Uncertainty Measure of Process Models using Entropy and Petri Nets
Martin Ibl
Institute of System Engineering and Informatics, University of Pardubice, Pardubice, Czech Republic
Keywords: Petri Nets, Entropy, Behavioural Analysis, Process Measure.
Abstract: In recent years, many measures of process models have been proposed to predict or quantify the number of
specific properties. These properties may include readability, complexity, cohesion or uncertainty of process
models. The content of this work is to propose a method that allows the measurement of uncertainty in the
process models, which can be expressed in the form of a Petri net. The actual method works by mapping the
set of all reachable marking of Petri net to Markov chain and subsequent quantification of steady-state
probabilities of its states. Uncertainty is then quantified as the entropy of states in the Markov chain.
Uncertainty can also be expressed as a percentage of the calculated entropy to the maximum entropy of a
Petri net.
1 INTRODUCTION
Currently, there are a large number of different
modelling languages, which serve to describe
business processes. They differ in their notation,
complexity, mathematical foundation and other
characteristics. From a comparative perspective, in
recent years, many measures of process models have
been proposed that aims to create a variety of
metrics for analysing process models in terms of
complexity (Lassen and van der Aalst, 2009);
(Rolón et al., 2009), uncertainty (Jung et al., 2011)
or cohesion (Reijers and Vanderfeesten, 2004).
These metrics are then used for various purposes,
such as evaluation of user-friendliness,
understandability, usability, maintainability and
other (González et al., 2010). In the following is
proposed the approach, which allow the analysis and
quantification of the uncertainty of any process
model, which has been modelled using the classic
P/T (Place/Transition) Petri net, or can be remap into
it (van der Aalst, 1998). Quantification of the
uncertainty of any process model implies the
predicted behaviour of the modelled process and
therefore its degree of predictability. Reducing
uncertainty in process models can lead to better
predictability of process behaviour and also improve
managerial efficiency.
Petri nets are a suitable tool for modelling
discrete event dynamic systems which feature
concurrency, parallelism and synchronization. Their
main advantage is the ability to precisely verify the
assumptions imposed on the model. Petri nets have
been defined by Carl Adam Petri in 1962 (Petri,
1962) and since then, their development evolves in
many directions. One direction is to define a new
model features that extend the verification power of
Petri nets. These are primarily properties of Petri
nets such as liveness, boundedness, reachability and
many more. Most of these properties also require a
number of assumptions that restrict the definition of
Petri net. Another direction of development is
expanding definition of Petri nets by adding new
elements to refine and simplify modelling (but
mostly with a lower degree of formality). Examples
are timed and stochastic Petri nets, which allow
refining the individual state changes, taking into
account time-consumption (deterministically or
stochastically). Another example is coloured Petri
nets, which combine the basic petri net with another
modelling language, thus dramatically expanding
(and mainly simplify) modelling capabilities Petri
nets. The main drawback of this second direction is
limited verification options.
The aim of this work is to define a method that
allows quantifying the uncertainty of Petri net
models. This objective is achieved using the
concepts of information theory (Shannon´s entropy
(Shannon, 1948)) and stochastic processes (Markov
chains).
This work is divided into 5 sections. The second
section presents the basic definition of Petri nets and
542
Ibl M..
Uncertainty Measure of Process Models using Entropy and Petri Nets.
DOI: 10.5220/0004584205420547
In Proceedings of the 8th International Joint Conference on Software Technologies (ICSOFT-PT-2013), pages 542-547
ISBN: 978-989-8565-68-6
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
other terms that relate to the issue. The third chapter
presents the basic issues of Markov chains
associated with the determination of steady-state
probabilities. The fourth chapter contains a
definition of Shannon entropy and the method of
uncertainty calculation over arbitrary Petri net. The
fifth section aims to illustrate the issues defined in
the previous section on simple example. The sixth
section discusses the advantages and disadvantages
of the presented method. The last section concludes
this paper and proposes possibilities for further
expansion of this issue.
2 PETRI NETS
Currently, there are a number of basic definitions of
petri nets, which are distinguished by its formality
(restrictions and verification force). The following is
the general definition of P/T (Place/Transition) Petri
nets, which allows quantifying the edges (arcs) with
positive integers.
Definition 2.1: Generalized P/T Petri net is a 5-
tuple, ,,,,
where:
,
,
,
,…,
– a finite set of places,
,
,
,…,
– a finite set of transitions,
∩∅ – places and transitions are
mutually disjoint sets,
⊆⨯∪⨯ – a set of edges
(arcs), defined as a subset of the set of all
possible connections,
:→
– a weight function, defines the
multiplicity of edges (arcs),
:→
– an initial marking.
Such a definition does not contain any implicit
restriction in terms of capacity of individual places.
If it is required to model capacity constraints on
some subset of places it is possible to use the so-
called complementary-place transformation to adjust
net as required (Murata, 1989). In practice, it is
advantageous to specify a priori capacity of
individual sites and thus simplifying subsequent
analysis of model (eliminates the problem of infinite
capacity).
Definition 2.2: Capacity P/T Petri net is a 6-tuple,
,,,,,
where:
,
,
,
,…,
– a finite set of places,
,
,
,…,
– a finite set of transitions,
∩∅ – places and transitions are
mutually disjoint sets,
⊆⨯∪⨯ – a set of edges,
defined as a subset of the set of all possible
connections,
:→
– a weight function, defines the
multiplicity of edges,
∶→
– capacities of places,
:→
– an initial marking.
Definition 2.3: Marking of Petri net
Let ,,,,,
is a Petri net.
Map:→
, is called marking of Petri net PN.
Marking represents the state of the network after
execution a specific number of steps, i.e. the firing a
specific number of enabled transitions. If a transition
is enabled (or not) depends on the net structure and
the actual marking.
Definition 2.4: Pre-set, Post-set
Let ,,,,,
is a Petri net. Pre-
sets and post-sets are defined as:
•
|,∈ – the set of input
transitions of ,
•
|,∈ – the set of input places
of,
•
|,∈ – the set of output
transitions of,
•
|,∈ – the set of output places
of.
Definition 2.5: Enabled transition
Let ,,,,,
is a Petri net.
Transition ∈ is called enabled withmarkingM
M‐enabled,if
∀∈
•
:
,
∀∈
•
:
,
Definition 2.6: Next marking
Let ,,,,,
is a Petri net and
is its marking. If a transition ∈ is enabled at
marking, then by its execution is obtained next
marking´, which is defined as follows:
∀∈:´
,
,∈
•
\
•
,
,∈
•
\
•
,
,
,∈
•
∩
•
The situation that the transition changes the
marking to´, is usually expressed as
´.
Definition 2.7: Sequence of transitions, reachability
Let ,,,,,
is a Petri net.
Sequence of transitions is the sequence of enabled
UncertaintyMeasureofProcessModelsusingEntropyandPetriNets
543
transition that lead from marking to another
marking´. This situation is denoted as
´. A
marking for which there is a sequence of transitions
from the initial marking is called reachable marking.
Definition 2.8: The set of all reachable marking
Let ,,,,,
is a Petri net and
is its marking. The set of all possible markings
reachable from initial marking
in a Petri net
is denoted by ,
or simply
.
⋯
|
|
⋯
|
|
⋮⋮⋱⋮
⋯
|
|
Definition 2.9: Boundedness
Let ,,,,,
is a Petri net.
Place ∈ is called -bounded if:
∃∈
:∀∈
:
Place ∈ is called bounded, if it is k-bounded for
some∈
. If every place in PN is bounded, then
this net is called bounded Petri net.
Definition 2.10: Live marking, live net
Let ,,,,,
is Petri net.
Marking ∈
is live, if ∀∈ exist some
marking
∈
such that transition is
-
enabled. If ∀∈
is live, then PN is live.
3 PROBABILITY OF MARKINGS
AND MARKOV CHAINS
The set of all reachable markings can be expressed
in terms of Markov chains. For the purposes of
defining the steady-state probability of each marking
∈
is necessary to define the transition
matrix.
Definition 3.1: Transition matrix
Let ,,,,,
is a Petri net and
its reachability set. Transition matrix A of
Petri net PN is defined as:
:
→0,1
Where values are made according following rule and
the matrix form right stochastic matrix:
,
0∄
∈
:
1
|
|
∃
∈
:
Where |
| represents the number of marking that
are reachable from
. In this way each branching in
the state space (graph) assigned uniform
probabilities between different paths. However,
explicitly chosen values of probabilities for various
branches can be used as well, subject to the
condition
∑
,
1
|
|
(right stochastic matrix).
Definition 3.2: Steady-state probabilities
Let ,,,,,
is a Petri net and
is its transition matrix. Steady-state distribution
vector is defined as left eigenvector of transition
matrix:
Vector then represents the probabilities of all
markings from
:
Pr
Pr
⋮
Pr
|
|
Definition 3.3: Long term probability of marking
∈
is defined as a corresponding element of
vector :
Pr
The probability of marking M can be seen as a joint
probability of markings of individual places:
Pr
,
,…,
When examining the steady-state probabilities it is
appropriate to place emphasis on liveness of
analysing model, since each dead marking of Petri
net corresponds to absorb state in terms of Markov
chains. Each absorption state can always occur, i.e.
its probability equal 1 and thus all live markings
have probability equal 0. This would lead to a
deterministic model without any uncertainty.
4 ENTROPY
Entropy can measure the amount of disorder, which
is associated with a random variable.
Definition 4.1: The entropy of the random variable
X is defined as:
log
With the assumption0∙log
0
≡0.
Definition 4.2: Joint entropy
The joint entropy of two discrete random
variables and is defined as
ICSOFT2013-8thInternationalJointConferenceonSoftwareTechnologies
544
,
,
log
,
Where and are particular values of and,
respectively Pr, is the probability of
these values occurring together and the general form
for random variables:
,..,
…
,..,
log
,…,
Definition 4.3: Entropy of Petri net
Let ,,,,,
is a Petri net and
is a vector of steady-state probabilities
Pr
,
∈
. Entropy of is defined as
log
|
|
Definition 4.4: Uncertainty of Petri net
Let ,,,,,
is a Petri net and
its entropy. Uncertainty of is defined as
log
|
|
Uncertainty value is then located in the interval
<0,1>, where 0 stands for fully deterministic model
and 1for absolute chaotic model. The more is the
uncertainty value close to 1, the less is predictable
the behaviour of the model.
5 EXAMPLE OF SIMPLE MODEL
As a simple example, consider a Petri net, which is
composed of 5 places and 5 transitions, see Figure 1.
The model contains some typical elements that are
abundant in classic process models. These elements
are for instance sequence (transition T4), AND-split
(transition T1), XOR (transition T2 and T3) and
cycle (transition T5). For more information on the
mapping of these (and other) elements into Petri net
can be found in (Jung et al., 2011).
Figure 1: Petri net example.
The set of all reachable markings
of the
Petri net contains five markings:
10000
01010
01100
00101
00011
The corresponding state space (graph) is shown in
Figure 2.
Figure 2: State space.
This state space corresponds to Markov chain, which
generates the following transition matrix:
01000
001/21/20
00001
00001
10000
The solution of this matrix is a vector of steady-state
probabilities:
0.250
0.250
0.125
0.125
0.250
It is then possible to quantify the entropy of the
presented example:
3∗0.250log
0.2502
∗0.125log
0.1252.25
For this example the upper bound of uncertainty
islog
52.3219. Degree of uncertainty itself can
be quantified as a percentage of the calculated
entropy to maximum entropy, according to the
UncertaintyMeasureofProcessModelsusingEntropyandPetriNets
545
formula /log
, i.e. 2.25/2.3219 = 0.969.
This result can be loosely interpreted as the fact that
uncertainty of sample Petri net reaches 96.9%,
which can be classified as a high degree of
uncertainty. And with that pose problems, such as
low readability, interpretability, predictability, and
other indicators.
6 DISCUSSION
Measurement of uncertainty in process models can
be an indicator for reasoning about the explanatory
power of these models. Mainly the ability to support
different managerial decisions associated with the
prediction of the system behaviour under defined
probabilities (transition matrix). Degree of
uncertainty is usually influenced by a number of
elements that contains a process model. These
elements include OR, XOR, AND, LOOP. Their
arrangement in the process model then implies its
uncertainty. Finally, the main influences for the
amount of uncertainty in the process model are the
probabilities associated witch each branching path
(e.g. OR-split). Another approach of uncertainty
measurement, which uses quantification of
individual substructures in model at different levels
of abstraction, is defined in (Jung et al., 2011). That
approach measures the structural uncertainty of
process models, depending on the location of the
above-mentioned components (OR, AND, etc.).
Approach defined in this work quantifies uncertainty
using concepts of Petri nets with relation to Markov
chains. This approach also allows the measurement
uncertainty in any model that can be modelled as a
Petri net. Thereby is for instance possible to use
multiple tokens in the model or implicitly defined
multiplicity of edges (arcs).
Advantages of this Approach
Universal metric for measuring the uncertainty of
process models that can be modelled using Petri
nets.
The possibility of using the verification features
of Petri nets.
Clearly defined boundaries of uncertainty
(interval0,1.
Possibility to set specific probabilities for
branching in the model.
Disadvantages of this Approach
Fundamental deficiencies of Petri nets in general,
i.e. state explosion, restrictions based on
definitions, etc.
7 CONCLUSIONS AND FUTURE
WORK
In this paper was defined method for calculating the
uncertainty of any process model, which can be
modelled by Petri net. The actual uncertainty
quantification is based on the measurement of
entropy on the set of all reachable marking of Petri
net and its steady-state probabilities. On the prime
example is presented the calculation of the
uncertainty.
One of the relative weaknesses of this approach
is non-implicit definition of branching probabilities,
i.e. the need to explicitly define these probabilities in
the transition matrix (or not consider probabilities at
all). Therefore, the future research will be focused
on defining this method using stochastic Petri nets,
which implicitly define probability rates of
transitions in its definition.
ACKNOWLEDGEMENTS
This work was supported by the project
No. CZ.1.07/2.2.00/28.032 Innovation and support
of doctoral study program (INDOP), financed from
EU and Czech Republic funds.
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