Enforcing Liveness in S
3
PR Nets by Specialization of Resources
Asaftei Timotei
1
and Jos
´
e-Manuel Colom
2
1
Department of Automatic Control and Industrial Informatics, Technical University “Gh. Asachi”, Iasi, Romania
2
Arag
´
on Insitute of Research Engineering (I3A), University of Zaragoza, Zaragoza, Spain
Keywords:
S
3
PR, Liveness, Pruning Graph, Minimal Siphons.
Abstract:
Siphon-based control methods are often used in deadlock prevention in Petri nets models for resource alloca-
tion systems caused by the shared resources. In this article we used the properties of the minimal siphons of
the S
3
PR and Pruning Graph to develop a new method to prevent the deadlock. This method consists in the
increasing of the number of copies of a given resource type, and in the splitting of the total copies of resources
in two new types of resources that will be used in a private way for each one of the two disjoint groups in which
the old holder places of the original resource type are divided. The algorithm uses only structural information
of the net.
1 INTRODUCTION
A Flexible Manufacturing System (FMS) contains
multiple concurrent flows of job processes that make
different products in the same time. To do this the
FMS use shared resources, and these shared resources
can cause the deadlock. In order to deal with this
deadlock problem, many article was written trying to
give a solution to this deadlock prevention problem
(Chao, 2010),(Chao and Liu, 2012).
The use of Petri Net models has been proven very
convenient for the formal modeling and further anal-
ysis and correction of the model of the Resource Al-
location Systems. In particular, two important sub-
classes of Petri Nets have been defined for modeling a
wide class of Resource Allocation Systems: the S3PR
nets (Ezpeleta et al., 1995) and the S4PR nets (Tri-
cas, 2003). For these classes of nets characterizations
of nonliveness have been obtained that are based in
the existence of certain structural objects named bad
siphons that are the structural cause of the appearing
of the deadlock states. A bad siphon is a set of places
such that the set of input transitions to these places is a
subset of the output transitions of these places. There-
fore, if this set of places becomes empty of tokens, it
remains empty forever and all output transitions are
dead.
In this article a new algorithm which prevent dead-
locks in S
3
PR will be developed. For this algorithm
we have to construct the Pruning Graph of the net, and
after that to split some siphons such that will does not
exist any more strongly connected components. For
this article some of the nodes of the Pruning Graph
will have special properties because we consider the
case with un-replicable resources. In (Asaftei and
Colom, 2013) we considered that all resources in the
system are replicable; i.e. a replicable resources has
the property that it can be replicate (will appear a ex-
act copy of this resource). For particular situations,
because of the lack of space, money or even time, nor
all resources in a systems are replicable. Neverthe-
less, in the logistic of an hospital, an un-replicable
resource is common: the hospital do not has enough
doctors or surgery rooms or medical equipment.
The article is organized as follows. Section 2 are
introduced the S
3
PR nets. Section 3 presents the new
RAS approach and in Section 4 the main results of
the paper are presented: the algorithm for splitting the
siphons and the liveness enforcing algorithm. Finally
some conclusions are given in section 4.
2 DEFINITIONS AND PREVIOUS
RESULTS ON S
3
PR
In this section we recall the basic definitions, nota-
tions and previous results about the Petri Net models
that we use to study the Resource Allocation Systems.
These nets belongs to the S
3
PR class. We assume that
the reader is familiar with the basic concepts and no-
tations of Petri Nets. Next, the definition of the S
3
PR
170
Timotei A. and Colom J..
Enforcing Liveness in S3PR Nets by Specialization of Resources.
DOI: 10.5220/0004275603180321
In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 318-321
ISBN: 978-989-8565-40-2
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

nets will be presented.
Definition 1. ((Ezpeleta et al., 1995)) An S
3
PR is de-
fined recursively as follows:
1. An S
2
PR is an S
3
PR
2. Let N
i
= hP
i
∪ P
0
i
∪ P
R
i
, F
i
i, i ∈ {1, 2} be two S
3
PR
so that (P
1
∪ P
0
1
) ∩ (P
2
∪ P
0
2
) =
/
0, P
R
1
∩ P
R
2
=
P
C
(6=
/
0) and T
1
∩ T
2
=
/
0 (in which case we will
say that N
1
and N
2
are two composable S
3
PR);
then, the net N = hP ∪ {P
0
} ∪ P
R
, T, Fi resulting
of the composition of N
1
and N
2
via P
C
(denoted
as N = N
1
◦ N
2
) defined as follows:
a. P = P
1
∩ P
2
,
b. P
0
= P
0
1
∩ P
0
2
,
c. P
R
= P
R
1
∩ P
R
2
,
d. T = T
1
∩ T
2
,
e. F = F
1
∩ F
2
,
is also an S
3
PR.
We will consider that for an acceptable marking,
the nets have at least one token in each idle place and
at least one token in every resource (there is at least
one copy of every resource in the system). In S
3
PR
there is no direct characterization of the liveness prop-
erty but some known results present characterizations
of the non-liveness of a net. In (Tricas, 2003) there are
presented two theorems characterizing non-liveness
of S
4
PR nets. These theorems are almost the same
for a S
3
PR only with some minor modifications.
A siphon in a Petri net is a structural object D
where D ⊆ P with the property:
•
D ⊆ D
•
. A siphon
is a non-empty set of places which are connected via
transitions. One property of the siphons is that once
there is no token inside the places, it is not possible to
gain tokens coming from the rest of the net. We say
that a siphon S is minimal if it does not contain an-
other siphon as a proper subset. In a minimal siphon
it must exists at least two places; otherwise the struc-
ture remained can not be considered a siphon.
For the algorithms that will be presented in this ar-
ticle, the concept of pruning relation between siphons
will be used. The siphons we are working in this ar-
ticle are the minimal siphons. In (Cano et al., 2012)
it is presented the pruning relation and the way the
Pruning Graph is built. In (Asaftei and Colom, 2013)
it is proven that the concepts of pruning and Pruning
Graph can be applied to S
3
PR nets too.
3 THE NEW APPROACH FOR
LIVENESS ENFORCING
The approach presented in this paper share with the
previously presented liveness enforcing techniques
that is based in the increasing of the number of avail-
able resources at the initial marking (Wang et al.,
2010). This family of approaches is consistent with
the fact that the deadlock states appearing in this kind
of RAS systems are produced by problems in the al-
location of resources. Taking into account that the re-
sources are used in a conservative way and the num-
ber of customers is bounded by the initial marking
of the idle places of the net, it is possible to increase
the number of resources in a quantity such that the re-
sources are not a constraint in the execution sequences
of the system. In other words, the resource places be-
come implicit places that can be removed from the
net.
Nevertheless, this strategy based in the indiscrim-
inate increase of resources, fails when the number of
customers is increased. That is, the reached solution
about the number of resources to make live the net for
a given number of customers, must be revised if this
number of customers change. The existence of this
latent problem that manifests itself with the increase
of customers is that the structural problems of the net
remain. These structural problems are the existence
of bad siphons that can be emptied after the execution
of some sequences.
Deadlock prevention (Hou et al., 2010) techniques
based in the addition of monitor places avoid this
problem. In effect, monitor places are places, that can
be interpreted as virtual resources of the system (in
fact a new type of resources), that are designed with
the goal of constraint the possible firing sequences
that can lead to empty the siphon. In this sense, these
techniques improves the techniques based in an in-
discriminate increasing of resources in two different
aspects. The first one is that the computation of the
monitor places is done from the structural defect of
the Petri Net giving rise to the appearing of dead-
locks: the bad siphons of the net. The second as-
pect, is that the solution is based in the addition of
constraints avoiding the emptiness of the siphon, i.e.
increasing the number of customers, the constraints
work well because the siphon cannot be emptied. But
in this case there are problems with the concurrency
that can be obtained in the system. In fact the moni-
tors sequentialize the processes sharing the resources
implied in the deadlock states, i.e. the concurrency
is reduced. Moreover, you can increase the number of
original resources but you cannot improve, in general,
the concurrency of the system and its corresponding
performance figures, because the constraints imposed
by the monitors.
The approach proposed in this paper save the two
previous drawbacks obtaining advantages of the two
approaches. That is, we increase the number of re-
EnforcingLivenessinS3PRNetsbySpecializationofResources-RationaleandFormalisms
171

sources in order to enforce the liveness without the
reduction of the concurrency of the system. Neverthe-
less, in order to propose a solution independent of the
number of customers, a new type of resource is intro-
duced (as in the case of the monitors) is such a way
that the original siphons are broken and then disap-
pear solving the structural defect of the original Petri
Net. In essence, the steps to be done are:
1. Identification of a resource belonging to a bad
siphon that can give rise to the appearing of a
deadlock state.
2. Classification of the holder places of the selected
resources in two disjoint groups. That is, the
method proceeds increasing the number of re-
sources but these new resources will be used in
a private way by only one of the classified groups.
This is the idea of specialization of the new re-
sources.
3. Definition of two new resource places and remov-
ing the old one. Each one of these new resource
places has one the previous sets of holder places
and they are connected to the input and output
transitions of the holder places in the same way
than the original resource place. Observe that now
there exist two different types of resources where
previously existed only one type of resources. The
copies of each type of resource are used in a
private way by the corresponding set of holder
places.
4. Introduction of the same number of tokens in each
new resource place than the number of copies of
resources in the original resource place.
The method can be considered as structural be-
cause the splitting of the original resource place into
two new places is done in such a way that the siphon is
broken or at least, in what concerns to the considered
resource place the siphon has been reduced. There-
fore, proceeding with all bad siphons in the same way
we can obtain a net without bad siphons and then by
the non-liveness theorem of S
3
PR nets, the admissibly
marked net must be live: the method enforces live-
ness.
Observe, that the method increases the number of
resources although they are used in a private way with
respect to the way they are used in the original net.
This means that the method does no cut bad markings,
the method introduces new states allowing to go out
from the old deadlock states. That is, the old deadlock
states are reached but now there are enough resources
(that they are used in a private way) to go out to a new
state saving the bad scenario. In classical control the-
ory, probably, this strategy is unsatisfactory because
the strategy there is to forbid all bad states and all
states that inevitably lead to a bad state. Nevertheless,
in other application domains as in the hospital logis-
tics the important goal is to reach a complete treat-
ment for the patient, adding all needed resources to
reach this goal.
For this new approach of liveness enforcing we
need an algorithm which describes how the splitting
is done in the S
3
PR. The splitting is needed in the
S
3
PR so there will appear new resources. With these
resources the deadlock state is avoided. The idea is
that after the construction of the initial Pruning Graph
of the S
3
PR, we need to find and eliminate all circuits
from the net. An elimination in the Pruning Graph
will be equivalent with a new resource in the S
3
PR
(resource added by splitting an existing resources).
Next, an algorithm of splitting the siphon will be
presented. For this algorithm we consider r ∈ P
R
is
the resource we are working on.
Algorithm 1: The splitting of a resource.
Input: resource r which belongs at least to one cir-
cuit, N, the Pruning Graph of N;
Output: N
0
, r is substituted with r
1
and r
2
;
1. Delete r and all arcs between r and the transitions
t
i
, t
i
∈ {r
•
∪
•
r};
2. Add places r
1
and r
2
in S
3
PR;
3. Connect place r
1
with the net such that all the in-
puts of v
i
are the inputs of v
1
i
: ∀(s, r) ∈ E and
∀p ∈ P
S
such that (t, p) ∈ L((s, r)) add an arc from
r
1
to all t ∈
•
p and an arc from each t ∈ p
•
to r
1
;
4. Connect place r
2
with the net such that all the out-
puts of v
i
are the outputs of v
2
i
: ∀(r, s) ∈ E and
∀p ∈ P
S
such that (t, p) ∈ L((r, s)) add an arc from
r
2
to all t ∈
•
p and an arc from each t ∈ p
•
to r
2
;
For the Algorithm 2 we use the fact that the multi-
plicity of the arcs from the Pruning Graph will be the
same. Using this algorithm, all strongly connected
components will be eliminated. One way for a graph
to became acyclic is to use reversed edges. The prob-
lem of finding a set of smallest number of feedback
edges is a problem called minimum feedback arc set
problem. This is used in our algorithm when we com-
pute the minimal set of arcs.
From the the two algorithms above, it can be seen
that if the pruning graph is acyclic then the S
3
PR net
is live. But in the most cases the pruning graph has
cycles(Strongly Connected Components) which have
to disappear. This can be done by removing(or change
the direction) arcs from the pruning graph. Once the
pruning graph is modified, some changes have to be
done in the S
3
PR net according to Algorithm 1.
ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems
172

Figure 1: The evolution of the Pruning Graph according to Algorithm 1 and Algorithm 2.
Algorithm 2: The liveness enforcing approach.
Input: the Pruning Graph - PG, N, the nodes V, the
edges E;
Output: the new Pruning Graph - PG
0
, N
0
, V
0
;
1. Compute the pruning graph (PG) G(V, E) of the
N;
2. While G contains some cycles do
3. Compute a set of arcs W S = e
j
using a mini-
mum feedback arc set algorithm;
4. While W S 6= 0 do
5. Take o node v
i
from V which is connected to
e
j
in PG;
6. Modify N according to the Algorithm1;
7. Remove all arcs between v
i
and the nodes
next to it in PG;
8. Remove v
i
and put v
1
i
and v
2
i
;
9. W S = W S \ {e
j
}
10. Endwhile
11. Endwhile
The Pruning Graph of the the S
3
PR net from Fig-
ure 1 from (Asaftei and Colom, 2013) is depicted in
Figure 1. The S
3
PR net has three siphons. The goal of
the approach is to split the siphons. The order we split
the siphons is random (until now it does not exist an
algorithm to describe the order the siphons have to be
split). In this example we consider that first we split
siphon S
1
and after that we split siphon S
3
. After these
two splits, the pruning graph become acyclic, so the
algorithm stops. This is not the only solution of this
problem. We can split siphon S
3
and after that siphon
S
1
or siphon S
2
and S
3
.
4 CONCLUSIONS
Exploiting the results of a previous work, where the
minimal siphons has to be diminished, a new dead-
lock prevention method was presented. This is done
by a set of process places that can be computed from a
pruning relation on the minimal siphons with only one
resource. This pruning relations are represented by a
pruning graph. The method consists in two phases:
one which work with the S
3
PR and one which work
with its pruning graph. The main idea is to mod-
ify (by splitting resource places) the petri net so in
the pruning graph will not left any strongly connected
components. The main advantage for this method is
that we work with high level objects (siphons). Using
these objects, we improve the time and space needed
to solve the deadlock problem.
REFERENCES
Asaftei, T. and Colom, J. (2013). Enforcing liveness in s3pr
nets by specialization of resources. In 2nd Int. Conf.
on Operation Research and Enterprise Syst.
Cano, E., Rovetto, C., and Colom, J. (2012). An algorithm
to compute the minimal siphons in s
4
pr nets. In Dis-
crete Event Dynamic Systems. Springer Verlag. Pub-
lished online.
Chao, D. Y. (2010). A simple modification of deadlock pre-
vention policy of s3pr based on elementary siphons.
In Transactions of the Institute of Measurement and
Control, volume 33, pages 93–115.
Chao, D. Y. and Liu, G. J. (2012). A simple suboptimal
siphon-based control model of a well-known s3pr. In
Asian Journal of Control, volume 14, pages 163–172.
Ezpeleta, J., Colom, J.-M., and Martinez, J. (1995). A petri
net based deadlock prevention policy for flexible man-
ufacturing systems. IEEE Transactions on Robotics
and Automation, 11:173–184.
Hou, Y., Liu, D., Li, Z., and Zhao, M. (2010). Deadlock pre-
vention using divide-and-conquer strategy for ws3pr.
In International Conference on Mechatronics and Au-
tomation (ICMA), pages 1635–1640.
Tricas, F. (2003). Deadlock analysis, prevention and avoid-
ance in sequential resource allocation systems. PhD
thesis, University of Zaragoza, Zaragoza.
Wang, S., Wang, C., and Yu, Y. (2010). An algorithm to find
the condition on initial markings of resource places
and job places for liveness of s3pmr. In 8th IEEE
International Conference on Control and Automation,
pages 1445–1449.
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