End yes
Figure10: Individual EvacuationProcess[46].
nodej whenthewhole systemisin stateX. Ifthenumberof walkways
connectedtonodeiisdenotedasÆ
i ,then
p
k
(i;j;X):=
1
Æ
i
Modi ed random choice. It includes the possibility that evacuees will
returntotheirpreviouswalkway. De nen
k
asthenode-numberofthelast
nodevisitedbyapersonkandintroduceanumber!,inwhich0!1.
Theparameter! isafactorshowinghowaperson iswillingto returnto
his/herpreviouswalkway. Usingthisparameter,theprobabilityp
k
(i;j;X)
canberede nedasfollows.
p
k
(i;j;X):=
8
>
<
>
:
!
Æi
;j=n
k
1
!
Æ
i
Æi 1
;j6=n
k
0 ;otherwise
Suppose thereis anevacuationrouteP determined apriori bythe
evac-uation planner (this is atypical situation, indicated by evacuation sign
attachedtothewallofroomsorcorridors).
P :=(s=origin;:::;i;j;:::;destination=d)
Supposethat Æ
i
>1forallnodesin thisroute,exceptpossiblytheorigin
anddestination. Whenanemergencysituationarisesthereisahigh
ten-dencythatevacueeswillusefamiliarroutesthatmaynotfollowtheroute
i
routeatanintersectionnode. Thenp
k
(i;j;X)canbewritten as
p
k
(i;j;X):=
1
i
;j=j
i
Æi 1
;j6=n
k
Pathchoice rules likethe ones presentedabovecanbe used together with
attributes of evacuees and building components to design a simulation of a
building evacuation (see [63]). Obviously, such simulation is a good tool to
model individual behaviour in evacuation planning and can also be used to
validateoptimizationmodelsaspresentedinthepreceding sections.
Inrecentyears,thereisanincreasingtrendtousesimulationbasedon
cellu-larautomata(CA).Eitherdeterministicorprobabilisticrulescanbeappliedto
modelthemovementpatternsbetweentimeperiodsrelativetothemovementof
other persons and/orphysical barriers. CA simulationo ersthepossibilityto
emulate theessential,diverse movementsofevacueesasbehaviouralresponses
to varying and uncertainlocal conditions. A cellular automata isde ned asa
regular n-dimensional latticepartitioned into discrete elements called cells or
siteswhichhasadicretesteptimeevolution. Usingcellularautomata,thespace
of theevacuation areais dividedinto accessible andnon accessiblecells, each
of xedandequalsize. Thestateofeachcellisoneof nitelymanyvaluesand
has adynamical behaviour. It is updatedsimultaneously basedon thevalues
ofthestatesin itsneighborhoodattheprecedingtimestepandaccordingtoa
speci c local rules . The set of local rulesis de ned to control themovement
of evacuees, orthestate transition of each cell. Since therules are neededto
governonlylocalrelationshipsamongtheneighboringcells,CAisconsideredto
beverye ectiveforsimulatingphysicalphenomena, therelationshipsofwhich
overthewholedomainareunknown. Inthe1-dimensionallattice,acellhasonly
twoneighbourhoodcells,theleftandrightcelladjacenttoit. For2-dimensional
lattice, thereare someneighbourhoodclassi cationsof acellasshownin
Fig-ure 11. The von Neumann neighbourhood consists of 4 cells, the cell above,
below, rightand left of the reference cell. The radius r of the von Neumann
neighborhood is 1,sinceit considers only thenextlayerof acell. TheMoore
neighborhood isan extensionof thevonNeumannneighborhood in which the
diagonalcellsare added. Itsradius isstill r=1. TheexpandedMoore
neigh-bourhoodfurtherextendstheMooreneighborhoodbyincludingtwolayers(i.e.,
r = 2). Anothertype of neigborhood is the Margolus neigborhood in which
22 cellsof latticesare considerd at once. Thetransition from onestateto
anotheris arrangedbyaset ofrules accordingtothe neighboringstate ofthe
preceding timestep. Ifx(t)isastateofcellxattimet thenthestateofcellx
attime t+1canbewritten as
x(t+1)=f(x(t);N
x (t))
whereN
x
(t)isstateoftheneighborhoodcellsofcellxattimetandwheref is
therule. Asimpleexampleofarulecanbegivenasfollows. Consideranarrow
corridor (assumedwide enough onlyfor one person) with a given lengththat
canbemodeledasaonedimensionallattice(seeFigure12). Acellrepresentsa
spacein thecorridorthatmaybeoccupiedby onepersonoritmaybeempty.
The walking speed of the i th
person v(i) corresponds to the number of cells
x x
x x
x x x x x x x x
x x x x x x x x
x x x x x x x
x x x x x x x x x
(a) (b)
(c)
Figure 11: (a) von NeumannNeighborhood (b) Moore Neighborhood (c)
Ex-panded MooreNeighborhoodofaCell(ShadedCell).
thatapersonadvancesinoneiteration(onetimestep). Byde ningx(i)asthe
position of thei th
person in thecorridor, thedistance betweenthe i th
person
andthepersonimmediatelyaheadisgivenby
g(i)=x(i+1) x(i) 1
Usingthese3variablesthetransitionrulecanbede nedasfollow:
Accelerationof freeperson:
Ifv(i)<v
max
andg(i)v(i)+1thenv(i)=v(i)+1,wherev
max isthe
maximumpossiblespeed.
Slowingdowndueto otherperson :
Ifv(i)>g(i) 1thenv(i)=g(i)
Movement: personisadvancedv(i)cells.
Asamethodofdiscretesimulation,CAisusedbysomeevacuationsoftwares
includingEGRESSandFlightSim.
EGRESSisaC++programdevelopedformodellingthebehaviourof
evac-ueesinemergencysituations,especiallyino shoreenvironments. Thephysical
structure oftheo shoreinstallationis representedbyusing hexagonalcellular
grids (see, e.g., Doheny and Fraser [20]). It models evacuation using cellular
automata where the movementsand interactions of the automata on the
cel-lulargridsimulate themovementsandphysical interactionsofevacueesonthe
platform. EGRESSintegratesMOBEDIC(ModellingBehaviouralDecisionsIn
Computer) forevacuees'sdecisionmakingandamovementmodelwhich
mod-t
t+1
2 3
2
1
1
2
2 x
Figure 12: CA Model of Evacuees Moving in a Corridor with v
max
= 3
cells/time-step.
representsthebrainof theautomatain themovementmodel. Itis responsible
formakingdecisionsandforinstructinganautomatontocarryoutsomeaction.
TheMovementModeldistributesinformationssuchasthepresenceofsmokeor
re,theoccurrenceofalarmsandthecurrentpositionofautomaton. These
in-formationsarecombinedwithsomepropertiesofevacuees,suchasreactionsto
alarmandhazards,familiaritywiththebuildingstructure,knowledgeand
expe-riences aboutthe emergencysituation. The combinedinformations determine
the movement of evacuees. In the movement model evacuees are represented
usingcellularautomatawhichcanmoveabouttheplanfromcelltocell.
Move-ment algorithms determine which cell each automaton should occupy at any
giventimeandmovetheautomataaccordingly.
FlightSimwasdevelopedattheUniversityofDuisburg,Germany(seeKlupfel
etal. [42]). Itwasoriginallydevelopedforanalyzingtheevacuationprocessofa
passengership. Someadjustedparametersareincludedinthemodel,for
exam-plethetimethateveryevacueeneedstoenterthesavingboats,thedistribution
ofthemaximumspeedamong theevacuee,andthedistributionofthenumber
of cellsaperson looks forwardfororientation. Insteadofusing parallel
updat-ing when updatingtheevacuees'positions, FlightSimusessequential updating
([11]). Inparallel updating, variablesofallevacuees(e.g., position,speed) are
changedatthesametime. Incontrast,thesequentialupdatingselectsthe
evac-ueesoneafter theotherand updates his/herattributes. Theselectiononwho
willbethe rstisdonerandomlyin ordernottogiveadvantagetoanyspeci c
person. Figures 13and 14 showthe di erence ofthese twoupdatingsystems
([11]). LookingatFigure13,duringthe rststepunderparallelupdating,
evac-ueeamustwaituntilthenextupdate,sinceheisblockedbyevacueeb. Theleft
columnofFigure14describesthesequentialupdatingwhenevacueeaisalways
chosen rst. Themovementpatternisthesameasonewithparallelupdate. If
evacueebischosen rst,thesituationchanges,asisshownin therightcolumn
ofFigure14.
Anothersimulationbasedevacuationsoftware,EXODUS,wasdevelopedby
the Fire Safety Engineering group at the university of Greenwich, United of
Kingdom(see[29], [32])anddesignedtosimulatealargenumberofoccupants
in a closed environment, for instance building and airplane. It is an expert
system-based software which has a set of heuristics or rules to determine the
progressivemotionand behaviourof each individual. Ittrackseach individual
eithermakinghis/herwayoutoftheevacuationareaorisbeingovercomeby re
hazards. A more complete survey onevacuationsoftwares using either macro
approachesormicroapproachesisdiscussedbyGwynne,et al. [32].
b a
a
a b
a b
Figure13: ParallelUpdate.
b a
a
a
b a
a b
a
a b
b
Figure14: SequentialUpdate.
12 Summary and Conclusion
In this paper, a reviewof models and algorithms forevacuation planninghas
beenpresented. Thereviewcoveredmacroscopicmodelsquiteextensivelyand
sketched microscopicmodels. Both approachesareable tomirror the owsof
evacuationsovertime. Theformerhasitsstrengthinitspossibilitytooptimize
thesystem(whileneglectingindividuals'behaviour),while thelatterisableto
captureandutilizepropertiesofeachoftheevacuees.
Underthemacroscopicapproach,minimumturnstilecostdynamicnetwork
owmodelscanbeappliedtoestimatetheaverageevacuationtimeperevacuee.
Maximumdynamic owsanduniversalmaximum owscanbeusedtoestimate
themaximumnumberofevacueeswhichcanreachsafetyduringanygiventime
horizon fortheevacuation. Quickest owmodelsallowtheestimatationofthe
minimumtimerequiredtobringagivennumberofevacueesto safety.
Consid-eringthe sourceand propagation of hazards, availability of emergencyservice
units and betterorganization,the evacuation regioncanbedividedinto some
regions withdi erentpriority levels. Therefore,multiple objectivemodels are
presentedto copewith this problem. Constanttraveltime ismostly assumed
in the literature. This time can beobtained by taking the travel time of the
average ow or travel time of aspeci c queuing level. In order to re ect the
congestion phenomenon, it was shown how the constant travel time
assump-tion can be strengthened by considering density dependent travel time. This
approachwill,however,signi cantlyincreasethecomplexityofthemodel.
In contrast to macroscopic models, the microscopic approach, usually
im-this approach. Succesful implementationsofthe macroapproach arebasedon
cellularautomata.
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