International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 4
Number 1
March 2010
49
The main goal of transport consists of movements of
various means of transport, carrying people and car-
go. This process shall be carried out maintaining a
high level of safety. To this end movements of
means of transport are based on ordered principles
of traffic organisation, which from the point of view
of technical functionality are fulfilled by systems of
vehicles movements’ control, so-called traffic con-
trol systems. These systems enable execution,
through equipment localised in various places of the
transport network, of appropriate control algorithms.
All components and systems of transport traffic
control are required to show operation certainty. The
operation certainty is understood as a probability of
defect non-occurrence. A defect consist in – at two-
state classification – a transition of a piece of
equipment (in defined operating conditions and at
defined time) from the state of availability (fitness)
to non-availability.
Transport traffic control systems work in diversi-
fied, frequently most critical, operating conditions.
The experience from such equipment operation con-
firms the dependence of proper systems’ functioning
on reliability of their components.
In transport traffic control equipment defects may
cause only traffic disturbances (e.g. delays), but also
occurrence of dangerous situations.
Traffic control system is a set of pieces of equip-
ment, which change their states between the state of
availability (i.e. the state of fitness to execute a task
or function in the system) and states of non-
availability at discrete moments in time, i.e. they are
dynamic states. The set of such systems’ states is a
discrete set. Transitions between the following states
are stochastic in nature and occur at random, in ac-
cordance with certain probability distribution.
The operation of traffic control systems to a large
extent shall focus on achieving appropriate availabil-
ity of traffic control equipment and on maintaining it
through a required period. This results in a need to
resolve problems of appropriate maintenance service
of traffic control equipment (repairs and inspections)
Most important features of safe systems include:
safety, availability, reliability, repairability.
Mutual links between main features of safe
systems are presented in Figure 1.
Figure 1. Links between features of safe systems
Availability of Traffic Control System Based on
Servicing Model
J. Mikulski
Silesian University of Technology, Katowice, Poland
ABSTRACT: Traffic control is a component of transport system, on which safety and efficiency of means of
transport movements substantially depend. It is not possible to achieve and maintain appropriate availability
of traffic control system unless issues of appropriate maintenance servicing of traffic control equipment have
been resolved. This will require applying a specific servicing policy, worked out on the basis of such system
availability model.
The servicing of a technical object is understood as any treatment, which results in
restoring the object’s state of availability. Servicing may consist in a repair of equipment or in its inspection.
Classification of servicing optimisation models, in respect of using appropriate mathematical methods, such
as Markov models, is presented.
.
50
The servicing of a technical object (equipment) –
Figure 2 – is understood as any treatment, which
results in restoring the object’s state of availability
(operational). Servicing may consist in a repair of
equipment or in its inspection, replacement of the
entire equipment with a new one or in replacement
of damaged components with new ones. Parameters
characterising the equipment at servicing must
ensure that it is operational, although they may differ
from a new object (in particular this refers to defects
intensity).
An inspection, maintenance and condition control
are comprised (apart from a repair) by so-called
technical service of equipment, which is opposite to
its use.
To prevent adverse effects of unpredicted defects
(failures), equipment which is still in the state of
availability (operational) is subject to servicing.
Such servicing is named preventive and
distinguished from emergency servicing.
Figure 2. Servicing of a technical object
Properties of any system (in the case considered,
of a transport traffic control system) indicate that
such system, from the point of view of its servicing,
may be presented (on certain level of generality) as a
set of states of using, repair and inspection (Fig. 3).
When analysing this diagram, the state of
availability (initial) and the state of effective control
(system transition between the basic state and the
control state under influence of introduction of
control command and after execution of the control
task) may be distinguished within the state of using.
But in servicing it is most important to distinguish
the state of repair and inspection.
Classification of servicing optimisation models,
both in respect of individual devices and systems,
also because of using appropriate mathematical
methods, such as inter alia linear and non-linear
programming, dynamic programming, but first of all
Markov models.
Game theory and stochastic processes theory,
mainly of Markov processes, are used to model the
process of technical objects operation (Koźniewska
& Włodarczyk 1978). Mass servicing theory
(referred to also as queuing theory) is strongly
related to technology and its development resulted
from practical demands.
In general form each queuing system may be
presented using a block diagram (Fig. 4).
System switching on
(starting)
Availability state
(initial)
Basic state
State
of control
State
of repair
State
of inspection
System switching off
State of effective
control
State
of using
Figure 3. States of the transport traffic control system
Requests stream
⇒
λ
Intensity of
requests stream
Queue
⇒
Process of servicing
Employee m
- - - - - - - - -
Employee 2
Employee 1
⇒
µ
Intensity of
servicing
stream
Output stream
Figure 4. Block diagram of a queuing system
A queuing system may be described using three
basic characteristics:
− Stream of requests – this is a statistical
description of process of requests arriving at the
system,
− Process of servicing – defines the process of
requests servicing performance,
− Queue regulation (discipline) – defines the
method of selecting the next request to be
serviced, if there is a queue in the system.
The case when these variables are subject to
exponential distribution is of great practical
importance.
The stream of requests is a statistical description
of the process of requests arrival at the servicing
system. It is usually described using distribution
functions for intervals between consecutive requests.
If this stream does not show variability, these
intervals are constant and the stream itself is of
deterministic nature. But if requests are arriving at
the system at random, then these intervals are a
random variable and then the function of their
distribution should be defined (Filipowicz 1997).
The following denotations are used:
Serviceability
PREVENTIVE SERVICE
Maintaining object’s functioning
Actions aimed at
reduction of defect probability:
REPAIR
Restoring object’s functioning
Actions carried out after defect’s emerging, to
restore required object’s function:
51
1
t
– average length of interval between two adjacent
requests,
λ – average intensity of requests stream (requests
intensity).
The relation between these values has the
following form
1
1
t
=
λ
Variable
1
t
stands for an important value in the
reliability technique – it is so-called average time
between failures. It is a measure of equipment
reliability.
In practice it happens very often that the time of
servicing is not constant and is subject to stochastic
fluctuations. In such a case it must be described
using appropriate distribution function. The time of
servicing is an important value characterising the
system of servicing. When considering the time of
servicing as a random variable, its distribution
function may be determined.
The following denotations are used:
2
t
– average time of request servicing,
μ – average intensity of request servicing.
The relation between these values has the
following form
2
1
t
=
µ
In the reliability technique the average time of
request servicing may mean, apart from repairs, also
servicing of inspections, and then inspections
intensity is an inverse of average time between
inspections.
A repair service of transport traffic control
equipment (group of m service employees carrying
out repairs of N pieces of equipment) is a typical
queuing system. Each piece of equipment is a source
of requests of intensity λ, while intensity of each
employee servicing is equal μ. Overall requests
intensity depends strictly on the number of damaged
equipment, i.e. it is a function of system states. Such
systems are named close queuing systems.
A visual diagram of traffic control equipment
servicing, as a mass servicing position, is presented
in Figure 5.
The simplest case of closed queuing system will
be considered. A two-state configuration of element
considered has been assumed:
− in state 0 the element fulfils its function (it is op-
erational) – state of using,
− in state 1 the element is damaged (it is not opera-
tional) – state of servicing.
System of repairs
Traffic Control
Equipment
queue of damaged equipment
to be repaired
1
2
m
1
2
N
Equipment
k
Servicing personnel
λ
λ
λ
μ
μ
μ
Figure 5. Visual diagram of transport traffic control equipment
servicing
The system consists of one piece of equipment
and one service employee. For the sake of considera-
tions clarity, the denotations have been given once
more:
− probability that the equipment is operational at a
given moment amounts to p
0
,
− probability that the equipment is damaged at a
given moment amounts to p
1
,
− intensity of equipment damage amounts to λ (the
ratio of the number of defects in a given interval
to the full time of equipment operation),
− intensity of servicing (performing repairs)
amounts to µ (ratio of the repairs number to the
full time of repairs duration).
The graph of states and transitions is presented in
Figure 6.
0
λ
µ
1
Fig. 6. Graph of transitions between the state of being opera-
tional and damage
1 piece of equipment,
1 service employee
where:
0 – means that the system is working properly, i.e. the equip-
ment does not require a repair (it is operational),
1 – means that the equipment has been damaged and requires a
repair.
When resolving this system we will obtain
( )
[ ]
( )
tt
eetp
µλµλ
µλ
λ
µλ
λ
µλ
λ
+−+−
+
−
+
=−
+
= 1)(
1
( )
[ ]
( )
tt
eetp
µλµλ
µλ
λ
µλ
λ
µλ
λ
+−+−
+
−
+
=−
+
= 1)(
1
Probability p
0
(t) defines so-called availability of
the system A(t). Availability A(t) is a probability that
the system is operational (usable) in the future, as-
suming it is operational at the initial moment. For
52
the exemplify system the course of availability is
presented
3
in Figure 7.
The limit value of availability is interesting
µλ
µ
+
=
∞→
)t(Alim
t
which also defines system fitness in a steady
state.
Fig. 7. Availability of the system
In example above only equipment repair was con-
sidered in servicing, while no preventive servicing
(periodic inspections) was taken into account. Pre-
ventive servicing is a planned undertaking, carried
out on an operational object to increase its reliabil-
ity.
Inspection service allows earlier finding of de-
fects (malfunctions), what enables preventing dam-
ages and increasing so-called availability of the sys-
tem.
Introducing a possibility to carry out (from time to
time) surveys and inspections of a given piece of equip-
ment the graph of transitions (similar to the graph in Fig-
ure 6) will look as in Figure 8. State 00 is the state of us-
ing (availability), and the other two states – servicing:
state 10 – a component is damaged and under repair, state
01 – the component is under inspection.
0,0
1,0
µ
0,1
µ
1
x λ
1
( 1 − x ) λ
Figure 8. Graph of transitions between states of fitness (opera-
tional), damage and inspection
1 piece of equipment,
1 service employee
3
for example for transport traffic control equipment, more spe-
cifically, for a signal
Intensity of equipment damage amounts to λ,
while intensity of repair request servicing amounts
to µ. Inspections intensity has been denoted by λ
1
,
while intensity of inspection request servicing, for
each service employee, amounts to µ
1
.
Resolving this system we obtain
( )
( )
( )
( )
[ ]
tt
tttt
ee
eeeetp
βα
βαβα
αββα
βααβ
δ
βα
γ
βα
βα
−−
−−−−
−+−
−
+
−
−
−−
−
=
1
)(
0,0
where:
C
B
A
BA
BA
C
CAB
A
=
−=−
=+
=
+=
+
=
−
=
++=
−=
+++=
αβ
βα
βα
µµδ
µµγ
β
α
µλλµµµ
µµλλ
1
1
111
2
11
2
2
4
Probability p
0,0
(t) defines also so-called availabil-
ity of the system A(t). For the exemplify system the
course of availability is presented in Figure 9.
Boundary availability, specifying system availa-
bility in a steady state
11111
1
1
1
)(lim
ρρµλλµµµ
µµ
αβ
δ
++
=
++
==
∞→
tA
t
Figure 9. Availability of the system
The case of system discussed in example above,
which task – apart from repair of damaged equip-
Time [days]
0,99
λ=0,0026548
3
Probability of fitness
p (t)
Time [days]
0,99888
λ=0,0026548
μ=3
λ
*
1
=0,0027397
Probability of fitness
p (t)
53
ment – consisted also of carrying out periodic in-
spections of the equipment, did not take into account
the fact that at the moment of switching the equip-
ment off for inspection this equipment was not
working. After all, an assumption is made (Zamojski
1980) that object’s reliability characteristics are
functions of working time that is the object may be
damaged only during work. Hence the time of defect
occurrence gets “elongated” and thereby in calcula-
tions the share of inspection in total intensity of de-
fects and inspections shall be considered. Percentage
of this share is determined by the selection coeffi-
cient
1
1
λλ
λ
+
=x
In other words, in the case analysed, the selection
coefficient specifies
average inspections number
(to) average number of repairs and inspections
This modification results in a change in the tran-
sitions graph model from Figure 8. Modified transi-
tions graph is presented in Figure 10.
0,0
1,0
µ
0,1
µ
1
x λ
1
( 1 − x ) λ
Fig. 10. Graph of transitions between states of fitness (opera-
tional), damage and inspection, taking into account the inspec-
tion time
1 piece of equipment,
1 service employee
Modified equation for probability that the system
is operational in such a case is
1
0,0
)1(1
1
ρρ
xx
p
+−+
=
or after substitution of coefficient x
1
1
1
1
0,0
1
1
ρ
λλ
λ
ρ
λλ
λ
+
+
+
+
=p
and after expansion of relative intensities ρ and ρ
1
( )
( )
µλµλµµλλ
µµλλ
µ
λ
λλ
λ
µ
λ
λλ
λ
2
11
2
11
11
1
1
1
1
1
0,0
1
1
+++
+
=
=
+
+
+
+
=p
The analysis of the course of probability function
of system availability (fitness) is interesting. Was
this function monotonously increasing, this would
mean full advisability of preventive actions (by the
way, for monotonously decreasing function preven-
tive actions would turn out “harmful”). In the event
that this function has an extremum, the introduction
of preventive actions affects object’s reliability in
different ways, depending on preventive actions fre-
quency (number) and on their duration.
Seeking for optimum inspections intensity that is
such inspections frequency for which probability of
correct operation would reach a maximum value, a
derivative of this expression shall be determined,
hence
( )
[ ]
( ) ( )
( )
[ ]
2
2
11
2
11
1111
2
11
2
111
1
0,0
2
µλµλµµλλ
µλµµµµλλµλµλµµλλµµ
λ
+++
++−+++
=
d
dp
This derivative is equal zero, if
02
1
2
1
2
1
=−+
µλλµλµλ
1
222
44
µµλµλ
+=∆
;
0 〉∆
always
hence
−
+
=
−+=
++−
=
1
11
2
22
1
1
1
2
1
µ
µµ
λ
µ
µ
λ
µ
µµµλ
λµ
λ
op
Checking, what is the condition for probability of
system availability (fitness) with inspections service
to be higher than in the case of only repair service,
consists in comparing appropriate expressions
( ) ( )
µ
λ
µλλ
λ
µλλ
λ
+
〉
+
+
+
+
1
1
1
1
11
2
1
1
2
The condition to satisfy this inequality is that
11
2
1
1
2
)()(
11
µλλ
λ
µλλ
λ
µ
λ
+
+
+
+〉+
µλµλµλλλ
2
11
2
11
)( +〉+
µλλµ
11
〉
54
that is that relative repairs intensity is higher than
relative inspections intensity
1
ρρ
〉
Calculations of system fitness in the event of in-
spections optimisation consist, having considered
the selection coefficient in calculations with “new”
defects intensity coefficients λ
*
and inspections in-
tensity λ
*
1
So the equipment defects intensity amounts then
to
1
2
λλ
λ
λ
+
=
∗
and the inspections intensity amounts then to
1
2
1
1
λλ
λ
λ
+
=
∗
On the other hand, the intensity of repair request
servicing amounts, as so far, to µ, and the intensity
of inspection request servicing for each service em-
ployee amounts, as so far, to µ
1
.
Resolving this system we obtain
( ) ( )
( )
( )
[ ]
tt
tttt
ee
eeeetp
βα
βαβα
αββα
βααβ
δ
βα
γ
βα
βα
−−
−−−−
−+−
−
+
−
−
−−
−
=
1
)(
0,0
where
C
B
A
BA
BA
C
CAB
A
=
−=−
=+
=
+=
+
=
−
=
++=
−=
+++=
∗
∗
αβ
βα
βα
µµδ
µµγ
β
α
µλµλµµ
µµλλ
1
1
*
111
2
1
*
1
2
2
4
Probability p
0,0
(t) defines also so-called availabil-
ity of the system A(t). The course of availability in
the event of application of the principle of inspec-
tions optimisation is presented in Figure 11, while in
the event of carrying out inspections with optimum
intensity – Figure 12.
Figure 11. The course of availability in the event of application
of the principle of inspections optimisation
Figure 12. The course of availability in the event of carrying
out inspections with optimum intensity
REFERENCES
Koźniewska I. & Włodarczyk M. 1978. Modele odnowy, nie-
zawodności i masowej obsługi (Models of replacement, re-
liability and mass servicing), Warszawa: Państwowe Wy-
dawnictwa Naukowe.
Filipowicz B. 1997. Modelowanie i analiza sieci kolejkowych
(Queuing networks modelling and analysis), Kraków: Wy-
dawnictwo Akademii Górniczo Hutniczej.
Zamojski W. 1980. Modele niezawodnościowo – funkcjonalne
systemów cyfrowych ze szczególnym uwzględnieniem sys-
temów jednoprocesorowych (Reliability-functional models
of digital systems with particular focus on one-processor
systems), Monografie No 10, Wrocław: Wydawnictwo Po-
litechniki Wrocławskiej.
Time [days]
0,99944
λ=0,0026548
μ=3
λ
*
=0,0013065
λ
1
=0027397
μ
1
=12
λ
*
1
=0,0013914
Probability of fitness
p (t)
0,99945
λ=0,026548
μ=3
λ
*
=0,0011872
λ
1
=0,0032815
Time [days]
Probability of fitness
p (t)
