WEBVTT
Kind: captions
Language: en
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Consider this example, the example which we
have considered earlier example 1.
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That means we have two places, place P on,
the other place P off.
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Initially, two tokens in the place P on, no
token in the place P off therefore, the marking
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will be a two tuple.
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Number of tokens in each place form a marking
therefore, the first place is P on, second
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place is P off with that assumption, suppose
if the first place is P on, second place is
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P off then the number of tokens are time zero
is 2, 0.
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After T failure firings the marking will be
1, 1.
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Again the T failure firing the marking will
be 0, 2.
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From 1, 1 if T repair fires then the marking
will be 2, 0.
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From 0, 2 if a T repair fires then the marking
will be 1, 1.
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Hence, the marking is the M tuple, number
of tokens in the place i at any given instant
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of time and the marking is reachable whenever
some sequence of a transition fires.
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And the reachability graph is a directed graph
whose nodes are the markings in the reachability
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set with directed arcs between the markings
represent the marking to marking transitions.
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Hence, the same example, the markings are
2, 0; 1,1; 0,2.
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The marking can be reachable from 2,0 to 1,1
by the transition T failure fires.
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The marking 1,1 is reachable to the marking
0,2 with the T failure firing.
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The marking 0,2 is reachable to the state
to the marking 1,1 by T repair fires.
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Similarly, 1,1 to 2,0 by firing T repair transition.
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So this is called reachability graph.
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Now we are moving extensions, arc extensions
in petri net.
00:03:03.470 --> 00:03:10.200
Till now we have considered very simple petri
nets.
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Now we are going for arc extensions.
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Both input and output arcs in the petri net
are assigned a weight or multiplicity or cardinality
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which is a natural number.
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If the multiplicity of an arc is not specified,
then it is taken to be unity.
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So in the previous examples, we have considered
as a multiplicity unity.
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The next extension in the petri net is inhibitor
arc.
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An inhibitor arc drawn from place to the transition
means that a transition cannot fire if the
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corresponding inhibitor place contains at
least as many tokens as the cardinality of
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the corresponding inhibitor arc.
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Usually, input arc and output arcs make a
transitions enabling and firing, removing
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the tokens depositing the tokens with the
multiplicity or cardinality.
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But the inhibitor arc drawn from place to
the transition means that the transition cannot
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fire if the corresponding inhibitor place
contains at least as many tokens as the cardinality
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of corresponding inhibitor arc.
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If there exist an inhibitor arc with the multiplicity
n between a place and a transition, and if
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the place has n or more tokens then the transitions
is inhibited even if it is enabled.
00:04:55.520 --> 00:05:05.189
The transition can enable based on the condition
through the input arcs but if there is a inhibitor
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arc then it may be inhibited based on the
number of tokens in the corresponding input
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place.
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Input arcs are represented graphically as
an arc ending in a small circle at the transition
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instead of arrowhead.
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Usually, input arcs as well as output arcs
drawn with arrow head but inhibitor arcs are
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represented graphically as an arc ending in
a small circle.
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We will see the example in this slide.
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In this example, we have a three places p1,
p2, p3, we have a one transition, one input
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arc, one output arc, one inhibitor arc.
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Here whenever the number is written next to
the arcs that means the multiplicity or cardinality
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of the output arc is number 3.
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If there is no number, natural number written
next to the arcs that means the default multiplicity
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is 1.
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Here also the default multiplicity is 1 that
means if one or more token in the place p1
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even the transition is enabled by the condition
through the input arcs for this place input
00:06:54.319 --> 00:07:05.340
place, this transition may not fire if one
or more tokens in the place p1.
00:07:05.340 --> 00:07:11.429
So here no token in the place p1 whereas one
token in the place p2, no token in the place
00:07:11.429 --> 00:07:24.969
p3 hence, the transition enables and then
fires by removing one token in the place p2
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and one token deposited in the place p3.
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Only the tokens will be removed from all the
input places which are connected, the places
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connected with the input arcs to the transition
and the tokens will be deposited to the all
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the output places which are connected from
transition to places through output arcs.
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So in this example, after the transition fires,
no token in the place p1, no token in the
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place p2 and three tokens in the place p3
because the multiplicity of the output arc
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is three.
00:08:08.469 --> 00:08:14.749
So even though one token is removed from the
place p2 because the multiplicity of the arc
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is three.
00:08:16.379 --> 00:08:24.459
Hence, three tokens will be multiplicity is
3, therefore, three tokens will be deposited
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at same time with the place p3.
00:08:31.869 --> 00:08:41.180
Now we are moving into the extension of petri
nets into stochastic petri nets.
00:08:41.180 --> 00:08:49.019
Petri nets are extended by associating time
with the firing of transitions, resulting
00:08:49.019 --> 00:08:52.300
in timed petri nets.
00:08:52.300 --> 00:09:00.540
A special case of timed petri nets is stochastic
petri nets.
00:09:00.540 --> 00:09:11.830
In other words SPN where the firing times
are exponential distributed.
00:09:11.830 --> 00:09:19.339
So whenever the firing times of all transitions
are exponential distribution then the corresponding
00:09:19.339 --> 00:09:26.759
timed petri nets are called stochastic petri
nets.
00:09:26.759 --> 00:09:36.759
Every petri net one can get the reachability
graph.
00:09:36.759 --> 00:09:43.589
The underlying reachability graph of a stochastic
petri net is isomorphic to a Continuous-time
00:09:43.589 --> 00:09:46.160
Markov Chain.
00:09:46.160 --> 00:09:55.959
For a stochastic petri net the underlying
reachability graph is isomorphic to a Continuous-time
00:09:55.959 --> 00:09:57.920
Markov Chain.
00:09:57.920 --> 00:10:04.350
Let us see through the examples.
00:10:04.350 --> 00:10:13.740
When two molecules of hydrogen is combined
with one molecule of oxygen then two molecules
00:10:13.740 --> 00:10:17.110
of water is formed.
00:10:17.110 --> 00:10:32.040
The balanced equation is written; the balanced
chemical equation is given by 2H2 plus O2
00:10:32.040 --> 00:10:39.089
gives 2 times H2 O.
00:10:39.089 --> 00:10:43.550
So this is the balanced chemical equation.
00:10:43.550 --> 00:10:53.480
For this scenario, we have made three places,
P hydrogen, P oxygen, P water and we have
00:10:53.480 --> 00:10:59.750
one transition that is called a T reaction.
00:10:59.750 --> 00:11:06.279
The multiplicity of an input arc from the
place of P hydrogen to the T reaction that
00:11:06.279 --> 00:11:17.860
is two whereas the multiplicity of the arc
from P oxygen to the T reaction that is one.
00:11:17.860 --> 00:11:26.720
Whereas the multiplicity of an output arc
from the transition to the place P water that
00:11:26.720 --> 00:11:30.070
is two.
00:11:30.070 --> 00:11:41.589
On firing of the transition T reaction, two
tokens because the multiplicity is two.
00:11:41.589 --> 00:11:45.970
Two tokens will be removed from the place
P hydrogen.
00:11:45.970 --> 00:11:51.670
The multiplicity is one therefore; one token
will be removed from the place P oxygen.
00:11:51.670 --> 00:12:04.399
Since, it is a rectangle bar that means the
timed transition follows the time of timed
00:12:04.399 --> 00:12:11.019
transition follows exponential distribution.
00:12:11.019 --> 00:12:16.480
So two tokens will be removed from the place
P hydrogen, one token will be removed from
00:12:16.480 --> 00:12:30.699
the place P oxygen on firing of the reaction
T reaction, two tokens will be deposited in
00:12:30.699 --> 00:12:34.420
the place P water because the multiplicity
is two.
00:12:34.420 --> 00:12:42.379
And here we made the assumption, the reaction
time is exponential distribution, hence we
00:12:42.379 --> 00:12:54.350
made a stochastic petri net
and we make the model using stochastic petri
00:12:54.350 --> 00:12:56.519
net of this chemical reaction.