpose of this chapter. Similar comments apply to the rule described
in the next subsection.
Arc-Ratio Rules for General Petri Nets
The synthesis rules of the knitting technique for ordinary PN should be modified for GPN in order to
meet the guidelines. This set of rules is referred to as connection rules since the absence of any may
cause some guidelines to be violated. The weight of arcs in the PN must satisfy some constraints. Consider
the guideline of no-disturbance, i.e., if the NP is a TT-path from to , then the firing ratio between
to of NP must equal that in N 1 . Otherwise, the firing behavior of in N1 �
is disturbed. Similarly,
if the NP is a PP-path from to , then the marking ratio between and of the NP must equal
that in . Otherwise, the marking behavior of in is disturbed. Similar but slightly different
conditions apply if the NP is a TP-path or a PT-path.
Now consider the second guideline of well-behaved NP. Examples are shown in Figure 8.16(a) and 8.16(b)
where the second guideline is violated; both NPs are not well-behaved. These additional rules are called the
arc-ratio Rules.
w
tg tj tg tj tj pg pj pg pj N 1
tj N 1
Arc-Ratio Rules
Again, one can develop the arc-ratio rules based on the two guidelines.
The following definitions of least ratios consider a pair of nodes, transitions, or places, and all the
paths between them in isolation (i.e., all nodes and arcs not in these paths are deleted).
f a
Definition: The least firing ratio between ti and tk , Rik � -- , where a is the least firing number of t
b
i
for tk to fire b times with no tokens left in paths between ti and tk .
m a
Definition: The least marking ratio between pi and pk, Rik � -- , where a is a least number of tokens in
b
pi for pk to get b tokens by firing all transitions between pi and pk with no tokens left in paths between
pi and pk .
mf a
Definition: The least marking-firing ratio between pi and tk, Rik � -- , where a is the least number of
b
tokens in pi such that tk fires b times with no tokens left in paths between pi and tk .
fm a
Definition: The least firing-marking ratio between ti and pk Rik � -- , where a is the least firing number
b
of ti such that pk can get b tokens by firing transitions between ti and pk, with no tokens left in paths
between and .
t i
p k