Quali y o T ansmission Es ima o Re aining o
Dynamic Op imiza ion in Op ical Ne wo ks
ANKUSH MAHAJAN,1,* KONSTANTINOS (KOSTAS) CHRISTODOULOPOULOS,2
RICARDO MARTÍNEZ,1 RAUL MUÑOZ,1 SALVATORE SPADARO3
1Cen e Tecnològic de Telecomunicacions de Ca alunya, CTTC/CERCA, Cas ellde els, 08860, Spain
2Nokia Bell Labs, S u ga , Ge many
3Uni e si a Poli ècnica de Ca alunya (UPC), Ba celona, Spain
*Co esponding au ho : [email p o ec ed]
Recei ed XX Mon h XXXX; e ised XX Mon h, XXXX; accep ed XX Mon h XXXX; pos ed XX Mon h XXXX (Doc. ID XXXXX); published XX Mon h XXXX
Op ical ne wo k op imiza ion in ol es an algo i hm and a Physical Laye Model (PLM) o es ima e he Quali y o
T ansmission (QoT) o connec ions while examining candida e op imiza ion ope a ions. In pa icula , he algo i hm
ypically calcula es in e media e solu ions un il i eaches he op imum which is hen con igu ed o he ne wo k. I
i uses a PLM ha was aligned once o e lec he s a ing ne wo k con igu a ion, hen he algo i hm wi hin i s
in e media e calcula ions can p ojec he ne wo k in o s a es whe e he PLM su e s om low accu acy, esul ing in
a subop imal op imiza ion. In his pape , we p opose o sol e dynamic mul i a iable op imiza ion p oblems wi h an
i e a i e closed con ol loop p ocess, whe e a e ce ain algo i hm s eps we con igu e he in e media e solu ion so
ha we moni o and ealign/ e ain he PLM o ollow he p ojec ed ne wo k s a es. The PLM is used as a Digi al
Twin (DT), a digi al ep esen a ion o he eal sys em which is ealigned du ing he dynamic op imiza ion p ocess.
Speci ically, we s udy he dynamic launch powe op imiza ion p oblem, whe e we ha e a se o es ablished
connec ions and we op imize hei launch powe s while he ne wo k ope a es. We obse ed subs an ial
imp o emen s in he sum and he lowes ma gin when op imizing he launch powe s wi h he p oposed app oach
o e op imiza ion using a one- ime ained PLM. The p oposed app oach achie ed nea o op imum solu ions as
ound by op imizing and con inuously p obing and moni o ing he ne wo k, bu wi h a subs an ial lowe
op imiza ion ime.
© 2019 Op ical Socie y o Ame ica
h p://dx.doi.o g/10.1364/JOCN.99.099999
1. INTRODUCTION
An accu a e and as physical laye model (PLM) is equi ed o
almos e e y op imiza ion ask o an op ical ne wo k [1], [2]. Today
mos op imiza ion asks a e s a ic, such as ne wo k se up and
upg ading, whe e calcula ions a e pe o med in ad anced. The PLM
used includes ma gins ha co e i s modeling unce ain ies and he
e olu ion o he physical laye condi ions o e he a ge ed li espan [3],
[4]. Mo eo e , as soon as he connec ion is p o isioned/ es ablished, he
endo can measu e i s quali y o ansmission (QoT), e.g. he signal o
noise a io (SNR), and co ec /imp o e he con igu a ion. No e ha
upg ades ha in ol e dynamic ope a ions such as he es ablishmen o
new o econ igu a ion o es ablished connec ions we e classi ied as
s a ic abo e, since ypically hey a e ca ied ou in main enance
windows and no on he ope a ing ne wo k. Dynamic econ igu a ions
o esiliency in ol e p o ec ed/ es o ed connec ions which we e
p obed be o ehand.
In any o hese op imiza ion asks he PLM needs o be accu a e;
howe e , he dynamic ope a ions a e no di ec ly applied on he
ne wo k, an indica ion o lack o ce ain y o such ope a ions. Recen ly
moni o ing and machine lea ning (ML) echniques ha e been p oposed
o accoun o he ac ual ne wo k condi ions, and imp o ing he
accu acy o he PLM [5]-[8]. This in u n imp o es he e iciency o s a ic
op imiza ion and pa es he way o educe o e p o isioning and ealize
some dynamic op imiza ion use cases [9]-[13].
Le us conside a ne wo k upg ade/inc emen al planning ask which
in ol es calcula ions o new es ablishmen s and possible
econ igu a ions o es ablished connec ions [8], [12]. T adi ionally a
PLM wi h high ma gins is used, e.g. conside s pessimis ic ibe
coe icien pa ame e s, ull spec al load, high modelling inaccu acy e c.
The op imiza ion will be qui e ine icien and esul in conside able
o e p o isioning. Using moni o ing eedback and e.g. ML [5]-[8] he
pa ame e s o he PLM can be i ed so ha i s es ima ed SNR alues a e
close o hose moni o ed in he ne wo k. Essen ially, eedback and ML is
used o unde s and he cu en s a e o he ne wo k and inc ease he
PLM es ima ion accu acy. We will e e o his p ocess as he alignmen
© 2019 Op ical Socie y o Ame ica. Use s may use, euse, and build upon he a icle, o use he a icle o ex o da a mining, so long as such
uses a e o non-comme cial pu poses and app op ia e a ibu ion is main ained. All o he igh s a e ese ed.
o he PLM o he physical laye o he ne wo k. The PLM accu acy is
e en mo e c i ical when i is used o dynamic op imiza ion asks, whe e
he a ge is o achie e high e iciency in an ope a ing ne wo k.
Today, op ical ne wo ks a e mo ing owa ds he so wa e de ined
ne wo king (SDN) concep , whe e a cen alized con olle handles he
p og ammabili y o all ne wo k elemen s. One o he main ad an ages
o SDN is i s in insic capabili y o enable dynamic op imiza ion
ope a ions [14], [15]. In his con ex , he SDN con olle implemen s he
op imiza ion logic, in e aces wi h a PLM, and can be ex ended o handle
closed con ol loops, which en ail he use o moni o ing da a as inpu o
eedback o conduc he a ge ed op imiza ion ask [7], [9], [16].
Simila p oblems a ise in almos e e y indus y. To keep up wi h he
apid ad ancemen s o he sys ems and ha es hei imp o emen s in
e ms o p oduc i i y, he Digi al Twin (DT) concep is gaining a lo o
a en ion. The DT is a digi al ep esen a ion o he eal / physical sys em,
used o unde s and and op imize he a ge ed sys em [17]. Acco ding o
he de ini ion o [18], he DT is mo e han a model o he sys em; i
includes an e ol ing se o da a, and a means o dynamically adjus ing
he model. The DT concep was o iginally in oduced in 2003 [18] and
i s pu o public by he NASA [20]. Di e en indus y sec o s a e aking
ad an age o DT’s abili y o simula e eal- ime wo king condi ions and
pe o m au onomous and in elligen decision-making ope a ions. DT
p o ides an al e na i e way in oday’s manual in e ac ion-based design,
ope a ion, and se ice pa adigms, o sol e he ela ed challenges
au onomously and in eal- ime [17], [19]. Depending on he dynamici y
o bo h he sys em and he op imiza ion p ocess, he DT needs o
ep esen he eal sys em wi h ce ain accu acy. To do his, he DT is
in eg a ed and ealigned wi h he physical sys em. Such a ealignmen
mechanism ypically in ol es moni o ing and ML schemes.
Tu ning ou a en ion back o he op ical ne wo k, he a ge is o use
he PLM as a DT, a model wi h app op ia e se o pa ame e s and a
mechanism o adjus hem o suppo he op imiza ion ask a hand. Fo
s a ic op imiza ion asks, such as inc emen al planning discussed abo e,
he only op ion is o ain he PLM once, jus be o e aking he decisions
o he en i e op imiza ion ask. This esul s in lowe ma gins and
inc eased ne wo k e iciency. Bu he main a ge and bene i s o DT
comes in dynamic op imiza ion. In dynamic op imiza ion, we would like
o squeeze he ma gins and achie e highe e iciency, making he
accu acy o PLM a c i ical ac o . Fo example, he accu acy o he PLM
de e io a es as connec ions a e es ablished/ eleased/ e ou ed/
change hei powe . Fo dynamic op imiza ion asks ha in ol e ew
such calcula ions and ac ions, e.g. he es ablishmen o econ igu a ion
o a single connec ion, he accu acy o he PLM would be accep able i i
was ealigned be o e he calcula ion. Howe e , ealignmen o he PLM
is expensi e; i equi es one o mo e con ol loops, including moni o ing
ha can be ime consuming and hus i migh no be easible. Mo eo e ,
o mo e complex/mul i a iable dynamic op imiza ion asks, ha
equi e mul iple econ igu a ions he accu acy o he PLM can become
c i ical. Algo i hms used in such cases a e ypically i e a i e, hey
calcula e se e al in e media e solu ions and imp o e o e hem o ind
he op imum, which is hen con igu ed in he ne wo k [11]. Howe e ,
he accu acy o he PLM de e io a es a e se e al in e media e
calcula ions and a e a poin i can ail o suppo he op imiza ion
calcula ions. The key ad an age is ha he ne wo k ope a es and hus
we can ealign he PLM/ e ain i s pa ame e s, so ha i ollows he
p ojec ions o s a es in e media ely calcula ed by he algo i hm.
In pa icula , we s udy he dynamic launch powe op imiza ion
p oblem, whe e we assume ha we ha e a se o es ablished
connec ions and we wan o op imize hei powe s while he ne wo k
ope a es. The op imum launch powe s can be ound wi h a con ex
op imiza ion algo i hm ha pe o ms se e al in e media e calcula ions.
To sol e he p oblem, h ee me hods a e explo ed: i) ha ing he
op imiza ion algo i hm p obe and moni o he ne wo k a each
in e media e i e a ion, ii) using a one- ime ained PLM o all
op imiza ion i e a ions, iii) implemen ing an i e a i e closed con ol
loop p ocess ha a e a numbe o in e media e i e a ions con igu es
he ne wo k, moni o s and e ains he PLM. We will e e o he las
op ion, he p oposed solu ion, as op imiza ion wi h a DT, since i
includes, apa om he PLM, e ol ing ne wo k condi ions, app op ia e
choice o pa ame e s o he PLM and he p ocess o align i o suppo
he dynamic op imiza ion a hand [18]. Al hough we applied ou
p oposed solu ion o he dynamic launch powe op imizing p oblem,
he p oposed i e a i e closed con ol loop which includes he
ealignmen o he PLM is gene ic. I can be applied o o he dynamic
mul i a iable op imiza ion p oblems such as dynamic esou ce
alloca ion, au oma ic ne wo k econ igu a ion, de agmen a ion, i ual
ne wo k econ igu a ion e c. [9], [16], [21]-[25]. I also p o ides ideas o
how o ealign he PLM in simple dynamic and e en s a ic op imiza ion
asks.
The emainde o his pape is o ganized as ollows. Sec ion 2
p esen s an o e iew o he ela ed wo k o exis ing powe
op imiza ions schemes, dynamic op imiza ion and closed con ol loops.
Sec ion 3 p esen s simula ions ha expose he op imiza ion misma ch
when using a one- ime ained PLM wi h espec o he eal
wo ld/op ical ne wo k. Then in Sec ion 4 we desc ibe he p oposed
(DT) op imiza ion concep . In sec ion 5, we e alua e he pe o mance o
he p oposed scheme. Finally, Sec ion 6 concludes he pape .
2. RELATED WORK
Op imiza ion in op ical ne wo ks is ypically classi ied as
planning/s a ic and online/dynamic. Dynamic op imiza ion e e s o
making changes while he ne wo k ope a es. Bo h s a ic and dynamic
op imiza ion in ol e algo i hms which ange om op imal o heu is ics
ha a e ypically i e a i e. They pe o m in e media e calcula ions un il
hey ind he inal solu ion. A hese in e media e calcula ions hey
gene ally ely on PLMs o ake in o accoun he physical laye . The PLMs
se e as es ima o s; hey es ima e he QoT o unes ablished o
econ igu ed connec ions [7], [8]. The PLM is a model ha has se e al
inpu pa ame e s, which a e known wi h ce ain accu acy, and hus
needs o use app op ia e ma gins o he op imiza ion ask a hand [3].
Fo example, o es ablishing connec ions ma gins a e gene ally used o
model he inaccu acy o he PLM and also o accoun o he e olu ion o
he physical laye o e he li e ime o he connec ions, inc eased
in e ence o upcoming connec ions, equipmen ageing, e c. Recen ly ML
has been used o imp o e he accu acy o he PLM by implemen ing i
wi h ML models [5] o i ing he pa ame e s o he exis ing PLM so ha
i s es ima ions ma ch hose moni o ed in he ne wo k [6], [8].
The op imiza ion p oblems in op ical ne wo ks a e mul idimensional
and combina o ial; a change in one a iable a ec s se e al o he s. Fo
example, an es ablishmen o a new connec ion o a change in a single
ansponde launch powe esul s in a ia ions in he QoT o all
in e e ed connec ions. The e ec o a ew econ igu a ions is ela i ely
easy o p edic . Howe e , in complex/mul i a iable op imiza ion
p oblems, i he algo i hm a in e media e s eps has assumed se e al
econ igu a ions, i can p ojec he ne wo k in o s a es whe e he PLM
su e s om low accu acy. This would mislead he subsequen
calcula ions and esul in poo op imiza ion.
Rega ding launch powe op imiza ion, se e al wo ks ha e appea ed
aiming he minimiza ion o non-linea sel - (o in a-channel) and mo e
impo an ly he c oss- (o in e -channel) in e e ence e ec s [2], [26]-
[30]. These c oss-channel nonlinea i ies (XCI) c ea e in e dependencies
among he launch powe s o he connec ions ha sha e a link, making
he p oblem mo e complex, as men ioned in he p e ious pa ag aph.
Au ho s in [2], [26] p esen ed se e al app oaches a ge ing he
op imiza ion o he launch powe s o all he channels be o e
es ablishmen (s a ic p oblem), wi h he objec i e o maximizing he
ne wo k spec al e iciency. Speci ically, [26] discussed he po en ial
ne wo k le el gains achie ed by op imizing he powe , cons ella ion
and ou e and wa eleng h alloca ions, conside ing he Gaussian Noise
(GN) model [27] as he PLM. The pa ame e s o he PLM such as ibe
non-linea i y, a enua ion, dispe sion coe icien s, ansponde
misma ch loss, ampli ie ( la ) gain e c. we e assumed o be ixed du ing
he op imiza ion ask. Also, o he p e ious wo ks we e based on a ixed
pa ame e PLM, and p oposed heu is ics o op imize all channels
launch powe s [1], [28]. No e ha a ixed pa ame e s PLM wo ks ine
o ew econ igu a ions, bu when he algo i hm decides on ex ensi e
econ igu a ions, and in pa icula , he adjus men o he launch powe s
o se e al connec ions, he accu acy o he PLM can become c i ical. The
abo e wo ks selec a di e en launch powe o each connec ion,
assumed o be se a each span ha he connec ion c osses. The local
op imiza ion leads o global op imiza ion (LOGO) me hod [27]
maximizes each span’s SNR assuming he same powe o all
connec ions c ossing i . D awbacks a e ha LOGO assumed spans wi h
ull load and canno ans e ma gins among he connec ions.
The au ho s o [29] o mula ed he p oblem o op imizing he launch
powe s o all connec ions o maximize he sum o he minimum channel
ma gin using a PLM based on he GN model (wi h ixed pa ame e s) as
a con ex op imiza ion p oblem. An ex ension o [29], ha imp o es he
SNR es ima ion accu acy om measu emen s ( hus assuming an
ope a ing ne wo k / dynamic op imiza ion) was p esen ed in [30]. The
au ho s p oposed o p obe (change he launch powe ) and moni o he
ne wo k, and use ha o calcula e he pa ial de i a i es needed by he
con ex op imiza ion algo i hm’s in e media e calcula ions. The limi ing
ac o s o ha wo k we e he assump ion on pe ec non-linea
impai men s moni o ing, which is gene ally conside ed e y ha d,
along wi h he ex ensi e in e ac ions wi h he ne wo k o p obing.
Addi ionally, he analysis was ocused on a single link.
Simila PLM accu acy issues a ise in o he mul i a iable dynamic
op imiza ion p oblems such as dynamic esou ce alloca ion, au oma ic
ne wo k econ igu a ion, de agmen a ion, i ual ne wo k
econ igu a ion e c. [9], [16], [21]-[25], whe e he op imiza ion
algo i hm elies on he PLM o pe o m calcula ions o candida e
econ igu a ions. The PLM in hose ela ed wo ks was assumed o ha e
ixed pa ame e s o was aligned be o e he op imiza ion ask and was
used o ake decisions which we e a e wa d con igu ed o he ne wo k.
Fo he ex ensi e econ igu a ions a ge ed in he abo e wo ks, he PLM
can ail since i s accu acy d ops as he algo i hm in i s in e media e
calcula ions p ojec s he ne wo k in o new s a es.
We he e p opose o use an i e a i e closed con ol loop o sol e
dynamic mul i a iable op imiza ion p oblems. A key pa o he
p oposed i e a i e closed con ol loop is ha a e a numbe o
op imiza ion algo i hm in e media e calcula ions we close he loop,
con igu e he ne wo k and ealign he PLM wi h he eal wo ld, ia
moni o ing and ML. By in oducing hese e aining cycles, he PLM
ep esen s he eal physical sys em wi h enough accu acy o pe o m
he op imiza ion ask a hand. The PLM becomes a digi al win (DT); a
model o he sys em wi h pa ame e s ha e ol e/adjus , and a means
o dynamically adjus ing i .
Closed con ol loops ha e been ex ensi ely s udied in con ol heo y
as discussed in [11], [31]. Howe e , con ol heo y ypically a ge s
in ini e ime ho izon p oblems, and conside s as loops wi h eal- ime
eedback. Also, ein o cemen lea ning has ecei ed a en ion on simila
opics [32]. Rein o cemen lea ning also a ge s in ini e ime ho izon
p oblems and a sys em desc ibed by a Ma ko Decision P ocess, which
is di e en om he con ex op imiza ion p oblem ha we ha e a hand.
To he bes o ou knowledge, he iden i ied issue o he lack o
accu acy o he PLM in dynamic op imiza ion p oblems has no been
s udied in he pas . No e ha , con ex op imiza ion algo i hms and hei
in e ac ion wi h a ool ha ep esen s eali y (in op imiza ion e ms his
is e e ed o as an ‘o acle’) ha e been s udied [33], including exac and
inexac o acles wi h a ying inaccu acies models. Ou p oposed
solu ion sha es ce ain ideas om his op imiza ion ield. We use con ex
algo i hms o sol e he launch powe op imiza ion p oblem, ollowing
[29] and [30], bu we a oid hea y moni o ing and apply he
op imiza ion o he ne wo k le el. We also sha e ideas wi h [7], [8], [12]
on he use o moni o ing and ML o ain/align he PLM wi h he
physical laye condi ions. Howe e , we ex end hose and
e ain/ ealign he PLM in a closed con ol loop, a ge ing dynamic
op imiza ion p oblems.
Finally, we would also like o no e ha ou s udy is qui e mo e
ealis ic han mos p e ious wo ks ha use he same PLM as bo h he
es ima o and he g ound u h, o gene a e he in o ma ion o ain he
es ima o . In pa icula , in ou simula ions we used VPI as he g ound
u h and he GN model as he es ima o . VPI is qui e mo e de ailed and
complex and close o a eal sys em han he GN model. This choice was
made o cap u e he misma ch be ween he eal ne wo k and he PLM
ha would be used in he op imiza ion p ocess, an addi ional di icul y
which is neglec ed in mos p e ious wo ks.
3. USE CASE AND MOTIVATION
In his pape we in es iga e how he PLM accu acy a ec s he
op imiza ion calcula ions. To do his, we ocus on a dynamic e sion o
he launch powe op imiza ion p oblem. We assume ha a se o
connec ions a e es ablished, and ou goal is o op imize hei launch
powe . Thus, no connec ions a e es ablished o eleased, bu he
exis ing connec ions a e econ igu ed ( hei launch powe s a e
adjus ed) as he ne wo k ope a es. To mo i a e and be e unde s and
he p oblem in his sec ion we discuss he op imiza ion o he launch
powe s o 25 channels ansmi ed o e a single link.
A. Physical Laye Model T aining
We c ea ed a single link wi h 6 iden ical spans se up in VPI
T ansmission Make [34] as shown in Fig. 1. On his link we simula ed
he ansmission o 25 channels a 32 Gbaud wi h PM-16QAM and
assumed SNR h eshold o 𝑆𝑁𝑅𝑡ℎ =13.9dB o each [1], [6]. No e ha
hese simula ions we e ime consuming due o he high compu a ional
complexi y, as VPI uses spli -s ep Fou ie p opaga ion simula ions o
model he nonlinea signal p opaga ion o he channels. We conside ed
he VPI se up as he ‘ eal-wo ld’, he ac ual op ical ne wo k.
Fig. 1. Simula ed single link se up in VPI wi h 25 x 32Gbaud, PM-16QAM,
50GHz spaced ansmi e s and 6 iden ical spans.
We also implemen ed a PLM, and in pa icula he GN-model [27],
wi h a simila se up o 6 iden ical spans and 25 channels. We ound
app oxima ely 1dB o max. SNR di e ence be ween he PLM and VPI,
when all pa ame e s o he PLM and VPI (dispe sion coe icien , slope,
a enua ion coe icien o ibe , non-linea i y coe icien e c.) we e se
equal. Then we aligned he PLM wi h he eal wo ld (VPI). This
alignmen can be done wi h a ious me hods. In ou case, we moni o ed
Gain=16dB
NF = 5.5dB
80 km
˟Nsspans
Independen
ecei e blocks wi h
impai men
compensa ion
•CD & PMD
compensa ion
•Clock Phase
eco e y
•Ca ie equency
eco e y
•Time Domain
equaliza ion e c….
SNR1
SNR2
SNR12
SNR24
SNR25
MUX DEMUX
ecei e
band pass
il e s bank
span
( ibe +EDFA)
25 pol-mux 16QAM
ansmi e s
(32Gbaud,
50GHz spaced)
he channels SNR alues (in VPI) and adjus ed he PLM pa ame e s so
ha i s SNR es ima ions ma ch wi h he eal wo ld (VPI), using ML.
To be mo e speci ic, we implemen ed he ollowing alignmen
p ocess o he GN model [5], [6]. We assume a ne wo k wi h N
connec ions. The GN PLM is a model ha akes as inpu se e al
pa ame e s and calcula es he SNR alues o he connec ions. Le
deno e he se o GN model i ed pa ame e s: i) ibe a enua ion
coe icien s, ii) ibe non-linea coe icien s, iii) ibe dispe sion
coe icien s, i ) a wa eleng h dependen penal y e m, implemen ed as
a 4 h o de polynomial, o co e ansponde loss misma ches, ampli ie
ipple e c., and ) a bias. Also le 𝒑 = [𝑝1, 𝑝2,… 𝑝𝑁] be he launch powe
ec o o he N connec ions, which a e he a iables ha will be
op imized la e , and le z ep esen s he unchanged inpu pa ame e s
o ou op imiza ion, such as ou es, used wa eleng hs, span leng hs e c.
We deno e by 𝑄𝑛(𝒑,𝒓, 𝒛) he GN model SNR es ima ion o connec ion
n, and wi h 𝑸𝑵(𝒑,𝒓,𝒛) he SNR ec o o all connec ions N. The SNR
calcula ion unc ion is non-linea in i s pa ame e s (and also p). Finally,
le 𝒀𝒏(p) deno e he moni o ed SNR alue o connec ion n and 𝒀𝑵(𝒑)
deno e he ec o o all he connec ions N. In his wo k such moni o ing
is assumed o be done a he cohe en ecei e s. The aining e o
ec o is gi en by 𝑸𝑵(𝒑,𝒓,𝒛) − 𝒀𝑵(p), and he objec i e o he i ing
is o iden i y he pa ame e s ha minimize he squa ed e o . To i
his, we elied on he Le enbe g- Ma qua d (LM) algo i hm which is
sui able o sol ing nonlinea leas squa es i ing p oblems [35]. The
LM algo i hm inds
𝒓𝟎= a gmin𝑟(𝑸𝑵(𝒑,𝒓,𝒛)− 𝒀𝑵(𝒑))2 (1)
When we pe o m his PLM alignmen once, be o e he op imiza ion
ask, he PLM e lec s wi h good accu acy he s a ing s a e o he
ne wo k p io o op imiza ion. We e e o his as one- ime ained PLM
and deno e i by 𝑸𝑵(𝒑,𝒓𝟎, 𝒛).
We s udied wo ypes o e bium doped ibe ampli ie s (EDFA): one
whose gain is pe ec ly la /ideal and ano he wi h a gain ipple p o ile
o a peak o peak (p2p) alue o ±0.2dB [6]. No e ha he EDFAs we e
assumed o be ope a ed in au oma ic gain con ol (AGC) mode wi h
a e age gain equal o he p e ious span loss. We call he se up wi h he
la span EDFAs as Case 1, and he se up wi h EDFAs ha ing gain ipple
as Case 2. Case 1 ep esen s an ideal ne wo k wi h ela i ely s able
physical laye condi ions. On he o he hand, Case 2, wi h ippled EDFAs,
ep esen s a mo e ealis ic scena io wi h mo e ola ile/ dynamic
physical laye condi ions. The physical laye dynamici y comes om he
ac ha an EDFA wi h a gain ipple in oduces SNR a ia ions when
changing he connec ions powe s. These a ia ions a e ha d o es ima e,
unless we exac ly know he gain p o ile o he EDFA. This p o ile is ha d
o be ound in an ope a ing ne wo k and i migh change o e long ime.
Fig. 2. Es ima ed SNR alues om he one- ime ained PLM and VPI a
uni o m 0dBm launch powe and he ela ed aining e o o (a) la
EDFA, and (b) EDFA wi h gain ipple.
Fig. 2 (a) and Fig. 2 (b) show he es ima ed SNR alues o he
connec ions om he one- ime ained PLM 𝑸𝑵(𝒑,𝒓𝟎,𝒛) and he eal
ne wo k (VPI) 𝒀𝑵(p) a uni o m launch powe o p=0dBm, o la and
ippled EDFAs, espec i ely. The co esponding aining e o s a e also
displayed in he same igu es. Wi h one- ime aining, he PLM
pa ame e s we e adjus ed qui e well and i s es ima ed SNR alues
ma ched hose o he ac ual op ical ne wo k/ eal wo ld a he ini ial
s a e. This is deduced by he e y low e o s, less han 0.1dB o bo h,
Case 1 and Case 2.
B. Dynamic Launch Powe Op imiza ion
We now u n ou a en ion o he dynamic launch powe
op imiza ion p oblem. Fo a gene ic opology we assume ha we ha e
a se o N es ablished connec ions. The objec i e is o op imize he
launch powe s o he N ansponde s o maximize
(i) Objec i e 1: sum o connec ions ma gins
max 𝑓(𝒑)=∑(𝑙𝑜𝑔 𝑆𝑁𝑅𝑛(𝒑)− 𝑙𝑜𝑔 𝑆𝑁𝑅𝑡ℎ,𝑛)
𝑁
𝑛=1
(ii) Objec i e 2: minimum ma gin
max 𝑓(𝒑)=𝑚𝑖𝑛𝑛∊[1,𝑁](𝑙𝑜𝑔 𝑆𝑁𝑅𝑛(𝒑)− 𝑙𝑜𝑔𝑆𝑁𝑅𝑡ℎ,𝑛)
subjec o:
𝑙𝑜𝑔 𝑆𝑁𝑅𝑛(𝒑)− 𝑙𝑜𝑔𝑆𝑁𝑅𝑡ℎ,𝑛 ≥ 0,∀𝑛 ∊ [1,𝑁]
𝒑𝒎𝒊𝒏 ≤ 𝒑 ≤ 𝒑𝒎𝒂𝒙
whe e 𝒑 = [𝑝1, 𝑝2,… 𝑝𝑁] is he launch powe ec o o he N
connec ions; 𝑝𝑚𝑖𝑛 and 𝑝𝑚𝑎𝑥 a e he lowe and uppe powe limi s o
he ansponde s’ launch powe ; 𝑆𝑁𝑅𝑛(𝒑) is he SNR o connec ion n
o he co esponding powe ec o 𝒑; and 𝑆𝑁𝑅𝑡ℎ,𝑛 is he SNR
h eshold equi ed o he modula ion o ma o n.
The op imiza ion o channels’ launch powe s wi h one o he abo e
objec i es is known o be con ex and o polynomial complexi y [29],
[30]. Hence, we used he in e io -poin algo i hm o sol e i . In gene al,
a con ex op imiza ion algo i hm pe o ms in e media e
calcula ions/i e a ions. A each in e media e i e a ion he algo i hm
decides on new ansponde s launch powe s o mo e owa ds he
op imum. Howe e , hese in e media e s eps a e in e nal, only he inal
(op imal) will be con igu ed in he ne wo k. The algo i hm decides
hese s eps by using he knowledge o he pa ial de i a i es ( i s and
some imes second o de , depending on he algo i hm) o he objec i e
and cons ain s wi h espec o he a iables (launch powe s 𝒑).
The i s op ion o calcula e such de i a i es is o in e ace he
op imiza ion algo i hm wi h a PLM. In his case, he 𝑺𝑵𝑹𝒏(𝒑) alues in
he algo i hm come om he PLM calcula ions 𝑸𝒏(𝒑,𝒓,𝒛). I he PLM
has closed o m pa ial de i a i es, hen he op imiza ion p ocess is
s aigh o wa d. Howe e , ypically he PLMs (e.g. GN model) do no
ha e closed o m de i a i es. Then we can use a de i a i e
iden i ica ion sub ou ine based on ini e di e ences. This sub ou ine
makes changes in he launch powe s and uses he PLM o calcula e he
ou comes (connec ions’ new SNR alues). In his sec ion we will assume
ha we use a PLM ha was aligned/ ained once be o e he
op imiza ion, as discussed abo e, so we use he i ed pa ame e s 𝒓 =
𝒓𝟎. We will e e o his as op imiza ion wi h one- ime ained PLM.
An al e na i e op ion is o p obe he eal ne wo k, ha is, o in e ace
he algo i hm and, in pa icula , he de i a i e iden i ica ion sub ou ine
wi h he ne wo k, bypassing he PLM. In his case he 𝑺𝑵𝑹𝒏(𝒑) alues
in he algo i hm come om he moni o s o he ne wo k 𝒀𝒏(𝒑). The
de i a i e iden i ica ion p ocess would con igu e h ough he con ol
plane he launch powe s o he ansponde s, and would moni o he
ou comes (connec ions’ SNRs) o calcula e he de i a i es. This would
be epea ed a each algo i hm’s i e a ion. We will e e o his op ion as
op imiza ion wi h moni o ing p obes.
No e ha he o me op ion is as . The PLM is ained once and used
he ea e o compu e he de i a i es. Al hough he PLM is called
se e al imes, i has low compu a ion complexi y (a leas he GN
model), esul ing in low o e all op imiza ion ime. Howe e , his op ion
su e s om accu acy issues. Speci ically, se e al pa ame e s such as he
-0.1
-0.05
0
0.05
0.1
21.4
21.6
21.8
22
22.2
22.4
22.6
0 7 14 21 28
SNR(dB) aining e o
SNR (dB)
channel ID
eal wo ld (VPI)
one ime ained PLM
aining e o
-0.1
-0.05
0
0.05
0.1
20.8
21.2
21.6
22
22.4
22.8
0 7 14 21 28
SNR(dB) aining e o
SNR (dB)
channel ID
eal wo ld (VPI)
one ime ained PLM
aining e o
(a). (b).
ampli ie gain ipple, non-linea in e e ence (NLIs), c oss alk a
swi ches, e c., change o di e en ne wo k con igu a ions/s a es. So,
he one- ime ained PLM which is qui e accu a e a he
beginning/ini ial s a e (Fig. 2(a) and (b)) beha es a he inaccu a ely as
he i e a i e op imiza ion algo i hm p ojec s he ne wo k in o s a es
ha a e away om he ini ial. The inaccu acy o he PLM esul s in
inaccu a e es ima ion o he de i a i es which in u n esul s in
subop imal op imiza ion o he launch powe s. This accu acy p oblem
is expec ed o be mo e p o ound when he ne wo k physical laye is
mo e dynamic, as in Case 2, whe e EDFA gain ipples esul in SNR
a ia ions as he launch powe s change.
On he o he hand, he la e op ion, op imiza ion wi h moni o ing
p obes, in ol es se e al in e ac ions wi h he ac ual ne wo k a each
in e media e s ep, which ypically ake long ime and is also suscep ible
o moni o ing e o s. No e ha , he e m moni o ing p obes e e s o he
capabili y o he ne wo k o change he launch powe s and moni o he
ou comes. In o he op imiza ion p oblems, e.g. in ol ing
es ablishmen / elease o connec ions, such capabili y will p obably no
be p esen . Finally, no e ha in he esul s p esen ed he e and in Sec ion
5 up o Fig. 12, he moni o ing e o was assumed o be ze o. Thus, he
esul s ob ained wi h moni o ing p obes and ze o moni o ing e o a e
op imal and se as he e e ence o all o he cases.
Fig. 3. (a) Op imized launch powe s, and (b) co esponding SNR and
obj#1 alue, e alua ed in he eal ne wo k (VPI), o Case 1 ( la EDFAs).
C. De ia ion o Op imizing wi h he One-Time T ained PLM
Fig. 3 (a) and (b) show he op imized launch powe s o obj#1 wi h
he one- ime ained PLM and he moni o ing p obes app oaches o
la EDFAs (case 1). We see ha he one- ime ained PLM did no
suppo well he op imiza ion, since he algo i hm using i iden i ied
qui e di e en powe le els. Tha is, al hough he algo i hm using he
one- ime ained PLM iden i ied he op imum, his was op imum o he
PLM and no close o he op imum in he eal ne wo k (VPI). The eason
o his is ha he PLM could no ollow/p edic wi h good accu acy he
eal SNR alues a he powe le els calcula ed by he algo i hm,
al hough he accu acy was e y good o he ini ial s a e o he ne wo k,
igh a e he (one ime) aining. A ma gin on he PLM could co e his,
bu again would esul in subop imal calcula ions. The maximized sum
o SNR ma gins (obj#1) was op imized o 204.96dB when he algo i hm
used moni o ing p obes and in e ac ed wi h he eal ne wo k, and o
199.31dB when i in e ac ed wi h he one- ime ained PLM. The e
exis s a misma ch o ~5.6dBs in obj#1 alue be ween hese wo
op imiza ion app oaches. No e ha he SNR alues and he objec i e
(Fig. 3 (b)) we e and should be e alua ed in he eal wo ld (VPI), so ha
we can see he de ia ion. This also explains he ipples in SNR seen in
Fig 3(b), since VPI models some wa eleng h dependen ac o s no
co e ed by GN. Simila beha io was obse ed o obj#2, no shown
he e o conciseness. No e ha op imiza ion wi h obj#2 esul s in
choosing he launch powe s ha esul in almos la SNR alues, since
maximizing he minimum ma gin i e a i ely pushes he lowes SNR
alue and educes he highe . The maximized minimum ma gin was
op imized o 7.64dB wi h moni o ing p obes, and o 7.36dB wi h he
one- ime ained PLM.
To emula e a mo e ealis ic scena io we assigned a gain ipple p o ile
o all EDFAs ha ing a p2p ipple alue o ~±0.2dBs (case 2). In such a
scena io when he algo i hm used he one- ime ained PLM i eached
an op imiza ion objec i e (e alua ed in he eal ne wo k - VPI) qui e
wo se han when i used moni o ing p obes and in e ac ed wi h he
ac ual ne wo k a in e media e op imiza ion i e a ions. Fig. 4 (a) and (b)
shows he op imized launch powe s and hei co esponding SNR
alues espec i ely, o obj#1. A maximum inpu powe di e ence o
~1.5dBm was obse ed, esul ing in ~8.4dB o SNR di e ence o
obj#1. Simila ly, o obj#2, we obse ed a maximum inpu powe
di e ence o ~1.2dBm, esul ing in ~0.62dB o SNR di e ence o
obj#2. No e ha he misma ch is highe han p e iously (case 1 / la
EDFAs). This is because we now ha e a mo e ola ile physical laye
(EDFA gain ipples a ec he SNRs) and he PLM we use does no co e
his addi ional ola ili y.
Fig. 4. (a) Op imized launch powe , and (b) co esponding SNR and
obj#1 alue, e alua ed in he eal ne wo k (VPI), o Case 2 (EDFA wi h
gain ipple o ±0.2dB).
Concluding, o any op imiza ion p oblem he PLM accu acy is
impo an . Fo planning/s a ic p oblems, we co e inaccu acy issue
wi h ma gins, while se e al pape s ha e a ge ed he educ ions o
ma gins, by aligning he PLM o he physical laye condi ions e.g.
h ough moni o ing and ML. Howe e , he e ha e been limi ed
discussions on dynamic op imiza ion p oblems; he disad an age is ha
o jus i y dynamic op imiza ion we should a ge o achie e high
e iciency, making he accu acy o he PLM mo e c i ical. Fo complex /
mul i a iable dynamic op imiza ion asks, such as he dynamic launch
powe op imiza ion p oblem discussed abo e, an i e a i e algo i hm is
ypically used ha calcula es se e al in e media e solu ions. One op ion
is o in e ace he algo i hm wi h he ne wo k o p obe and moni o i in
o de o ca y ou he in e media e s eps un il i achie es he op imum.
This, howe e , is cumbe some and e y slow. On he o he end, we can
ain he PLM be o e he op imiza ion and use i in all in e media e
calcula ions. Since PLM calcula ions a e as he op imiza ion will inish
quickly. Howe e , he accu acy o he PLM can de e io a e and esul o
subop imal op imiza ion as seen in he p elimina y esul s discussed
abo e. This mo i a ed us o add ess he limi a ions o he op imiza ion
wi h one- ime ained PLM by explo ing he ope a ing ne wo k and i s
eedback. Ou goal is o app op ia ely ealign he PLM a in e media e
op imiza ion calcula ions, so ha he di e ence be ween he
op imiza ion objec i e achie ed wi h moni o ing p obes (in e ac ing
wi h he eal-wo ld) and wi h he e ained PLM is negligible, while he
whole op imiza ion is much as e .
(a). (b).
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 7 14 21 28
obj#1 launch powe (dBm)
channel ID
moni o ing p obes
one ime ained PLM
21.7
21.9
22.1
22.3
22.5
22.7
0 7 14 21 28
SNR (dB)
channel ID
moni o ing p obes
one ime ained PLM
obj#1 alue:
•204.96dB
•199.31dB
(a). (b).
21
21.3
21.6
21.9
22.2
22.5
22.8
23.1
0 7 14 21 28
SNR(dB)
channel ID
moni o ing p obes
one ime ained PLM
-2.5
-2
-1.5
-1
-0.5
0
0 7 14 21 28
obj#1 launch powe (dBm)
channel ID
moni o ing p obes
one ime ained PLM
obj#1 alue:
•202.97dB
•194.55dB
4. NETWORK DYNAMIC OPTIMIZATION AND PLM RE-
TRAINING
PLMs, which can be analy ical, semi-analy ical, ML models, e c., ha e
ce ain accu acy. The modeling assump ions impac he es ima ion
accu acy. Fo example, many PLMs neglec EDFA gain ipples, pa ially
model NLIs (e.g. conside ull load), il e s’ (inside ROADMS) c oss alk,
esidual dispe sion, speci ic pa ame e s o ansponde s, e c. No e ha
a de ailed PLM is slowe in he calcula ions and equi es mo e inpu
pa ame e s. Then, a second ac o comes in play: he inpu pa ame e s
migh no be known wi h good accu acy, which e en ually educes he
accu acy o a de ailed PLM.
Op imiza ion asks esul in ne wo k changes and ypically use a PLM
o es ima e he e ec o such changes. Howe e , hese changes also
modi y he physical laye i sel , hey mo e he ne wo k o a new s a e.
Depending on he PLM, such changes a e co e ed o a ce ain deg ee.
Fo example, when changing he powe o a connec ion, NLIs, c oss alk,
bu also he penal ies due o EDFAs’ gain ipple p o iles change. The
PLM model could o example, co e he e ec o NLIs and c oss alk, bu
no he e olu ion o gain ipples. Fo hese easons ma gins a e used.
Howe e , in dynamic use cases he aim is o be mo e e icien and hus
he accu acy o he PLM is a c ucial ac o . To imp o e ha we can ake
ad an age o he ope a ing ne wo k.
Hence a basic need o dynamic op imiza ion is o ha e a PLM ha
ollows he ne wo k changes. In he AI/ML e a, a way o do his is o
choose an app op ia e se o pa ame e s and e ain he PLM a ce ain
poin s. Howe e , e aining is cumbe some and hus we canno e ain
i be o e e e y dynamic ask. On he o he end, aining he PLM once
be o e a mul i a iable op imiza ion ask can esul in subop imal
op imiza ion, since he accu acy o he PLM de e io a es a e se e al
in e media e calcula ions.
In his pape we ocus on dynamic mul i a iable op imiza ion
p oblems, and, in pa icula , we s udy he launch powe op imiza ion
p oblem o es ablished connec ions as in oduced in he p e ious
sec ion. No e, howe e , ha he p oposed solu ion is gene ic and
applicable o o he dynamic simple o mul i a iable op imiza ion
p oblems as well. We p opose o use an i e a i e closed con ol loop
p ocess o sol e such dynamic mul i a iable op imiza ion p oblems. A
ce ain in e media e i e a ions o he algo i hm we close he loop,
con igu e he ne wo k and moni o o e ain he PLM (wi h ML) o
ollow he p ojec ed ne wo k condi ions. The a ge is o make he PLM
a digi al eplica, ha is, a digi al win (DT), o he op ical physical laye
o he dynamic op imiza ion ask a hand. The equency o he PLM
e aining depends on he PLM model and on he op imiza ion ask. As
discussed in he p e ious sec ion al e na i e op ions o he algo i hm
a e o a oid using a PLM and ha e he algo i hm in e ac wi h (p obe
and moni o ) he ne wo k o use a one- ime ained PLM. All h ee
op ions a e o mally desc ibed in he ollowing subsec ions.
A. Op imiza ion wi h Moni o ing P obes
The scheme ha is conside ed in his subsec ion assumes ha he
op imiza ion algo i hm in e ac s di ec ly wi h he ac ual ne wo k and
ollows a closed con ol loop p ocess. The algo i hm employs a
sub ou ine o speci y he p obes, he con igu a ions ha a e applied o
he ne wo k. Then i moni o s he ou comes o iden i y he in o ma ion
ha i needs o an in e media e op imiza ion s ep. A ep esen a ion o
his scheme is shown in Fig. 5. (a).
To be mo e speci ic, we ocus on he dynamic launch powe
op imiza ion p oblem wi h a ypical objec i e such as maximizing he
sum o SNR ma gins, o min. ma gin, as discussed in Sec ion 3. This
p oblem is known o be con ex and o polynomial complexi y. The
con ex op imiza ion algo i hms, such as (sub)g adien me hods,
in e io poin , us - egion- e lec i e e c., a e i e a i e; a each i e a ion
hey need o calcula e Jacobians and/o Hessians o he objec i e and
cons ain unc ions [36], [37]. Ac ually, he ela ed algo i hms a e
classi ied in o i s o second o de depending on he o de o he pa ial
de i a i es hey use. Fo op imiza ion p oblems ha in ol e PLMs
wi hou closed o m pa ial de i a i es a way o calcula e hem is o use
a sub ou ine ha implemen s ini e di e ences [37].
Fo example, o he powe op imiza ion p oblem a hand, o
calcula e he g adien o an objec i e unc ion we need o
ind/moni o he changes in he SNR alues o all connec ions, assumed
o be done h ough he cohe en ecei e s, wi h espec o changes in
he powe s o he ansponde s. To gi e an example, assuming a
ne wo k wi h a se o N es ablished connec ions wi h a launch powe
ec o 𝒑. We deno e by 𝜹𝒑𝒏 he ec o wi h all ze os apa om elemen
n whose alue we se o 𝑝𝑠𝑡𝑒𝑝, wha we e e o as he powe p obe s ep.
As a ma e o ac , he change in launch powe o he single connec ion
n esul s in changes in he SNR alues o all in e e ed connec ions
( hose ha sha e a common link). So, we deno e by 𝑺𝑵𝑹𝑵(𝒑) and
𝑺𝑵𝑹𝑵(𝒑 + 𝜹𝒑𝒏) he SNR ec o o all N connec ions o he espec i e
powe ec o s. I is he objec i e unc ion, hen he i s o de pa ial
de i a i e o n is gi en by
𝑔𝑛 = 𝑓(𝒑)− 𝑓(𝒑 + 𝜹𝒑𝒏)/𝑝𝑠𝑡𝑒𝑝 (2)
Depending upon he unc ion his in ol es ce ain ope a ions wi h he
ec o s 𝑺𝑵𝑹𝑵(𝒑) and 𝑺𝑵𝑹𝑵(𝒑 + 𝜹𝒑𝒏). The g adien g is he ec o o
all pa ial de i a i es, ha is, 𝑔𝑛 o all n. To calcula e he g adien wi h
he ini e di e ence me hod we need o p obe wi h 𝒑 + 𝜹𝒑𝒏 and
moni o 𝑺𝑵𝑹𝑵(𝒑 + 𝜹𝒑𝒏), and epea his p obe/moni o ing p ocess
o all connec ions n=1,…,N.
Gene alizing his, he op imiza ion algo i hm calcula es ( i s o
second o de ) pa ial de i a i es h ough a ini e di e ences sub ou ine
a each in e media e i e a ion. Le us assume ha he algo i hm calls he
ini e di e ences me hod di imes a i e a ion i. Fo he abo e example
wi h he g adien , we ha e 𝑑𝑖= 𝑁. This is he simples case; we
ypically ha e 𝑑𝑖≥ 𝑁 depending on he algo i hm. No e also ha we
migh ha e di e en numbe o p obes pe i e a ion, ha is di e en 𝑑𝑖
pe i. Howe e , o simpli y ou analysis we assume ha his is cons an ,
𝑑𝑖= 𝐷, o each i e a ion i.
Fig. 6. Pseudo-code o op imiza ion wi h moni o ing p obes.
The con ex op imiza ion algo i hm wi h moni o ing p obes
pe o ms 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 i e a ions o ind he op imum. We also deno e by
𝑡𝑚𝑜𝑛 he moni o ing ime, assumed he e o moni o simul aneously all
N es ablished connec ions. The moni o ing ime can ange om
minu es o hou s, depending upon he ne wo k size, he moni o ing
plane, he a ge ed moni o ing e o e c. [38]. Howe e , once he
moni o ing in o ma ion is o wa ded o he algo i hm, he ime 𝑡𝑐𝑎𝑙𝑐 o
calcula e he g adien s/ Hessian and also he nex launch powe s is
Pseudo-code - 1
S a wi h ini ial launch powe ec o 𝒑𝟎, i e a ion numbe i=-1
While no con e ged
Inc ease i
Fini e di e ences p ocess, ime: 𝑡𝑚𝑜𝑛 pe p obe o moni o all
connec ions 𝒀𝑵
P obe di imes (con igu e new launch powe s and moni o )
(e.g. o he i s o de pa ial de i a i e o connec ion n, p obe
wi h 𝒑𝒊+ 𝜹𝒑𝒏 and moni o 𝒀𝑵(𝒑𝒊+ 𝜹𝒑𝒏))
Calcula e he de i a es and nex launch powe ec o 𝒑𝑖, ime:
𝑡𝑐𝑎𝑙𝑐
E alua e con e gence (e.g. compa e objec i e imp o emen wi h
a h eshold), when con e ged: 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 = 𝑖
Apply he calcula ed powe ec o 𝑝𝑖 o he ne wo k
qui e lowe (msec ange) compa ed o he moni o ing ime (𝑡𝑚𝑜𝑛 ≫
𝑡𝑐𝑎𝑙𝑐). So, wi h he moni o ing p obes-based app oach, unde he
assump ion ha N connec ions a e moni o ed in pa allel, he o al
op imiza ion ime 𝑇𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 is gi en by:
𝑇𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 = 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ∙ (𝐷 ∙ 𝑡𝑚𝑜𝑛 + 𝑡𝑐𝑎𝑙𝑐) ≈ 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ∙ 𝐷 ∙
𝑡𝑚𝑜𝑛 (3)
Op imizing wi h moni o ing p obes is desc ibed wi h pseudo-code 1 in
Fig. 6.
In gene al, he op ical moni o s ha e ce ain measu ing accu acy. In
ou p oposal we assume ha we moni o he SNR om he cohe en
ecei e s which a e qui e accu a e. No e ha highe accu acy can be
achie ed h ough ime a e aging; o educe he e ec o sho e m ime
impai men s, e.g. pola iza ion, he moni o ing measu emen s could be
a e aged o e ime esul ing in highe accu acy bu also highe
moni o ing ime. Depending on he moni o ing e o , we migh end up
o a di e en and wo se objec i e alue ins ead o he op imum. Ano he
ac o o be accoun ed o in a eal ne wo k is ha he powe p obe s eps
(𝑝𝑠𝑡𝑒𝑝) canno be e y small because ine- uning o he equipmen is no
easible. Thus, in a eal ne wo k, he e a e wo ac o s ha hinde he
moni o ing p obe op imiza ion p ocess:
(i) he moni o ing e o s
(ii) he minimum powe p obe s ep 𝑝𝑠𝑡𝑒𝑝 ha can be con igu ed
We call he SNR ec o ob ained om moni o s wi h e o s as he
noisy moni o ed ec o , and deno e i by
𝒀
𝑵(𝒑) = 𝒀𝑵(𝒑) + 𝒗 (4)
whe e, 𝒗 is a ec o ha ep esen s he moni o ing e o (o noise).
S ochas ic subg adien me hods [39] o ze o mean e o s p o ably
ind he op imum solu ion wi h speci ic s ep sizes bu migh equi e a
e y la ge numbe o i e a ions. Howe e , in a eal ne wo k whe e he
use o small s eps is no suppo ed by he ansponde s and i e a ions
a e expensi e since hey in ol e se e al moni o ing phases, such
me hods a e ha dly applicable.
In his moni o ing p obes op imiza ion app oach he algo i hm
op imizes he launch powe s and checks a each s ep he ac ual
condi ions o he ne wo k. Fo ze o e o his app oach iden i ies he
op imum 𝑜𝑏𝑗𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 bu equi es a high op imiza ion ime
𝑇𝑚𝑜𝑛_𝑝𝑟𝑜𝑏. So, we will use his as he e e ence o all o he app oaches.
Also no e ha he moni o ing p obes, which a e used in his me hod
o iden i y he pa ial de i a i es, is no a uni e sal solu ion. A
moni o ing p obe in he s udied use case e e s o he con igu a ion o
new launch powe (s) o one (o mo e) ansponde s o he es ablished
connec ions and moni o ing o all connec ions SNRs a hei ecei e s.
So, he de ini ion is speci ic o he p oblem; di e en op imiza ion
p oblems equi e di e en moni o ing p obes de ini ions. Fo some
asks, moni o ing p obes migh no be a ailable, e.g. asks in ol ing he
es ablishmen / elease o connec ions. Fo such asks, we migh need
spa e ansponde s o ex ac he in o ma ion equi ed o he
op imiza ion [12], which imply highe cos and complexi y.
B. Op imiza ion wi h one- ime ained PLM
In his subsec ion, we conside he me hod whe e he PLM is aligned
only once, a he beginning o op imiza ion. We pe o m he alignmen
o he PLM using moni o ing in o ma ion 𝒀𝑵(𝒑𝟎) om he ac ual
ne wo k, assumed o ake ime 𝑡𝑚𝑜𝑛 as abo e. We hen use ML o i he
pa ame e s o he PLM 𝑸𝑵(𝒑𝟎,𝒓,𝒛) o he physical laye condi ions,
so as o iden i y 𝒓𝟎. This is assumed o ake ime 𝑡𝑡𝑟𝑎𝑖𝑛. Then he
op imiza ion algo i hm in e ac s wi h his one- ime ained PLM,
𝑸𝑵(𝒑𝟎,𝒓𝟎,𝒛), a each in e media e s ep, o es ima e he QoT (SNR) o
he connec ions as shown in Fig. 5(b). In pa icula , since he e a e no
closed o m de i a i es equa ions o he GN model, we use a simila
de i a i es iden i ica ion sub ou ine ( ini e di e ences), as in he
p e ious me hod, bu his ime we in e ace ha wi h he PLM ins ead
o he ac ual ne wo k. We deno e by 𝑡𝑃𝐿𝑀 he ime ha he PLM akes o
calcula e he SNR alues o all connec ions. As be o e, his sub ou ine is
assumed o be called D imes a each algo i hm in e media e i e a ion.
We assume ha he ime 𝑡𝑐𝑎𝑙𝑐 ha he algo i hm needs o calcula e he
g adien s/Hessian and also he nex launch powe s is he same as he
p e ious me hod. We also deno e by 𝐿𝑃𝐿𝑀 he numbe o i e a ions ha
he algo i hm pe o ms. Wi h he one- ime ained PLM, he o al
op imiza ion ime 𝑇𝑃𝐿𝑀 is gi en by
𝑇𝑃𝐿𝑀 = 𝑡𝑚𝑜𝑛 + 𝑡𝑡𝑟𝑎𝑖𝑛 + 𝐿𝑃𝐿𝑀 ∙ (𝐷 ∙ 𝑡𝑃𝐿𝑀 + 𝑡𝑐𝑎𝑙𝑐) (5)
I s ands o eason ha he PLM aining and es ima ion calcula ions
and he algo i hm calcula ions a e subs an ially as e han moni o ing
(𝑡𝑚𝑜𝑛 ≫ 𝑡𝑡𝑟𝑎𝑖𝑛,𝑡𝑃𝐿𝑀,𝑡𝑐𝑎𝑙𝑐). We also expec a simila numbe o
i e a ions (𝐿𝑃𝐿𝑀 ≈ 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏), because he PLM/GN model sa is ies
he con exi y p ope ies [29]. So, we ha e 𝑇𝑃𝐿𝑀 ≈ 𝑡𝑚𝑜𝑛. By compa ing
his o Eq. (3) we can see ha he one- ime ained PLM based
op imiza ion app oach equi es subs an ially less op imiza ion ime
han he p e ious app oach, 𝑇𝑃𝐿𝑀 ≈ 𝑡𝑚𝑜𝑛 ≪ 𝑇𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ≈
𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ∙ 𝐷 ∙ 𝑡𝑚𝑜𝑛. In pa icula , he speedup we ob ain is in he
Fig. 5. Op imiza ion wi h (a) Ac ual deployed moni o s, (b) One- ime ained PLM, and (c) P oposed PLM e aining (digi al win) app oach.
Digi al Twin based P oposed App oach
Op imiza ion Task
(launch powe op .)
op ical
ne wo k
moni o ing
op imiza ion algo.
(con ex, g ad. descen e c. )
(a). (b). (c).
Moni o ing based PLM based
op imiza ion p obes,
pa ial de i a i es wi h
ini e di e ence sub ou ine
Op imiza ion Task
(launch powe op .)
op ical
ne wo k
PLM
≈
op ical
ne wo k
Digi al Twin
Moni o ing
ML e aining
PLM
pa ame e s adap wi h
e ol ing condi ions ia
( e) alignmen / aining
Op imiza ion Task
(launch powe op .)
op imiza ion algo.
(con ex, g ad. descen e c. )
op imiza ion p obes,
pa ial de i a i es wi h
ini e di e ence sub ou ine
Lk
op imiza ion
I e a ions
𝒑 , = 𝒑
new launch powe s
o ansponde s
k+1
e aining
cycle
op imiza ion algo.
(con ex, g ad. descen e c. )
op imiza ion p obes,
pa ial de i a i es wi h
ini e di e ence sub ou ine
o de o 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ∙ 𝐷. This happens because he one- ime ained
PLM p o ides as all he necessa y in o ma ion o he op imiza ion
algo i hm a each in e media e s ep, a oiding moni o ing. Op imizing
wi h a one- ime ained PLM is desc ibed wi h pseudo-code 2 in Fig. 7.
The op imiza ion algo i hm using he one- ime ained PLM
iden i ies he launch powe s ha yield he op imum 𝑜𝑏𝑗
𝑃𝐿𝑀, bu his is
Fig. 7. Pseudo-code o op imiza ion wi h one- ime ained PLM.
iewed h ough he PLM. Howe e , he PLM has ce ain accu acy, and
was ained a ini ial condi ions. So he iden i ied launch powe s yield
he objec i e 𝑜𝑏𝑗𝑃𝐿𝑀 in he eal ne wo k, which is wo se han he
objec i e o he moni o ing p obe me hod which is always e alua ed in
he eal ne wo k, 𝑜𝑏𝑗𝑃𝐿𝑀≤𝑜𝑏𝑗𝑚𝑜𝑛_𝑝𝑟𝑜𝑏. This p oblem was iden i ied
in Sec ion 3 and shown in Fig. 3 and 4.
C. Op imiza ion wi h a Digi al Twin (DT)
Following he discussions o he wo abo e app oaches, and he
esul s p esen ed in Sec ion 3, we obse e a clea adeo be ween
op imiza ion ime and pe o mance. The moni o ing p obes-based
app oach (in Fig. 5(a)) implemen s closed con ol loops which a e no
as , due o he complex p obing and slow moni o ing sub ou ine.
Howe e , i achie es he eal op imal as i acks he ne wo k e ol ing
condi ions/s a es by con igu ing and moni o ing. On he o he hand, he
one- ime ained PLM app oach (in Fig. 5(b)) is subs an ially as e since
i uses he PLM o quickly ind he de i a i es a in e media e s a es.
Howe e , he PLM is ained only once, a he beginning o he
op imiza ion. So i he algo i hm p ojec s he ne wo k o subs an ially
di e en physical condi ions, hen he op imiza ion is subop imal, since
he PLM di e s om eali y, as seen in Fig. 3 and 4. The ollowing
p oposed scheme keeps he bene i s o bo h app oaches: i inds a nea
o op imal solu ion, bu wi h an o e all low op imiza ion ime.
We p opose o use an i e a i e closed con ol loop p ocess o sol e
he dynamic op imiza ion p oblem. A ce ain in e media e i e a ions o
he algo i hm we con igu e he in e media e solu ion o he ne wo k
and moni o o ealign he PLM (wi h ML) o ollow he p ojec ed
ne wo k condi ions. As shown in Fig. 5(c). The idea is o make he PLM
a digi al win (DT), o ha e a PLM model which is pa ame ic and de ine
he me hod o eadjus / ealign i o ep esen s he physical sys em wi h
enough accu acy o pe o m he dynamic op imiza ion calcula ions a
hand. Fo ealigning he PLM, many echniques can be used. We he e
use ML aining. In his s udy we used as PLM he GN model [27], which
conside s he launch powe s and wa eleng h occupancy. Thus, i
models qui e accu a ely linea and NLI ansmission impai men s. We
also ex ended i and added a wa eleng h dependen penal y on op o
he GN SNR calcula ion o co e e.g. he EDFA ipple penal ies [6]. The
GN alignmen p ocess was desc ibed in Sec ion 3, and ex ended he e o
be pe o med i e a i ely.
As abo e, we deno e by 𝑸𝑵(𝒑,𝒓,𝒛) he calcula ion o he SNR
alues ec o o all N connec ions by he GN PLM, whe e p is he launch
powe ec o (op imiza ion a iables), ep esen s he PLM i ed
pa ame e s, he ibe coe icien s and he wa eleng h dependen ipple
penal y, and z ep esen s he unchanged inpu pa ame e s o ou
op imiza ion such as ou es, used wa eleng hs, e c. The dynamic
op imiza ion p ocess s a s wi h he con igu ed launch powe ec o 𝒑𝒐
o he es ablished connec ions (e.g., all 0dBm). Fo his ini ial powe
ec o 𝒑𝒐 he PLM is ained wi h he moni o ed SNR ec o 𝒀𝑵(𝒑𝒐).
To be mo e speci ic, we use ML and in pa icula he Le enbe g-
Ma qua d (LM) algo i hm o ind
𝒓𝟎= a gmin𝑟(𝑸𝑵(𝒑,𝒓,𝒛)− 𝒀𝑵(𝒑))2. Now, le us assume ha a he
end o he k- h PLM aining cycle, he op imiza ion algo i hm has
pe o ms Lk in e media e i e a ions and iden i ied he launch powe s
𝒑𝑘
𝐿𝑘. We hen s a he nex cycle k+1 by con igu ing he ne wo k wi h
he ou come so 𝒑 = 𝒑𝑘
𝐿𝑘 . To e ain he PLM o he k+1 cycle we
con igu e he ne wo k wi h 𝒑 and moni o o ob ain 𝒀𝑵(𝒑 ).
Then ML is used o i he pa ame e s 𝒓 , ha is 𝒓 =
a gmin𝑟(𝑸𝑵(𝒑 ,𝒓,𝒛)− 𝒀𝑵(𝒑 ))2. This PLM is hen used in he
op imiza ion algo i hm i e a ions o cycle k+1. No e ha , a each
e aining cycle o he PLM, we can make use o he p e iously
moni o ed SNRs, including hus he his o y, he ne wo k e olu ion
condi ions. This ends o imp o e he PLM accu acy as he algo i hm
i e a es, whe e he accu acy is mo e c i ical.
Fig. 8. Schema ic showing he wo nes ed o loops, ou e o e aining
PLM cycles and he inne o he op imiza ion algo i hm i e a ions.
We assume ha in o al we e ain K e ain imes he PLM. Al hough we
can ha e di e en numbe o algo i hm i e a ions pe cycle, o simpli y
ou analysis in he ollowing we assume ha he algo i hm uns Li e
i e a ions a e each PLM e- aining. So, Lk=Li e o all e aining cycles
k=1…K e ain. I is easy o isualize he o e all concep as wo nes ed o
loops, as shown in Fig. 8. The ou e one pe ains o he PLM e aining,
and he inne o he op imiza ion algo i hm in e media e i e a ions wi h
he e ained PLM/DT. The ime o each e aining cycle is equal o
𝑇𝑃𝐿𝑀 o Li e i e a ions, ha is 𝑡𝑚𝑜𝑛 + 𝑡𝑡𝑟𝑎𝑖𝑛 + 𝐿𝑖𝑡𝑒𝑟 ∙ (𝐷 ∙ 𝑡𝑃𝐿𝑀 +
𝑡𝑐𝑎𝑙𝑐). We deno e, he o e all op imiza ion ime wi h his DT based
app oach as 𝑇𝐷𝑇, which is gi en by:
𝑇𝐷𝑇 = 𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 ∙ (𝑡𝑚𝑜𝑛 + 𝑡𝑡𝑟𝑎𝑖𝑛 + 𝐿𝑖𝑡𝑒𝑟 ∙ (𝐷 ∙ 𝑡𝑃𝐿𝑀 + 𝑡𝑐𝑎𝑙𝑐)) (6)
The p oposed me hod o op imizing wi h a DT is desc ibed wi h
pseudo-code 3 in Fig. 9.
#k e aining cycle
con igu e pk, moni o 𝑺𝑵𝑹𝑵𝒑 and i PLM (Le enbe g-
Ma qua d algo i hm): 𝑺𝑵𝑹𝑵𝒑 ≈ 𝑸𝑵(𝒑 ,𝒓 ,𝒛)
ou e loop: k=1,…, K e ain
a e Li e , p oduce he powe
pk+1 o (k+1) h cycle
cycle#2
cycle#k
cycle#1
Digi al Twin
l
Li e
k
K e ain
op imiza ion algo.
(con ex, g ad. descen e c. )
op imiza ion p obes,
pa ial de i a i es wi h
ini e di e ence sub ou ine
T ansponde s
launch powe
adjus men om
𝒑 o 𝒑 ec o
a cycle k
op ical ne wo k
𝑺𝑵𝑹𝑵𝒑
moni o ing
new
powe
ec o
inne loop: l=1,…., Li e i e a ions
PLM e ained a
each ou e loop
execu ion
( e aining cycle)
1
k
k-1
𝑸𝑵(𝒑 ,𝒓 ,𝒛)
i e a i e closed con ol loop
2
Pseudo-code - 2
S a wi h ini ial launch powe ec o 𝒑𝟎, i e a ion numbe i=-1
Align PLM o ini ial / p io - o-op imiza ion s a e (moni o 𝒀𝑵(𝒑𝟎) and
ain he PLM 𝑸𝑵(𝒑𝟎,𝒓,𝒛) o ind 𝑟0), ime: 𝑡𝑚𝑜𝑛 + 𝑡𝑡𝑟𝑎𝑖𝑛
While no con e ged
Inc ease i
Fini e di e ences p ocess, ime :𝑡𝑃𝐿𝑀 pe SNR ec o calcula ion by
he PLM 𝑸𝑵
P obe di imes he PLM: change he launch powe s and
calcula e he SNR ec o o all connec ions wi h he PLM
(e.g. o he i s o de pa ial de i a i e o connec ion n, se
𝒑𝒊+ 𝜹𝒑𝒏 and calcula e 𝑸𝑵(𝒑𝒊+ 𝜹𝒑𝒏,𝒓𝟎,𝒛)),
Calcula e he de i a es and nex launch powe ec o 𝒑𝑖, ime: 𝑡𝑐𝑎𝑙𝑐
E alua e con e gence (e.g. compa e objec i e imp o emen wi h a
h eshold), when con e ged: 𝐿𝑃𝐿𝑀 = 𝑖
Apply he las calcula ed powe ec o 𝒑𝑖 (= 𝒑 𝑷 𝑴) o he ne wo k
No e ha in o al he op imiza ion algo i hm pe o ms 𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 ∙
𝐿𝑖𝑡𝑒𝑟 i e a ions, and e ains he PLM 𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 imes. Ou a ge is o
choose he e aining pe iod 𝐿𝑖𝑡𝑒𝑟 app op ia ely so ha he PLM would
ollow wi h good accu acy he physical laye in he algo i hm’s
in e media e calcula ions. I his is achie ed he PLM es ima ed
objec i e ha is calcula ed a each i e a ion and he inal one 𝑜𝑏𝑗
𝐷𝑇
would be e y close o he eal alue in he eal ne wo k 𝑜𝑏𝑗𝐷𝑇. Also, he
achie ed objec i e would be e y close o he op imum, as calcula ed by
he moni o ing p obes me hod 𝑜𝑏𝑗𝑚𝑜𝑛_𝑝𝑟𝑜𝑏. So, we would ha e
𝑜𝑏𝑗
𝐷𝑇 ≈𝑜𝑏𝑗𝐷𝑇 ≈𝑜𝑏𝑗𝑚𝑜𝑛_𝑝𝑟𝑜𝑏. Mo eo e , o an app op ia e
e aining pe iod he i e a ions o he op imiza ion algo i hm would be
close o hose o he moni o ing p obes, ha is 𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 ∙ 𝐿𝑖𝑡𝑒𝑟 ≈
𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏. Looking a he o al op imiza ion imes, and assuming ha
𝑡𝑚𝑜𝑛 is he dominan ac o , we ha e 𝑇𝐷𝑇 ≈ 𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 ∙ 𝑡𝑚𝑜𝑛. Thus we
ob ain a speedup o 𝐿𝑚𝑜𝑛_𝑝𝑟𝑜𝑏 ∙ 𝐷/𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 = 𝐿𝑖𝑡𝑒𝑟 ∙ 𝐷 wi h espec
o he moni o ing p obes op imiza ion app oach.
Fig. 9. Pseudo-code o op imiza ion wi h he p oposed PLM e aining/
Digi al Twin.
5. RESULTS AND DISCUSSIONS
To quan i y he bene i s o he de ised DT based powe op imiza ion
app oach, we ca ied ou simula ions using bo h VPI T ansmission
Make and MATLAB. The ac ual ne wo k was implemen ed in VPI, and
he PLM ( elying on he GN model) and he con ex op imiza ion
algo i hm we e de eloped in MATLAB. No e ha VPI is qui e mo e
de ailed and complex and close o a eal sys em han he GN model.
This choice was made o cap u e he misma ch be ween he eal
ne wo k and he PLM used in he op imiza ion p ocess. This is a
conside able imp o emen in e ms o ealism compa ed o many p io
s udies (lis ed in Sec ion 2), whe e au ho s used he same PLM o bo h
he eal ne wo k/g ound u h and hei p oposed solu ion.
To be mo e speci ic, we implemen ed in MATLAB he GN model and
he launch powe op imiza ion algo i hm o maximize: (obj#1) he sum
o SNR ma gins, o (obj#2) he lowes ma gin, as discussed in Sec ion 3.
This op imiza ion p oblem is known o be con ex and o polynomial
complexi y. Hence, we implemen ed an in e io -poin algo i hm o
sol e i . The algo i hm was un un il i ound he op imum (op imali y
ole ance 10-6). The GN model was in e aced wi h he op imiza ion
algo i hm and bo h we e in eg a ed in an au oma ed sys em in VPI. Fo
each simula ion, VPI implemen s he ou e loop (PLM e aining cycle).
I akes as inpu he launch powe s coming om he algo i hm o he
in eg a ed MATLAB module, pe o ms he de ailed ansmission
simula ions and calcula es he SNRs o he channels. These a e passed
as inpu o he in eg a ed MATLAB module. Wi h ha inpu he
in eg a ed PLM ge s ained and his ained PLM is hen used by he
con ex algo i hm o 𝐿𝑖𝑡𝑒𝑟 in e media e i e a ions. In hose i e a ions
he algo i hm uses he PLM o iden i y he pa ial de i a i es, using he
ini e di e ences sub ou ine, and hen he new launch powe s. A e he
𝐿𝑖𝑡𝑒𝑟 i e a ions (inne cycle), a new se o launch powe s a e
au oma ically ed o VPI ansponde s as a closed con ol loop o he
nex e aining cycle. No e ha , a each e aining cycle o he PLM, we
e ained wi h he cu en and p e iously moni o ed SNRs, including
hus he his o y, he ne wo k e olu ion condi ions.
Fig. 10. (a) VPI se up wi h (a) Single link o 6 iden ical spans, (b) 3 nodes
and 15 connec ions wi h di e en pa hs/ ou es, added/d opped poin s
o emula e a small ne wo k.
The moni o ing p obes app oach (Fig. 5(a)) was used as e e ence in
his wo k. To implemen his, we implemen ed ano he (mo e equen )
closed con ol loop wi hou using a PLM: he moni o ing p obes om
he ini e di e ences sub ou ine (in MATLAB) we e ca ied di ec ly o
VPI and he SNR alues we e hen passed back o ha sub ou ine. The
one- ime ained PLM app oach (Fig. 5(b)) ep esen s he adi ional
op imiza ion scheme ia a PLM. Fo ha , we ain he PLM only once
wi h he SNR da a om VPI a he beginning o he simula ion and used
ha PLM o he powe op imiza ion (in ol ing in e media e i e a ions
no con igu ed in he ne wo k). No e ha in all cases we s a by
assigning 0dBm uni o m powe o all ansponde s in VPI.
Fo all he op imiza ion schemes, he objec i e alue was calcula ed
in VPI, so in he eal ne wo k. As discussed, he e exis s a di e ence
be ween he iew o he PLM/op imiza ion p ocess ha uses i and he
eal objec i e. Finally no e ha we e alua ed he bene i s o ou
p oposed scheme o ela i ely small channel coun (=25) and up o wo
links, due o he slow execu ion ime o VPI (spli -s ep Fou ie
simula ions). Ac ually, his can be conside ed as an indica ion o he long
ime o in e ac ing wi h/moni o ing he eal ne wo k.
To be speci ic, we made wo ully au oma ed se ups in VPI:
(i) single link o 6 iden ical spans wi h 25 channels (Fig. 10 (a))
(ii) wo links wi h 15 channels which we e added/d opped a he
in e media e node (Fig. 10 (b))
Fo he i s se up, 25 WDM channels wi h pol-mux 16QAM
modula ion o ma a 32Gbaud, leading o 200 Gbps da a a e pe
channel we e launched. We assumed SNR h eshold o 13.9dB [1], [6].
The wa eleng h spacing be ween he channels was assumed o be
50GHz. We s a ed wi h uni o m 0dBm o launch powe s o all
1 2
Gain=16dB
NF = 5.5dB
80 km
(a).
Ns= 6
1 , 2 , …. , 24 , 25
2
(b).
Gain=16dB
NF = 5.5dB
d op add
1 , 2
1 3
Pseudo-code - 3
S a wi h ini ial powe s 𝒑𝟎, ou e loop i e a ion numbe k=0
While no con e ged
Align he PLM o cu en ne wo k s a e (moni o 𝒀𝑵(𝒑 ) and
ain he PLM 𝑸𝑵(𝒑 ,𝒓,𝒛) o iden i y 𝒓 , ime: 𝑡𝑚𝑜𝑛 + 𝑡𝑡𝑟𝑎𝑖𝑛
Inc ease k, 𝒑
𝟎= 𝒑
Fo l=0,…, Li e -1(Inne loop i e a ions)
Fini e di e ences p ocess, ime :𝑡𝑃𝐿𝑀 pe SNR ec o
calcula ion by he PLM 𝑸𝑵
P obe di imes he PLM: change he launch powe s and
calcula e he SNR ec o o all connec ions wi h he PLM
(e.g. o he i s o de pa ial de i a i e o connec ion n,
se 𝒑
𝒍+ 𝜹𝒑𝒏 and calcula e 𝑸𝑵(𝒑
𝒍+ 𝜹𝒑𝒏,𝒓 ,𝒛))
Calcula e he de i a es and nex powe ec o 𝒑
𝒍 ,
ime: 𝑡𝑐𝑎𝑙𝑐
E alua e con e gence (e.g. compa e objec i e
imp o emen wi h a h eshold), when con e ged:
𝐾𝑟𝑒𝑡𝑟𝑎𝑖𝑛 = 𝑘
Apply he calcula ed powe ec o 𝒑𝑘
𝐿𝑖𝑡𝑒𝑟 o he ne wo k
