EUROPEAN COOPERATION
IN THE FIELD OF SCIENTIFIC
AND TECHNICAL RESEARCH
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EURO-COST
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COST 2100 TD(10)11032
Aalbo g, Denma k
2010/June/02-04
SOURCE: UPC - Uni e si a Poli `ecnica de Ca alunya
I2CAT Founda ion
Me aheu is ic P oposal o Minimize Sel -In e e ence in Single F equency
Ne wo ks
M. Ga c´ıa-Lozano, S. Ruiz-Boqu´e, M.A. Lema, E. To as, J.J. Olmos, F. Mine a?
UPC. C/ Es e e Te adas 6, EPSC-214. Cas ellde els, Spain.
?I2CAT Founda ion. C/ G an Capi `a 2-4. Nexus I. 08034, Ba celona, Spain.
Phone: +34 93 413 72 13
Fax: +34 93 413 70 07
Email: [email p o ec ed]c.edu
Me aheu is ic P oposal o Minimize Sel -In e e ence in Single
F equency Ne wo ks
M. Ga c´ıa-Lozano, S. Ruiz-Boqu´e, M.A. Lema, E. To as, J.J. Olmos, F. Mine a.
May 26, 2010
Abs ac
This pape concen a es on he maximiza ion o co e age in OFDM ne wo ks wi h a single equency
deploymen . The case-s udy is pa icula ized o DVB-T ne wo ks. To achie e his objec i e, in e nal
delays a he ansmi e s a e join ly op imized so ha sel -in e e ed a eas can be educed as much
as possible be o e ha ing o deploy new cen e s. A me aheu is ic app oach is p oposed and alida ed.
Resul s show ha he s a egy succeeds in i s objec i e and cons i u es a p ac ical ool ha helps o
assess adio planning decisions.
1 In oduc ion
Digi al Video B oadcas ing Te es ial (DVB-T) ope a o s commonly exploi (pa o ) hei ne wo ks using
a single- equency ne wo k (SFN) scheme. This is possible because he physical laye is based on O hogo-
nal F equency Di ision Mul iplexing (OFDM) and he in oduc ion o a cyclic p e ix be ween consecu i e
symbols. This allows ha ing a gua d in e al (GI) ha copes wi h in e symbol in e e ence (ISI) induced by
he mul ipa h channel.
SFNs allow a mo e e icien use o a ailable bandwid h han classic Mul iple F equency Ne wo ks (MFNs).
They also simpli y he adio-planning p ocess since equency alloca ion s a egies a e no equi ed. Indeed,
co e age holes can be easily sol ed by jus he ins alla ion o a new ansmi e (TX) o gap- ille wi hou
ha ing o e ise a p e-exis en equency plan.
On he o he hand, du ing he design o he ne wo k, he adioplanning enginee has o gua an ee he GI
condi ion since signals ha all ou side i cause sel -in e e ence. As a gene al ule-o - humb, TXs should be
placed a a dis ance equal o ha co e ed by he signal in one GI. Fo classic DVB-T ne wo ks ope a ing in
8K mode wi h a GI o 1/4 o he du a ion o he OFDM symbol, his yields o a sepa a ion o a ound 67 km
which implies ha he ea h cu a u e will also con ibu e o isola e ecei e s om in e e e TXs.
I is clea ha he longe he GIs, he easie he educ ion o sel -in e e ence, howe e his also implies
a less e icien ansmission since no new in o ma ion is con ained in he added in e al and so he e ec i e
da a a e is educed. Besides, mobile ele ision is gaining ocus pa icula ly in he con ex o he DVB-T2
s anda d, and long symbols wi h la ge GIs a e much mo e sensi i e o dopple e ec . A good sys em design
implies as sho as possible GIs while main aining su icien mul ipa h p o ec ion. Thus s a egies o minimize
sel -in e e ed a eas a ise as pa amoun . In his sense, he a iables wi h a highe impac can be g ouped in
hose ha a e uncon ollable by he ope a o and hose ha a e suscep ible o op imiza ion:
•In he i s g oup one can ind he p opaga ion en i onmen and he con igu a ion o OFDM ecei e s.
Di e en comme cial equipmen posi ion he beginning o he GI (and so he FFT window) ollowing
di e en c i e ia ha imply di e en co e age and in e e ence oo p in s [1]. Gi en his, di e en
ecei e op ions should be conside ed and assessed when op imizing he ne wo k planning, o a leas
he wo s case should be s udied.
1
•The second g oup o a iables a e he geog aphic posi ion o he TXs, hei ansmission powe , he
con igu a ion o hei adian sys em (diag am pa e n, down il , null- illing echniques, pola iza ion...)
and hei s a ic in e nal delays. Among hese, he las one is o special in e es because changes can
be done wi h cos ze o. Mo eoe e , in a con ex o ope a i e DVB-T ne wo ks and in some cases in
he beginning o a ansi ion owa ds DVB-T2, powe s, an ennas and posi ions (in his o de ) a e
inc easingly mo e s a ic and unlikely o be d ama ically changed.
Gi en his, his wo k deals wi h he op imal adjus men o s a ic delays o he TXs in SFNs. The inal aim
is o econ igu e his pa ame e so ha sel -in e e ed a eas a e minimized, wi h he co esponden inc ease
in co e age o a gi en layou o TXs.
This ac ion is ypically done manually by he adio planning enginee as TXs a e ins alled, ha means
one-by-one sequen ially. A manual join adjus men o all he nodes in a ce ain a ea is e y complex because
o in e dependencies among hem. In his sense, he e a e no many p oposals in he li e a u e o au oma e
his p ocess. The ecen e e ences [2; 3] a e an example on how add essing his p oblem success ully wi h
he help o a pa icle swa m op imiza ion algo i hm. The au ho s in [4] simpli y he p oblem easonably and
add ess i sma ly om an analy ical iewpoin . Also, [5] p oposes a echnique unde he assump ion ha
he posi ion o use s is known, who equi e a GPS ecei e . O he examples o adio planning op imiza ion
ocus on he adjus men o pa ame e s such as he numbe o TXs, hei loca ion, emi ed powe and an enna
heigh [6; 7].
The no el y o his pape is he p oposal o a new echnique o join ly op imize he s a ic in e nal delays
o a ce ain se o TXs in a SFN. I ac i ely sea ches o he se o delays ha minimize he a eas a ec ed by
ISI and so ha inc ease he inal ne wo k co e age. The p oposal sol es he op imiza ion p oblem making
use o he me aheu is ic Simula ed Annealing, al eady applied wi h success in o he a eas o he wi eless
communica ions ield, as de ailed a e wa ds. Se e al ealis ic scena ios ha e been success ully sol ed and
compa ed wi h usual manual esolu ion.
The es o he pape is o ganized as ollows. Nex sec ion desc ibes he sys em model and i s assump ions,
he impac o delays adjus men on sel -in e e ed a eas is also add essed. Subsequen ly, Sec ion 3 explains
he basis o he op imiza ion algo i hm. Resul s a e p esen ed in Sec ion 4 and also some ema ks on he
implemen a ion o he s a egy and i s execu ion a e included. Finally, he pape is closed wi h he conclusions
and some inal ema ks.
2 Sys em Model
2.1 Sel -In e e ence Modelling
We assume a gene ic SFN deploymen co e ing a ce ain se ice a ea wi h NTXs b oadcas ing in a synch o-
nized manne , o example making use o a GPS e e ence. In e nal delays in each TX can be econ igu ed o
modi y he ime o ansmission. Conside ing ha he p og am does no eaches all TXs a he same ime,
some o hem ha e a ma gin o nega i e adjus men s.
The quali y o he se ice a a gi en loca ion is gi en by he Ca ie - o-In e e ence-plus-Noise Ra io
(CINR), deno ed by Γ. No e ha he minimum equi ed CINR is ound be o e compu ing he co e age
a eas. One possibili y is o combine link-budge s wi h link le el simula ions wi h an app op ia e adio channel
model. In gene al, dense SFNs imply many pa hs inside he GI wi hou a clea dominan , which leads o a
pu ely Rayleigh beha io wi h impo an equency selec i e adings.
The i h pa h is conside ed o be use ul o in e e ing conside ing i s delay ∆τiwi h espec o he begin-
ning o he FFT window. The inal alue depends on he ecei e because di e en s a egies o synch onize
his window a e possible [1]. Besides, echoes alling ou he GI bu wi h an impo an o e lapping wi h he
FFT window, con ibu e pa ly o he use ul signal and pa ly o he in e e ence pe cei ed in he nex
symbol [1]. The weigh ing unc ion w(∆τi) ha is used o compu e he inal con ibu ion in Γ is gi en nex .
w(∆τi) = 




1 i 0 ≤∆τi≤TGI
TU−∆τi+TGI
TU2i TGI <∆τi≤Te
0 o he wise
(1)
2
Locus o poin s whe e he di e ence o
he dis ances o he wo oci is a cons an
CIR h
IG
Figu e 1: Di e en ecei ing si ua ions in a canonical scena io.
(a) Equal s a ic delay (b) Posi i e delay on he le
Figu e 2: Reduc ion o le in e e ed a ea by means o delays modi ica ion
Being TGI he GI alue, TU he use ul symbol pa and Te he equaliza ion in e al. P e-echoes and signals
alling ou o TU+Tea e conside ed as ull in e e ence.
Gi en his, he o mal exp ession o Γ is:
Γ = PN
i=1 w(∆τi)Pi
PN
i=1 [1 −w(∆τi)] Pi+PN
(2)
Whe e, Piis he powe ecei ed om he i h TX and PNis he he mal noise powe .
2.2 On he Impac o Time O se Adjus men s
Le conside a canonical scena io in which wo TXs (L on he le and R on he igh ) a e deployed in a la
e ain. Unde hese ci cums ances, he bo de be ween he a eas wi h he second con ibu ion alling inside
o ou side he GI is gi en by he locus o poin s whe e he di e ence o he dis ances o he wo TX is a
cons an , ha is a hipe bola (Fig. 1) wi h bo h TXs as oci. Howe e , no he ull a ea on he le o he le
semi-hipe bola and on he igh o he igh semi-hipe bola a e necessa ily ou o co e age. As long as he
CINR is good enough, o he con ibu ions can be ecei ed ou o he GI, as i is g aphically poin ed ou .
In his sense, he con igu a ion o he OFDM ecei e plays and impo an ole, as p e iously men ioned.
Sel -in e e ed a eas can be modi ied by means o changes on s a ic delays. Thus, o example, i he in e nal
delay o L is inc eased, hen R has i ually go close and consequen ly he le semi-hipe bola is educed
(e en ually elimina ed). Con e sely, his ac ion has a nega i e e ec on R, because now L has been i ually
mo ed u he away and so he sel -in e e ed a ea on he igh is inc eased. This is g aphically ep esen ed on
Fig. 2 whe e he be o e and a e si ua ion a e plo ed. No e ha he blue a ea ep esen s hose poin s wi h a
p obabili y o co e age o 90% o highe , he ed one ep esen s he opposi e. Thus, his simple modi ica ion
could be use ul o example in an en i onmen in which R ansmi s wi h a highe powe and so can cope
wi h he signal om L causing in e e ence.
I he numbe o nodes is inc eased o 3, c ossed e ec s s a o make di icul he adjus men . Tha is why
he di e en delays a e ypically se in a manual manne bu jus one-by-one. Whene e a new node is added
3
o he ne wo k, he new sel -in e e ence is e alua ed and ac ions a e aken o e he new TX, commonly
espec ing he exis ing ne wo k o wi h mino changes on i . No e ha his p ocedu e is indeed a Local
Sea ch (LS), because i is jus an i e a i e sea ch p ocedu e ha , s a ing om an ini ial easible solu ion S,
p og essi ely imp o es i wi h a se ies o modi ica ions. In pa icula , he se o new solu ions ha can be
gene a ed om he cu en one is he solu ion neighbo hood N(S) and all he possible solu ions con o m he
solu ions space. This p ocedu e implies subop imal solu ions ha could be imp o ed i he whole a ge a ea
was op imized a a ime. The poin is ha he complexi y o he p oblem inc eases exponen ially wi h he
numbe o TXs and in gene al, i canno be join ly sol ed manually i mo e han 4 nodes a e o be op imized.
Gi en his, he p oposed echnique is able o op imize a andom numbe o nodes and ind a se o
op imized delays pe o ming a join analysis.
3 Basis o he Op imiza ion Algo i hm
The esolu ion me hod is basically o ien ed o he minimiza ion o a cos unc ion Fcos ha ga he s he
ope a o ’s equi emen s and exp esses he global alue o a ce ain adio planning solu ion S. In pa icula ,
Fcos ep esen s he summa ion o he pixels, in he digi al ele a ion model, ha a e no co ec ly se ed
and weigh ed by a ac o ep esen ing he popula ion densi y in ha pa icula pixel. The op imiza ion
can be subjec o se e al cons ain s, as o example no modi ying he exis ing co e age o a pa icula
a ea. Because o non-linea i ies and dependencies among di e en TXs, he p oblem can be conside ed a
combina o ial op imiza ion one wi h a e y high numbe o solu ions when a signi ican g oup o TXs is
conside ed.
In 1983, Ki kpa ick, Gela and Vecchi desc ibed in [8] a new heu is ic app oach called Simula ed
Annealing (SA) wi h he ou s anding ea u e ha con e ged o he op imal solu ion o a combina o ial
p oblem, al hough in ini e compu ing ime was equi ed. Ne e heless, he appea ance o SA showed ha
o he ways o ackle combina o ial op imiza ion p oblems we e possible and i boos ed he in e es o he
esea ch communi y. O he examples a e Gene ic Algo i hms and he An Colony Algo i hm. All hese
me hods a e now collec i ely known as me aheu is ics.
Me aheu is ics also equi e a p ocedu e o gene a e a new combina ion (o solu ion, s a e...), usually
de i ed om he cu en one. This is usually a p obabilis ic ac ion ha mu a es he p esen solu ion. In
mos o hem, i is in e es ing o no e ha mo es in he space o solu ions can be bo h uphill o downhill
and ha means accep ing solu ions wi h a wo s cos a pa icula momen s o he sea ch. In ac , his is
one o he main di e ences wi h espec o LS since i is in ended o a oid ge ing apped in local minima.
As p e iously men ioned, SA is one o he algo i hms ha enjoys mo e popula i y in he esolu ion o
combina o ial op imiza ion p oblems and i is he one ha has been adop ed in his wo k. SA is being widely
used a many le els o elecommunica ions enginee ing as o example:
•F equency alloca ion p oblem: [9; 10; 11].
•Loca ion o TXs and ansmission powe s: [6; 12].
•Hub loca ion p oblem: [13; 14].
The name and inspi a ion comes om he cooling p ocess o a liquid and i s con e sion in o a solid which
SA a emp s o ma hema ically cap u e. The cooling p ocess is o mula ed as he sea ch o he solu ion
implying a lowe cos (ene gy). E e y new solu ion is gene a ed by applying a sligh pe u ba ion o e he
cu en one. Likewise, om a physical iewpoin , he e is some non-ze o p obabili y o eaching a highe
ene gy s a e. As a consequence he accep ance o wo se solu ions is allowed wi h a ce ain p obabili y oo, as
i usually happens wi h all me aheu is ics. The p ocess is summa ized in Table 1, whe e p e iously de ined
no a ion Fcos and N(S) has been used.
The quali y o he inal solu ion depends on aspec s such as he ini ial hea ing o gene a ion o he
s a ing solu ion, he cooling s a egy, he c i e ia o gene a e he neighbo hood o solu ions, e c. In gene al,
a ade-o is always p esen be ween he quali y o he esul and execu ion ime.
In o de o make he algo i hm obus , i is desi able ha he quali y o he inal solu ion is independen
o he ini ial one. Thus, he pa ame e ha con ols he p obabili y o accep ing wo se solu ions ( empe a u e
4

Table 1: Basic SA schema.
1. Ob ain ini ial solu ion Sand empe a u e T
2. Ob ain ini ial cos : C←Fcos (S)
3. Gene a e new solu ion S0∈N(S)
4. C0←Fcos (S0)
5. Accep S0as cu en solu ion Swi h p obabili y P
P=exp [(C−C0)/T] i C0≥C
P= 1 i C0< C
6. I equilib ium condi ion no eached, go o 3
7. Upda e empe a u e T
8. I e mina ion c i e ion no eached, go o 3
o he algo i hm, T) mus be high enough, o he wise he algo i hm could be condi ioned o be apped in
a local minima. In he p oposed design an ini ial hea ing p ocess is execu ed un il he a io o accep ed
solu ions is highe han 85%.
The gene a ion o a new solu ion consis s o a sligh pe u ba ion o e he cu en one. This modi ica ion
is done acco ding o wo andom elec ions: one delay in he ange o possible alues and one TX in he a ea
o be op imized. The new alue o Fcos is ecalcula ed and possible ope a o cons ain s a e e alua ed, in
his sense he scena ios e alua ed in his wo k a e no subjec o es ic ions.
Finally, he upda e o Tis done acco ding o equa ion 3 because i is ma hema ically demons a ed ha
i p ese es he con e gence heo y o he algo i hm owa ds op imum solu ions as much as possible [15].
Tn+1 =Tn
1 + Tnln(1+δ)
3σn
(3)
The speed in he educ ion o Tcan be con olled wi h δso ha simula ion ime can be adjus ed a will.
On he o he hand, σn ep esen s he s anda d de ia ion o he cos e olu ion wi h he p e ious empe a u e
Tn.
4 Resul s
Resul s a e p esen ed o h ee di e en scena ios co e ing di e en a eas o Ca alonia, in he no heas
o Spain. A digi al ele a ion model wi h a esolu ion o 100 ×100 m2has been used and pa h-losses ha e
been compu ed ollowing he ecommenda ion ITU-R 526. Only hose pixels ecei ing a leas one signal
con ibu ion wi h a signi ican le el a e e alua ed by he algo i hm. Fo he sake o cla i y Fcos is exp essed
in km2and no in numbe o pixels.
Fig. 8 shows how he p oposed solu ion is modi ied as he algo i hm ad ances in he op imiza ion. I
is no iceable how he cu es a e e y noisy a he beginning; he algo i hm explo es he solu ion space
andomly and as i ad ances in i s sea ch, a de ined end in he op imal delays is obse ed. By he end o
he simula ion i can be seen how he close he algo i hm o he solu ion, he mo e co ela ed he changes
a e. This is logical, because SA is posi ioned in an in e es ing a ea o he space o solu ions and so i is
no mal ha a e a new change, se e al delays a e eadjus ed o keep he ela i e ime o se , which in ac
is he impo an me ic, a he han absolu e alues.
Simila ly, Fig. 4 ep esen s an example o he e olu ion o he cos unc ion. I can be obse ed how he
algo i hm succeeds in i s commi men and he unco e ed a ea is e ec i ely educed. The ini ial si ua ion
is ha in which he delays a e no adjus ed. Howe e , as s a ed be o e, some ype o manual adjus men
is usually pe o med e e y ime a new node is ins alled. In his sense, in o de o cap u e and ep esen
he esul o hese ac ions, a LS based op imiza ion was also assessed . The p ocess is as ollows, TXs a e
andomly o de ed and e alua ed sequen ially conside ing all possible delays, i a be e solu ion is ound,
i subs i u es he p e ious one. Di e en uns ha e been done conside ing di e en o de s and esul s a e
plo ed in Fig. 5 o wo o he scena ios. Gi en ha 10 es s whe e pe o med, he a e age cos alue is
shown along wi h he maximum and minimum esul s. F om he e, i can be obse ed how SA ou pe o ms
5
020 40 60 80 100 120
0
20
40
60
80
I e a ions
E olu ion o delays [μs]
Figu e 3: E olu ion o p oposed delays.
020 40 60 80 100 120
1400
1600
1800
2000
2200
2400
2600
I e a ions
Unco e ed a ea [km2]
Figu e 4: E olu ion o unco e ed a ea.
No LS SA
1200
1300
1400
1500
1600
1700
1800
Unco e ed a ea [km2]
No LS SA
350
400
450
500
550
600
2287
Figu e 5: A e age, Max. and Min. unco e ed a ea wi h no op imiza ion (No), LS and SA o wo di e en s udy
cases.
all possible LSs. Besides, he o de o e alua ion in he LS showed a signi ican impac on he inal esul
and ha is why he de ia ion o esul s is clea ly highe .
Finally, Fig. 6 illus a es he co e age be o e and a e he op imiza ion o he h ee conside ed scena ios.
No e ha he maps ha e been scaled o he same size bu hei eal dimensions a e 90×90 km2, 100×90 km2
and 50 ×50 km2 o scena ios 1, 2 and 3 espec i ely. Al hough in all cases he e is a co e age gain, di e en
le els o imp o emen a e ob ained and his ob iously depends on he layou o he ne wo k and he o og aphy.
Besides, i can be obse ed ha gains a e no a cos ze o. A eas a e modi ied and in some pixels he ini ial
co e age is los . O cou se, in his poin is whe e he popula ion weigh in he cos unc ion plays i s
impo ance, also c i ical a eas can be p o ec ed by means o cons ain s o be espec ed.
6
(a) Scena io 1. Be o e.
(b) Scena io 1. A e .
(c) Scena io 2. Be o e.
(d) Scena io 2. A e .
(e) Scena io 3. Be o e.
( ) Scena io 3. A e .
Figu e 6: Compa ison o co e ed a eas be o e and a e he op imiza ion.
7
5 Implemen a ion issues and o he esul s
This sec ion cons i u es a complemen o he desc ip ion o he algo i hm. Al hough, nex g aphs and igu es
we e ob ained p io o he inal esul , o he sake o cla i y, i was conside ed mo e in e es ing o keep his
sec ion a he end o he pape .
Since a lo o e alua ions a e equi ed, one o he d awbacks o he p oposal could be i s execu ion ime,
howe e he me hod was easily pa allelized because each pixel can be e alua ed independen ly o o he s. Ou
pa icula implemen a ion was p og ammed in C++ and OpenMP [16] o achie e he pa allel execu ion. In
a compu e wi h ou 3 GHz p ocesso s (quad-co e), he op imiza ion o 10 TXs in an a ea o 90 ×90 km2
ook less han wo hou s.
E en hough SA is a me aheu is ic wi h e y ew pa ame e s o adjus , some p elimina y simula ions
whe e equi ed o gua an ee ha CPU ime was minimized. In pa icula , i was analyzed he impac o
he δ ac o in he cooling equa ion, he leng h o he equilib ium condi ion o each empe a u e alue and
he maximum ange o delays o e alua e. No e ha all subsequen esul s a e no malized o he maximum
ob ained alue in each case.
5.1 Leng h o equilib ium condi ion
Equilib ium is de ined as he numbe o i e a ions ha mus be e alua ed in each empe a u e alue. I can
be demons a ed ma hema ically ha he highe his numbe , he highe he p obabili y o eaching he
op imum solu ion [8]. Howe e , an uppe bound is needed o p ac ical op imiza ions. A gene al accep ed
ule-o - humb is a numbe o imes a ound he numbe o neighbo ing solu ions. This is es ima ed as he
p oduc o he numbe o TXs and he numbe o delays o be e alua ed. No e ha we assume a disc e e
ange, wi h delay alues ounded o he second decimal place when exp essed in µs.
Fo a ine adjus o his empi ical adjus men , he ule was mul iplied by a pa ame e βwi h he inal
aim o ob aining as e simula ions. Resul s pe ec ly ma ch he heo y, bu ma ginal cos gains a e ob ained
whe eas execu ion ime inc eases exponen ially. Following he esul s in Fig. 7, e en 0.5 can be an app op ia e
alue o β, so simula ion ime can be d ama ically educed in he scena io o op imizing e y la ge ne wo ks.
5.2 Maximum ange o delays o e alua e
Rega ding he selec ion o he maximum ange o delays o e alua e, i is ema kable ha wha i is impo an
is he ela i e alue be ween he minimum and maximum delay and no he absolu e ones. In ac , as s a ed
be o e, da a does no eaches all TXs a he same ime in la ge SFN ne wo ks and so some TXs ha e a
ma gin o nega i e adjus men s.
Wi h some quick p e-simula ions he ope a o can ob ain an idea o he bes ange o delays. Adjus ing
his ma gin o a oo low alue educes he solu ion space, and so SA is limi ed o ind he bes solu ion. On
he o he hand, an indisc imina e inc ease o his ange hinde s he no mal ope a ion o SA, which is o ced
o e alua e edundan solu ions. Fig. 8 quan i ies hese ac s o an easy scena io wi h 4 TXs.
5.3 Impac o δ
Finally, in o de o compu e he mos adequa e alue o δ, se e al alues we e simula ed and again compa ed
in e ms o cos and execu ion ime. The mos clea impac o δappea s on execu ion ime which inc eases
p ohibi i ely o he smalles simula ed alues (Fig. 9(b)). Rega ding he impac on cos (Fig. 9(a)), gains
a e e y modes , al hough i was obse ed a close dependence wi h he size o he scena io. In his es case,
wha would de e mine he inal alue o δis ha dwa e capabili ies and he a ailable simula ion ime.
8