analysis of duration of the selected operational states of warsaw

ZESZYTY NAUKOWE INSTYTUTU POJAZDÓW
1(105)/2016
Andrzej Szymanek1, Marzenna Dębowska–Mróz2, Marta Orlińska3, Piotr
Kozłowski4
ANALYSIS OF DURATION OF THE SELECTED OPERATIONAL STATES OF
THE WARSAW METRO PASSENGER INFORMATION SYSTEM
1. Introduction
In recent years we have witnessed significant technological progress in
implementation of intelligent transport systems (ITS) – telematics systems which should
exhibit the following characteristics [6]:
 application for the physical systems which are spatially distributed and consist
of a large number of elements,
 application where communication between the environment and the user is
needed,
 integration of many electronic technique functions (including hardware and
software),
 capability of instant response to possible changes in functioning,
 data transfer, gathering and processing,
 high reliability for the sake of users’ safety,
 possibility of the intelligent system extension by introducing new functions or
elements.
The above mentioned features of telematics systems account for their wide range of
applications. A crucial dependence describing the telematics system operation in view of
its application seems to be its availability to perform the task which is assigned to it.
This concept is used for an analysis and assessment of the telematics system functioning
at random moments
. An example of such an application may be the
systems supporting the technical measure operational process which is represented,
among others, by the Warsaw Metro Passenger Information System discussed below.
One of the important features which characterize the above mentioned system is the fact
that it is capable of starting the task performance on time within the required time
interval. In the case discussed, the timely starting of a task accomplishment in the
required time interval shall mean that the system together with its technical resources
and the supervising team will begin the task accomplishment in the required time , and
the task will be completed in the pre-determined time
[2].
Getting acquainted with the characteristics of the telematics system operational
process, first of all, it is necessary analyse its properties which describe the given
dr hab. inż. Andrzej Szymanek, prof. UTH - Wydział Transportu i Elektrotechniki, Uniwersytet
Technologiczno Humanistyczny w Radomiu
2
dr inż. Marzenna Dębowska - Mróz, Wydział Transportu i Elektrotechniki, Uniwersytet Technologiczno
Humanistyczny w Radomiu
3
mgr inż. Marta Orlińska - Wydział Transportu i Elektrotechniki, Uniwersytet Technologiczno Humanistyczny
w Radomiu
4
mgr inż. Piotr Kozłowski - Dyrektor do spraw przewozów pasażerskich w Warszawskim Metrze
1
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process [1]. To do this one has to distinguish its particular operational states. The correct
identification of the actual research object and the operational process accomplished in it
together with an analysis of the significance of their individual interconnections or their
components is the foundation of solving complex research problems. By observation of
the occurring operational events and processes accompanying them, identification of the
complex systems and operational processes occurring in them is accomplished. In this
way it is possible to correctly determine the most important operational states and the
values of parameters describing a given process [2].
This paper outlines the methodology of determining particular operational states of
the Warsaw Metro Passenger Information System.
2. Duration of the operational states of the Warsaw Metro Passenger Information
System
For the purposes of an analysis of the characteristics describing the operational
process of the Warsaw Metro Passenger Information System, the following operational
states , have been distinguished in which the described system may be at any moment
in only one of the distinguished subsets of the operational states:

– wait state of the operable system to start the carrying out of the assigned task,

– state of task performance,

– wait state of the system for a repair - treatment,

– state in which the system is effectively repaired.
To analyses the properties of the Warsaw Metro Passenger Information System
correctly it is not enough to distinguish particular operational states. They must be
considered in a more comprehensive way and additionally all possible transitions
between them must be identified. An event-based model was used for modelling the
operational process of the described system. It was to present the analyzed process
graphically. Owing to the construction of a graph of transition between the distinguished
operational states it will be possible to map the operational process of the system in
which the operational states will be the vertices, and arcs will be the possible transitions
between them [2]. Therefore the operational process model was presented graphically by
constructing a transition graph between particular operational states (Fig. 1).
Fig. 1. Directed graph operation process Passenger Information System of the Warsaw
Metro , where: – wait state of the operable system to start the carrying out of
the assigned task, – state of task performance, – wait state of the system for
a repair - treatment, – state in which the system is effectively repaired [4, 5]
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The outlined model of the operational process of the Warsaw Metro Passenger
Information System in a graphic representation is implemented in the actual transport
system. Thanks to it we can determine a mathematical model of the duration time
distribution for particular operational states. It must be assumed, however, that
a homogenous semi-Markov process
, is a model of the analysed process, whereas
a random process
with a finite set of states . is a model describing the operational
process of the examined system. The process described is a discrete-state process and
continuous in time. It was also assumed that the duration time of operational states
is a random variable of any probability distribution defined on the space of positive real
numbers , and transition of a subsystem to a specific operational state does not depend
on the sequence of states but only on the current state. The operational state of such
a subsystem for
is treated as a semi-Markov process with a finite number of
states [3, 4, 5].
The process is defined by a homogenous Markov chain with a state-transition matrix
, and conditional probability distribution of random variables [4, 5]:
(1)
and the initial distribution [4, 5]:
(2)
(3)
where:

indexes of possible operational process states

probability of the process transition from state to state ,

cumulative distribution function of random variable

random variable defining the state’s duration time
in the next stage it will move to state ,

a set of operational process states.
,
on condition that
The model of the telematics system operational process does not take servicing into
account [4]. It is also assumed that random variables indicating the time in which the
system is in the state
on condition that a transition to state
will occur, are
defined by [4, 5]:
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Additionally described random variables also have positively defined functions of
probability density
, finite expected values
and finite second moments
).
Therefore, the knowledge of the following functions and numerical characteristics is
assumed [4, 5]:
),
),
),
),
),
),
),
)
),
),
),
),
),
),
),
)
Owing to the determination of duration times of particular operational states of the
Warsaw Metro Passenger Information System, irrespective of the state to which its
operational process moved, partial evaluation of its functioning will be possible.
The operational process of the described system will be described by random
variables , for i=1, 2, 3, 4, which successively mean [4]:

– the time in which the operable system remains in the wait state to start
accomplishing a task,

– the time in which the system described is carrying out the task,

– the time in which the system waits for repair,

– the time in which the system is under repair.
Using dependence (4) describing the expected value of a particular operational state
duration, the expected values for the Warsaw Metro Passenger Information System were
determined represented by dependences (5), (6), (7) and (8) respectively, which describe
random variables
for i=1, 2, 3, 4.
( )=
(4)

for random variable
the expected value of the time in which the system waits to
start performing the task [2]:
(5)

for random variable
the expected value of the time in which the system is
carrying out the task assigned to it [2]:
(6)

for random variable
repair [2]:
the expected value of the system being in the wait state for
(7)
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
for random variable
repair [2]:
the expected value of the system being in the state under
(8)
Due to co-operation with the Warsaw Metro it was possible to determine the
analysed expected values. The summary of results is depicted in Figure 2.
Fig. 2. The expected values for the Warsaw Metro Passenger Information System
3. Conclusions
The Warsaw Metro Passenger Information System is operated in the hours when
the metro trains are running. Taking into consideration the metro working hours and the
time the system is waiting for a task accomplishment, both before and after work, as well
as prolonged working hours due to circumstances, e.g. mass events organized by the city
of Warsaw, it has been assumed that the system described operates ca. 21 hours per day.
Figure 3 includes a percentage summary of particular operational state duration of
the Warsaw Metro Passenger Information System. It indicates that the system described
is used in ca. 67%, in other words duration of carrying out the task amounts to 13.87 h.
Another parameter discussed is the wait time for carrying out the task assigned which
takes ca. 23% of the system operation, i.e. 4.67 h. On the other hand, the system’s wait
time for repair took ca. 8% and the repair time ca. 2%, which gives 1.68 h and 0.32 h,
respectively.
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Fig. 3. Percentage summary of particular operational state duration of the Warsaw Metro
Passenger Information System
References
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eksploatacyjnych. Oficyna Wydawnicza Polskiego Towarzystwa Zarządzania
Produkcją, t. 2, str. 152–161, 2010 r.
[2]. Migawa K.: Sterowanie gotowością w systemach eksploatacji środków transportu.
Rozprawa habilitacyjna nr. 168, Bydgoszcz 2013 r.
[3]. Niziński S., Kolator B.: Ocena efektywności funkcjonowania systemów
eksploatacji pojazdów i maszyn z wykorzystaniem procesów Markowa.
Motoryzacja i Energetyka Rolnictwa, MOTROL nr. 8, str. 156–168, 2006 r.
[4]. Siergiejczyk M.: Efektywność eksploatacyjna systemów telematyki transportu,
Prace Naukowe Politechniki Warszawskiej z. 67, Warszawa 2009 r.
[5]. Siergiejczyk M.: Metoda oceny efektywności eksploatacyjnej systemów telematyki
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i Techniki Informacyjne nr. 1–2, str. 116–130, 2005 r.
Abstract
This publication is devoted to the analysis of the distribution of the duration of the
particular operating statuses of the Passenger Information System of the Warsaw Metro.
Firstly, I specify the operating statuses that the system can reach. Secondly, I present
a directed graph of the operation process, which enables to identify all the possible
transitions between the specified states, and, ultimately, to determine their expected
duration time. This analysis was performed using semi-Markov processes. Finally, the
article contains the evaluation of the Passenger Information System of the Warsaw
Metro based on the results.
Keywords: telematics systems, passenger information system of the Warsaw Metro,
semi-Markov processes, schedule durations operating conditions.
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ANALIZA CZASÓW TRWANIA WYBRANYCH STANÓW
EKSPLOATACYJNYCH SYSTEMU INFORMACJI PASAŻERSKIEJ METRA
WARSZAWSKIEGO
Streszczenie
Niniejsza publikacja przedstawia zagadnienia związane z analizą rozkładów
czasów trwania poszczególnych stanów eksploatacyjnych Systemu Informacji
Pasażerskiej Warszawskiego Metra. W pierwszej kolejności wyszczególniono stany
eksploatacyjne w których może znaleźć się opisywany system, następnie przedstawiony
został graf skierowany procesu eksploatacji, który pozwolił na określenie wszystkich
możliwych przejść pomiędzy wyszczególnionymi stanami, by w konsekwencji
wyznaczyć wartości oczekiwane ich czasów trwania. Analiza ta dokonana została przy
wykorzystaniu procesów semi-Markowa. W dalszej kolejności przedstawiono ocenę
Systemu Informacji Pasażerskiej Warszawskiego Metra na podstawie uzyskanych
wyników.
Słowa
kluczowe: systemy telematyczne, System Informacji Pasażerskiej
Warszawskiego Metra, procesy semi-Markowa, rozkład czasów
trwania stanów eksploatacyjnych.
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