A petri net model for hardware software codesign

Design Automation for Embedded Systems, 4, 243–310 (1999)
c 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

A Petri Net Model for Hardware/Software Codesign
PAULO MACIEL
Departamento de Inform´ tica Universidade de Pernambuco CEP. 50732-970, Recife, Brazil
a

EDNA BARROS
Departamento de Inform´ tica Universidade de Pernambuco CEP. 50732-970, Recife, Brazil
a

WOLFGANG ROSENSTIEL
Fakultaet fuer Informatik Universitaet Tuebingen D-7207 Tuebingen, Germany

Abstract. This work presents Petri nets as an intermediate model for hardware/software codesign. The main
reason of using of Petri nets is to provide a model that allows for formal qualitative and quantitative analysis
in order to perform hardware/software partitioning. Petri nets as an intermediate model allows one to analyze
properties of the specification and formally compute performance indices which are used in the partitioning
process. This paper highlights methods of computing load balance, mutual exclusion degree and communication
cost of behavioral description in order to perform the initial allocation and the partitioning. This work is also
devoted to describing a method for estimating hardware area, and it also presents an overview of the general
partitioning method considering multiple software components.

Keywords: Petri nets, hardware/software codesign, quantitative analysis, estimation

1. Introduction

Because of the growing complexity of digital systems and the availability of technologies,
nowadays many systems are mixed hardware/software systems. Hardware/software code-
sign is the design of systems composed of two kinds of components: application specific
components (often referred to as hardware) and general programmable ones (often referred
to as software).
Although such systems have been designed ever since hardware and software first came
into being, there is a lack of CAD tools to support the development of such heterogeneous
systems. The progress obtained by the CAD tools at the level of algorithm synthesis,
the advance in some key enabling technologies, the increasing diversity and complexity
of applications employing embedded systems, and the need for decreasing the costs of
designing and testing such systems all make techniques for supporting hardware/software
codesign an important research topic [21, 22, 23, 24, 25].
The hardware/software codesign problem consists in implementing a given system func-
tionally in a set of interconnected hardware and software components, taking into account
design constraints.
In the case where an implementation on a microprocessor (software component), cheap
programmable component, does not meet the timing constraints [2], specific hardware
devices must be implemented. On the other hand, to keep cost down, an implementation of
components in software should be considered.

244 MACIEL, BARROS, AND ROSENSTIEL

Furthermore, a short time-to-market is an important factor. The delay in product launching
causes serious profit reductions, since it is much simpler to sell a product if you have little
or no competition. It means that facilitating the re-use of previous designs, faster design
exploration, qualitative analysis/verification in an early phase of the design, prototyping, and
the reduction of the required time-to-test reduce the overall time required from a specification
to the final product.
One of the main tasks when implementing such systems is the partitioning of the descrip-
tion. Some partitioning approaches have been proposed by De Micheli [20], Ernst [18],
Wolf [4] and Barros [17].
The hardware/software cosynthesis method developed by De Micheli et al. considers that
the system-level functionality is specified with Hardware as a set of interacting processes.
Vulcan-II partitions the system into portions to be implemented either as dedicated hardware
modules or as a sequence of instructions on processors. This choice must be based on the
feasibility of satisfaction of externally imposed data-rate constraints. The partitioning is
carried out by analyzing the feasibility of partitions obtained by gradually moving hardware
functions to software. A partition is considered feasible when it implements the original
specification, satisfying performance indices and constraints. The algorithm is greedy and
is not designed to find global minimums.
Cosyma is a hardware/software codesign system for embedded controllers developed by
Ernst et al. at University of Braunschweig. This system uses a superset of C language, called
C∗ , where some constructors are used for specifying timing constraints and parallelism.
The hardware/software partitioning in Cosyma is solved with simulated annealing. The
cost function is based on estimation of hardware and software runtime, hardware/software
communication time and on trace data. The main restriction of such a partitioning method
is that hardware and software parts are not allowed for concurrent execution.
The approach proposed by Wolf uses an object-oriented structure to partition functionality
considering distributed CPUs. The specification is described at two levels of granularity.
The system is represented by a network of objects which send messages among themselves to
implement tasks. Each object is described as a collection of data variables and methods. The
partitioning algorithm splits and recombines objects in order to speed up critical operations,
although the splitting is only considered for the variable sets and does not explore the code
sections.
One of the challenges of hardware/software partitioning approaches is the analysis of a
great varity of implementation alternatives. The approach proposed by Barros [14, 17, 16,
15] partitions a description into hardware and software components by using a clustering
algorithm, which considers the distinct implementation alternatives. By considering a
particular implementation alternative as the current one, clustering is carried out. The
analysis of distinct implementation alternatives in the partitioning allows for the choice of
a good implementation with respect to time constraints and area allocation. However, this
method does not present a formal approach for performing quantitative analysis, and only
considers a single processor architecture.
Another very important aspect is the formal basis used in each phase of the design process.
Layout models have a good formal foundation based on set and graph theory since this deals
with placement and connectivity [94]. Logic synthesis is based on boolean algebra because

A PETRI NET MODEL 245

most of the algorithms deal with boolean transformation and minimization of expressions
[95]. At high-level synthesis a lot of effort has been applied to this topic [96, 97, 98].
However, since high-level synthesis is associated to a number of different areas, varying
from behavioral description to metric estimation, it does not have a formal theory of its
own [9]. Software generation is based on programming languages which vary from logical
to functional language paradigms. Since hardware/software codesign systems take into
account a behavioral specification in order to provide a partitioned system as input for high-
level synthesis tools and software compilers, they must consider design/implementation
aspects of hardware and software paradigms. Therefore, a formal intermediate format able
to handle behavioral specification that would also be relevant to hardware synthesis and
software generation is an important challenge.
Petri nets are a powerful family of formal description techniques able to model a large
variety of problems. However, Petri nets are not only restricted to design modeling. Several
methods have been developed for qualitative analysis of Petri net models. These methods
[28, 29, 37, 26, 27, 30, 31, 1] allow for qualitative analysis of properties such as deadlock
freedom, boundedness, safeness, liveness, reversibility and so on [42, 37, 87, 39].
This work extends Barros’s approach by considering the use of timed Petri nets as a
formal intermediate format [61, 59] and takes into account a multi-processor architecture.
In this work, the use of timed Petri nets as an intermediate format is mainly for computing
metrics such as execution time, load balance [72], communication cost [70, 69], mutual
exclusion degree [73] and area estimates [75]. These metrics guide the hardware/software
partitioning.
The next section is an introduction to the PISH codesign methodology. Section 3 in-
troduces the description language adopted. Section 4 describes the hardware/software
partitioning approach. Section 5 is an introduction to Petri nets [1, 64, 56, 58]. Section 6
presents the occam-time Petri net translation method [64, 59, 61, 62, 40]. Section 7 de-
scribes the approach adopted for performing qualitative analysis. A method for computing
critical path time, minimal time and the likely time based on structural Petri net methods
is presented in Section 8. Section 9 presents the extended model and an algorithm for esti-
mating the number of processors needed. Section 10 describes the delay estimation method
adopted in this work. Sections 11, 12, 13 and 14 describe methods for computing com-
munication cost, load balance, mutual exclusion degree and area estimates, respectively.
Section 15 presents an example and finally some conclusions and perspectives for future
works follow.

2. The PISH Co-Design Methodology: An Overview

The PISH co-design methodology being developed by a research group at Universidade
de Pernambuco, uses occam as its specification language and comprises all phases of the
design process, as depicted in Figure 1. The occam specification is partitioned into a set
of processes, where some sets are implemented in software and others in hardware. The
partitioning is carried out in such a way that the partitioned description is formally proved to
have the same semantics as the original one. The partitioning task is guided by the metrics
computed in the analysis phase. Processes for communication purposes are also generated


Figure 1. PISH design flow.

in the partitioning phase. A more detailed description of the partitioning approach is given
in Section 4.
After partitioning, the processes to be implemented in hardware are synthesized and the
software processes are compiled. The technique proposed for software compilation is also
formally verified [93]. For hardware synthesis, a set of commercial tools has been used. First
system prototype has been generated by using two distinct prototyping environments: The
HARP board developed by Sundance and a transputer plus FPGAs boards, both connected
through a PC bus. Once the system is validated, either the hardware components or the
whole system can be implemented as an ASIC circuit. For this purpose, layout synthesis
techniques for Sea-of-gates technology have been used [94].

3. A Language for Specifying Communicating Processes

The goal of this section is to present the language which is used both to describe the appli-
cations and to reason about the partitioning process itself. This language is a representative
subset of occam, defined here by the BNF-style syntax definition showed below, where
[clause] has the usual meaning of an optional item.
The optional argument rep stands for a replicator. A detailed description of these con-
structors can be found in [62]. This subset of occam has been extended to consider two
new constructors: BOX and CON. The syntax of these constructors is BOX P and CON
P, where P is a process. These constructors have no semantic effect. They are just anno-
tations, useful for the classification and the clustering phases. A process included within
a BOX constructor is not split and its cost is analyzed as a whole at the clustering phase.
The constructor CON is an annotation for a controlling process. Those processes act as an
interface between the processes.


The occam subset is described in BNF format:
P: : = SKIP STOP x: = e nop, deadlock and assignment

ch?x ch!e input and output
IF(c1 p1 , . . . , cn pn ) ALT(g1 p1 , . . . , gn pn ) conditional and non-deterministic
conditional
SEQ( p1 , . . . , pn ) PAR( p1 , . . . , pn ) sequential and parallel combiners
WHILE(c P) while loop
VAR x: P variable declaration
CHAN ch: P channel declaration
Occam obeys a set of algebraic laws [62] which defines its semantics. For example, the
laws PAR( p1 , p2 ) = PAR( p2 , p1 ) and SEQ( p1 , p2 , . . . , pn ) = SEQ( p1 , SEQ( p2 , . . . , pn ))
define the symmetry of the PAR constructor and the associativity of the SEQ constructor,
respectively.

4. The Hardware/Software Partitioning

Due to the computational complexity of optimum solution strategies, a need has arisen for
a simplified, suboptimum approach to the partitioning system.
In [20, 14, 18, 10, 11] some heuristic algorithms to the partitioning problem are presented.
Recently, some works [12, 13] have suggested the use of formal methods for the partitioning
process. However, although these approaches use formal methods to hardware/software
partitioning, neither of them includes a formal verification that the partitioning preserves
the semantics of the original description.
The work reported in [16] presents some initial ideas towards a partitioning approach
whose emphasis is correctness. This was the basis for the partitioning strategy presented
here. As mentioned, the proposed approach uses occam [62] as a description language
and suggests that the partitioning of an occam program should be performed by applying a
series of algebraic transformations into the program. The main reason to use occam is that,
being based on CSP [60], occam has a simple and a elegant semantics, given in terms of
algebraic laws. In [63] the work is further developed, and a complete formalization of one
of the partitioning phases, the splitting, is presented.
The main idea of the partitioning strategy is to consider the hardware/software partition-
ing problem as a program transformation task, in all its phases. To accomplish this, an
extended strategy to the work proposed in [16] has been developed, adding new algebraic
transformation rules to deal with some more complex occam constructors such as replica-
tors. Moreover, the proposed method is based on clustering and takes into account multiple
software components.
The partitioning method uses Petri nets as a common formalism [41] which allows for a
quantitative analysis in order to perform the partitioning, as well as the qualitative analysis
of the systems so that it is possible to detect errors in an early phase of the design [33].


Figure 2. Task diagram.

4.1. The Partitioning Approach: An Overview

This section presents an overview of our proposed partitioning approach. The partitioning
method is based on clustering techniques and formal transformations; and it is guided by
the results of the quantitative analysis phase [61, 69, 70, 72, 73, 75]. The task-flow of the
partitioning approach can be seen in Figure 2. This work extends the method presented
in [15] by allowing to take into account a more generic target architecture. This must
be defined by the designer by using the architecture generator, a graphical interface for
specifying the number of software components, their interconnection as well as the memory
organisation. The number and architecture of each hardware component will be defined
during the partitioning phase.
The first step in the partitioning approach is the splitting phase. The original description
is transformed into a set of concurrent simple processes. This transformation is carried out
by applying a set of formal rewriting rules which assures that the semantics is preserved
[63]. Due to the concurrent nature of the process, communication must be introduced


among sequential data dependent processes. The split of the original description in simple
processes allows a better exploration of the design space in order to find the best partitioning.
In the classification phase, a set of implementation alternatives is generated. The set
of implementation alternatives is represented by a set of class values concerning some
features of the program, such as degree of parallelism and pipeline implementation. The
choice of some implementation alternative as the current one can be made either manually
or automatically. When choosing automatically, the alternatives lead to a balanced degree
of parallelism among the various statements and the minimization of the area-delay cost
function.
Next, the occam/timed Petri net translation takes place. In this phase the split program is
translated into a timed Petri net.
In the qualitative analysis phase, a cooperative use of reduction rules, matrix algebra
approach and reachability/coverability methods is applied to the Petri net model in order
to analyze the properties of the description. This is an important aspect of the approach,
because it allows for errors to be detected in an early phase of the design and, of course,
decreases the design costs [33].
After that, the cost analysis takes place in order to find a partition of the set of processes,
which fulfills the design constraints. In this work, the quantitative analysis is carried out
by taking into account the timed Petri net representation of the design description, which is
obtained by a occam/timed Petri net translation tool. Using a powerful and formal model
such as Petri nets as intermediate format allows for the formal computation of metrics
and for performing a more accurate cost estimation. Additionally, it makes the metrics
computation independent of a particular specification language. In the quantitative analysis,
a set of methods for performance, area, load balance and mutual exclusion estimation have
been developed in the context of this work. The results of this analysis are used to reason
about alternatives for implementing each process.
After this, the initial allocation takes place. The term initial allocation is used to describe
the initial assignment of one process to each processor. The criteria used to perform
the initial allocation is the minimization of the inter-processor communication, balancing
of the workload and the mutual exclusion degree among processes. One of the main
problems in systems with multiple processors is the degradation in throughput caused
by saturation effects [68, 66]. In distributed and multiple processor systems [65, 67], it
is expected that doubling the number of processors would double the throughput of the
system. However, experience has showed that throughput increases significantly only for
the first few additional processors. Actually, at some point, throughput begins to decrease
with each additional processor. This decrease is mainly caused by excessive inter-processor
communication. The initial allocation method is based on a clustering algorithm.
In the clustering phase, the processes are grouped in clusters considering one implemen-
tation alternative for each process. The result of the clustering process is an occam descrip-
tion representing the obtained clustering sequence with additional information indicating
whether processes kept in the same cluster should be serialized or not. The serialization leads
to the elimination of the unnecessary communication introduced during the splitting phase.
A correct partitioned description is obtained after the joining phase, where transformation
rules are applied again in order to combine processes kept in the same cluster and to eliminate


communication among sequential processes. In the following each phase of the partitioning
process is explained with more detail.

4.2. The Splitting Strategy

As mentioned, the partitioning verification involves the characterization of the partitioning
process as a program transformation task. It comprises the formalization of the splitting
and joining phases already mentioned.
The goal of the splitting phase is to permit a flexible analysis of the possibilities for com-
bining processes in the clustering phase. During the splitting phase, the original description
is transformed into a set of simple processes, all of them in parallel. Since in occam PAR
is a commutative operator, combining processes in parallel allows an analysis of all the
permutations. The simple processes obey the normal form given below.

chanch 1 , ch 2 , . . . , ch n : PAR(P1 , P2 , . . . , Pk )

where each Pi , 1 < i < k is a simple process.

Definition 4.1 (Simple Process). A process P is defined as a simple if it is primitive
(SKIP, STOP, x: = e, ch ? x, ch ! e), or has one of the following forms: (i.) ALT(b, &gk ck : =
true) (ii.) IF(bk ck : = true) (iii.) BOX Q, HBOX Q and SBOX Q, where Q is an arbitrary
process. (iv.) IF(c Q, TRUE SKIP) where Q is primitive or a process as in (i), (ii) or (iii).
(v.), where Q is simple and are communication commands, possibly combined in sequence
or in parallel. (vi.) WHILE c Q, where Q is simple. (vii.) VAR x: Q, where Q is simple.
(viii.) CON Q, where Q is simple.

This normal form expresses the granularity required by the clustering phase and this
transformation permits a flexible analysis of the possibilities for combining processes in
that phase. In [63] a complete formalization of the splitting phase can be found.
To perform each one of the splitting and joining phases, a reduction strategy is necessary.
The splitting strategy developed during the PISH project has two main steps.

1. the simplification IF and ALT processes

2. the parallelisation of the intermediary description generated by Step 1.

The goal of Step 1 is to transform the original program into one in which all IFs and ALTs
are simple processes. To accomplish this, occam laws have been applied. Two of these
rules can be seen in Figure 3. The Rule 4.1.1 deals with conditionals. This rule transforms
an arbitrary conditional into a sequence of IFs, with the aim to achieve the granularity
required by the normal form. The application of this rule allows a flexible analysis of each
sub-process of a conditional in the clustering phase.
Note that the role of the first IF operator on the right-hand side of the rule above is to make
the choice and to allow the subsequent conditionals to be carried out in sequence. This is
why the fresh variables are necessary. Otherwise, execution of one conditional can interfere


Figure 3. Two splitting rules.

in the condition of a subsequent one. After Step 1, all IFs and ALTs processes are simple,
but not necessarily PAR, SEQ and WHILE processes. To be simple, the sub-process of a
conditional is either a primitive process or can include only ALT, IF and BOX processes.
Rule 4.1.2 distributes IF over SEQ and guarantees that no IF will include a SEQ operator
as its internal process.

Figure 4. Splitting rule.


The goal of Step 2 is to transform the intermediate description generated by Step 1 into
a simple form stated by the definition given above where all processes are kept in parallel.
This can be carried out by using four additional rules. Rule 2 is an example of these rules,
which puts in parallel two original sequential processes.
In this rule, z is a list of local variables of S E Q(P1 , P2 ), x1 is a list of variables used and
assigned in Pi and xi is a list of variables assigned in Pi . Assigned variables must be passed
because one of Pi may have a conditional command including an assignment that may or
not be executed.
The process annotated with the operator CON is a controlling process, and this operator
has no semantic effect. It is useful for the clustering phase. This process acts as an
interface between the processes under its control and the rest of the program. Observe
that the introduction of local variables and communication is essential, because processes
can originally share variables (the occam language does not allow variable sharing between
parallel processes). Obviously, there are other possible forms of introducing communication
than that expressed in Rule 2. This strategy, however, allows for having control of the
introduced communication, which makes the joining phase easier.
Moreover, although it may seem expensive to introduce communication into the system,
this is really necessary to allow a complete parallelisation of simple processes. The joining
phase must be able to eliminate unnecessary communication between final processes. After
the Step 2, the original program has been transformed into a program obeying the normal
form.

4.3. The Partitioning Algorithm

The technique for hardware/software process partitioning is based on the approach proposed
by Barros [14, 15], which includes a classification phase followed by clustering of processes.
These phases will be explained with more detail in this section. The considered version
of such an approach did not cope with hierarchy in the initial specification, and only fixed
size loops had been taken into account in the cost analysis. Additionally, the underlying
architecture template considered only one software component (i.e. only one microprocessor
or microcontroller), which limits the design space exploration. This work addresses some
of these lacks by using Petri nets as an intermediate format, which support abstraction
concepts and provide a framework for an accurate estimation of communication, area and
delay cost, as well as load balance and mutual exclusion between processes. Additionally,
the partitioning can taken into account more complexes architectures, which can be specified
by the designer through a graphical environment. The occam/Petri net translation method
and the techniques for a quantitative analysis will be later explained, respectively.

4.3.1. Classification

In this phase a set of implementation possibilities represented as values of predefined
attributes is attached to each process. The attributes were defined in order to capture the
kind of communication performed by the process, the degree of parallelism inside the


Figure 5. Classiﬁcation and clustering phases.

process (PAR and REPPAR constructors), whether the assignments in the process could
be performed in pipeline (in the case of REPSEQ constructor) and the multiplicity of each
process. As an example, Figure 5a illustrates some implementation alternatives for the
process, which has a REPPAR constructor. Concerning distinct degrees of parallelism, all
assignments inside this process can be implemented completely parallel, partially parallel
or completely sequential.
Although a set of implementation possibilities is attached to each process, only one is
taken into account during the clustering phase and the choice can be done automatically
or manually. In the case of an automatic choice, a balance of the parallelism degree of all
implementations is the main goal.

4.3.2. The Clustering Algorithm

Once a current alternative of implementation for each process has been chosen, the clustering
process takes place. The ﬁrst step is the choice of some process to be implemented in the
software components. This phase is called initial allocation [74] and may be controlled by
the user or may be automatically guided by the minimization of a cost function. Taking
into account the Petri net model, a set of metrics is computed and used for allocating


one process to each available software component. The allocation is a very critical task,
since the partitioning of processes in software or hardware clusters depends on this initial
allocation. The developed allocation method is also based on clustering techniques and
groups processes in clusters by considering criteria such as communication cost [69, 70],
functional similarity, mutual exclusion degree [73] and load balancing [72]. The number
of resulting clusters is equal to the number of the software components in the architecture.
From each cluster obtained, one process is chosen to be implemented in each software
component. In order to implementing this strategy, techniques for calculating the degree of
mutual exclusion between processes, the work load of processors, the communication cost
and the functional similarity of processes have been defined and implemented.
The partitioning of processes has been performed using a multi-stage hierarchical clus-
tering algorithm [74, 15]. First a clustering tree is built as depicted in Figure 5.b. This
is performed by considering criteria like similarity among processes, communication cost,
mutual exclusion degree and resource sharing. In order to measure the similarity between
processes, a metric has been defined, which allows for quantifying the similarity between
two processes by analyzing the communication cost, the degree of parallelism of their cur-
rent implementation, the possibility of implementing both processes in pipeline and the
multiplicity of their assignments [15]. The cluster set is generated by cutting the clustering
tree at the level (see Figure 5.b), which minimizes a cost function. This cost function
considers the communication cost as well as area and delay estimations [75, 72, 61, 59, 15].
Below a basic clustering algorithm is described. First, a clustering tree is built based on
a distance matrix. This matrix provides the distance between each two objects:
p1 p2 p3 p4
0 d1 d2 d3 p1
0 d4 d5 p2
D =
0 d6 p3
0 p4

Algorithm

1. Begin with a disjoint clustering having level L(0) = 0 and sequence number m = 0.
2. Find the least dissimilar pair r , s of clusters in the current clustering according to
d(r, s) = min{d(oi , o j )}.
3. Increment the sequence number (m ← m + 1). Merge the clusters r and s into a
simple one in order to form the next clustering m. Set the level of this clustering to
L(m) = d(r, s).
4. Update the distance matrix by deleting the row and the column corresponding to
the clusters. The distance between the new cluster and an old cluster k may be de-
fined as: d(k, (r, s)) = Max{d(k, r ), d(k, s)}, d(k, (r, s)) = Min{d(k, r ), d(k, s)} or
d(k, (r, s)) = Average{d(k, r ), d(k, s)}.
5. Whether all objects are in one cluster, stop. Otherwise, go to the step 2.


After the construction of the clustering tree, this tree is cut by a cut-line. The cut-line
makes clusters according to a cut-function.
In order to allow the formal generation of a partitioned occam description in the joining
phase, the clustering output is an occam description with annotations, which reflects the
structure of the clustering tree, indicating whether resources should be shared or not.

4.4. The Joining Strategy

Based on the result of the clustering and on the description obtained in the splitting, the
joining phase generates a final description of the partitioned program in the form:

chan: ch 1 , ch 2 , . . . , ch n : PAR(S1 , . . . , Ss , H1 , . . . , Hr )

where each Si and Hj are the generated clusters. Each of these is either one simple process
(Pk ) of the split phase of a combination of some (Pk )’s. Also, none of the (Pk )’s is left out:
each one is included in exactly one cluster. In this way, the precise mathematical notion of
partition is captured. The Si , by convention, stands for the a software process and each Hj
for a hardware process.
Like the splitting phase, the joining phase should be designed as a set of transforma-
tion rules. Some of these rules are simply the inverses of the ones used in the splitting
phase, but new rules for eliminating unnecessary communication between processes of a
same cluster have been implemented [63]. The goal of the joining strategy is to elim-
inate irrelevant communication in the same cluster and after joining branches of condi-
tionals and ALT processes as well as reducing sequences of parallel and sequential pro-
cesses.
The joining phase is inherently more difficult than splitting. The combination of simple
processes results in a process which is not simple anymore. Therefore, no obvious normal
form is obtained to characterize the result of the joining in general. Also, one must take
a great care for not introducing deadlock when serializing processes in a given cluster.
Also, in this phase some sequential processes are carried out in parallel, by applying some
transformation rules.

5. A Brief Introduction to Petri Nets

Petri nets are a powerful family of formal description techniques with a graphic and mathe-
matical representation, and have powerful methods which allow for performing qualitative
and quantitative analysis [37, 28, 29, 64].
This section presents a brief introduction to Place/Transitions nets and Timed Petri nets.

5.1. Place/Transition Nets

Place/Transition Nets are bipartite graphs represented by two types of vertices called places
(circles) and transitions (rectangles), interconnected by directed arcs (see Figure 6).


Figure 6. Simple Petri net.

Place/Transition nets can be defined in terms of matrix as follow:

Definition 5.1 (Place Transition Net). is defined as 5-tuple N = (P, T, I, O, M0 ), where
P is a finite set of places which represents the state variables, T is a set of transitions which
represents the set of actions, I : P × T → N is the input matrix which represents the
pre-conditions, O: P × T → N is the output matrix which represents the post-conditions
and M0 : P → N is the initial marking which represents the initial state.

Definition 5.2 (Firing Rule). One transition t j is enabled to fire if, and only if, all its input
places ( p ∈ P) has M( p) ≥ I ( p, t j ). The transition firing changes the marking of the net
(M0 [t j > M ). The new marking is obtained as follows: M ( p) = M0 ( p) − I ( p, t j ) +
O( p, t j ), ∀ p ∈ P

The execution of actions is represented by the transition firing. However there are two
particular cases where the firing rule is different: the first case is represented by source
transitions. One source transition does not have any input places. This transition is always
enabled. In the second case the transition does not have any output place. This transition
is called of sink transition. Its firing does not create any token.
Using the matrix representation, the structure of the net is represented by a set of places,
a set of transitions, an input matrix (pre-conditions) and an output matrix (post-conditions).
When one transition t fires, the difference between the markings is represented by O( p, t)−
I ( p, t), ∀ p ∈ P. The matrix C = O − I is called incidence matrix. This matrix represents
the structure of the net and if the net does not have self-loop.

Definition 5.3 (Incidence Matrix). Let a net N = (P, T, I, O). The incidence matrix rep-
resents the relation C: P ×T → Z , ∀ p ∈ P defined by: C( p, t) = O( p, t)− I ( p, t), ∀ p ∈
P.

A net that has self-loop may be represented by the incidence matrix if the self-loop is
refined using dummy pair [37].
The state equation describes the behavior of the system, as well as allows for the analysis
of properties of the models. Using matrix representation, the transition t j is represented by


a vector s, where the components of the vector are zero except for the j − th component,
which is one. So it is possible to represent a marking either as

M ( p) = M0 ( p) − I ( p, t j ) + O( p, t j )

or as

M ( p) = M0 ( p) − I.s(t j ) + O.s(t j )
= M ( p) = M0 ( p) − C.s(t j )T , ∀ p ∈ P.

Applying the sequence sq = t0 , . . . , tk , . . . , t0 , . . . , tn of transitions to the equation above,
the following equation is obtained

M ( p) = M0 ( p) + C.¯ ,
s

where s = s(t0 )T , s(t1 )T , . . . , s(tn )T is called Parikh vector.
¯
The Place/Transition nets can be divided into several classes [28]. Each class has distinct
modeling power. The use of subclasses improves the decision power of the models, although
without excessively decreasing the modeling power.

5.2. Analysis

The methods used to analyze Petri nets may be divided into three classes. The first method
is graph-based and it builds on the reachability graph (reachability tree). The reachability
graph is initial marking dependent and so it is used to analyse behavioral properties [39].
The main problem in the use of reachability tree is the high computational complexity [43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] even if some interesting techniques are
used such as: reduced graphs, graph symmetries, symbolic graph etc [35, 36]. The second
method is based on the state equation [34, 37, 84, 86]. The main advantage of this method
over the reachability graph is the existence of simple linear algebraic equations that aid
in determining properties of the nets. Nevertheless, it gives only necessary or sufficient
condition to the analysis of properties when it is applied to general Petri nets. The third
method is based on reduction laws [37, 38]. The reduction laws based method provides a
set of transformation rules which reduces the size of the models, preserving the properties.
However, it is possible that for a given system and some set of rules, the reduction can not
be complete.
From a pragmatic point of view, it is fair to suggest that a better, more efficient and more
comprehensive analysis can be done by a cooperative use of these techniques. Nevertheless,
necessary and sufficient conditions can be obtained by applying the matrix algebra for some
subclasses of Petri nets [33].


5.3. Timed Petri Nets

So far, Petri nets have been used to model a logical point of view of the systems; no
formal attention has been given to temporal relations and constraints [71, 78, 76]. The
first temporal approach was proposed by Ramchandani [57]. Today, there are at least three
approaches which associate time to the nets: stochastic, firing times specified by intervals
and deterministic. In the stochastic nets a probability distribution to the firing time is
assigned to the model [80, 79].
Time can be associated with places (Place-Timed Models) [83], tokens (Token-Timed
Models) [82] and transitions (Transition-Timed Petri Models). The approach proposed in
[71] associates to each transition a time interval (dmin , dmax ) (Transition-Time Petri Nets).
In deterministic timed nets the transitions firing can also be represented by policies: the
first one is the three phase policy firing semantics and the second model is the atomic firing
semantics approach. In the deterministic timed net with three phase policy semantics, the
time information represents the duration of the transition’s firing [57]. The deterministic
timed net with atomic firing may be represented by the approach proposed at [71] where
time interval bounds are the same (di , di ) (Transition-Time Petri Nets).
This section deals with the Timed Petri Nets, or rather, Petri net extensions in which the
time information is expressed by duration (deterministic timed net with three phase policy
firing semantics) and it is associated to the transitions.

Definition 5.4 (Timed Petri Nets). Let a pair N t = (N , D) be a timed Petri net, where
N = (P, T, I, O, M0 ) is a Petri net, D: T → R+ ∪ 0, is a vector which associates to each
transition ti the duration of the firing di .

A state in a Timed Petri Net is defined as 3-tuple of functions, one of which is the marking
of the places; the second is the distribution in firing transitions and the last is the remaining
firing time [77], for instance, if the number of tokens in a firing transition ti is mt (ti ) = k,
then the remaining firing time is represented by a vector RT (t) = (r t (t)1 , . . . , r t (t)k ).
More formally:

Definition 5.5 (State of Timed Petri Net). Let N t = (N , D) a Timed Petri Net, a state
S of N t is defined by a 3-tuple S = (M, M T, RT ), where M: P → N is the marking,
M T : T → N is the distribution of tokens in the firing transitions and RT : K → R+ is the
T
remaining firing time function which assigns the remaining firing time to each independent
firing of a transition for each transition that mt (t) = 0. K is the number of tokens in a
firing transition ti (mt (ti ) = K ). RT is undefined for the transitions ti which mt (ti ) =
0.

A transition ti obtaining concession at a time x is obliged to fire at the time, if there is no
conflict.

Definition 5.6 (Enabling Rule). Let N t = (N , D) be a Timed Petri Net, and S =
(M, M T, RT ) the state of N t. We say that a bag of transitions BT is enabled at the instant


x if, and only if, M( p) ≥ ∀ti ∈BT #BT (ti ) × I ( p, ti ), ∀ p ∈ P, where BT (ti ) ⊆ BT and
#BT (ti ) ∈ N.

If a bag of transitions BT is enabled at the instant x, thus at that instant it removes from
the input places of the bag BT the corresponding number of tokens (#BT (ti ) × I ( p, ti ))
and, at the time x + di (di is the duration related to the transition ti in the bag BT ), adds the
respective number of tokens in the output place. The number of tokens stored in the output
place is equal to the product of the output arc weight (I ( p, t)) by the module of the bag
regarding each transition t which has the duration equal to di (#BT (t), BT (t) ⊆ BT ) plus
the number of tokens already “inside” the firing transitions, such that their duration have
elapsed, during the present bag firing.

Definition 5.7 (Firing Rule). Let N t = (N , D) be a Timed Petri Net, and S =
(M, M T, RT ) the state of N t. If a bag of transitions BT is enabled at the instant
x, then at the instant x ∀ti ∈BT #BT (ti ) × I ( p, ti ), ∀ p ∈ P number of tokens is re-
moved from the input places of the bag BT . At the time x + d, the reached marking is
M = M − ∀ti ∈BT #BT (ti )× I ( p, ti )+ ∀ti ∈BT |di =d #BT (ti )× O( p, ti )+ ∀tj ∈T,M T (tj )>0
M(t j ) × O( p, t j ), ∀ p ∈ P, if no other transition t ∈ T has been fired in the interval
(x, x + di ).

The inclusion of time in the Petri net models allows for performance analysis of systems.
Section 8 introduces performance analysis in a Petri net context paying special attention to
structural approaches of deterministic models.

6. The Occam—Petri Net Translation Method

The development of a Petri net model of occam opens a wide range of applications based
on qualitative analysis methods. To analyze performance aspects one requires a timed net
model that represents the occam language.
In our context, an occam program is a behavioral description which has to be implemented
combining software and hardware components. Thus, the timing constraints applied to
each operations of the description is already known, taking into account either hardware or
software implementation.
This section presents a timed Petri net model of occam language. This model allows
for the execution time of the activities to be computed using the methods described in
Sections 10 and 8.
The occam-Petri net translation method [61] receives an occam description and translates
it into a timed Petri net model. The occam programming language is derived from CSP
[60], and allows the specification of parallel algorithms as a set of concurrent processes.
An occam program is constructed by using primitive processes and process combiners as
described in Section 3.
The simplest occam processes are the assignment, the input action, the output action, the
skip process and the stop process. Herewith, due to space restriction, only one primitive


Figure 7. Communication.

process as well as one combiner will be described. A more detailed description can be
found in [59, 61, 64].

6.1. Primitive Communication Process: Input and Output

Occam processes can send and receive messages synchronously through channels by using
input (?) and output (!) operations. When a process receives a value through a channel,
this value is assigned to a variable.
Figure 7.b gives a net representing the input and the output primitive processes of the
example in Figure 7.a. The synchronous communication is correctly represented by the
net. It should be observed that the communication action is represented by the transition
t0 and is only ﬁred when both the sender and the receiver processes are ready, which are
represented by tokens in the places p0 and p1 . When a value is sent by an output primitive
process through ch 1 , it is received and assigned to the variable x, being represented in the
net by the data part of the model. Observe that the transition t1 can only be ﬁred when the
places p2 and p3 have tokens. After that, both processes become enabled to execute the
next actions.


Figure 8. Parallel.

6.2. Parallelism

The combiner Par is one of the most powerful of the occam language. It permits concurrent
process execution. The combined processes are executed simultaneously and only finish
when every one has finished.
Figure 8.a shows a program containing two processes t1 and t2 . Figure 8.b shows a net
that represents the control of this program.
One token in the place p0 enables the transition t0 . Firing this transition, the tokens
are removed from the input place ( p0 ) and one token is stored in the output places ( p1
and p2 ). This marking enables the concurrent firings of the transitions t1 and t2 . After
the firing of these transitions, one token is stored in the places p3 and p4 , respectively.
This marking allows the firing of the transition t3 , which represents the end of the parallel
execution.

7. Qualitative Analysis

This section depicts the proposed method to perform qualitative analysis. The quantitative
analysis of behavioral descriptions is described in following sections.
Concurrent systems are usually difficult to manage and understand. Therefore, misun-
derstanding and mistakes are frequent during the design cycle [87].
Therefore, a need arises to decrease the cost and time-to-market of the products. Thus,
it is fair to suggest the formalization of properties of the systems so that it is possible to
detect errors in an early phase of the design.


There are so many properties which could be analyzed in a Petri net model [37]. Among
these properties, we have to highlight some very important ones in a control system context:
boundedness, safeness, liveness and reversibility. Boundedness and safeness imply the
absence of capacity overflow. For instance, there may be buffers and queues, represented
by places, with finite capacity. If a place is bounded, it means that there will be no overflow.
One place bounded to n means that the number of tokens that will be stored in it is at most n.
Safeness is a special case of boundness: n equal to one. Liveness is related to the deadlock
absence. Actually, if a system is deadlock free, it does not mean that the system is live,
although if a system is live, it is deadlock free. This property guarantees that a system can
successfully produce. One deadlock free system may also be not live, which is the case
when a model does not have any dead state, but there exists at least one activity which is
never executed. Liveness is a fundamental property in many real systems and also very
complex to analyze in a broad sense. Therefore, many authors have been divided liveness
in terms of levels in such a way to make it easy to analyze. Another very important property
is reversibility. If a system is reversible, it means that this system has a cyclic behavior and
that it can be initialized from any reachable state.
The analysis of large dimensions nets is not a trivial problem, therefore in the prag-
matic point of view, a cooperative use of reduction rules, structural analysis and reachabil-
ity/coverability methods is important.
The first step of the adopted analysis approach is the application of the ‘closure’ [32] by the
introduction of a fictitious transition t, whose the time duration is 0. The second step should
be the application of transformation rules which preserves properties like boundedness and
liveness of the models. Such rules should be applied in order to reduce the model dimension.
After that, the matrix algebra and the reachability methods can be applied in order to obtain
the behavioral and the structural properties related to the model. Below, the sequence of
steps adopted to carry out the qualitative analysis is depicted.

• Application of a ‘closure’ to the net by the introduction of the transition t,

• Application of reduction rules to net,

• Application of structural methods and

• Application of reachability/coverability methods.

8. Performance Analysis

The calculation of execution time is a very important issue in hardware/software codesign. It
is necessary for performance optimization and for validating timing constraints [3]. Formal
performance analysis methods provide upper and/or lower (worst/best) bounds instead of a
single value.
The execution time of a given program is data dependent and takes into account the
actual instruction trace which is executed. Branch and loop control flow constructs result
in an exponential number of possible execution paths. Thus, specific statistics must be
collected by considering a sample data set in order to determine the actual branch and


loop counts. This approach, however, can be used only in probabilistic analysis. In some
limited applications it is possible to determine the conditionals and loop iteration counts by
applying symbolic data flow analysis.
Another important aspect is the cost of formal performance analysis. Performance anal-
ysis does not intend to replace simulation; instead, besides a model for simulation, an
accurate model for performance analysis must be provided.
Worst/best case timing analysis may be carried out by considering path analysis techniques
[19, 6]. In worst/best case analysis, it has been assumed the worst/best case choices for each
branch and loop. An alternative method allows the designer to provide simple execution
number of certain statements. It helps to specify the total execution number of iteration of
nested loops. Methods based on Max-Plus algebra have also been applied for performance
evaluation, but for mean case performance it suffers of a great complexity and only works
for some special classes of problems [81].
This section presents one technique for static performance analysis of behavioral descrip-
tions based on structural Petri net methods. Static analysis is referred to as methods which
use information collected at compilation time and may also include information collected
in profiling runs [90].
Real time systems can be classified either as a hard real time system or a soft real time
system. Hard real time systems cannot tolerate any missed timing deadlines. Thus, the
performance metric of interest is the extreme case, typically the worst case, nevertheless
the best case may also be of interest. In a soft real time system, a occasional missing timing
deadline is tolerable. In this case, a probabilistic performance [80, 79, 89] measure that
guarantees a high probability of meeting the time constraints suffices.
The inclusion of time in the Petri net model allows for performance analysis of systems
[85]. Max-Plus algebra has been applied to Petri net models for system analysis [92], but
such approaches can only be considered for very restricted Petri nets subclasses [5]. One
very well known method for computing cycle time in Petri net is based on timed reachability
graph with path-finder algorithm. However the computation of the timed reachability graph
is very complex and for unbounded systems is infinite. The pragmatic point of view suggests
exploring the structure (to transform or to model the systems in terms of Petri nets subclasses)
of the nets and to apply linear programming methods [86].
Structural methods eliminate the derivation of state space, thus they avoid the “state
explosion” problem of reachability based methods; nevertheless they cannot provide as
much information as the reachability approaches do. However, some performance measures
such as minimal time of the critical-path can be obtained by applying structural methods.
Another factor which has influenced us to apply structural based methods is that every other
metric used in our proposed method to perform the initial allocation and the partitioning is
computed by using structural methods, or rather, the communication cost and the mutual
exclusion degree computation algorithms are based on t-invariants and p-invariants methods.
Firstly, this section presents an algorithm to compute cycle time for a specific deterministic
timed Petri net subclass called Marked Graph [1]. After that, an extended approach is
presented in order to compute extreme and average cases of behavioral description.
First of all, it is important to define two concepts: direct circuit and direct elementary
circuits. A direct circuit is a directed path from one vertex (place or transition) back to


Figure 9. Timed marked graph.

itself. A directed elementary circuit is a directed circuit in which no vertices appear more
than once.

THEOREM 8.1 For a strongly connected Marked Graph N , M( pi ) in any directed circuit
remains the same after any ﬁring sequence s.

THEOREM 8.2 For a marked graph N , the minimum cycle time is given by Dm =
maxk {Tk /Nk }, for every circuit k in the net. Where Tk is the sum of every transition
delays in a circuit k and Nk = M( pi ) in a circuit.
To compute all directed circuit in the net, ﬁrst we have to obtain all minimum p-invariants,
then use them to compute the direct circuits.
Figure 9 shows a marked graph, adopted from [1], in which we apply the method described
previously. The reader should observe that each place has only one input and output
transition.
Applying very well known methods to compute minimum p-invariants [37, 84], these
supports are obtained: sp1 = { p0 , p2 , p4 , p6 }, sp2 = { p0 , p3 , p5 , p6 }, sp3 = { p1 , p2 , p4 }
and sp4 = { p1 , p3 , p5 }. After obtaining the minimum p-invariants, it is easy to compute
the directed circuits: c1 = { p0 , t1 , p2 , t2 , p4 , t4 , p6 , t5 }, c2 = { p0 , t1 , p3 , t3 , p5 , t4 , p6 , t5 },
c3 = { p1 , t1 , p2 , t2 , p4 , t4 } and c4 = { p1 , t1 , p3 , t3 , p5 , t4 }. The cycle time associated
to each circuit is: Dm (N ) = max{T (1), T (2), T (3), T (4), T (5), T (6)}. Therefore, the
minimum cycle time is the Dm = maxk {12, 16, 10, 12} = 16.
If instead of using a Petri net model in which the delay related to each operation is attached
to the transition, a Petri net model was adopted in which each transition has a time interval
(tmin , tmax ) attached to it (see Section 5.3), we may compute the lower bound of the cycle
time. Note that if the delay related to each transition is replaced by the lower bound (tmin )
the value obtained means the lower bound of the cycle time [87].


Herewith a structural approach is presented to computing extreme and average cases of
behavioral description. The algorithm presented computes the minimal execution time,
the minimal critical-path time and likely minimal time related to branch probabilities in
partially repetitive or consistent strongly connected nets covered by p-invariants. This
approach is based on the method presented previously, however it is not restricted only to
marked graphs.
First, we present some definitions and theorems in Petri net theory which are important
for the proposed approach [37].
A Petri net N is said to be repetitive if there is a marking and a firing sequence from this
marking such that every transition occurs infinitely often. More formally:

Definition 8.1 (Repetitive net). Let N = (R; M0 ) be a marked Petri net and firing sequence
s. N is said to be repetitive if there exists a sequence s such that M0 [s > Mi every transition
ti ∈ T fires infinitely often in s.

THEOREM 8.3 A Petri net N is repetitive if, and only if, there exists a vector X of positive
integers such that C · X ≥ 0, X = 0.
A Petri net is said to be consistent if there is a marking and a firing sequence from this
marking back to the same marking such that every transition occurs at least once. More
formally:

Definition 8.2 (Consistent net). Let N = (R; M0 ) be a market Petri net and firing sequence
s. N is said to be consistent if there exists a sequence s such that M0 [s > M0 every transition
ti ∈ T fires at least once in s.

THEOREM 8.4 A Petri net N is consistent if, and only if, there exists a vector X of positive
integers such that C · X = 0, X = 0.
The proofs of such theorems can be found in [37].
The method presented previously is based on the computation of the direct circuit by using
the p-minimum invariants. In other words, first the p-minimum invariants are computed,
then, based on these results, the direct circuits are computed (sub-nets). The cycle time
related to a marked graph is the maximum delay considering each circuit of the net.
This method cannot be applied to nets with choices because the circuits which are com-
puted by using the minimum p-invariants will provide sub-nets such that the simple summa-
tion of the delays, attached to each transition, gives a number representing all those branches
of the choice. Another aspect which has to be considered is that concurrent descriptions
with choice may lead to a deadlock. Thus, we have to avoid these paths in order to compute
the minimal execution time of the model.
In this method we use the p-minimum invariants in order to compute the circuits of the
net and the t-minimum invariants to eliminate from the net transitions which do not appear
in any t-invariant. With regard to partially consistent models, these transitions are never


fired. We also have to eliminate from the net every transition which belongs to a choice,
in the underlined untimed model, it does not model a real choice in the timed model. For
instance, consider a net in which we have the following output bag related to the place
pk : O( pk ) = {ti , t j }. Thus, in the underlined untimed model these transitions describe
a choice. However, if in the timed model these transition have such time constraints: di
and d j and d j > di , a conflict resolution policy is applied such that transition ti must
be fired. After obtaining this new net (a consistent net), each sub-net obtained by each
p-minimum invariant must be analyzed. In such nets, their transitions have to belong to
only one t-minimum invariant. The sub-nets which cannot satisfy this condition have to be
decomposed into sub-nets such that their transitions should belong to only one t-invariant.
The number of sub-nets obtained has to be the minimum, or rather, the number of transitions
of each sub-net in each t-invariant should be the maximum.

Definition 8.3 (Shared Element). Let N = (P, T, I, O) be a net, two subnets S1 , S2 ⊂ N ,
a set of places P such that ∀ pz ∈ P , pz ∈ S1 , pz ∈ S2 . The set of places P is said to be a
shared element if, and only if, its places are strongly connected and ∃ pm ∈ P , ti ∈ S1 , tk ∈
S2 such that ti , tk ∈ O( pm ) and ∃ pl ∈ P , t j ∈ S1 , tr ∈ S2 such that t j , tr ∈ I ( pl ).

THEOREM 8.5 Let N = (P, T, I, O) be a pure strongly connected either partially con-
sistent or consistent net covered by p-invariants without shared elements, S Nk ⊂ N
a subnet obtained from a p-minimum invariant I pi and covered by one t-minimum in-
variant I t j , S Nc ⊂ N a subnet obtained from a p-minimum invariant I pi and does not
covered by only one t-minimum invariant I t j and if ∃ pk such that #O( pk ) > 1 then
l j > h i , ∀ti , t j ∈ O( pk ). The time related to the critical path of the net N is given
by C T (N ) = max{d S Nk , d S N j }, ∀S Nk , S N j ⊂ N , where S N j is obtained by a decomposi-
tion of S Nc such that each S N j ⊂ S Nc is covered by only one t-minimum invariant.

THEOREM 8.6 Let N = (P, T, I, O) be a pure strongly connected either partially con-
sistent or consistent net covered by p-invariants without shared elements, S Nk ∈ N be
a subnet obtained from a p-minimum invariant I pi and covered by one t-minimum in-
variant I t j , S Nc ∈ N be a subnet obtained from a p-minimum invariant I pi and not
covered by only one t-minimum invariant I t j and if ∃ pk such that #O( pk ) > 1 then
∃l j > h i , ∀ti , t j ∈ o( pk ). Each S N j ⊂ S Nc is covered by only one t-minimum invariant,
where S N j is obtained by a decomposition of S Nc . The minimal time of the net N is given by
M T (N ) = max{d S Nk , d S Nc }, ∀S Nc ⊂ N such that each d S Nc = min{d S N j }, ∀S N j ⊂ S Nc .
The proof of both theorems can be found in [72].
It is important to stress that the main difference among these metrics is related to the
execution time of pieces of program in which choices must be considered. The computation
of the minimal time related to the whole program takes into account the choice branch which
provides the minimum delay.
Another important measure is the likely minimal time. This metric takes into ac-
count probabilities of branches execution. Therefore, based on specific collected statis-


tics, the designer may provide, for each branch in the description, the probability of
execution.

Deﬁnition 8.4 (Strongly Connected Component Probability Matrix). Let a net N =
(P, T, I, O, D), a strongly connected component S Nk ⊂ N be a branch bi related to a
choice-net. SC P M(S Nk , t j ) is the execution probability of the transition t j in the strongly
connected component S Nk .

0 if t j ∈
/ S Nk .
SC P M(S Nk , t j ): N × N → pri (0 ≤ R ≤ 1) if t j ∈ S Nk , t j ∈ bi .

1 if t j ∈ S Nk , t j ∈ bi , ∀bi ∈ S Nk .
/

Deﬁnition 8.5 (Probability Execution Vector). Let a net N = (P, T, I, O, D) and a strongly
connected component probability matrix SC P M related to N . P E V (t j ): N → 0 ≤ R ≤ 1,
#P E V (t j ) = T . Each vector component pev(t j ) = ∀S Nk SC P M(S Nk , t j ), ∀t j ∈ T .

The likely minimal time related to the net N is given by M L T = max{mlt S Nk }, ∀S Nk ,
where mlt S Nk = d(t j ) × pev(t j ), ∀t j ∈ S Nk . d(t j ) is the lower bound time related to the
transition t j and S Nk is a strongly connected component.
Similar results are obtained for either partially repetitive or repetitive nets covered by
p-invariants. In nets with these properties, instead of computing the t-minimum invariants
we have to compute the support of the equation systems C · X ≥ 0, X = 0.
The algorithm to compute the minimal time, the minimal critical path time and likely
minimal time is given in the following:
• Input:
a net N = (P, T, I, O, D).
branch probabilities bi = { pri ; t j }, ∀bi ⊂ N and ∀t j ∈ bi .
• Output:
the minimal time.
the minimal critical path time.
the likely minimal time.

• Algorithm:
1. Compute the t-minimum invariants (I t) of the net N .
2. Remove from the net N the transition tl and the places pg ∈ I (tl ) or pg ∈ O(tl ),
if it1 = 0, ∀I t, such that: N = (P , T , I , O ), where P = P pg , pg ∈ I (tl )
and/or pg ∈ O(tl ) and T = T tl .
3. Compute the p-minimum invariants (I p) of the net N .
4. Compute the probability execution vector (P E V ).
5. Compute the likely execution time of each component S Nk ⊂ N (mlt S Nk =
d(t j ) × pev(t j ), ∀t j ∈ S Nk ).


Figure 10. Example.

6. Compute the strongly connected components S Nk obtained from I pk .
7. Compute the strongly connected components S Nk obtained from I pk and covered
by one t-minimum invariant I t.
8. For each strongly connected component (obtained from I pk ) not covered by a t-
minimum invariant (S Nc ), do:
Decompose S Nc into nets (S N j s) in such a way that a t-minimum invariant
I t j covers each subnet S N j .
9. Compute the execution time of each component S Nk , S N j ⊂ S Nc ∀S Nk , S N j ∈
N.
10. Compute the critical path time C T (N ) = C T (N ) = max{d S Nk , d S N j } ∀S Nk , S N j ∈
N.
11. Compute the minimal time M T (N ) = M T (N ) = max{d S Nk , d S NC } ∀S Nk , S NC ∈
N and d S Nc = min{d S N j }, ∀S N j ⊂ S Nc .
12. Compute the likely minimal time M L T (N ) = max{mlt S Nk }, ∀S Nk ∈ N .
The method presented is applied to a small example (see Figure 10). This example is a
concurrent description in which there are two choices. It is important to emphasize that one
of these leads to a deadlock. In the qualitative analysis phase, we found out that the model is
strongly connected, partially consistent, covered by semi-positive p-invariants, structurally
bounded and safe.


Figure 11. Transformed net.

If the system is partially consistent, it means that some of its transitions either are not
able to be fired or, if they were fired, they would lead to a deadlock.
The first step of the algorithm is the computation of the t-minimum invariant. The support
of the t-minimum invariants are st1 = {t0 , t2 , t6 , t7 , t8 , t9 , t10 , t11 , t12 , t13 , t14 , t15 , t17 } and
st2 = {t0 , t2 , t6 , t7 , t8 , t9 , t10 , t11 , t12 , t13 , t14 , t16 , t18 }. By analyzing these invariants it may
be observed that the transitions t1 , t3 , t4 , t5 do not belong to any t-minimum invariant.
The second step is the elimination of the transitions which do not belong to any t-minimum
invariant and also the elimination of the places that are input and/or output of such transitions.
The resulting net (obtained by this transformation) is presented in Figure 11.
Considering the transitions of the branches of the choice (transitions t11 , t16 , t17 and t18 ),
an execution probability of 0.5 was assigned.
The following step is the computation of the p-minimum invariants of the transformed
net. The support of the p-minimum invariants are: sp1 = { p0 , p1 , p5 , p8 , p16 , p18 },
sp2 = { p0 , p3 , p12 , p15 , p17 , p18 , p19 , p20 }, sp3 = { p0 , p1 , p5 , p10 , p18 }, sp4 = { p0 , p2 ,
p9 , p10 , p18 }, sp5 = { p0 , p2 , p8 , p9 , p16 , p18 } and sp6 = { p0 , p3 , p11 , p13 , p14 , p17 , p18 }.
After that, the strongly connected components (Subneti ) are obtained from these in-
variants. Each component has the following transitions as its elements: Subnet1 =
{t0 , t2 , t6 , t7 , t13 , t14 }, Subnet2 = {t0 , t10 , t11 , t12 , t13 , t14 , t16 , t17 , t18 }, Subnet3 = {t0 , t2 , t6 ,


t13 , t14 }, Subnet4 = {t0 , t6 , t8 , t13 , t14 }, Subnet5 = {t0 , t6 , t7 , t8 , t13 , t14 } and Subnet6 =
{t0 , t9 , t10 , t11 , t12 , t13 , t14 , t15 }.
Then, the likely minimal time for each strongly connected component mlt (S Nk ) =
d(t j ) × pev(t j ), ∀t j ∈ S Nk ): mlt (1) = 12, mlt (2) = 21, mlt (3) = 7, mlt (4) = 10,
mlt (5) = 15 and mlt (6) = 8 is computed
Each component which is not covered by only one t-minimum invariant is decomposed
into subnets in such a way that each of its subnets is covered by only one t-minimum
invariant. Therefore, the Subnet2 = {t0 , t10 , t11 , t12 , t13 , t14 , t16 , t17 , t18 } is decomposed
into Subnet21 = {t0 , t10 , t11 , t12 , t13 , t14 , t17 } and Subnet22 = {t0 , t10 , t12 , t13 , t14 , t16 , t18 }.
After that, we compute the time related to each subnet of the whole model. The result-
ing values are: T (1) = 12, T (21) = 20, T (22) = 22, T (3) = 7, T (4) = 10, T (5) = 15 and
T (6) = 8. Then, we have C T (N ) = max{T (1), T (21), T (22), T (3), T (4), T (5), T (6)} =
22, M T (N ) = max{T (1), min{T (21), T (22)}, T (3), T (4), T (5), T (6)} = 20 and finally
M L T (N ) = max{mlt (1), mlt (2), mlt (3), mlt (4) mlt (5), mlt (6)} = 21
For execution time computation in systems with data dependent loops, the designer has to
assign the most likely number of iterations to the net. This number could be determined by
a previous designer’s knowledge, therefore he/she may directly associate a number using
annotation in the net. Another possibility is by simulating the net on several sets of samples
and recording how often the various loops were executed.
Such a method has been applied to several examples and compared to the results obtained
by the petri net tool INA. The results are equivalents, but INA only provides the minimal
time.

9. Extending the Model for Hardware/Software Codesign

This section presents a method for estimating the necessary number of processors to execute
a program, taking into account the resource constraints (number of processors) provided
by the designer, in order to achieve best performance, disregarding the topology of the
interconnection network.
First, let us consider a model extension in order to capture the number of proces-
sors of the proposed architecture. The extended model is represented by the net N =
(P, T, I, O, M0 , D), which describes the program, a set of places P in which each of
its places ( p ) is a processor type adopted in the proposed architecture; the marking of
each of these places represents the number of processors of the type p ; the input and the
output arcs that interconnect the places of the set P to the transitions which represents the
arithmetic/logical operations (ALUop ⊂ T ).
In the extended model the number of conflicts in the net increases due to the competition
of operations for processors [32]. These conflicts require the use of a pre-selection policy
by assigning equal probabilities to the output arcs from processors places to the enabled
transitions t j ∈ ALUop (O( p, t j ), p ∈ P ) in each reachable marking Mz . Thus, more
formally:

Definition 9.1 (Extended Model). Let a net N = (P, T, I, O, M0 , D) a program model, a
set of places P the processor types adopted in the architecture such that P ∩ P = ∅ and


M0 ( p), p ∈ P the number of processors of the type p. Let a net Ne = (Pe , Te , Ie , Oe , M0 , e

De , f ) the extended model such that Pe = P ∪ P , Te = T, Ie ( p, t j ) = 1 and Oe ( p, t j ) =
1, ∀t j ∈ ALUop , ∀ p ∈ P , otherwise Ie ( p, t j ) = I ( p, t j ) and Oe ( p, t j ) = O( p, t j ).
M0 : N P∪P → N and De = D. Let Mz a reachable marking from M0 , f : Oe ( p, t j ) → 0 ≤
e

R ≤ 1 a probability attached to each output arc Oe ( p, t j ) where ∀tj ∈T f (Oe ( p, t j )) = 1
such that Mz [t j > and p ∈ P .

In such a model, the concurrence is constrained by the number of available processors
(M0 ( p), p ∈ P ) provided by the designer. The main goal of the proposed approach is to
estimate the minimal number of processors that can achieve best performance taking into
account the upper bound of available processors already specified. Therefore, the designer,
in the architecture generator, provides the number of available processors, then the execution
time (C T ) is computed by reachability based methods. The following step comprises the
reduction in the number of processors (M( p) = M( p) − 1, p ∈ P ) in order to compute
a new execution time (C T ). If C T > C T , the necessary number of processors has been
reached. This number of processors is used in the proposed method for initial allocation.
The proposed algorithm to estimate the needed number of processor is:

• Input:
a net Ne = (Pe , Te , Ie , Oe , M0 , De , f ).
e

the number of available processors (M0 ( p), ∀ p ∈ P ).

• Output:
the optimum number of processors Mopt ( p), p ∈ P taking into account the re-
sources constraints provided by the designer.
the minimal execution time (CT) regarding to Mopt ( p).

• Algorithm:

1. Compute the execution time C T (Ne )
2. C T = C T (Ne )
3. For each place p ∈ P , do:
M( p) = M( p) − 1
Compute a new execution time C T (Ne )
if C T (Ne ) ≤ C T
C T = C T (Ne )
Mopt ( p) = M( p), ∀ p ∈ P
else or if M( p) = 0, ∀ p ∈ P
end.

The number of necessary processors can also be reached by taking into account either the
speed up, the efficiency or the efficacy provided by the use of multiple processors.
The gain in terms of execution time by a parallel execution of a program (Ne ) can be
defined as the ratio between the execution time of Ne taking into account only one processor
and the execution time concerning the use of # pr processors to execute it.


Definition 9.2 (Speed up). Let Ne be an extended model, P be a set of places representing
processors of a given type adopted in the architecture, M( p), p ∈ P , be the number
of processors of the type p, C T (Ne , 1) the execution time of Ne carried out by only
one processor and C T (Ne , M( p)), ∀ p ∈ P , be the execution time of Ne considering
M( p), ∀ p ∈ P , processors. S(Ne , M( p)) = C T (Ne , 1)/C T (Ne , M( p)), ∀ p ∈ P is
defined as the speed up due to M( p)), ∀ p ∈ P , processors.

The speed up of S(Ne , M( p)) provided by the use of M( p), ∀ p ∈ P , processors to
execute Ne divided by the number of processors defines the efficiency. More formally:

Definition 9.3 (Efficiency). Let Ne be an extended model and S(Ne , M( p)) the speed
up provided by M( p), ∀ p ∈ P , processors to execute Ne . The efficiency is defined by
E(Ne , M( p)) = S(Ne , M( p))/ M( p), ∀ p ∈ P .

Considering the execution time C T (Ne , M( p)) as a cost measure and the efficiency
E(Ne , M( p)) as a benefit, the cost-benefit relation and its inverse can be defined as the
efficacy.

Definition 9.4 (Efficacy). Let Ne be an extended model, E(Ne , M( p)) the efficiency,
S(Ne , M( p)) the speed up, C T (Ne , 1) the execution time of Ne carried out by only one
processor and C T (Ne , M( p)), ∀ p ∈ P the execution time of Ne carried out by M( p), ∀ p ∈
P , processors. E A(Ne , M( p)) = CE(Nee,M( p)) × C T (Ne , 1) = S(Ne , M( p)) ×
T (N ,M( p))
S(Ne ,M( p))2
E(Ne , M( p)) = M( p)
, ∀p ∈ P is defined as efficacy.

A small example follows in order to illustrate the proposed method. A Petri net represents
the control flow of an occam program obtained by the translation method proposed in [64,
59, 61]. This net describes a program composed of three subprocesses (see Figure 12).
The process P R1 is represented by the set of places PP R1 = { p1 , p4 , p5 , p6 , p7 }, the
set of transitions TP R1 = {t1 , t2 , t3 , p4 , p5 }, their input and output bags (multi-sets) and
the respective initial markings of its places. The process P R2 is composed of PP R2 =
{ p2 , p8 , p10 } as its set of places, TP R2 = {t6 , t8 }, its input and output bags and the markings
of its places. Finally, the process P R3 is composed of PP R3 = { p3 , p9 , p11 } as its set of
places, TP R3 = {t7 , t8 }, its input and output bags as well as the markings of its places. Each
transition has the duration (d) attached to it.
The extended model is represented by the net in Figure 13. The place p13 represents
a processor type. Its marking represents the number of available processors. The only
transitions that describe the ALU operations (t ∈ ALUop ) are interconnected in order to
find out the number of functional units (ALUs—we are supposing that each processor has
only one ALU) needed to execute the program and achieve best performance. These arcs
are expressed by dotted lines. Due to the fact that these transitions are connected to the
place p13 by an input and an output arc, for short these are represented by bidirectional arcs.
Suppose, for instance, that the designer has specified, in the architecture generator, the
upper bound of available processor as n = 4. Thus, the execution time of the extended


Figure 12. Description.

Table 1. Critical
path time.

M( p13 ) C T (Ne )
4 7
3 7
2 7
1 10

model should be analyzed taking into account n = 4, n = 3, n = 2 and n = 1 and then the
conﬁguration with small execution time and small number of processors must be chosen.
By applying the proposed algorithm, the results depicted in Table 1 are obtained. Thus,
the necessary number of processors to execute the program is Mopt ( p13 ) = 2. This number
should be used in the initial allocation process.
In the following sections techniques for computing others useful metrics for hardware/
software codesign are presented. Next section describes a method for delay estimation.
A method for computing communication cost is described in Section 11. Workload of
processors is calculated by the method detailed in Section 12. The degree of mutual


Figure 13. The extended model.

Table 2. Speed up, efﬁciency and efﬁcacy.

Processors S E EA
1 1 1 1
2 1.4286 0.7143 1.0204
3 1.4286 0.4762 0.6803
4 1.4286 0.3571 0.5102

exclusion among processes is calculated by the method described in Section 13. The
silicon area in terms of logic blocks for hardware and software implementation of each
process is computed by the technique described in Section 14.

10. Delay Estimation

Previous sections described methods to compute execution time (cycle time) associated
to occam behavioral description translated into timed Petri nets by using the translation
method proposed in [64, 61].


However, the method used to estimate the delay of the arithmetic and logical expressions
performed in the assignments still needs to be defined.
The delay of expressions is estimated in terms of the control steps needed to perform
the expression. The expression execution time (delay) is given as a function composed
of two factors: the delay related when performing the arithmetic and logical operations
of assignments and the delay when reading and writing variables. It was assumed that
operations in one expression are sequentially executed.

Dex (e) = D RW (e) + D O P (e)

 ∀vu ∈e D Rh (vu ) + vd ∈e DW h (vd )


if e is implemented in hardware
D RW (e) =
 ∀vu ∈e D Rs (vu ) + vd ∈e DW s (vd )


if e is implemented in software
The variables vu and vd are the used and defined variables in the expression e.

 ∀opi ∈e D O Ph (opi ) × #opi


if e is implemented in hardware
D O P (e) =
 ∀opi ∈e D O Ps (opi ) × #opi


if e is implemented in software

11. Communication Cost

This section presents the method proposed for computing communication cost between
processes by using Petri nets [69, 70]. The communication cost related to a process depends
on two factors: the number of transferred bits by each communication action and how many
times the communication action is executed (here referred as number of communication).
Considering that we are dealing with behavioral descriptions that are translated into Petri
nets, we have already defined, in each communication action, the number of transferred
bits in each communication action execution.

Definition 11.1 (Number of Transferred Bits in a Communication Action). N bb : nbc → N,
where #N bb = T and T is the set of transitions. Each component (nbc), associated to a
transition that represents a communication action, defines the number of transferred bits in
the respective communication action, otherwise is zero.

However, we have to define a method to compute how many times the communication
action is executed the process, the communication cost for each process, the communication
cost of the whole description, the communication cost between two sets of processes and
finally to compute the normalized communication cost.
The communication cost for each process (CC( pi )) is the product of the number of
transferred bits in each communication action (N bc ) and the number of communication
(N C( pi )).


Definition 11.2 (Communication Cost for each Process). Let N bc be the number of
transferred bits in the communication actions and N C( pi )T a vector that represents the
number of communication. The communication cost for each process pi is defined by
CC( pi ) = N bc × N C( pi )T .

The Number of Communication (N C( pi )) is a vector, where each component (nc( pi )),
associated to a transition that represents a communication action in the process pi , is the
execution number related to the respective communication action, otherwise, that is, the
component vector associated to the transition which does not represent the communication
action in the process pi is zero. More formally:

Definition 11.3 (Number of Communication). N C( pi ): nc( pi ) → N, where: #N C( pi ) =
T and T is the set of transitions. Each component nc( pi ) = max(nc X k ( pi )), ∀X k ,
where X k is a vector of positive integers such that either C · X = 0 or C · X ≥ 0.
N C X k ( pi ): nc X k ( pi ) → N, where #N C X k ( pi ) = T and T is the set of transitions.

Each vector X k is the minimum support which can be obtained by solving either C · X = 0
(in this case X k are minimum t-invariants) or C · X ≥ 0. The number of communication of
an action ai (represented by a transition ti ) is the respective value obtained in the component
xi for the correspondent X k . The other components, which do not represent communication
actions in the process pi , are equal zero.
According to the results obtained in the qualitative analysis, it is possible to choose
methods to compute the number communication (N C pi ) considering the complexity of the
method used.
If the net (system) is consistent, first we have to compute the minimum t-invariants then
the N C( pi ), ∀ pi ∈ D, are obtained. However, if the net is not consistent, but is repetitive,
first the minimal support to X k , which is obtained using X in the system C · X ≥ 0, where
X = 0, has to be obtained, then the N C( pi ), ∀ pi ∈ D are computed. In the case of the
net not being repetitive and if it is possible to transform it into a repetitive or consistent net
by inserting one transition t f such that I (t f ) = { p f } and O(t f ) = { p0 }, we apply the same
method to compute X and then to obtain the N C( pi ), ∀ pi ∈ D. These places ( p0 and p f )
are well defined, because one token in the place p0 (initial place) enables the execution of the
process and when one token arrives in the place p f (final place), it means that the execution
has already finished. Otherwise, if it is not possible to transform the net into a repetitive
or consistent one, although this system seems not to have interesting properties and even
so the designer does not intend to modify it, we can compute the X and then N C( pi ) by
using either the reachability graph or by solving the system C · X = M f inal − M0 , where
M f inal and M0 are the final and the initial markings, respectively. However, the reader has
to remember that the state equation could provides spurious solutions for some Petri net
sub-classes [33].

THEOREM 11.1 Let N be a consistent net and X k a minimum t-invariant in the net. Con-
sidering every minimum t-invariant in the net (∀X k ∈ N ), the maximum value obtained for
each component vector is the minimum transition firing number for each transition.


THEOREM 11.2 Let N be a repetitive net and X k a minimum support in the net which
can be obtained by using X which solves the equation C · X ≥ 0. Considering every
minimum support X k in the net, the maximum value obtained for each component vector is
the minimum transition firing number for each transition.
The proof for both theorems can be found in [70].
The communication cost between two processes ( pi and p j ) is the product of the num-
ber of communications between the processes by the number of transferred bits in each
communication action. More formally:

Definition 11.4 (Communication Cost between Processes). Let N bc : nbc → N be the
number of transferred bits in a communication action and N C( pi , p j ) a vector that repre-
sents the number of communication between the processes pi and p j . The communication
cost between processes is defined by CC( pi , p j ) = N bc × N C( pi , p j )T .

The communication execution number between two processes ( pi and p j ) is represented
by a vector, where each vector component, associated to a transition which represents a
communication action between both processes, defines the execution number related to the
respective action.

Definition 11.5 (Number of Communication Between two Processes). N C( pi , p j ):
nc( pi , p j ) → N, where #N C( pi , p j ) = T and nc( pi , p j ) = min(nc( pi ), nc( p j )).

In order to compute the communication cost in systems with data dependent loops, the
designer has to assign the more likely number of iteration for each loop of the net. This
number can be determined in two ways: the designer may either directly associate a number
using annotation in the model based on previous knowledge or by simulating the model on
several sets of samples in order to record how often the various loops were executed.
Therefore, each component of the vector N C( pi , p j ) which represents a communication
action between the processes pi and p j and that is inserted in the loop lk has to be multiplied
by the more likely number of iteration (L V (lk )) of the loop lk . A similar procedure has to
be taken for the vector N C( pi ). In this case, each component of the vector which represents
the communication action of the process pi and which is inserted in the loop lk has to be
multiplied by the more likely number of iteration (L V (lk )) of the loop lk .
In the following, we present the algorithms to compute the number of communications
between pairs of processes (N C( pi , p j )) and the the number of communication of each
process (N C( pi )) taking into account data dependent loops.

• Input:
set of processes {. . . , pi , . . .},
set of loops {. . . , lk , . . .},
set of communication action (SC A = {. . . , t j , . . .}),
more likely number of iteration of each loop (L V (lk )),
N C( pi ) of each process.


• Output:
N C( pi ) of each process taking into account the data dependent loops.
• Algorithm:
• For each process pi ∈ D, do:
For each loop lk ∈ pi , do:
For each transition t j ∈ lk , do:
L V (t j ) = L V (lk )
For each loop lm ∈ pi , do:
If lm ⊆ lk , do:
For each transition t j ∈ SC A, do:
L V (t j ) = L V (t j ) × L V (lm )
Else
L V (t j ) = min{L V (t j ), L V (L m )}
N C( pi , t j ) = N C( pi , t j ) × L V (t j )T
• Input:
set of processes {. . . , pi , . . .}
set of communication action (SC A = {. . . , t j , . . .}),
N C( pi ) of each process taking into account the data dependent loops,
N C( pi , p j ) of each pair of processes.

• Output:
N C( pi , p j ) of each pair of processes taking into account the data dependent loops.
• Algorithm:

• For each pair of processes ( pi , p j ) ∈ D, do:
For each transition tk ∈ SC A, do:
If N C( pi , p j , t j ) = 0, do:
N C( pi , p j , tk ) = min{N C( pi , tk ), N C( p j , tk )}
The behavioral description communication cost is given by summing the communication
cost between each pair of processes in the description. More formally:

Deﬁnition 11.6 (Communication Cost). CC(D) = ∀( pi , p j )∈D CC( pi , p j )

In order to be able to compare distinct metric values, we have adopted two kinds of
normalization: local and global normalization. The normalization refers to scaling of
metric values to a number between 0 and 1. The normalization provides a possibility of
combining different units, such as communication cost and mutual exclusion degree, into a
single value and it also provides an absolute closeness measure between objects. Thus, we
know that a closeness value between two objects of 0.95 is high, nevertheless we can not
assert about the number 25 [7].


The global normalized communication cost between two processes is defined by the
communication cost between these processes divided by the communication cost for the
whole description.

Definition 11.7 (Global Normalized Communication Cost). Let CC( pi , p j ) be the com-
munication cost between processes pi and p j , and CC(D) the communication cost in the
whole behavioral description. The global normalized communication cost is defined by
CC( pi , p )
N CC( pi , p j ) = CC(D)j .

The local normalized communication cost between two processes is defined by the com-
munication cost between both processes divided by the summation of the communication
cost for each process.

Definition 11.8 (Local Normalized Communication Cost). Let CC( pi , p j ) be the commu-
nication cost between processes pi and p j , and CC( pi ) and CC( p j ) the communication
cost of the processes pi and p j , respectively. The local normalized communication cost is
defined by LCC( pi , p j ) = CC( pi p j )/(CC( pi ) + CC( p j )).

The global normalized communication cost must also be defined for each process. This
metric is defined by the communication cost related to the process divided by the commu-
nication cost of the whole description.

Definition 11.9 (Global Normalized Communication Cost of a Process). Let CC( pi )
be the communication cost for each process and CC(D) the communication cost of the
whole behavioral description. The global normalized communication cost of each process
is defined by N CC( pi ) = CC( pi ) .
CC(D)

The local normalized communication cost for each process pi is defined by the communi-
cation cost of the process divided by the summation of the communication cost of processes
p j s which has some communication (CC( pi , p j ) = 0). In the following we present the
formal definition:

Definition 11.10 (Local Normalized Communication Cost of a Process). Let CC( pi ) be the
communication cost for each process pi and p j one process which CC( pi p j ) = 0. The local
CC( pi )
normalized communication cost of each process is defined by LCC( pi ) = CC( p )
.
∀ p j ∈D j

The algorithms below compute the the number communication between pairs of processes
(N C( pi , p j )) and the communication execution number of each process (N C( pi )) taking
into account the data dependent loops.

1. Compute the communication execution number for each process (N C( pi ))

2. Compute the communication cost of each process pi (CC( pi ) = N bc × N C( pi )T )


Table 3. Communication cost CC( pi , p j ).

Processes CC N CC LCC
p1 , p2 32 0.333333 0.333333
p1 , p3 32 0.333333 0.500000
p2 , p3 32 0.333333 0.333333
Description 96 — —

3. Compute the communication execution number between each pair of processes of the
whole the description (N C( pi , p j )).

4. Compute the communication cost between each pair of processes (CC( pi , p j ) = N bc ×
N C( pi , p j )T )

5. Compute the communication cost of the whole description (CC(D) =
∀( pi , p j )∈D CC( pi , p j ))

6. Compute the global normalized communication cost of each process (N CC( pi ) =
CC( pi )
CC(D)
).

7. Compute the local normalized communication cost of each process (LCC( pi ) =
CC( pi )
CC( p j )
)
∀ p j ∈D

8. Compute the global normalized communication cost of each pair of processes
CC( pi , p )
(N CC( pi , p j ) = CC(D)j )

9. Compute the local normalized communication of each pair of processes (LCC( pi , p j ) =
CC( pi , p j )/(CC( pi ) + CC( p j ))).

Applying the proposed algorithm to the example shown in Figure 14.a, the results de-
scribed in Table 3 and in Table 4 are obtained (considering that an integer is represented
by 32 bits). Figure 14.b shows the net that represents the control ﬂow of the algorithm
described.

Table 4. Communication cost of pro-
cesses.

Process CC N CC LCC
p1 32 0.333333 0.333333
p2 64 0.666667 1
p3 32 0.333333 0.333333


Figure 14. Example.

It is important to stress that the algorithm proposed is based on the well known struc-
tural Petri net methods, that is, it is a formal approach that takes into account the com-
putation of semiflows (invariants) in order to find the number of transition firings re-
lated to communication action in the processes. One common measure for quantifying
communication cost is through the delay related to communication actions between pro-
cesses [68, 91]. In the method proposed in this section, communication cost is calcu-
lated in terms of the amount of data transferred between processes. This approach allows
one to compute the communication cost without considering assignments in a particular
processor, since it is unnecessary to estimate the exact time for the communication ac-
tions.

12. Load Balancing

In this section a method for computing the static load balancing among simple occam pro-
cesses across processors [72] is presented. This metric is an important aspect of performing
the initial allocation as well as the partitioning.
The occam simple processes may be either sequential or concurrent. Parallel processes
may have distinct parallelism degrees. The load balance is defined in terms of loading
related to processes in the processors. For instance, if two processes pi and p y are assigned
to two processors (processor K, processor L), the load balance among this pair of processors
related to that specific pair of processes is defined in terms of the process loading in the
processors.

A petri net model for hardware software codesign

A petri net model for hardware software codesign

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