Modeling producer-consumer
example using Petri Nets
Neelam Gupta
The University of Arizona
Producer-consumer example
write
consume
P
1
P
2
C
C1
produce
read
separate nets
for the subsystems
read
0
read
1
write
2
write
2
Producer-consumer example
consume
C1
C2
read
read
one net for the
entire system
0
1
2
write
P1
produce
write
P2
Petri nets with priorities and
Timed Petri nets
Neelam Gupta
The University of Arizona
Extension 1
specifying priorities
• A priority function pri from transitions to
natural numbers:
• pri: T ‡ N
• When several transitions are enabled, only
the ones with maximum priority are allowed
to fire
• Among those transitions that are allowed to
fire at the same time, the one to fire is
chosen non-deterministically
Extension 2
Timed Petri nets
• A pair of constants <tmin, tmax> is
associated with each transition
• Once a transition is enabled, it must wait
for at least tmin to elapse before it can
fire
• If enabled, it must fire before tmax has
elapsed, unless it is disabled by the firing
of another transition before tmax
Example: combining priorities
and time
P
1
P
2
T1
tmin = 1
tmax = 4
priority = 1
P
4
P
3
T2
tmin = 2
tmax = 3
priority = 3
T3
tmin = 0
tmax = 5
priority = 2
At t=0, i.e, when the above system begins to execute, all the
transitions above are enabled. However, only T3 can fire at
t=0. At t=1, transition T1 is also ready to fire. If T3 has not
fired before t=1, then T1 cannot fire until T3 has fired, since
T3 has higher priority than T1. If T3 does not fire till t=4,
then T1 can never fire since T1 must fire before t=4.
Example: combining priorities
and time (cont.)
P
1
P
2
T1
tmin = 1
tmax = 4
priority = 1
P
4
P
3
T2
tmin = 2
tmax = 3
priority = 3
T3
tmin = 0
tmax = 5
priority = 2
If T3 fires before t=1, T1 can fire at t=1 or anytime after
t=1 but before t=2. This is because at t=2, T2 can fire and
T2 has higher priority than T1, therefore it will prevent T1
from firing. If T1 fires before t=2, then T2 gets disabled
and can never fire.
Example: combining priorities
and time (cont.)
P
1
P
2
T1
tmin = 1
tmax = 4
priority = 1
P
4
P
3
T2
tmin = 2
tmax = 3
priority = 3
T3
tmin = 0
tmax = 5
priority = 2
If T2 fires before T1 fires, it disables T1. Therefore, T1
can never fire after T2 fires. If T3 does not fire before
t=2, then it cannot fire till t=3 since T2 can fire at t=2 and
T2 has higher priority than T3. If T2 fires before t=3,
then T1 is disabled, but now T3 can fire anytime before
t=5. After t=3, if T3 has not fired yet, it can fire anytime
before t=5 irrespective of whether T2 has fired or not.
Example: combining priorities
and time (cont.)
P
1
T1
tmin = 1
tmax = 4
priority = 1
P
4
P
3
P
2
T2
tmin = 2
tmax = 3
priority = 3
T3
tmin = 0
tmax = 5
priority = 2
In the example given in the textbook, P4 does not contain
any token at t=0. The discussion in the book considers the
case “if at time t=1, a token is produced into P4”. This
assumes that P4 may be a part of a subsystem which can
produce tokes into P4. This assumption results in discussion
of many different situations, such as what if token in P4
appears at t=2, or at t=3 or at any other time? Therefore,
in the example I have discussed in these slides, I have
assumed the initial marking of the system in which T3 is
enabled t=0.