© 2001 by CRC Press LLCmore complex the design of the deadlock avoidance controller.. The r-reduced G is a Petri net Gr constructed from G by the following steps.. Remove the place r fro
Trang 1© 2001 by CRC Press LLC
more complex the design of the deadlock avoidance controller To lower the complexity of controller
Definition 3
Let G (P R, T, F, m0) be an and r be a resource place Let H(r) {p P R(p) r} The
r-reduced G is a Petri net G(r) constructed from G by the following steps
1 Remove the place r from G and all arcs which are incidental to or from r
2 Repeat the following steps for each place p H(r) Select and remove p H(r) and all transitions
• if {r1} in G, then add a new transition, denoted by t1 t2, and some arcs which
{p, r, r1}
We will call t1 t2-transition, and let T denote the set of all -transitions in G(r) Notice that in
Let G be an and r1 and r2 be two resource places For the r1-reduced G(r1), we can do an r2
-reduced procedure for G(r1) and obtain an {r1, r2}-reduced G(r1, r2) in the same way G(r1, r2) is an
model In general, for any set of resource places R , we can construct R-reduced , denoted by G(R)
Example 4
G contains at least ten D-structures and G(M2) has only three D-structures
Let R K be the set of key resource places and G(R K) (P R, T, F, m0) be the R K-reduced
Then G(R K) is an model in which there are no key resources The set of transitions in G(R K) can
be divided into two parts, T0 and T1, where T0 is a set of -transitions and every -transition t1 t2
… t k in T0 corresponds to a maximal key path t1p1t2, p2…p k t k , K 2, in G T1 TT0 Let T2
denote the set of transitions of G which are in some key path Then T T1 T2
The complexity for reducing an by the set of key resources R K is linear with |{p P | R(p)
R K }| Since {p P | R(p) R K} is finite, the procedure for reducing an model is efficient, and
Optimal Deadlock Avoidance Petri Net Controllers for a Class of s
contains no key resources For such a special class of s, we can first present the following deadlock avoidance Petri net controller:
Definition 4
Let G (P R, T, F, m0) be a marked , R K 0/ A controller for G is a marked Petri net defined by
,
R2PN
R2PN
R2PN
R2PN
p
t1( )r ( )rt2
t1 t2
•
t
•
1•t2
( ) ( t1 t2)•
t1• t2•
t1( )r ( )rt2
t1 t2
•
t
•
1 t2
t1• t2•
t1 t2
r
( )
t1 t2
( )( )r
R ( )p( t1 t2) ( ) R t ( ( 1 t2)( )p )
t1 t2
p
( )
t1 t2
( )( )p
R2PN
R2PN
R ( )p( t2 t3) ( ) R t ( 2 t3)( )p
t2 t3
p
( )
t2 t3
( )( )p
R2PN
R2PN
R2PN
R2PN
R2PN
R2PN
R2PN
R2PN
C ( Pc, , , T Fc m0c)