937
1 INTRODUCTION
According to International Maritime Organization
(IMO) Resolution A.893(21) “Guidelines for voyage
planning”, adopted on 25 November 1999, for crewed
ships, the travel time, time of arrival at the destination
port and the time in which the trip will be completed
should be determined after completing the work
related to the sea voyage planning. The same
standards should be maintained in Maritime
Autonomous Surface Ships (MASS) operation.
The implementation of MASS sea voyage plan is a
subject to monitoring operational, technical and
environmental parameters, control of a safety level
and risk assessment by MASS operator in Remote
Operation Centre (ROC). The responsibilities of the
ROC operator are also making decisions related to
necessary changes in the plan, changing the stages of
the planned voyage or withdrawing from the plan.
Preparation for MASS sailing out to sea should
meet the functional requirements to safely navigate in
accordance with an appropriate voyage plan
determining safe routes. The voyage plan should be
approved by the responsible person to ensure safe
navigation of MASS. A detailed voyage or passage
plan should cover the entire voyage or passage from
berth to berth. The voyage plan should be developed
taking into account the following issues:
− the voyage plan should ensure that sufficient
information is provided to operators to enable
operations to be conducted with due regard to the
safety of the ship and persons,
− navigational charts and publications should be
updated with the latest available information [8, 9],
− a comprehensive information should be provided,
including operational design domain (ODD) for
autonomous navigation,
− it should be possible to define and update an
itinerary, describing the complete voyage from
Semi-Markov Model of MASS Voyage
T. Abramowicz-Gerigk, Z. Burciu, J. Soszynska-Budny & A. Weintrit
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The paper presents the readiness/safety model for the sea voyage of Maritime Autonomous
Surface Ship (MASS), based on Semi-Markov process. The states of MASS during the voyage were defined as
triplets of intentional type of MASS operations, reliability state and safety state, dependent on weather
conditions. The determined states were aggregated, and the disjoint subsets were used to build a macro-model
of MASS voyage process. An example of changes of MASS states during the sea voyage, for Gdynia - Port
Everglades connection, is presented and discussed in the paper. The Semi-Markov process was used for the
analysis of MASS reliability and safety during the sea voyage. The obtained matrix of transition probabilities
between the states of MASS during the voyage can be used by MASS operator in Remote Operations Centre to
make decisions related to voyage planning. The proposed model can constitute the basis of a computer program
supporting the decision-making process of the operator.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 18
Number 4
December 2024
DOI: 10.12716/1001.18.04.21
938
departure to arrival at the destination port, at any
time,
− the crew and ROC operator should check the
correct entry of the travel plan into ANS.
The model of MASS voyage safety proposed in the
paper makes it possible to obtain information on the
level of MASS safety at any stage of the sea voyage.
The concept of a stochastic (random) process with a
discrete set of states and a continuous set of
parameters can be used to mathematically describe
MASS during the sea voyage.
Constructing a model describing MASS during sea
voyage should begin with defining the states. The
state of MASS is determined by the state of its
components at the moment t. The final form of the set
of states of MASS during a sea voyage depends on the
modeling purpose. The model should facilitate the
ROC operator in making decisions leading to the safe
completion of the MASS sea voyage. The weather [2,
3], port infrastructure [4] conditions and
psychophysical condition of the ROC operator [1]
plays an important role in this process. In the
presented model only the weather conditions are
considered.
The most important issue in constructing a random
process, is to define the possible states of the MASS
unit during the sea voyage.
2 STATES, PARAMETERS AND
CHARACTERISTICS OF MASS
The states of MASS during the voyage should be
defined including:
− information on the intentional type of operation at
a given moment,
− information on the technical condition of MASS,
i.e. reliability and safety status depending on
hydrometeorological conditions.
Each state of MASS can be described as a triplet
(1):
( )
,,=
ijk
S i j k
(1)
where:
i - intentional operational state of MASS,
j - reliability state of MASS,
k - safety state of MASS.
The possible operating, reliability and safety states
of MASS are presented in Tables 1 - 3.
Table 1. Operating states of MASS
________________________________________________
i Intentional operational state of MASS
________________________________________________
1 Completion of loading operations, preparation for
departure
2 Port maneuvers, leaving the port
3 Sea voyage
4 Approach maneuvers, entering the port, port maneuvers,
berthing and mooring
________________________________________________
Table 2. Reliability states of MASS
________________________________________________
j Reliability state of MASS
________________________________________________
0 Unsuitability
1 Partial suitability
2 Suitability
________________________________________________
Table 3. The possible safety states of MASS
________________________________________________
k Intentional safety state of MASS
________________________________________________
0 Inability to complete the voyage
1 Relative hazard to safety
2 Complete safety
________________________________________________
The states defined above combined in triplets
describe MASS state, for example the state Sijk=(3,2,1)
means that MASS is on a sea voyage, in full suitability
state (in fully technically fit) and is in a state of
relative safety hazard, while the state Sijk =(3,1,2)
means that the MASS is on a sea voyage, in partial
suitability state (in partially technically fit) and in
state of complete safety.
The set of all possible states S in this case consists
of: 4 x 3 x 3 = 36 elements.
Such a number of states for a single unit is too
large to build a macro-model of the sea voyage
process. Therefore, states are aggregated, creating
disjoint subsets (2):
(2)
3 MODEL OF THE SEA VOYAGE PROCESS OF
THE MASS UNIT
The subsets Z0, Z1, ..., Z8 are taken as the final states
of MASS during her sea voyage. To facilitate the
description, we can replace symbols denoting states
with their numbers Z_k\leftrightarrow k,\vthicksp
k=0,1,2,...,8.
The process states are defined in Table 4.
Table 4. States of MASS sea voyage process
________________________________________________
k State of MASS
________________________________________________
0 Total unsuitability to perform sea voyage
1 Completion of loading operations, preparation for
departure of MASS in partial suitability state and/or with
relative hazard to safety
2 Port maneuvers, leaving the port of MASS in partial
suitability state and/or with relative hazard to safety
939
3 Sea voyage of MASS in partial suitability state and/or
relative hazard to safety
4 Port maneuvers, entry to the port, berthing and mooring
of MASS in partial suitability state and/or with relative
hazard to safety
5 Completion of loading operations, preparation for
departure of MASS with total technical suitability and/or
complete safety state
6 Port maneuvers, leaving the port of MASS with total
technical suitability and/or complete safety state
7 Sea voyage of MASS with total technical suitability
and/or complete safety state
8 Port maneuvers, entry to the port, berthing and mooring
of MASS with total technical suitability and/or complete
safety state
________________________________________________
An example of changes of MASS states during the
sea voyage is presented for Gdynia - Port Everglades
connection for the assumed four variants of this sea
route.
The variants of Gdynia - Port Everglades route are
presented in Figure 1.
Figure 1. Gdynia - Port Everglades variants of MASS sea
routes
The MASS states changes can be presented in a
graph, showing all possible assumed states and
transitions between the states.
The graph of changes in MASS states during
Gdynia - Port Everglades sea voyage, computed for
the assumed sea routes is presented in Figure 2.
The symbols used in the graph, including four
MASS route variants are as follows:
− A – Gdynia
− B – Hammerodde Fyr (Bornholm)
− C – Skagen
− D – Kiel Canal
− E – English Channel, Dover Strait
− F – Port Everglades - Pentland Firth
− G – Port Everglades – Falmouth (Great Circle)
− H – Port Everglades – Falmouth (rhumb line)
− I – Port Everglades – Falmouth via Acores
Constructing the MASS sea voyage process, the
probabilistic characteristics, generating the process
implementation are assigned to the directed arcs of
the graph.
If the process describing MASS during a sea
voyage takes values in the subset of states S2={5,6,7,8},
this means that the MASS sea voyage goes normally.
If this process takes values in the set S1={1,2,3,4}, this
means that the MASS sea voyage is subject to
disruptions caused by her technical condition and/or
hydrometeorological conditions resulting, i.a in
increased hazard to safety during the sea voyage. The
process state defined by the single-element subset
S0={0}, means that MASS is not conducting a sea
voyage and may be in a state of failure and/or with
relative hazard to safety.
5
0
4
8
3
76
21
B
G
H
I
F
ED
C
B
I
E
D
Warianty tras podróży morskiej MASS:
A – Gdynia
B – Hammerodde Fyr (Bornholm)
C – Skagen
D – Kiel Canal
E – English Channel, Dover Strait
F – Port Everglades – Pentland Firth
G – Port Everglades – Falmouth (ortodroma)
H – Port Everglades – Falmouth
(loksodroma)
I – Port Everglades – Falmouth via Acores
X
Y
Figure 2. Graph of changes in MASS unit states during
Gdynia - Port Everglades sea voyage, developed for the
assumed four (F ‒ I) Gdynia - Port Everglades sea routes
To construct the MASS sea voyage process, the
directed arcs of the graph (Fig. 2) should be assigned
probabilistic characteristics that generate realizations
of the process. The arc (i, j) corresponds to the
probability of changing state from i to j, which is
denoted by the symbol pij and the distribution
function Fij(t) of the random variable Tij denoting the
duration of state i when the next state will be state j.
Then the function Q_{ij}(t) (3) is the probability of
changing state from i to j in time no longer than t.
( ) ( )
,0=
ij ij ij
Q t p F t t
(3)
Determining the transition probabilities functions
for individual arcs of the graph from equation (3), we
can determine a functional matrix Q(t), called the
kernel [5, 6, 7], which, together with the initial
distribution, defines the Semi-Markov process {X(t): t
0} with the set of states S={0,1,2,…,8}.
The kernel of the Semi-Markov process has the
following form (5):
( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
01 05
10 12 15 16
20 23 26 27
30 34 37
40
51 56
62 67
73 78
80 84
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
Q t Q t
Q t Q t Q t Q t
Q t Q t Q t Q t
Q t Q t Q t
Qt
Qt
Q t Q t
Q t Q t
Q t Q t
Q t Q t







=









(4)
The sea voyage process of MASS defined in this
way allows the use of Semi-Markov process to
analysis of the reliability and safety of the unit.
The theory of Semi-Markov processes [5, 6, 7]
results in the following property:
( )
lim , 0
→
=
ij ij
t
p Q t t
(5)
940
In the presented model, using equations (3), (4)
and property (5), we can obtain a matrix of transition
probabilities between states for the considered MASS
during her sea voyage (6).
01 05
10 12 15 16
20 23 26 27
30 34 37
40
51 56
62 67
73 78
80 84
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0







=







pp
p p p p
p p p p
p p p
Pp
pp
pp
pp
pp
(6)
The determined probabilities can be used by the
ROC operator to make decisions related to the
planning and implementation of MASS sea voyage.
The readiness/safety model for the sea voyage of
MASS is presented in Figure 3.
In order for the ROC operator to make decisions,
the critical values of probabilities were determined in
the following ranges RMASS<0.5, 0.5RMASS0.8 and
RMASS>0.8.
Depending on the situation from which range we
received the values of the probability of transition
between states generated in the matrix, we have a
different decision scenario.
If RMASS>0.8, it means that the task can be carried
out without disruptions and that the unit is in the
highest reliability state and there are no threats to its
safety. Then the operator can, for example, choose any
variant of route form X variants (Figure 2).
If 0.5RMASS0.8, it means that the unit is in a state
of intermediate airworthiness and in an intermediate
state of safety, which means that the operator must
identify the reasons of this state and decide to
continue the task and choose the route from variants
Y, e.g. the more safe route variant, or one of the states:
reliability or safety should be improved by for
example repairing or changing the course in order to
improve the ship's performance in difficult weather
conditions.
When RMASS<0.5, it means that there is little chance
of completing the task or continuing the journey,
MASS is at risk and a decision should be made either
to interrupt the journey or to increase the probability
by increasing the reliability and safety status to allow
the MASS voyage to continue.
4 DISCUSSION AND CONCLUSIONS
The model of safety and reliability of an autonomous
unit proposed in the paper, during the
implementation of a transport task, using semi-
Markov processes, can support ROC operator
decision-making related to voyage planning and
conducting.
It can also contribute to ensuring an appropriate
level of safety.
The final effect of the model is a matrix of
transition probabilities between states. In reliability
models, the characteristics of Semi-Markov process
translate into the reliability characteristics of the
modeled object MASS.
Figure 3. Readiness/safety model of MASS sea voyage
941
The determined probabilities give us information
about the probability with which MASS is able to
perform a given task under current weather
conditions and in its current technical condition. In
other words, we can say how reliable it is to perform a
given task.
The proposed model can constitute the basis of a
computer program supporting the decision-making
process of the operator of an autonomous unit.
ACKNOWLEDGEMENT
This work was supported by the projects of Gdynia
Maritime University No. WN/2024/PZ/03 and No.
WN/2024/PZ/08.
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