Table of Contents
bcg_steady - steady-state numerical analysis of (extended) continuous-time
Markov chains encoded in the BCG format
] [-sol solution_file
] [-mat matrix_file
] [-log log_file
where bcg_options is defined below (see GENERAL OPTIONS).
performs steady-state analysis on filename.bcg
, which is an (extended) continuous-time
Markov chain encoded in the BCG format.
bcg_steady first transforms filename.bcg
into a numerical matrix indexed by states. Then, it reduces this matrix
by normalizing probabilistic transitions, removing unreachable states and
"vanishing" states, keeping "tangible" states only (see section INPUT below
for details about the BCG graphs accepted by bcg_steady and the definition
of tangible and vanishing states). As a result, the reduced matrix obtained
is the generator matrix of a continuous-time Markov chain. Finally, bcg_steady
computes the corresponding equilibrium ("steady-state") probability distribution
on the long run using the Gauss/Seidel algorithm (see e.g. [Ste94]). It can
also compute throughputs for the transitions of the system.
are currently supported: -version
, and -tmp
. See the bcg
manual page for a description of
Taking as input filename.bcg
, on the form
of which various restrictions apply (see section INPUT below), bcg_steady
can produce five kinds of output files, depending on the command-line options
The optional list of "parameter=value" arguments at the end of
the command-line (where parameter is any character string that neither contain
blanks nor the "=" character, and where value is any character string that
does not contain blanks) is only meaningful to option -thr. These arguments
have no influence on the actual numerical computations, they only serve
to add columns in throughput tables (see section OUTPUT-2 below).
options are supported:
- -epsilon eps
- Set the precision of certain floating-point
comparisons to eps, where eps is a real value. When eps is out of [0..1[,
bcg_steady reports an error. Default value for eps is 1E-6.
- -sol solution_file
- Write the probability vector, computed at the equilibrium state, to file
solution_file (see section OUTPUT-1 below for a description of the file
format). If solution_file already exists, its contents will be overwritten.
If solution_file is equal to the special string `-', the probability vector
is displayed on the standard output. Not a default option.
- -thr [-append] throughput_file
- Write the transition throughputs, computed at the equilibrium state, to
file throughput_file. The format of this file is determined by the suffix
(i.e., file extension) of throughput_file (see section OUTPUT-2 below for
a description of the available file formats). If throughput_file already
exists, its contents will be overwritten, unless the -append option is specified,
in which case the transition throughputs will be added at the end of throughput_file
so as to form a proper data table. If the -thr option is missing or if throughput_file
is equal to the special string `-', the transition throughputs are displayed
on the standard output. Option "-thr -" is the default.
- -mat matrix_file
the transposed "raw" matrix (prior to matrix reduction) to file matrix_file.
The format of this file is determined by the suffix (i.e., file extension)
of matrix_file (see section OUTPUT-3 below for a description of the available
file formats). If matrix_file already exists, its contents will be overwritten.
If matrix_file is equal to the special string `-', the matrix is displayed
on the standard output. Not a default option.
- -red reduced_matrix_file
the reduced transposed matrix to file reduced_matrix_file. The format of
this file is determined by the suffix (i.e., file extension) of reduced_matrix_file
(see section OUTPUT-3 below for a description of the available file formats).
If reduced_matrix_file already exists, its contents will be overwritten.
If reduced_matrix_file is equal to the special string `-', the matrix is
displayed on the standard output. Not a default option.
- -log log_file
various information about data structures and computations to file log_file.
The format of this file is undocumented but self-understandable, and might
change in future releases of bcg_steady. If log_file already exists, its
contents will be overwritten. If log_file is equal to the special string
`-', information is displayed on the standard output. Not a default option.
The files solution_file, throughput_file, matrix_file, reduced_matrix_file,
and log_file should be pairwise different; otherwise, the result is undefined.
The input of bcg_steady
is an extended Markovian model
combining features from discrete-time and continuous-time Markov chains. In
order to be accepted by bcg_steady
must satisfy several conditions,
otherwise an error message will occur. All transition labels of filename.bcg
must have one of the following forms:
rate %f" (called a stochastic transition),
; rate %f" (called a labelled stochastic transition),
prob %p" (called
a probabilistic transition), or
; prob %p" (called a labelled probabilistic
%f denotes a strictly positive floating-point number,
%p denotes a floating-point number in the range ]0..1], and label denotes
a character string that does not contain the "
;" character (label may be
equal to the internal action, often noted "i" or "tau").
labelled with only "label" are always forbidden by bcg_steady, including
the case where "label" denotes the internal action.
When constructing the
"raw" matrix, all labels occurring in labelled probabilistic transitions
If there exists a (labelled) probabilistic transition from
a state S1 to a state S2, then all (labelled) stochastic transitions from
S1 to any state (including S2) are discarded when constructing the "raw"
matrix. This reflects that probabilistic transitions are instantaneous,
while stochastic transitions are not.
We classify states as being either
vanishing if at least one (possibly labelled) probabilistic transition
goes out of these states, or tangible otherwise.
The input BCG graph should
contain at least one tangible state, and it should not contain cycles of
vanishing states connected by (possibly labelled) probabilistic transitions.
Loops, which are transitions from and to identical states, are simply ignored.
Note: The sum of
%p values on all (possibly labelled) probabilistic transitions
leaving a vanishing state needs not be equal to 1; if this sum is different
from 1, then probabilistic values will be normalized (i.e., divided by this
To build the reduced matrix, bcg_steady eliminates all vanishing states,
so that this matrix contains tangible states only. The input BCG graph
should be such that, after reduction, each tangible state possesses at
least one outgoing transition (i.e., there is no deadlock state).
for a discussion about the various probabilistic and stochastic
models present in the literature.
of the file generated using the -sol
option of bcg_steady
is the following.
There is one line per tangible state. Each line contains two numbers: an
integer corresponding to the state index in the reduced matrix (see section
OUTPUT-3 below) and a real number corresponding to the probability of being
in this state on the long run (i.e., at the equilibrium).
State indexes, both
in the solution vector and in the matrices, start from 1 (and not from
The throughput table has two (possibly
empty) groups of columns:
- The first group contains one column for each
option parameter=value given on the command-line. These columns, if any,
are useful when evaluating the performance of a system parameterized with
one or more variables, namely to aggregate in the same table the different
throughputs measures corresponding to different values of the parameters.
Columns of the first group appear in the same order as the corresponding
options on the command-line.
- The second group contains one column per labelled
stochastic transition label present in the input BCG graph, precisely,
one column for each different label occurring on a transition of the form
; rate %f".
The throughput table starts with a first "header" line
followed by one or several "subsequent" lines.
- The header line contains
the "titles" of columns. For a column of the first group associated to a
parameter=value option, the corresponding title is parameter. For a column
of the second group associated to a label, the corresponding title is the
- The subsequent lines contain values. For a column of the first
group associated to a parameter=value option, the corresponding cell contains
value. For a column of the second group associated to a label, the corresponding
cell contains the throughput for this label, i.e., the sum, for each stochastic
; rate %f", of the rate value %f weighted with the probability
of being in the tangible source state of this transition, in the long run.
If the -append option is absent, or if the throughput file is equal to the
special string `-', or if the throughput file does not exist, or if it is
empty, bcg_steady will generate automatically the header line and one single
Otherwise, the first line of the throughput file is expected
to contain the titles of columns and will be parsed to identify the correspondance
between labels and columns. In particular, bcg_steady checks that the first
group of columns corresponds to the parameters given on the command-line.
After parsing the header line, bcg_steady will append one single subsequent
line at the end of the throughput file. As regards the second group of columns,
if the label of a given column title does not occur in filename.bcg, a zero
throughput will be reported in the corresponding column; conversely, labels
of filename.bcg for which there is no corresponding column title will be
Throughputs can be displayed in two different formats, which are
determined according to the suffix (i.e., file extension) of the throughput
- If the file name has the ".csv" extension, the throughput table
will be displayed in the CSV (Comma-Separated Values) exchange format understood
by most relational data base applications and spreadsheet software (such
as Microsoft Excel, etc.).
- Otherwise, if the file name has a different extension,
or no extension, or if it is the standard output, throughputs will be displayed
in a human-readable format that is essentially the same format as CSV with
commas replaced by spaces so as to align columns properly. Note that this
format is also understood by some data visualization tools such as Gnuplot.
Both the (transposed) "raw" matrix produced
using option -mat
and the (transposed) reduced matrix produced using option
follow the same format conventions. The essential difference is that
the former contains vanishing and tangible states, whereas the latter only
contains tangible states. Also, the reduced matrix is a generator of a continuous-time
For two different indexes i and j, the element (i,j) of the
matrix, located at the i-th row and the j-th column, is the sum of all the
floating-point numbers associated to the (labelled) stochastic or probabilistic
transitions going from the j-th state to the i-th state, where floating-point
numbers associated to (labelled) stochastic transitions are interpreted
as positive numbers whereas floating-point numbers associated to (labelled)
probabilistic transitions are interpreted as negative numbers between -1
and 0. Note that rates and probabilities are never mixed since, between
two states, there cannot be stochastic and probabilistic transitions at
the same time.
The diagonal elements (j,j) are defined to be the negative
sum of all matrix elements (i,j) with i different from j.
be displayed in three different formats, which are determined according
to the suffix (i.e., file extension) of the matrix file name.
- If the file
name has the ".csv" extension, the matrix will be displayed in the CSV (Comma-Separated
Values) format mentioned above. Each raw of the matrix is displayed on one
line of the output file, and on each line, the matrix elements are separated
- If the file name has the ".spm" extension, the matrix will be
displayed in the format used by the Sparse 1.3 software library (see the
CREDITS section below). This format is the following. The first line of
the file contains the file name. The second line contains the number of
states, followed by the "
real" keyword. Then, there is one line per each
non-zero element (i,j) in the matrix. Each line contains two integers followed
by one real number: the value of i, the value of j, and the value of matrix
element (i,j). The file ends with a "sentinel" line consisting of three
- Otherwise, if the file name has a different extension, or no extension,
or if it is the standard output, the matrix will be displayed in a human-readable
format. The columns of the matrix are split into "packets" so that the text
fits on the size of the display. The indexes of rows and columns are indicated
and null elements of the matrix are displayed as "..." instead of "0". Statistics
(such as matrix size, density, etc.), are displayed after the matrix.
for graphs with many states, whatever the chosen matrix format, the matrix
files can be large and writing them to disk may take time.
See the bcg
manual page for a description of the environment
variables used by all the BCG application tools.
is 0 if everything is all right, 1 otherwise.
The first version of
was written by Christophe Joubert (INRIA/VASY) and Holger Hermanns
(Saarland University and University of Twente). The algorithm for steady-state
analysis is based on a former implementation by Vassilis Mertsiotakis (University
of Erlangen). Bruno Ondet (INRIA/VASY) ported the tool to various architectures.
David Champelovier and Hubert Garavel (both at INRIA/VASY) deeply revised
code and manual page to allow their integration within
CADP. Holger Hermanns and Frederic Lang (INRIA/VASY) proof-checked the manual
uses (an extended version of) the Sparse 1.3 package
from the University of California, Berkeley, developed by Kenneth S. Kundert
under the supervision of Alberto Sangiovanni-Vincentelli.
- BCG graph (input)
- dynamic library (input or output)
- ``bcg_steady'' program
See the bcg
page for a description of the other files.
Additional information is available
from the CADP Web page located at http://cadp.inria.fr
Directives for installation
are given in files $CADP/INSTALLATION_*.
Recent changes and improvements
to this software are reported and commented in file $CADP/HISTORY.
report bugs to email@example.com
[GH02] H. Garavel and H. Hermanns.
On Combining Functional Verification and Performance Evaluation using
CADP. In proceedings of FME'2002, LNCS 2391, pages 410-429, Springer Verlag.
Full version available as INRIA Research Report 4492. Available from http://cadp.inria.fr/publications/Garavel-Hermanns-02.html
[HJ03] H. Hermanns and Ch. Joubert. A Set of Performance and Dependability
Analysis Components for CADP. In proceedings of TACAS'2003, LNCS 2619, pages
425-430, Springer Verlag. Available from http://cadp.inria.fr/publications/Hermanns-Joubert-03.html
[Mer98] V. Mertsiotakis. Approximate Analysis Methods for Stochastic Process
Algebras. Ph.D Thesis, University of Erlangen (Germany), 1998.
[Ste94] W. J.
Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton
University Press, 1994.
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