Real-Time Systems
Lecture 9, 10
Modeling Real-Time
systems :
Petri Nets
[email protected]
Contents
Introduction
Petri Nets
Modeling with Petri nets
Timed Petri Nets
Introduction
Petri Nets provide for a mathematical
approach to modelling and analysis of
the systems. We can model the
following aspects of the HRT system :
• Concurency
• Synchronization
• Non-determinism
Introduction
Petri nets are applicable to:
Communication protocols
Manufacturing systems
Computer networks
Industrial process control
Introduction
References:
1. Nissanke N., Realtime Systems,
Prentice Hall, 1997
2. Szmuc T., Zaawansowane metody
tworzenia oprogramoweania
systemów czasu rzeczywistego,
Katedra Automatyki AGH, Kraków
1998, Zeszyt nr 15.
Petri Nets
Definition:
A Petri net is a bipartite graph which
consist of two sets of vertices(places
and transitions) and a set of arcs
between pairs of vertices.
Petri Nets
Places ( P )
{p1, p2, ..., pi}
Transitions ( T )
{t1, t2, ..., tj}
Petri Nets
• Input arcs :
Set i  P x T , (pi, tk)
• Output arcs :
Set o  T x P , (tk, pi)
• Weight of the arc:
W : ((P x T)  (T x P))  N,
where: dom W = i  o
Petri Nets
The simplest Petri net :
p1
1
t1
1
p2
Petri Nets
The simplest Petri net :
p1
1
t1
1
p2
Petri Nets
The simplest Petri net :
p1
1
t1
1
p2
Petri Nets
The simplest Petri net :
p1
1
t1
1
p2
transition
Petri Nets
„time invariant”
„time variant”
(P, T, i, o, W)
Marking function:
M:PN
{M0, M1, M2 ... }
PN = (P, T, i, o, W, M0)
Petri Nets
Firing the transition:
• Only enabled transition may fire.
• When transition fires, W((pi, t)) tokens
are removed from each pi and W((po, t))
tokens are deposited at each po.
Petri Nets
• Only enabled transition may fire.
Transition is enabled only when every
input place p has at least W((p, t))
tokens.
1
2
4
Petri Nets
• Only enabled transition may fire.
Transition is enabled only when every
input place p has at least W((p, t))
tokens.
1
2
4
Petri Nets
• When transition fires ...
W((pi, t)) tokens are removed from each
pi and W((po, t)) tokens are deposited at
each po
1
2
4
Petri Nets
• When transition fires ...
W((pi, t)) tokens are removed from each
pi and W((po, t)) tokens are deposited at
each po
1
2
4
Petri Nets
• When transition fires ...
Transition function :
M0
t1
M1
t2
M2
t3
M3 ...
Mi+1 = (Mi , ti)
Marking Mi+1 is directly rechable from
the marking Mi
Modeling with Petri Nets
The model
The system
Places
Potential conditions
of the system
Transitions
Potential events in
the system
Input arcs
Relating events to
preconditions
Modeling with Petri Nets
The model
The system
Output arcs
Relating events with
postconditions
Tokens
Fulfillment of the
conditions
Firing
Occurence of the
given event
Modeling with Petri Nets
The model
The system
Marking
Current system
state
Initial marking
Initial system state
Change of
marking
Transition between
system states
Modeling with Petri Nets
Concurency
t1
2
p3
p1
p4
p2
t2
p5
Modeling with Petri Nets
Conflict
t1
p1
p4
p2
p5
p3
t2
Modeling with Petri Nets
Symmetric confusion
t1
p1
p3
t2
p4
p2
p5
t3
Modeling with Petri Nets
Asymmetric confusion
p4
p1
p5
t1
t2
p2
p6
p3
t3
Modeling with Petri Nets
Properties: Reachability
Marking Mn is reachable from marking
M0 iff n = 0 or there exists a sequence of
transitions l=<t1, t2, ..., tn> altering the
marking M0 into Mn.
R(PN, M0) – set of all possible markings
for the given Petri net PN
Modeling with Petri Nets
Properties: Reachability
Marking Mn is reachable from marking
M0 iff n = 0 or there exists a sequence of
transitions l=<t1, t2, ..., tn> altering the
marking M0 into Mn.
L(PN, M0) – set of all possible sequences
of transitions for the given
Petri net PN
Modeling with Petri Nets
Properties: k-Boundness
A Petri net is k-bounded when
M(p)  k for some kN, MR(PN, M0)
and pP
Modeling with Petri Nets
Properties: Boundness
A Petri net is bounded when
exists kN, that PN is k-bounded.
Modeling with Petri Nets
Properties: Safety
A Petri net is safe if it is 1-bounded.
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN. We can say,
that the transition tT is
L0-live (dead) iff we cannot fire it for any
sequence lL(PN, M0)
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN. We can say,
that the transition tT is
L1-live iff we are able to fire it at least
once for some sequence lL(PN, M0)
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN. We can say,
that the transition tT is
L2-live iff we are able to fire it k-times
(k>0) for some sequence lL(PN, M0)
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN. We can say,
that the transition tT is
L3-live iff transition t appears infinite
times inside some sequence lL(PN,
M0)
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN. We can say,
that the transition tT is
L4-live iff transition t is L1-live for every
marking MR(PN, M0)
Modeling with Petri Nets
Properties: Liveness
Let be given a Petri net PN.
We can say, that PN is Lk-live iff its
every trasnsition is Lk-live, where
k = 0, 1, 2, 3, 4.
A Petri net is said to be live if it is L4live. It means, that there is no deadlock
in the net.
Modeling with Petri Nets
Properties: Persistence
A Petri net PN is persistent iff for every
pair of enabled transitions, firing one of
those transitions doesn’t disable the
other one.
Modeling with Petri Nets
Analysing properties: Coverage tree
Coverage tree consists of the finite
number of tree-nodes and branches,
where:
• Tree-node: a marking MnR(PN, M0)
• Tree-branch: a transition tiT
If Mi+1 = (Mi, tj) then there is an arc inside
the tree from Mi to Mi+1 labeled by tj.
Modeling with Petri Nets
Analysing properties: Coverage tree
t1
M0
M1
t2
t1
M3
M2
t3
M4
Modeling with Petri Nets
Analysing properties: Coverage tree
Repeated sequence of transitions
p1
t2
t1
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
Repeated sequence of transitions
(1,0)
t1
(0,1)
t2
(1,0)
t1
(0,1) ...
Modeling with Petri Nets
Analysing properties: Coverage tree
Repeated sequence of transitions
(1,0)
t1
(0,1)
t2
(1,0)
t1
(0,1) ...
Stop –
repeated state !
Modeling with Petri Nets
Analysing properties: Coverage tree
Increasing number of tokens
p1
t1
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
Increasing number of tokens
(1,0)
t1
(1,1)
t1
(1,2)
t1
(1,3) ...
Modeling with Petri Nets
Analysing properties: Coverage tree
Increasing number of tokens
(1,0)
t1
(1,)
where  has the following properties:
 > n,   n =  ,    (for any nN)
Modeling with Petri Nets
Analysing properties: Coverage tree
Algorithm for creating the coverage tree:
Step 1: Let marking M0 be the root-node
of the tree. Mark it as a new-node.
Step 2: If exists any new-node do
the following:
Modeling with Petri Nets
Analysing properties: Coverage tree
Step 2.1: Take any new-node Mi.
Step 2.2: If there is a node Mj on the path
from M0 to Mi and Mj = Mi then
mark Mi as repeated-node and
return to step 2.
Step 2.3: If there is no enabled transition for
the node Mi, mark the node as the
end-node and return to step 2.
Modeling with Petri Nets
Analysing properties: Coverage tree
Step 2.4: For every enabled transition tj
related to the node Mi do
the following:
Step 2.4.1: Calculate Mi+1=(Mi, tj)
Step 2.4.2: if there exists marking Mk on the
path from M0 to Mi, such that:
Modeling with Petri Nets
Analysing properties: Coverage tree
pP : Mk(p)  Mi+1(p) 
pP : Mk(p) < Mi+1(p)  Mi+1(p) = 
Step 2.4.3: Insert node Mi+1 into the tree (it
is connected to the node Mi
via the branch labeled by tj)
and mark it as a new-node.
Return to step 2.
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t1
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t1
1, 1 ,0
t1
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t1
1,  ,0
t1
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t1
1,  ,0
t1
1,  ,0
t2
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t1
1,  ,0
t1
1,  ,0
t2
0,  ,1
t3
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t2
t1
1,  ,0
t1
1,  ,0
t2
0,  ,1
t3
0,  ,1
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t2
t1
0, 1 ,1
1,  ,0
t1
1,  ,0
t2
t3
0,  ,1
t3
0,  ,1
Modeling with Petri Nets
Analysing properties: Coverage tree
p1
t2
t1
p3
t3
p2
Modeling with Petri Nets
Analysing properties: Coverage tree
1, 0 ,0
t2
t1
0, 1 ,1
1,  ,0
t1
1,  ,0
t2
t3
0,  ,1
0, 0 ,1
t3
0,  ,1
Modeling with Petri Nets
Analysing properties: Coverage tree
The properties of the Petri net:
1. PN is bounded iff there is no  symbol
inside the coverage tree.
2. PN is safe iff only 0,1 digits are used
inside all the markings Mi.
Modeling with Petri Nets
Analysing properties: Coverage tree
The properties of the Petri net:
3. Transition tj is dead iff there is no arc
labeled by tj inside the coverage tree.
4. Marking Mi is reachable from the marking
M0 iff there is a node Mi in the coverage
tree and there exists a path inside the tree
from the node M0 to the node Mi.
Modeling with Petri Nets
Analysing properties: Example
Try to model the „4 readers and a writer”
problem using Petri nets. Then build the
coverage tree for this problem.
Timed - Petri Nets
Augmenting Petri nets with time :
• Assign time to transitions
• Assign time to places
• Time Basic Nets (TBN)
Timed - Petri Nets
TBN model
• Time offsets (Ti) related to the transitions
• Time-stamps related to the tokens
• Time intervals related to the transitions as
a function of time-stamps and time offsets
(time conditions)
Timed - Petri Nets
TBN model – firing the transition
Transition is enabled iff:
1. There are necessary number of tokens in
all input places
2. Time values in time interval related to the
given transition are greater than, or equal
to, the largest time stamp on input tokens.
Timed - Petri Nets
TBN model – firing the transition
Enabling time - the greatest value of the
tokens’ time-stamp in all the input places
related to the particular transition.
Firing period – modified time interval related
to the given transition (its values are
greater than, or equal to the enabling
time).
Timed - Petri Nets
TBN model – firing the transition
1. Only enabled transitions may fire. The
firing can happen somewhere during the
firing period.
2. The proper number of tokens is removed
from the input places and populated to the
output places with the time-stamp equal
to the firing time (the weights of arcs are
preserved during this process).
Timed - Petri Nets
TBN model – firing the transition
3. The other rules are exactly the same like
in the untimed Petri net.
Timed - Petri Nets
TBN model – example
A server :
There is given a server with 1 CPU. When a task
comes to the system, its execution must be started
within T3=5 seconds, otherwise the request is
discarded. The task’s processing time lasts from
T1=4 to T2=10 seconds. The system becomes ready
to accept a new task after T4=2 seconds.
Timed - Petri Nets
TBN model – example
t1
p1
t3
p3
p2
t2
Timed - Petri Nets
TBN model – example
request
start
discard
ready
being
processed
finish
Timed - Petri Nets
TBN model – example
Tran. Offsets
Input
places
Time conditions
t1
T3, T4
p1, p3
[time(p3)+T4, time(p1)+T3)
t2
T1, T2
p2
[time(p2)+T1, time(p2)+T2)
t3
T3
p1
[time(p1)+T3, )
time(p3) – time-stamp of the token stored in place p3.
Timed - Petri Nets
TBN model – example
t1
p1
t3
p3
p2
t2
Timed - Petri Nets
TBN model – example
time(p1) = 2 (let it be a time of the request)
time(p3) = 0
Enabled Time conditions
transitions
t1
[0+2, 2+5) = [2, 7)
t3
[2+5, ) = [7, )
Timed - Petri Nets
TBN model – example
t1
p1
t3
p3
p2
t2
Timed - Petri Nets
TBN model – example
After firing t1 at time 4 (the task is started)
time(p2) = 4
Enabled Time conditions
transitions
t2
[4+4, 4+10) = [8, 14)
Timed - Petri Nets
TBN model – example
t1
p1
t3
p3
p2
t2
Timed - Petri Nets
TBN model – example
After firing t2 at time 10 (the task is finished)
time(p3) = 10
No enabled transitions – system is waiting for
another request.
At last .....