169
1 INTRODUCTION
Crisis management has become last years an
importantpartofmanyaspectsofoureverydaylife.
This is coming out of both: increasing menace of
terrorist attacks, and increasing number of different
kindsofelementaldisasterstaking place in the near
past.
Oneofcrisismanagementmostimportant
features
isprotectionofcriticalinfra structure,thatisdefined
as: systems, and included within them, interconnected‐
objects,devices,installations,services,essentialforstate’s
safetyanditscitizens, serving for efficient functioning of
publicadministration,institutionsandbusiness.
Protection of critical infrastructure systems
becomes even more important, if considering
significant incidents,
that took place last years –
terrorist attacks (New York 2001, Madrid 2004,
London 2005), earthquakes resulting with tsunami
waves, causing huge destructions of large areas,
including sensitive objects placed inside them
(Japan2011), and floods caused by tropical cyclones
(Katrina – New Orleans 2005, Sandy – New York
2012).
This
paper is undertaking issues connected to
modeling of critical infrastructure systems safety,
basing on maritime transportation system, being
essential system for both: critical infrastructure, and
European critical infrastructure. The model of the
safety states transitions processes of critical
infrastructureisintroduced.
Further,paperincludesawayofapplicationofthe
modelintheevaluationandpredictionofthesafetyof
real process, concerned with determining the
unknownparametersoftheproposedmodel.
Particularly, concerning the safety states
transitions process of critical infrastructure, the
probabilitiesofthisprocessstayingatthesafetystates
attheinitialmoment,theprobabilitiesofthisprocess
transitions between the system operation states and
thedistributionsoftheconditionallifetimesofthisat
theparticularoperationstates,arealsoshown.
Maritime Transportation System Safety – Modeling
and Identification
P.Dziula,K.Kołowrocki&J.Soszyńska‐Budny
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT: The article is showing a concept of critical infrastructure systems’ safety states model. Model
constructionisbasingon:popula rtechnicalsystems’safetystatesmodels,andnotionsspecifiedinactsoflaw
andother studiesconcerningcrisismanagement.Paperisincludingsomeconcept ofproposedmodel
usage
possibilities‐methods and proceduresfor estimating unknownbasic parameters of safety states transitions
process:identifying thedistributionsofitsconditionallifetimeatsafetystates, estimatingprobabilitiesofits
staying at safety states at the initial moment, probabilities of its transitions between safety states and
parametersofthedistributionfor
thedescriptionofitsconditionallifetimesatsafetystates.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 2
June 2013
DOI:10.12716/1001.07.02.02
170
2 MARITIMETRANSPORTATIONSYSTEM
MODELING
2.1 Maritimetransportation systemasasubsystemof
criticalinfrastructureandEuropeancritical
infrastructure
Maritime transportation system is one of the most
important components of critical infrastructure.
disturbances to its functioning can cause significant
negative results for surrounding systems, including
naturalenvironment.
Act of Law
on Crisis Management (2007) is
indicatingalltransportsystemsingeneral,asapartof
critical infrastructure. Council Directive 2008/114/EC
goesevenfurther–indicatingseparately:Road,Rail,
Air, Inland waterways transport, andOcean and
short‐seashippingand ports,assectorsofEuropean
criticalinfrastructure.Europeancriticalinfrastructure
is
there defined as: critical infrastructure located in
Member States the disruption or destruction of which
would have a significant impact on at least two Member
States.
TheDirectiveisalsodemandingspecialeffortsand
activitiestobeundertakentoprotectEuropeancritical
infrastructure.
2.2 Maritimetransportation systemsafetystatesmodel
reflectingcrisismanagementphases
Implementation of crisis management issues and
problems into the technical systems’ safety states
models commonly known, has resulted in
formulating of critical infrastructure systems’ safety
statesmodel,illustratingprocessesconnectedtotheir
transitions, corresponding to particular crisis
managementphases(Figure1).
Figure1.Criticalinfrastructuresystems’safetystatesmodel
S0 state is seen as corresponding to threats zero
level.StateS
1standsforincreasedlevelofthreats,but
below level causing transition to crisis situation.
StatesS
0andS1canbeunderstoodasoneaggregated
state.Stayofsystematoneofthesestatescanbeseen
asonewidernothreatsstate
Aggregatednothreatsstatecanbeinterpretedasa
state,coveringintensiveeffortsofcrisismanagement
services, aiming to stand up for threats, meaning
increasingrate
oftransitionfromstateS1toS0.
The efforts are corresponding to following crisis
managementphases:
Prevention – analyzes of potentially possible
crisis situations, and undertaking activities
loweringprobabilityoftheirappearance,
Preparation – planning of actions (procedures),
thatshouldbeperformedincase of appearing of
foreseencrisissituations.
Crisissituationstateis
interpretedasaggregationof
two minor states shown in Figure 1: S
2 state,
illustrating threats level trespassing border of crisis
situation, but not causing damages to critical
infrastructure systems, and S
3 state, reached when
damagesmentionedaretakingplace.
Crisis management services efforts, undertaken
whencrisissituationoccurs,aimingtomovesystem
from crisis situation state into no threats state, are
namedReaction:
Reaction – undertaking of previously planned,
coordinated activities, leading to stop crisis
situation expanding, support casualties, and
restrictdamagesandlosses.
Transition between states S
2 and S3 reflects the
fourth, not mentioned until now, phase of crisis
management,namedRecovery(Reconstruction):
Recovery (Reconstruction) – restoration of
previousconditionsofcriticalinfrastructure.
Thus, the model is representing all four crisis
managementphases.
3 PROBABILISTICDESCRIPTIONOFSAFETY
STATESTRANSITIONS’PROCESS
According to outcome of chapter 2
above, critical
infrastructuresafetystatestransitionsprocessS(t),t∈
<0,+∞),canstayatoneoffourparticularsafetystates
S0,S1,S2, S3, already defined. Furthermore, it can be
assumed that critical infra structure safety states
transitions process S(t) is a semi‐Markov process,
withtheconditionalsojourntimesT
ijattheoperation
statesS
iwhenitsnextoperationstateisSj,i,j=0,1,2,
3i≠j.
The critical infrastructure safety states transitions
process can be described by its following basic
parameters:
thevector[p
i(0)]1x4oftheinitialprobabilities
),)0(()0(
ii
SSPp
,3,2,1,0
i
(1)
of the critical infrastructure safety states
transitionsprocessS(t)stayingatparticularsafety
statesatthemomentt=0;
thematrix[p
ij]4x4ofprobabilitiespij,i,j=0,1,2,3i≠
j, of the critical infrastructure safety states
transitions process S(t) transitions between the
safetystatesS
iandSj;
the matrix [F
ij(t)]4x4 of conditional distribution
functions
)()( tTPtF
ijij
,
,3,2,1,0,
ji
,
j
i
(2)
of the critical infrastructure safety states
transitions process S(t) conditional sojourn times
T
ij at the operationstates, and the corresponding
matrixofthedensityfunctions[f
ij(t)]4x4,where
171
)],([)( tF
dt
d
tf
ijij
,3,2,1,0, ji
j
i
;
By means of above mentioned parameters
following characteristics of critical infrastructure
safetystatestransitionsprocesscanbedetermined:
mean values of the critical infrastructure safety
states transitions process S(t) conditional sojourn
timesT
ij,attheparticularsafetystates:
][
ijij
TEM
0
)(ttdF
ij
0
),(ttf
ij
,3,2,1,0, ji
;
j
i
(3)
rates of critical infrastructure safety states
transitionsprocessS(t)betweenthesafetystates:
,
)(1
)(
)(
tF
tf
t
ij
ij
ij
,3,2,1,0, ji
;
j
i
(4)
unconditionaldistributionfunctionsofthecritical
infrastructuresafetystatestransitionsprocessS(t)
staytimeT
iatparticularsafetystates:
)(tF
i
=
3
0
),(
j
ijij
tFp
;3,2,1,0i
(5)
themeanvaluesofthecriticalinfrastructuresafety
states transitions process S(t) unconditional
sojourntimesT
iatthesafetystates:
][
ii
TEM
3
0
,
j
ijij
Mp
,3,2,1,0i
(6)
whereM
ijisgivenby(3);
thelimitvaluesofthecriticalinfrastructuresafety
states transitions process S(t) transient
probabilitiesattheparticularsafetystates
)(tp
i
=P(Z(t)=
i
z ),
),,0 t ,3,2,1,0i
(7)
aregivenby:
i
p =
)(
lim
tp
i
t
=
,
3
1
j
jj
ii
M
M
,3,2,1,0
i
(8)
whereM
i,i=0,1,2,3,aregivenby(6),whilethe
steady probabilities π
i of the vector [ πi]1x4 satisfy
thesystemofequations
1
[][][ ]
1;
iiij
j
j
j
p
(9)
OtherinterestingcharacteristicsoftheprocessS(t)
possibletoobtainare:
totalsojourntimesT
iattheparticularsafetystates
S
i,i=0,1,2,3,duringthefixedsystemopetation
time Θ, having approximately normal
distributionswiththeexpectedvaluegivenby
,]
ˆ
[
ˆ
iii
pTEM
,3,2,1,0
i
(10)
wherep
i,i=0,1,2,3,aregivenby(7);
the total cost (loss)
C
ˆ
concerned with critical
infrastructure exploitation at fixed exploitation
timeΘ,thatareaproximately
3
0
,
ˆ
i
ii
CpC
(11)
wherep
i,i=0,1,2,3,aregivenby(7),whileCi,i=
0,1,2,3,areaveragecosts(losses)ofexploitation
atparticularsafetystatesS
i,i=0,1,2,3,withinthe
time frame, at which exploitation time Θ is
measured.
In special circumstances, when critical
infrastructure safety states transitions process
conditional sojourn times T
ij at the particular safety
states, are having Weibull’s distribution with the
densityfunction
)(tf
ij
,],)(exp[)(
,0
1
ijijijijijij
ij
xtxtxt
xt
ijij
(12)
where
,0
ij
,0
ij
,3,2,1,0,
ji
,
j
i
its two main characteristics given by (3)
and(4)are:
the mean values of critical infrastructure safety
states transitions process S(t) conditional sojourn
timesT
ijattheparticularsafetystates
ij
M
][
ij
TE ),
1
(1
1
ij
ij
ijij
Γx
(13)
where
,)(
0
1
dtetuΓ
tu
,0u
is the gamma
function;
rates of critical infrastructure safety states
transitionsprocessS(t)betweenthesafetystates:
,)()(
1
ij
ijijijij
xtt
,
ij
xt
,3,2,1,0, ji
j
i
(14)
Described above determination of critical
infrastructure safety states transitions process’ basic
parameters, can beused formaritime transportation
system(treatedaspartofcriticalinfr astructure)safety
states transition process’ description. Further
evaluationofformulatedrelationsishoweverneeded,
leadingto:obtainratingsofactualsystemparameters’
influence on crisis
situation appearance probability;
possibilitiesofcrisissituationexpansioninhibiting,in
case if its appearance; and backward system
transitiontothestatefrombeforecrisissituation.
172
4 SAFETYSTATESTRANSITIONSMODEL
PARAMETERSIDENTIFICATION
Below chapter is showing general methodology of
criticalinfrastructuresafetystatestransitionsprocess’
parametersidentification.Thiswillbeusedinfurther
research works for maritime transportation system
safetystatesprocess’parametersidentification.Then
itwillmakepossiblediagnosingofsystemparameters
influence on
transitions’ rates between particular
system safety states, consequently allowing to
investigate studies possibilities on influencing on
maritimetransportationsystemincrisissituations.
4.1 Basic assumptions
Tomaketheestimationoftheunknownparametersof
the critical infrastructure safety states transitions
process, the experiment delivering the necessary
statisticaldatashouldbe
preciselyplanned.
First,beforetheexperiment,followingpreliminary
stepsshouldbeperformed:
1 toanalyzetheprocess;
2 to fix or to define the process following general
parameters:
thenumberofthesafetystatesoftheprocessv;
thesafetystatesofthesystemoperationprocess
z
1,z2,…,zv;
3 to fix the possible transitions between the safety
states;
4 to fix the set of the unknown parameters of the
processsemi‐Markovmodel.
Next,toestimatetheunknownparametersofthe
process,basedontheexperiment,necessarystatistical
data should be collected, performing the following
steps:
1 to
fix and to collect the following statistical data
necessary to evaluating the probabilities p
i(0) of
theprocessstayingatthesafetystatesattheinitial
momentt=0:
thedurationtimeoftheexperimentΘ,
the number of the investigated (observed)
realizationsoftheprocessn(0),
thevectoroftherealizationsn
i(0),i=1,2,…,v,
of the numbers of staying of the process
respectivelyatthe safety statesz
1,z2,…,zv, at
the initial moments t = 0 of all n(0) observed
realizationsoftheprocess
)]0(),...,0(),0([)]0([
21
nnnn
i
(15)
wheren
1(0)+n2(0)+nv(0)=n(0);
2 to fix and to collect the following statistical data
necessary to evaluating the probabilities p
ij(0) of
theprocesstransitionsbetweenthesafetystates:
thema trixoftherealizationsofthenumbersn
ij,
i, j= 1,2,..., v, i j, of the transitions of the
process from the safety state z
i intothe safety
state z
j at all observed realizations of the
process
nnn
nnn
nnn
n
ij
...
...
...
...
21
22221
11211
(16)
wheren
ii=0fori=1,2,…,v;
thevectoroftherealizationsofthenumbersn
i,
i=1,2,...,v,ofdeparturesoftheprocessfrom
thesafetystatesz
i(thesumsofthenumbersof
thei‐therowsofthematrix[ n
ij])
],...,,[][
21
nnnn
i
(17)
wheren
1= n11+n12+...+ n1v,n2=n21+n22+...+
n
2v,...,nv=nv1+nv2+...+nvv;
3 to fix and to collect the following statistical data
necessary to evaluating the unknown parameters
of the distributions F
ij(t) of the conditional
lifetimesT
ijofthesystemoperationprocessatthe
particularsafetystates:
the numbers n
ij, i, j = 1,2,...,v, i≠j, of
realizationsoftheconditionalsojourntimesΘ
ij,
i, j = 1,2,...,v, i≠j, of the system operation
process at the safety state z
i when the next
transition is to the safety state z
j during the
observationtimeΘ,
the realizations
,
k
ij
k = 1,2, …, nij, of the
conditional sojourn times Θ
ij of the system
operationprocessatthesafetystatez
iwhenthe
nexttransitionistothesafetystatez
jduringthe
observationtimeΘforeachi,j=1,2,...,v,ij.
4.2 Criticalinfrastructuresafety statestransitions
process’basicparametersestimating
After collecting the statistical data, it is possible to
estimate the unknown parameters of the system
operationprocessperformingthefollowingsteps:
1 to
determinethevector
12
[ (0)] [ (0), (0), . . ., (0)]ppp p
(18)
oftherealizationsoftheprobabilitiesp
i(0),i=1,2,
... , v, of the system operation process staying at
the safety states at the initial moment t = 0,
accordingtotheformula
)0(
)0(
)0(
n
n
p
i
i
for
,,...,2,1
i
(19)
where
1
),0()0(
i
i
nn
(20)
is the number of the realizations of the system
operation process starting at the initial moment
t=0;
2 todeterminethematrix
173
,
...
...
...
...
][
21
22221
11211
vvvv
v
v
ij
ppp
ppp
ppp
p
(21)
of the realizations of the probabilities p
ij, i, j =
1,2,...,v,ofthesystemoperationprocesstransitions
from the safety state z
i to the safety state zj
accordingtotheformula
i
ij
ij
n
n
p
fori,j=1,2,...,v,i≠j,
p
ii=0fori,j=1,2,...,v, (22)
where
ji
iji
nn
,
,,...,2,1
i
(23)
istherealizationofthetotalnumberofthesystem
operationprocessdeparturesfromthesafetystate
z
iduringtheexperimenttimeΘ.
4.3 Estimatingdistributionparametersofcritical
infrastructuresafetystatestransitionsprocess’
lifetimesatsafetystates
Priortoestimatingtheparametersofthesafetystates
transitionsprocess’conditionallifetimesdistributions
attheparticularsafetystates,thefollowingempirical
characteristicsoftherealizationsofthelifetimesof
the
criticalinfrastructuresafetystatestransitionsprocess
attheparticularsafetystateshavetobedetermined:
the realizationsof the empirical mean values
ij
T
oftheconditionallifetimesT
ijoftheprocessatthe
safety state z
i when the next transition is to the
safetystatez
j,accordingtotheformula
,
1
1
ij
n
k
k
ij
ij
ij
n
T
,,...,2,1,
ji
ij, (24)
the number
ij
r of the disjoint intervals
),,
j
ij
j
ijj
baI
ij
rj ,...,2,1 , that include the
realizations
k
ij
,
,,...,2,1
ij
nk
oftheconditional
sojourn times
ij
at the safety state zi when the
nexttransitionistothesafetystatez
j,accordingto
theformula
ijij
nr
,
the length
ij
d
of the intervals
),,
j
ij
j
ijj
baI
,
ij
rj ,...,2,1 ,accordingtotheformula
1
ij
ij
ij
r
R
d
,
where
,minmax
1
1
k
ij
nk
k
ij
nk
ij
ij
ij
R
the ends
,
j
ij
a
,
j
ij
b
of the intervals ,),
jj
jijij
I
ab
1, 2,...,
ij
jr
,accordingtotheformulae
},0,
2
minmax{
1
1
ij
k
ij
nk
ij
d
a
ij
,
1
ijij
j
ij
jdab
ij
rj ,...,2,1
,
,
1
j
ij
j
ij
ba
ij
rj ...,3,2
,
insuchawaythat
),...
1
21
ij
ij
r
ijijr
baIII
and
ji
II
forall
,ji
},,...,2,1{,
ij
rji
the numbers
j
ij
n
of the realizations
k
ij
in the
intervals
),,
j
ij
j
ijj
baI
ij
rj ,...,2,1 , according
totheformula
}},,...,2,1{,:{#
ijj
k
ij
j
ij
nkIkn
ij
rj ,...,2,1 ,
where
,
1
ij
r
j
ij
j
ij
nn
whereas the symbol
# means the number of
elementsoftheset;
Toestimatetheparametersofthedistributions of
the conditional lifetimes of the process at the
particular safety states, it has to be proceeded