International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 6
Number 3
September 2012
357
1 INTRODUCTION
Transport is a complex system combining advanced
technical systems, operators and procedures. All
these elements work in a large spatial dispersion, but
are closely interrelated. They interact, and the time
horizon of these interactions is very short. In sea, air
or railway transport, the risk is traditionally identi-
fied with the accidents, which typically produce a
high number of deaths and huge financial losses. Se-
verity of the consequences is the reason why the
safety was always a key value in transport.
For instance Polish aviation regulations define
three categories of events (Aviation Law, 2002):
− accident - as an event associated with the opera-
tion of the aircraft, which occurred in the pres-
ence of people on board, during which any person
has suffered at least of serious injuries or aircraft
was damaged,
− serious incident - as an incident whose circum-
stances indicate that there was almost an accident
(such as a significant violation of the separation
between aircraft, without the control of the situa-
tion both by the pilot of the aircraft and the con-
troller),
− incident - as an event associated with the opera-
tion of an aircraft other than an accident, which
would adversely affect the safety of operation
(e.g. a violation of separation, but with the control
of the situation).
In this paper, traffic incidents in transport are sub-
ject of interest. A method for modelling these inci-
dents with use of Petri nets theory is presented. This
method allows the analysis of the causes of incidents
as well as assessing the probability of transformation
of incidents into accidents. The method uses col-
oured, stochastic, timed Petri nets, with the time as-
signed to markers.
The first part of the paper presents basic infor-
mation about Petri nets. The next discusses the spec-
ificity of the analyzed transport systems and a meth-
od of modelling those using coloured, timed Petri
nets. The next two chapters contain examples of
analysis using the proposed method. The first exam-
ple comes from the air traffic and presents the calcu-
lation of the possibility of transforming a serious in-
cident into an accident (Skorupski 2010). The
Modelling of Traffic Incidents in Transport
J. Skorupski
Warsaw University of Technology, Faculty of Transport, Warsaw, Poland
ABSTRACT: Safety is one of the most important criteria for assessing the transport process. The traffic pro-
cess in available traffic space are partly organized and planned. However, these plans are subject to numerous
disturbances of probabilistic nature. These disturbances, contribute to the commission of errors by the opera-
tors of vehicles and traffic managers. They lead to traffic incidents, which under certain circumstances may
transform into accidents.
In the paper the method of modelling traffic incidents, using different types of Petri nets is presented. Exam-
ple of the serious air traffic incident shows the opportunities offered by the application of this modelling tech-
nique. In addition, the possibility of its use in maritime transport, for example, modelling of traffic at the wa-
terways intersection is presented.
358
second example concerns the maritime traffic and
demonstrates the applicability of the method for
modelling conflict at the intersection of the water-
ways.
2 THE BASICS OF PETRI NETS
Petri nets provide a convenient way to describe
many types of systems. Especially a lot of applica-
tions they found in software engineering, where they
are used particularly to describe and analyze concur-
rent systems. There is a rich literature in this subject,
e.g. (Jensen, 1997, Szpyrka, 2008), which also con-
tains an extensive bibliography of the topic. In this
paper it was shown that Petri nets can also be used
for modelling transport systems, particularly the traf-
fic processes. The examples concern the analysis of
traffic safety problems in air and maritime transport,
but a similar approach can be applied to other modes
of transport.
2.1 Types of Petri Nets
Depending on the needs, one can define different Pe-
tri nets with certain properties. However, there is a
set of characteristics that are common to such net-
works. The basis for building a Petri net is a bipartite
graph containing two disjoint sets of vertices called
places and transitions. Arcs in this graph are directed
and single, and therefore it is a Berge graph. A char-
acteristic feature of the graph used in Petri nets is
that the arcs have to combine different types of ver-
tices. Below are presented brief definitions of basic
types of Petri nets: first low, then a high level (Mar-
san et al. 1999). Detailed analysis of the properties
of various types of nets is included in the literature
and will not be discussed here.
2.2 Generalised Petri net
Generalised Petri net (GPN) is described as:
=
{
, , , ,
}
(1)
where:
P - set of places,
T - set of transitions, = ,
I, O, H, are functions respectively of input, output
and inhibitors:
I, O, H: T → B(P)
where B(P) is the superset over the set P.
Given a transition
Tt ∈
it can be defined:
=
{
:
(
,
)
> 0
}
- input set of transition t
=
{
:
(
,
)
> 0
}
- output set of transition t
=
{
:
(
,
)
> 0
}
- inhibition set of transi-
tion t
GPN is characterized by the fact that the func-
tions described on arcs: I(t,p), O(t,p) and H(t,p), can
take values greater than 1, which is equivalent to the
presence of multiple arcs between nodes.
2.3 Marked Petri net
Marked Petri net (MPN) is described as:
=
{
, , , , ,
}
(2)
where: =
{
, , , ,
}
- generalised Petri net,
:
is the initial marking, i.e. a function as-
signing an integer to each place.
We also say that the marking specifies the num-
ber of markers assigned to each of the places.
Initial marking, along with the rules governing
the dynamics of the net, that is rules of marking
changes, determine all possible reachable markings.
The same network but with different initial markings
will describe different systems.
Transition t is called active in marking M if and
only if:
,
(
)
(
,
)
,
(
)
<
(
,
)
(3)
Firing of transition t, active in marking M re-
moves from any place p belonging to the set
, as
many markers as function
(
,
)
determines. At the
same time it adds to any place p from the set
, as
many markers as determined by the
(
,
)
function.
This means firing of transition t will change actual
marking to
M
′
such that
= +
(
)
(
)
(4)
This relationship is written briefly
[
. We
then say that M’ is reachable directly from M. If the
transformation requires firing a sequence of
transitions σ, then we say that M’ is reachable from
M and denote
[
.
2.4 Place-transition Petri net
Place-transition net (PTN) is a generalized, marked
Petri net, supplemented by the characteristics of
places interpreted as their capacity, i.e. the maxi-
mum number of markers that can accommodate any
of the places. Thus, a place-transition net can be
written as
=
{
, , , , , ,
}
(5)
where: =
{
, , , ,
}
- generalised Petri net,
:
{
}
– capacity of places, and the sym-
bol ∞ means that a place has unlimited capacity,
359
:
:
(
)
(
)
– initial
marking.
2.5 Timed Petri net
With timed Petri net (TPN) we have to do, when fir-
ing a transition is not immediate, but it takes a cer-
tain time. This means that definition of such net
would take into account the timed characteristics de-
scribed on the transitions
=
{
, , , , ,
,
}
(6)
where
=
{
, , , , ,
}
– marked Petri net,
:
– delay function, specifying static delay
τ(t) of transition t.
Characteristics on transitions may determine time
associated with firing of the transition in different
ways. In particular, this value may be described by a
deterministic or a random variable with a given
probability distribution. In the latter case, we may
talk about the stochastic network. In addition to stat-
ic delay it is sometimes convenient to use dynamic
delay δ(t), defined as the rest of the time remaining
until the firing of the transition t.
In timed Petri nets, the problem of verifying the
conditions required for activation of transitions is
closely related to treatment of transitions that have
not been fired due to the expiration of the time less
than τ(t), and which had lost activity. Depending on
the specific system being modelled, there are three
approaches possible:
− lack of memory – after firing of any transition,
dynamic delays for all transitions are set back to
the initial value, i.e. ,
(
)
=
(
)
,
− active memory – in case of firing any transition t,
all other transitions, which lost activity as a re-
sult, shall take the value of dynamic delay equal
to the initial value (as in the lack of memory
case), and the transitions that remain active - will
retain their existing value of δ(t),
− absolute memory – no matter which transition
fires, all other transitions retain their dynamic de-
lay value, and at next activation, countdown of
the time remaining for firing continues.
2.6 Coloured Petri net
The main difference between generalised and col-
oured nets is the ability to define markers of differ-
ent types. This is possible in coloured Petri nets
(CPN). Marker type is called a colour. Each place in
the coloured net is assigned a set of colours that it
can store. Expressions are assigned to arcs and tran-
sitions that allow manipulating various types of
markers. Coloured Petri net can be written as
=
{
, , , , , , , , ,
}
(7)
where
=
{
, , , , ,
}
– marked Petri net,
Г – nonempty, finite set of colours,
C – function determining what colour markers can
be stored in a given place: : ,
G - function defining the conditions that must be sat-
isfied for the transition, before it can be fired; these
are the expressions containing variables belonging
to Г, for which the evaluation can be made, giving as
a result a Boolean value,
E – function describing the so-called weight of arcs,
i.e. expressions containing variables of types belong-
ing to Г, for which the evaluation can be made, giv-
ing as a result a multiset over the type of colour as-
signed to a place that is at the beginning or the end
of the arc.
2.7 Coloured, timed Petri net
It is possible to combine the idea of CPN and TPN.
In this case the following structure of coloured,
timed Petri net (CTPN) is formed
=
{
, , , , , , , , ,
, ,
}
(8)
where:
=
{
, , , , ,
}
– marked Petri net,
Г – nonempty, finite set of colours, each of which
can be timed, that means whose elements are pairs
consisting of colour and a timestamp,
C, G, E – have the same meaning as in the case of
CPN, but taking into account the fact that certain
sets of colours can be timed,
R - set of timestamps (also called time points),
closed under the operation of addition, ,
r
0
– initial time, .
In the TPN it is necessary to implement a model
clock, which defines the local time flow. This is
achieved usually by using timestamps, which are
generally associated with the markers. This clock is
used to determine which of the transitions can be ac-
tivated. The condition for activation is the existence,
for all input places of the transition, markings, in
which all timestamps are smaller than local time.
The timed coloured Petri net changes the meaning
of the marking M, in relation to the timed colours. In
this case, the marking consists of a number of mark-
ers together with their timestamps, which may be
different for each of the markers.
State of the system modelled by coloured, timed
Petri net is called the pair (M,r), where M is the
marking and is a timestamp.
2.8 Petri nets properties
For each Petri net we can determine among others:
the reachability graph, reachability set, evaluate the
360
reversibility, the presence of deadlock, liveness, and
boundedness. In the presented method of analysis,
the most important property of the net (modelling a
traffic incident) is the reachability of selected states
(markings) from initial marking M
0
. It allows as-
sessing the probability and time of transition to those
selected markings. Particularly important are the
dead markings, because they illustrate the situations
in which we can assess whether the traffic process
results in an incident or in an accident.
In many cases, the reachability graph is very
complex and difficult to study, especially with the
analytical methods. In those cases, methods to re-
duce the graph will be extremely useful (Sistla A.P
& Godefroid P., 2004). The transport applications
will use mostly the reduction related to stable sets of
transitions. Reduction using symmetry will be used
much less frequently.
3 MODELLING OF TRAFFIC INCIDENTS
WITH THE USE OF PETRI NETS
As it is widely known, the traffic incidents in
transport systems are almost always a result of a
combination of many different factors. During the
development of a dangerous situation in time, there
are also inhibitory factors that hinder or prevent this
process.
Transport system includes:
− passive components, namely infrastructure, in-
cluding its characteristics,
− active elements, namely transport vehicles, per-
forming tasks and creating a traffic flow,
− organisation, i.e. the relations between the ele-
ments of the transport system, aimed at realisa-
tion of transport tasks.
In this paper active elements of the transport sys-
tem are studied, dealt dynamically, during the reali-
sation of their task - that is, the traffic processes. In-
frastructure and organisation are limitations to this
process and must be, to some extent considered dur-
ing its modelling.
The traffic process is ordered and designed to
reach a specific destination of vehicles using the
road (suitably organised in various branches of
transport), including the organisational rules, regula-
tions and standards to ensure the safety of all traffic
participants. In this process, there are time periods in
which vehicles move in a planned manner, in ac-
cordance with standard procedures. These fragments
of the traffic process are characterized by its dura-
tion. The process is dynamic, because there is a
change of position of vehicles in time, but from the
point of view of the purpose of analysis, which is
posed in this paper, it can be regarded as static. It is
possible because in those time periods there are no
events influencing the level of safety, and proce-
dures such as changing speed or direction are
planned, in accordance with the constraints resulting
from characteristics of infrastructure components
and tailored to the exploitation characteristics of ve-
hicles.
Between these fragments there are traffic events
which are extracted whereas the scope of the analy-
sis. In the case of an analysis designed to assess the
safety of the traffic process, these events are defined
as having an impact on safety of traffic. For such
events, one can include:
− occupation of conflicting point of the road (streets
junction, runway, waterways crossing) character-
ised by the fact that there may be only one vehi-
cle on it, or they may be few, but it is necessary
to specify the order of passing this point by vehi-
cles, as movement continued by each of them in-
dependently can lead to collisions,
− decision by the vehicle operator to continue the
movement, or to change its parameters (direction,
speed), in particular the decision to stop, or to re-
alise an emergency manoeuvre to avoid collision,
− decision by the traffic dispatcher (air traffic con-
troller, the railway station dispatcher, coordinator
of traffic in seaport) of a similar nature,
− decision by the vehicle operator to take action
that is inconsistent with the decisions (recom-
mendations) of traffic dispatcher,
− occurrence of dynamic and intensive meteorolog-
ical phenomena (storm, heavy fog), or other phe-
nomena of an environmental nature that may af-
fect the traffic process,
− occurrence of events (failures) associated with the
vehicle or traffic control system, which cause
hazard to vehicles.
The above mentioned events may have the nature
of conditions, which logical value can be evaluated.
In this case they are represented by a Boolean true
or false. They may also have a nature of a certain
process, mostly short-term. In this case, the event
will be represented by its type, but also by duration.
Such an approach to the traffic process allows the
use of Petri nets for modelling it. Stable traffic situa-
tions correspond to places in the net, traffic events –
to transitions. Markers in places can be identified as
traffic participants or states of environment. Partici-
pants may have different traffic characteristics. For
example, we may consider several types of vehicles
of varying size and performance. We may also con-
sider objects constituting the disturbances, affecting
the traffic process, such as pedestrians on the road,
ground service cars on taxiways at the airport. Simi-
lar interpretation can be applied to states of envi-
ronment or external events. Typically, these are log-
ical conditions, and therefore existence of a marker
361
in corresponding place represents the occurrence of
the event.
As one can see the markers are of different types,
which suggest the need to use coloured Petri nets.
This is obviously a universal solution, but in simpler
cases, the model of traffic incident can use a simpler
place-transition net. This is possible if parts of the
net using different types of markers are mostly dis-
joint. In cases where the same places are used by dif-
ferent types of markers CPN must be used.
Unlike other typical applications of Petri nets, in
modelling traffic processes in transport, in most cas-
es, it is necessary to use the timed Petri net. This re-
sults from the fact that time and the associated dy-
namic phenomena are often crucial in the analysis in
this area. For example, while modelling traffic inci-
dents, it is usually necessary to examine the time se-
quence of individual traffic situations, resulting in
specific sequence of occupation of conflicting
points. This sequence may decide about the occur-
rence of the accident or its avoidance. In specific
cases, sometimes it is preferable to use timed charac-
teristics associated with transitions, and sometimes
associated with markers.
There is also a class of applications of Petri nets
for modelling the traffic processes in transport,
where it is sufficient to use non-timed nets. This is
possible when considering only the sequence of
events leading up to the situation of interest, or se-
quence of events as a consequence of certain initiat-
ing event. This is in fact a study of an event tree
analysis, fault tree analysis, or bow-tie analysis. An-
alytical techniques derived from the theory of Petri
nets, applied in this case, can produce very interest-
ing results; in particular, can accelerate obtaining
satisfactory results with high accuracy.
This paper describes two examples traffic inci-
dents examination. First one is air traffic incident,
with particular emphasis on modelling the process of
transformation from the incident to an accident. Sto-
chastic TPN with time associated with transitions
was used. The second one is a model of waterway
intersection, where two conflicting traffic flows oc-
cur. In this case stochastic CTPN was used. Term
“stochastic” means that the time delays occurring in
the net are partially random values of given proba-
bility distributions. Network structure itself, howev-
er, is deterministic.
4 EXAMPLE ANALYSIS – SERIOUS AIR
TRAFFIC INCIDENT 344/07
As an example illustrating the method a serious air
traffic incident, which occurred in August 2007 at
Warsaw airport will be presented. Its participants
were Boeing 767 and Boeing 737 aircraft, and its
cause was classified as a "human factor" and the
causal group H4 - "procedural errors" (Civil Avia-
tion Authority 2009).
4.1 Circumstances of the serious incident
In the incident on 13
th
of August 2007 participated
two aircraft – Boeing 737 (B737) and the Boeing
767 (B767), which more or less at the same time
were scheduled for take-off from the Warsaw-
Okęcie airport. As the first, clearance for line-up and
wait on runway RWY 29 was issued to B737. As a
second, clearance for line-up and wait on runway
RWY 33 was given to B767 crew. The latter aircraft
was the first to obtain permission to take-off. A
moment after confirmation of permission to take-off,
both aircraft began start procedure at the same time.
B737 crew assumed that the start permission was
addressed to them. They probably thought that since
they first received permission to line up the runway,
they are also the first to be permitted to start. An air
traffic controller (ATC) did not watch planes take-
off, because at this time he was busy agreeing heli-
copter take-off. The situation of simultaneous start
was observed by the pilot of ATR 72, which was
standing in queue for departure. He reacted on the
radio. After this message, B767 pilot looked right
and saw B737 taking-off. Then, on his own initia-
tive, broke off and began a rapid deceleration, which
led to stopping the plane 200 meters from the inter-
section of the runways. Assistant controller heard
the ATR 72 pilot radio message and informed the
controller that B737 operate without authorization.
A controller, who originally did not hear the infor-
mation by radio, after 16 seconds from the start, rec-
ognized the situation and strongly ordered B737 to
discontinue take-off procedure. B737 crew per-
formed braking and stopped 200 m from the inter-
section of the runways.
4.2 Model of serious incident
This air traffic incident almost led to collision be-
tween the two aircraft, it means to accident. As in
most such situations, there were many factors con-
tributing to the creation of this dangerous situation.
The most important are:
− lack of situational awareness at the B737 crew,
− inadequate monitoring of radio communications
and, consequently, wrong acceptance of permis-
sion for the start, in fact directed to another plane,
− lack of the crew cooperation in the B737 cockpit,
− lack of proper monitoring of the take-off by the
controller,
− controller's lack of response to the information
from the pilot of ATR 72 transmitted by radio.
The factors impeding the development of the ac-
cident, which resulted in preventing it, include:
362
− good assessment of dangerous situation by the
crew of B767 and decision to immediately dis-
continue take-off,
− good recognition of the hazard by the crew of the
ATR 72 and immediate sending a message by ra-
dio,
− good weather conditions for visual observation of
the runways,
− proper response of assistant controller.
TPN model representing this serious incident is
shown in Figure 1.
Figure 1. The basic model of a serious air traffic incident
344/07
4.3 Model of air traffic accident
Analysis of the factors leading to the incident may
give an answer to the question what is the probabil-
ity of such incident. For example, one may check
how the situation would change if it was B767 the
first aircraft to obtain permission to line up the run-
way.
In the presented example, however, a goal is to
find a probabilistic dependence between the serious
incident and an accident that could result from it. In
this case, it is necessary to notice that it is sufficient
that there exists only one additional factor, and inci-
dent would in fact be an accident. There are several
scenarios that lead to an accident.
1 B767 crew, busy with their own take-off proce-
dure does not pay attention to the message trans-
mitted by radio by the ATR 72 pilot.
2 B767 crew takes a wrong decision to continue the
take-off, despite noting B737 aircraft.
3 ATR 72 pilot does not watch the situation on the
runways, just waiting for permission to line-up
the runway.
4 ATR 72 pilot observes a dangerous situation, but
does not immediately inform about it on the ra-
dio, instead discusses it with other members of
his own crew.
5 Assistant controller does not pay attention to the
information given by radio by the ATR 72 pilot,
or does not respond to it properly - does not in-
form the controller.
6 Weather conditions (visibility) are so bad that it is
impossible to see the actual traffic situation. This
applies to B767, ATR 72 crews, and the air traffic
controller.
All these scenarios will lead with certainty (or
with great probability) to transformation of the inci-
dent into an accident, and can be analyzed using Pe-
tri net model. In this analysis one should take into
account the possibility of occurrence of each scenar-
io separately, as well as several of them at once.
4.4 Probability of incident-accident transformation
Analysis of the probability of transformation of inci-
dent into an accident must take into account the
probability of each scenario mentioned above. In the
case of scenario 6 we can use statistical data on me-
teorological conditions (visibility) in the airport. But
in other scenarios, it is necessary to refer to experts'
evaluation.
Taking into account the objectives of the analysis,
it is possible to eliminate certain states without loss
of accuracy, while simplifying the analyzed model.
This applies, for example, to almost all the places
and transitions associated with the process of taxiing
and lining up the runway. For example, change the
set of places is determined as follows.
= (
)
(9)
where: P
w
- a set of places in the modelled accident,
P
r
- a set of reduced places,
P
d
- a set of places added to the model, to reflect the
above-mentioned scenarios.
In this case (Figure 1) P
r
= {“B767 awaiting per-
mission to start”, “B767 can line up RWY 33”,
“B767 on the RWY 33 threshold”, “B767 ready for
take-off”, “B737 awaiting permission to start”,
“B737 can line up RWY 29”, “B737 on the RWY 29
threshold”, “B737 ready for take-off”, “ATC not
busy”, “ATC busy”, “ATR observes a simultaneous
start”}.
On the other hand P
d
= {„ATR warns?”, “B737
continues to start”, “B737 at the crossing”, “B767
hears the warning?”, “B767 continues to start”,
“B767 at the crossing”, “B767 interrupts start?”,
“B767 begins deceleration”, “weather?”, “good visi-
bility}.
A similar modification was made in regard to
transitions, input, output and inhibition functions.
An additional issue to consider is change of transi-
tion type – from timed to immediate or vice versa.
()
() ()
()
()
()
()
()
()
()
()
()
()
()
()
2`()
()
()()
()
()
2`()
()
()
() () () ()
()
2`()
()()
()()
()
()
()
()
()
()
B737 - take-off
phase I
@+8
@+13
FD warns ATC
@+3
ATR warns about
simultaneus start
@+3
B737 interrupts
take-off
@+20
ATC orders to
stop take-off
@+2
Clearance for B737
to take-off
@+20
B737 lines up
RWY 29
@+100
Clearance for B737
to line-up RWY 29
@+20 @+10
B767 interrupts
take-off
@+12
B767 - take-off
phase I
@+8
Clearance for B767
to take-off
@+10
B767 lines up
RWY 33
@+100
Clearance for B767
to line-up RWY 33
@+20
B767 asks for
clearance
@+10
B737 ready
for take-off
A
ATC busy
A
ATC not busy
A
ATC sees
conflict
A
FD accepts
warning
A
ATR sees
simultaneus
take-off
A
B737 stops
A
B737 accepts
order to stop
A
A
A
B737 may
line-up RWY 29
A
A
B737 ready
for take-off
1`()@5
A
B767 stops
A
B767
accelerates
A
B767 ready
for take-off
AA
B767 may
line-up RWY 33
A
A
B767 ready
for take-off
1`()@0
A
B767 waits
for clearance
B767 at RWY 33
threshold
B737 at RWY 29
threshold
B737
accelerates
B737 waits
for clearance
B737 asks for
clearance
ATC agrees
helicopter
take-off
363
Petri net to model the transformation of the incident
into accident, after reduction is shown in Figure 2.
Figure 2. Model of serious incident 344/07 transformation into
air traffic accident (after reduction of the states).
This network may be treated as a stochastic timed
Petri net. Its analysis allows observing some inter-
esting relationships between a serious incident and
the air traffic accident. It also allows determining
some quantitative dependencies.
Assume the following places designations: p
1
–
„B767 ready for take-off”, p
2
– „B737 ready for
take-off”, p
3
– „B767 accelerates”, p
4
– „B737 ac-
celerates”, p
5
– „weather?”, p
6
– „good visibility”, p
7
– „ATR warns?”, p
8
– „B737 continues take-off”, p
9
– „FD accepts warning?”, p
10
– „B767 hears warn-
ing?”, p
11
– „ATC sees conflict”, p
12
– „B767 inter-
rupts take-off?”, p
13
– „B737 accepts order to inter-
rupt take-off”, p
14
– „B767 continues take-off”, p
15
–
„B767 begins braking”, p
16
– „B767 stops”, p
17
–
„B737 at crossing”, p
18
– „B767 at crossing”, p
19
–
„B737 stops”.
The set of all states, called a reachability set, for
model of accident is presented in Table 1.
The most important markings, from the perspec-
tive of the analysis presented in this article, are giv-
en in Table 2. Other states as well irrelevant places –
were omitted.
Table 1. The reachability set for the model of accident arising
from incident 344/07
M
0
p
1
+p
2
+p
5
M
1
p
1
+p
2
M
2
p
1
+p
2
+p
6
M
3
p
2
+p
3
M
4
p
1
+p
4
M
5
p
2
+p
3
+p
6
M
6
p
4
+p
6
M
7
p
3
+p
4
M
8
p
3
+p
4
+p
6
M
9
p
8
+p
14
M
10
p
7
M
11
p
6
+p
8
+p
14
M
12
p
6
+p
7
M
13
p
8
+p
18
M
14
p
14
+p
17
M
15
p
8
+p
18
M
16
p
6
+p
14
+p
17
M
17
p
9
+p
10
M
18
p
17
+p
18
M
19
p
6
+p
17
+p
18
M
20
p
10
+p
11
M
21
p
8
+p
10
M
22
p
9
+p
12
M
23
p
9
+p
14
M
24
p
11
+p
12
M
25
p
11
+p
14
M
26
p
8
+p
12
M
27
p
9
+p
15
M
28
p
11
+p
15
M
29
p
11
+p
18
M
30
p
13
+p
14
M
31
p
8
+p
15
M
32
p
11
+p
16
M
33
p
13
+p
15
M
34
p
13
+p
18
M
35
p
14
+p
19
M
36
p
8
+p
16
M
37
p
15
+p
17
M
38
p
13
+p
16
M
39
p
15
+p
19
M
40
p
18
+p
19
M
41
p
16
+p
17
M
42
p
16
+p
19
Table 2. Selected states of the system (model of accident)
M
18
M
19
M
40
M
41
M
42
p
6
– good visibility
0
1
0
0
0
p
16
- B767 stops
0
0
0
1
1
p
17
- B737 at crossing
1
1
0
1
0
p
18
- B767 at crossing
1
1
1
0
0
p
19
- B737 stops
0
0
1
0
1
States M
40
, M
41
, M
42
(called safe states) illustrate
situations in which there is no accident. States M
18
and M
19
represent the situation that analysed serious
incident transforms into accident. The joint probabil-
ity of finding system in one of these states is the
searched probability of incident-accident transfor-
mation. It can be determined both analytically and
by simulation using a suitable software tool. Analyt-
ical method for determining the sought probabilities
will be presented on the example of the final state
M
19
. Partial subgraph of the reachability graph, for
reaching M
19
from initial state M
0
is shown in Figure
3.
Figure 3. Partial subgraph of reachability of final state M
19
.
Let’s assume the following transitions designa-
tions: t
1
– „bad weather”, t
2
– „good weather”, t
3
–
„B767 take-off phase I”, t
4
– „B737 take-off phase
I”, t
5
– „ATR not watches”, t
6
– „ATR watches”, t
7
–
„ATR warns”, t
8
– „ATR not warns”, t
9
– „B767
hears”, t
10
– „B767 not hears”, t
11
– „FD not ac-
cepts”, t
12
– „FD accepts”, t
13
– „B767 interrupts”,
t
14
– „B767 not interrupts”, t
15
– „B767 decelerates”,
t
16
– „ATC orders B737 to interrupt”, t
17
– „B737 in-
INT
1
1
P16
INT
T15
1
INT
1
P15
1
T13
1
INT
1
1
INT
1
1
INT
1
1
INT
1
INT
1
INT
1
INT
INT
1
1
1
1
INT INT
1
1
1
1
1 1
INT
1
1
P2
T4
T3
P3
P4
T5
1
1
INT
1
T18
1
T7
P9
P10
T9
P12
T10
P14
T12
1
P13 T17
P19 P18
INT
1
1
T14
T16
P11
1
1
INT
1
1
INT
1
1
P7
T6
1
1
INT
1
T2
P5
T1
1
P1
1
P8
1
T11
P6
T8
T19
P17
364
terrupts take-off and stops”, t
18
– „B767 take-off
phase II”, t
19
– „B737 take-off phase II”.
Immediate transitions t
1
, t
2
, t
5
, t
6
, t
7
, t
8
, t
9
, t
10
, t
11
,
t
12
, t
13
, t
14
are assigned weights, respectively: α
1
, α
2
,
α
5
, α
6
, α
7
, α
8
, α
9
, α
10
, α
11
, α
12
, α
13
, α
14
. These weights
are used to determine the probability of firing transi-
tions in a situation of a conflict. Timed transitions t
3
,
t
4
, t
15
, t
16
, t
17
, t
18
, t
19
are assigned the intensities of
realisation, respectively: μ
3
, μ
4
, μ
15
, μ
16
, μ
17
, μ
18
, μ
19
.
Also for this type of transitions in the event of a con-
flict, it is necessary to determine the probability of
firing one of the conflicting transitions.
Because of the purpose of analysis, it is possible
to reduce the reachability graph. Reduction consists
of the removal of states that do not affect the proba-
bility of finding the system in the state M
19
. Reacha-
bility graph after reduction is shown in Figure 4.
Figure 4. Reduced reachability graph for state M
19
.
In this case, the probability that the system will
move from the state M
0
to M
19
depends on the prob-
abilities of firing of immediate transitions t
2
, t
5
, t
6
and t
8
, and is described by the two sequences σ
1
and
σ
2
, and after reduction of intermediate states is as
follows:
=
[
,
[
,
(10)
=
[
,
[
[
,
(11)
(
[
)
=
+
(12)
It is worth noting that in this case the probability
of transforming incident into accident is not affected
by intensities of timed transitions, and only the
weights of immediate transitions.
5 EXAMPLE ANALYSIS – VESSEL TRAFFIC
AT WATERWAYS INTERSECTION
Majzner & Piszczek (2010) formulated the interest-
ing problem of analysis of traffic safety at the inter-
section of the waterways. This problem can be mod-
elled using the presented method.
Two streams of traffic are studied: longitudinal
moving along the fairway with the speed v
w
and
crossing stream moving with the speed v
p
. It was as-
sumed that the ships moving in the longitudinal
stream have the right of way to the ships in the
crossing stream. The study analyses the average
waiting time for ships of crossing stream and the
probability of avoiding a premise for a collision, as a
function of intensity of longitudinal stream. Example
of Petri net for modelling this kind of problem is
presented in Figure 5. The net is the coloured, sto-
chastic, timed Petri net with priorities.
Figure 5. CPN modelling incidents at waterway intersection.
Places designations are: p
0L
– “unit from longitu-
dinal stream arrives at intersection”, p
0C
– “unit from
crossing stream arrives at intersection”, p
1
– “vessel
occupies intersection”, p
1-limit
– “anti-place for limit-
ing the number of vessels at intersection to 1”, p
wait
–
“vessel waits in the queue”.
Transitions designations are: t
1
– “unit from lon-
gitudinal stream enters the intersection”, t
2
– “unit
leaves the intersection”, t
3
– “unit from crossing
stream enters the intersection”, t
4
– “unit from cross-
ing stream enters the waiting area”, t
5
– “unit from
waiting area enters the intersection”,
Assuming the figures from discussed example we
obtain similar results. For example, for the parame-
ters shown in Figure 5 (longitudinal traffic - 4 units
per hour, crossing traffic - 3 units per hour) con-
sistency of results from simulation experiments with
the results of the sample model is above 90% for
mean delay time.
e
2
e
2
1
1`e
E
2@1
VESSEL
1@0
VESSEL
VESSEL
VESSEL
2
2
@+(rnd(120))
P_LOW
P-wait
T4
P0-C
2@+(rnd(1200))
T5
2
@+(rnd(120))
2@+(rnd(1200))
T1
T2
P1
P0-L
1
@+(rnd(200))
e
T3
2
(v)
1@+(rnd(900))
P1-limit
e
365
This indicates the usefulness of the proposed
modelling method to analyze safety and traffic ca-
pacity problems in the fairways. We may also expect
good results while researching other problems in the
field of maritime traffic engineering.
6 SUMMARY AND CONCLUSIONS
In the paper the method of modelling traffic inci-
dents and accidents was presented.
Petri nets are used for modelling. Type of net
used, depends on individual case and objective of
analysis. Presented examples show the applicability
of the proposed method for analysis of traffic pro-
cesses in various modes of transport. The use of Pe-
tri nets allows to easily generating the reachability
graph, which is the basic tool of analysis. This graph
is often large and the effective application of the
method depends on its reduction. The problem of ef-
fective reduction constitutes a different research
problem.
Method can be used in practice for improvement
of the transport safety. The case described as avia-
tion example is a part of analysis that is necessary
before any new equipment or procedure can be in-
troduced. The simple model described as maritime
example may be used as a part of more complex op-
timisation models for marine traffic engineering.
REFERENCES
Aviation Law 2002. Act of 3 July 2002 (Journal of Laws of
2002, No. 130, item. 1112) (in Polish)
Civil Aviation Authority 2009. Statement No. 78 of President
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air event No. 344/07, Warsaw. (in Polish)
Jensen K. 1997. Coloured Petri Nets. Basic Concepts, Analysis
Methods and Practical Use. Monographs in Theoretical
Computer Science. Springer Verlag.
Majzner P. & Piszczek W. 2010. Investigation of vessel traffic
processes at waterway intersections, Scientific Journals
Maritime University of Szczecin, vol. 21(93)/2010, p. 62-
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Marsan M.A., Balbo G., Conte G., Donatelli S., Franceschinis
G. 1999. Modelling with Generalized Stochastic Petri Nets,
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ca.
Sistla A.P. & Godefroid P. 2004. Symmetry and reduced sym-
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