indicates this problem in the translation process as
the most difficult (Mauw, 1995).
Resulted PN is analysed automatically to verify
the properties of the system using linear algebra.
Systems of linear equations over the set of natural
numbers are resolved using TSS method (Kryvyy,
2002) developed by one of the authors of this paper,
and which shows very high performance on large-
scale systems comparatively to the existing methods.
Given work is based on well-known definitions
of PNs theory, and classical definitions of Incidence
Matrix, State Equation and PN Invariants (Murata,
1989).
3 TELECOM EXAMPLE
Let’s consider the example of translation and
analysis of the real telephone system with basic
services traditionally called Plain Old Telephony
Service, POTS. The formal model of POTS is
presented as ordinary PN. One of the advantages of
building a formal model is to ensure the design is
correct and meets certain requirements. A correct
design of POTS at least has the following required
properties: must be a limitation on connection
channels resource usage; the telephone network
restores to its initial state after a talk of two
subscribers; the subscribers can call each other
indefinite number of times irrespectively given
network configuration; the telephone network can’t
get in deadlock state.
The set of MSC-diagrams presented in the
Figure 2 describes the work of POTS. Note that
MSC-diagrams Nº3, 4 and 8 on the Figure 2 shall be
repeated symmetrically relative to m
th
/n
th
instances.
m
th
(n
th
) instance corresponds to m
th
(n
th
) subscriber.
Let’s apply to the given set of MSCs the
algorithm of automatic translation (refer to section
2), and simplify obtained PN using net reduction
(Murata, 1989). The resulting ordinary PN is
illustrated in the Figure 3. The initial PN marking is
M
0
=(1,0,0,0,0,0,0,1,0,0,k), where k is the number of
connection channels. The 11-th place models pair
connections of the all network subscribers, and
corresponds to connection channels resource. So, we
have built the formal model aimed at analysis and
verification of the real system. In the given PN,
transitions are respectively interpreted as events of
data message transitions in the MSCs (Figure 2) in
the following way: t1=offhook(m), t2=dial_n,
t3=onhook(m), t4=busy, t5=onhook(m),
t6=ring(m,n), t7=offhook(n), t8=onhook(n),
t9=onhook(m), t10=onhook(m), t11=offhook(n),
t12=onhook(n); where offhook(m) means that the
m
th
subscriber hang off the phone,
onhook(n)/onhook(m) means that the n
th
/m
th
subscriber hang on the phone, ring(m,n) means that
m
th
is calling n
th
. The PN’s places in the Figure 3 are
named respectively as conditions of the given
MSCs: P1=“m free”, P2=“m busy”, P3=”dial state”,
P4=”NW_dial”, P5=”busy state”, P6=”ringing
state”, P7=”connected”, P8 =”n free”, P9=”n busy”,
P10=”dial state”, P11=”NW_free”.
To verify the correctness of the given model for
POTS with respect to the above properties it’s
necessary to calculate the PN’s S- and T-invariants.
The following invariants of the PN are obtained
automatically using TSS method (Kryvyy, 2002):
S-invariants — s
1
=(0,1,1,0,0,0,0,0,0,0,0),
s
2
=(0,0,0,0,0,1,0,1,0,0,0), s
3
=(1,0,0,1,1,1,1,0,0,0,0),
s
4
=(0,1,0,1,1,0,1,0,1,1,0), s
5
=(1,0,1,0,0,0,0,0,0,0,1,),
s
6
=(0,0,0,0,0,0,0,1,1,1,1), s
7
=(1,0,0,1,1,0,1,0,1,1,1);
T-invariants — t
1
=(1,0,1,0,0,0,0,0,0,0,0,0),
t
2
=(0,0,0,0,0,0,0,0,0,0,1,1), t
3
=(1,1,0,1,1,0,0,0,0,0,0,0),
t
4
=(1,1,0,0,0,1,0,0,0,1,0,0), t
5
=(0,1,0,0,0,1,1,1,0,0,0,0),
t
6
=(1,1,0,0,0,1,1,0,1,0,0,1).
Automatic analysis of invariants of the PN
proves the following properties of this model.
Boundedness: The given PN is structurally
A u tom a tic
tran slationofMSCs
intoPetriNet
Automaticanalysisof
th eformalmodel
PetriNetwhi c h
representtheformal
modelofthesy ste m
Theset ofMSCs
whi chdesc ribes
th esy st em
Ve rdictofthe
sy stem’sanalysis
(e rr orsan d
problemsofthe
sy stemrepresented
Fi
ure1
Figure 1
AUTOMATIC ANALYSIS AND VERIFICATION OF MSC-SPECIFIED TELECOMMUNICATION SYSTEM
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