Compu e Ne wo ks 226 (2023) 109689
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Compu e Ne wo ks
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A no el p edic i e app oach o mobili y ac i eness in mobile wi eless
ne wo ks
Peppino Fazio a,b,∗,Mi alem Mehic b,c,Mi osla Voznak b,Flo iano De Rango d,Mau o T opea d
aDepa men o Molecula Sciences and Nanosys ems, Ca’ Fosca i Uni e si y, Via To ino 155, Mes e (VE), 30172, I aly
bVSB – Technical Uni e si y o Os a a, 17. lis opadu 2172/15, Os a a, 70833, Czechia,
cDepa men o Telecommunica ions, Facul y o Elec ical Enginee ing, Uni e si y o Sa aje o, Zmaja od Bosne bb, Sa aje o, 71000, Bosnia and He zego ina
dDIMES, Uni e si y o Calab ia, ia P. Bucci 39/C, A ca aca a di Rende (CS), 87036, I aly
ARTICLE INFO
MSC:
0000
1111
Keywo ds:
Mobile ne wo ks
Mobili y
Rou ing
Ne wo king
Me ic
S abili y
ABSTRACT
Nowadays, mobile compu ing has become a key componen o elecommunica ion sys ems, and he Open
Sys ems In e connec ion (OSI) laye ope a ions a e a ec ed by he e ec s o node mo emen s along he oads,
om he physical o he ou ing/ anspo laye s. In pa icula , ou ing app oaches ha e been in es iga ed
om many yea s, ying o op imize he pe o mance o he whole conside ed sys em, unde di e en poin s
o iew. In his pape we a e ocusing he a en ion on he analysis o he mobili y g ade end o a mobile
ad-hoc ne wo k en i onmen , as well as on he way i can be a-p io i known, in o de o ha e he possibili y o
s udy how he dynamics o mobile nodes can be desc ibed and in-ad ance known, wi h a p edic ed knowledge
o nodes s abili y (in e ms o mobili y). Ou simula ions conside ed mobili y in eal geog aphical maps, and
he ob ained esul s con i med he goodness o ou p oposed s udy.
1. In oduc ion
The eme gence o he In e ne o Things (IoT), au onomous and ae ial
ehicles has led o an inc eased demand o ne wo k connec i i y and,
a he same ime, an inc easing o mobili y le el (mo e de ices a e
ela ed o ehicles which mo e wi h less physical cons ain s, such as
d ones, ae ial de ices and mobile senso s). In addi ion, pee o pee
communica ions wi h no in as uc u e pose special p oblems and some
pa ame e s o be aken in o accoun , such as ou ing pe o mance,
ene gy consump ion, scalabili y and secu i y. These issues ha e al eady
been add essed wi hin ad hoc ne wo ks, whe e ou es be ween wo
hos s may consis o hops h ough o he ne wo k hos s which, due
o dynamic na u e o ne wo k nodes, can cause equen and unp e-
dic able opology changes [1]. As he numbe o nodes inc eases, he e
is a g owing need o e ec i e ne wo k managemen and o ganiza ion,
whe e he ques ions o scalabili y and obus ness become i al.
In his pape we analyze wha happens o mobile nodes in e ms o
Mobili y Ac i eness (MA), o mobili y g ade, conside ed as a pa ame e
e e ing o he way he en i e mobile sys em e ol es in ime, wi h
some well-known consequences, such as high call d opping p obabili y,
huge signal e anescence and, abo e all, pee - o-pee link in e mi ence
and he consequen ne wo k uns abili y [2], [3]. Fi s o all, a de ailed
analysis o he MA in mobile scena io is gi en and, hen, a possible
app oach o he p edic ion o i s end is desc ibed. The ad an ages o
∗Co esponding au ho a : Depa men o Molecula Sciences and Nanosys ems, Ca’ Fosca i Uni e si y, Via To ino 155, Mes e (VE), 30172, I aly.
E-mail add ess: [email p o ec ed] (P. Fazio).
his kind o p edic i e s udy will be clea ly unde lined, also by he help
o a deep simula ion campaign, able o show some in e es ing esul s
abou he MA e olu ion.
In dynamic ne wo ks (whe e he adjec i e dynamic e e s o some
aspec s o he ne wo k, such as mobili y, opology, ene gy, ansmission
powe , e c.), ha ing he possibili y o apply p edic i e app oaches
o he enhancemen o he o e all ne wo k pe o mance is always
desi able, especially i we e e o he nex on ie s o mobile commu-
nica ions, i.e. 6G [4]. Le us hink, o example, o a ou ing p o ocol,
whe e mul i-objec i e me ics can be used o eac o equen changes
in ne wo k opology [5], o whe e ( ypically) me ics a e de ined
and e alua ed a he momen a which he decision should be aken
(e.g. packe o wa ding, ou ing able building, bes pa h e alua ion,
a ic eques s, e c.): i he ne wo k condi ion in he immedia e u u e
could be known in-ad ance, ou ing decisions could an icipa e a u u e
ne wo k con igu a ion, a oiding undesi able pe o mance [6]. I we
e e o nodes mobili y, as illus a ed in [7], many ideas ha e been
in oduced by he esea che s and all o hem may help ne wo k
adminis a o s, p o ocols and algo i hms o beha e di e en ly, on he
basis o he knowledge u u e condi ions.
In his wo k, we ocused ou a en ion on Mobile Ad-hoc NETwo ks
(MANETs) en i onmen s: he wo ks in [8,9] conside ed he way ad-
hoc nodes mo e in o he conside ed ne wo k and he au ho s ake
h ps://doi.o g/10.1016/j.comne .2023.109689
Recei ed 8 June 2022; Recei ed in e ised o m 12 Feb ua y 2023; Accep ed 6 Ma ch 2023
Compu e Ne wo ks 226 (2023) 109689
2
P. Fazio e al.
ad an age om nodes mobili y beha io o op imize ou ing ope a ions
h ough p edic ions and gua an ee a gi en le el o Quali y o Se ice
(QoS). In addi ion, in [9], he concep o ene gy in ad-hoc ne wo ks is
also desc ibed. Fi s ly, he au ho s conside he ‘‘in o ma ion amoun ’’
which is exchanged h ough packe signaling; secondly, he au ho s
a gue abou he mobile ene gy, de ining i as a unc ion o node
dis ibu ion and he p opaga ion powe law exponen . In [10–13] he
au ho s p opose some deep analysis o he way he s abili y o a poin -
o-poin connec ion can be p edic ed, in ad-hoc en i onmen s. The
main con ibu ions o his p oposal can be summa ized as ollows:
•A deep analysis and de ini ion o he mobili y g ade concep o
dis ibu ed ne wo ks, aimed o de ine a new pa ame e which
can help he sys em o enhance he o e head and he o e all
pe o mance;
•A new p edic ion app oach o conside ing he u u e alues o
mobili y g ade o mobile nodes;
•A deep analysis o he nume ical esul s, in o de o es ablish he
s ochas ic p ope ies o he p oposed model.
Be o e concluding he in oduc ion, we would like o unde line ha
we de ined he MA by ela ing i o he Shannon’s en opy de ini ion.
So in he ollowing, we e e o he concep o en opy e e ing o
MA (in he case o mobile nodes). We can a i m ha , he i s main
di e ence wi h he exis ing wo ks is ha mos o hem a e ela ed o
he en opy con ained in o he exchanged in o ma ion (in o ma ion-
en opy), while we ocused on he mobili y g ade o he nodes which
compose he ne wo k opology. In addi ion, mos o he exis ing wo ks
ake in o accoun only ne wo ks in which nodes a e comple ely mobile,
dis ega ding he ac ha he ou ing able o a s a ic node (wi h
no mobili y) is a ec ed by he en opy gene a ed om neighbo s, as
hey mo e and in luence he comple e ne wo k opology (Recip ocal
Mobili y Ac i eness, RMA, de ined in nex sec ions). Mo eo e , he e
a e al eady some p edic i e app oaches o MANET ou ing, bu ou
p oposal o e s he lowes compu a ional complexi y, because i is
based on an o de -1 Au o-Reg essi e (AR) model which is, om ou
poin o iew, he simples analy ical me hod om p edic ing da a
o a ime-se ies (we disco e ed ha mobili y ac i eness in a eal
mobile ne wo k, based on eal pa hs, can be modeled as an o de -1
p ocess). The las conside a ion abou he no el y and enhancemen
in oduced by ou con ibu ion ega ds he ype o mobili y conside ed
o simula ions: mos o he exis ing wo ks conside a syn he ic mobili y
model, which models mobili y by s ochas ic and analy ical equa ions.
In his way, node mo emen s may be unna u al (e.g. high s ee ing
deg ees wi h high speed). We c ea ed mobili y, ins ead, by conside ing
eal maps wi h Ci y4Roadmaps (C4R) [14,15], which is based on he
OpenS ee Map co e ( o ex ac ing oad-maps om he eal wo ld)
and SUMO co e ( o c ea ing eal nodes mo emen s in unc ion o he
ex ac ed map). In his way, we a e su e ha ou simula ions conside
eal na u al mobili y.
In Table 1 he main abb e ia ions used in he pape a e illus a ed.
As ega ds he s uc u e o he pape , he nex sec ion gi es a de ailed
o e iew o he main scien i ic wo ks exis ing in li e a u e, Sec ion 3
in oduces he mobili y g ade concep o s a ic and mo ing nodes.
Sec ion 4desc ibes he deploymen o adap i e il e ing o empo al
p edic ion o nodes e olu ion, while Sec ion 5p o ide de ails abou he
main ob ained esul s, in e ms o mobili y g ade alues in unc ion o
di e en sys em pa ame e s and p edic ion possibili ies, discussing he
b oade aspec s o ou app oach. A he end, a compa ison wi h he
AODV p o ocol wi h and wi hou ou p oposed me ic is illus a ed.
Sec ion 6concludes he pape .
2. S a e o he A
P edic i e app oaches ha e been o en conside ed in mobile ne -
wo ks, in o de o enhance he o e all pe o mance o he conside ed
sys em. Clea ly, hey depend on he accu acy o he p oposed idea, as
Table 1
Lis o ac onyms.
Ac onym Desc ip ion
ACF Au oCo ela ion Func ion
AF Adap i e Fil e ing
AF-LMS Adap i e LMS il e
AR Au o Reg essi e
FOA Fil e Op imiza ion Algo i hm
IoT In e ne o Things
LMS Leas Mean Squa e
MA Mobili y Ac i eness
MANET Mobile Ad-hoc NETwo k
OSI Open Sys ems In e connec ion
PACF Pa ial ACF
QoS Quali y o Se ice
RA Recip ocal Ac i eness
RMA Recip ocal and Mobili y Ac i eness
SMA Simple Mo ing A e age
SNR Signal- o-Noise Ra io
UKF Unscen ed Kalman Fil e
VANET Vehicula Ad-hoc NETwo ks
WSN Wi eless Senso s Ne wo k
well as on he in insic a ic/mobili y dynamics. In eg a ing a ou ing
p o ocol wi h a p edic i e app oach leads always o he enhancemen
o he o e all pe o mance [6]. In ac , as illus a ed in [7], many
ideas ha e been in oduced by he esea che s and all o hem may
beha e di e en ly. In pa icula , when e e ing o ad-hoc ne wo ks,
node mobili y is one o he key aspec s ha ha e been in es iga ed and
p edic ed, gi en ha i is c ucial o MANETs.
2.1. The concep o en opy in dynamic ne wo ks
In he wo ks [8,9], he main ocus is a ge ed on he way mobile
nodes mo e in o he conside ed ad-hoc ne wo k. The au ho s base hei
p oposal on he ‘‘en opy’’ concep o imp o e ou ing ope a ions by
p edic ing use s mo emen s, e lec ing se e al enhancemen s on he
QoS. In addi ion, in [9], he concep o in o ma ion ene gy in ad-hoc
ne wo ks is also desc ibed: i s o all, he au ho s e e o Shannon’s in-
o ma ion heo y, conside ing he ‘‘amoun o in o ma ion’’ exchanged
h ough packe exchanges, hen he au ho s conside node communi-
ca ions om he ene gy poin o iew, modeling hem as unc ions o
nodes dis ibu ion and he p opaga ion powe law exponen . The a icle
in [16] a gues abou he concep o opology changes measu emen s o
MANETs, conside ed as he unce ain y o changes in ne wo k opology.
I is s ic ly ela ed o he minimum o e head equi ed by nodes, du ing
ou ing ope a ions, o each he ‘‘con e ged’’ s a us ( ha is o say he
comple e opology is known by all nodes). The a icle in [17] a gues
abou he condi ional en opy in wi eless ne wo ks, by cha ac e izing
he opological unce ain y using he o malism o g aph en opy, while
he au ho s o [18] ake in o accoun he us iness be ween e es ial
and sa elli e nodes, by analyzing he packe s en opy o he exchanged
in o ma ion.
2.2. The concep o link s abili y/li e ime in dynamic ne wo ks
Ano he pa ame e ha can be op imized in ad-hoc ne wo ks is he
link li e ime (o link s abili y), which is hea ily a ec ed by mobili y
o esidual ene gy. In [10–13,19] he au ho s p opose some deep
analysis o he way he s abili y o a poin -2-poin connec ion can be
p edic ed, in ad-hoc en i onmen s. In [10,13] he au ho s make use
o he in e pola ion concep o p edic he ime in e al o e which a
conside ed node can be conside ed as us ed, de ining a new me ic
o ou ing able cons uc ion based on he bes Signal- o-Noise Ra io
(SNR) alue. Gi en ha ou ing p o ocols a e esponsible o sea ching
and main aining he bes ou es om a gene ic sou ce o a gene ic
des ina ion, in [11] a no el o wa ding app oach is p oposed, based
on pa h s abili y. Also in his case, he au ho s based he choice o he
Compu e Ne wo ks 226 (2023) 109689
3
P. Fazio e al.
Table 2
Symbols used in he p oposal.
Symbols Desc ip ion
𝐺Geog aphical a ea
𝑔𝑖𝑗 Sub-a eas
𝑁, 𝑂 Dimension o G in me e s
𝑙𝑥, 𝑙𝑦Side size o a gene ic 𝑔𝑖𝑗 ∈𝐺
𝑛 𝑛 =⌈𝑁∕𝑙𝑦⌉
𝑚 𝑚 =⌈𝑂∕𝑙𝑥⌉
𝑀𝑂𝐵 Se o mobile nodes wi h size 𝑀
𝑀Size o se 𝑀𝑂𝐵
𝑣𝑘𝑘- h mobile nodes
𝑊Obse a ion window size
𝑝𝑊
𝑘(𝑔𝑘
𝑖𝑗 )P obabili y o isi ing 𝑔𝑘
𝑖𝑗
𝑉∗
𝑘Numbe o dis inc 𝑔𝑖𝑗 isi ed by 𝑣𝑘
𝐼𝐷𝑘Iden i ie o 𝑘- h node
𝑛𝑔𝑘Numbe o one-hop neighbo nodes o 𝑣𝑘
𝑅𝐴𝑠Recip ocal Ac i eness con ibu ion
𝑅𝑀𝐴 Recip ocal and Mobili y Ac i eness
𝛾smoo hing ac o ([0..1])
𝑊𝑗𝑗- h obse a ion window
𝐼𝑅𝑊𝑗Impulse esponse a 𝑊𝑗
𝑃 𝑅𝐸𝑅𝑀𝐴 P edic ed ou pu
𝐷𝐸𝑆𝑅𝑀𝐴 Desi ed ou pu
𝑐𝑓 Con e gence ac o
𝜖𝑗Di e ence be ween he p edic ed RMA and he desi ed RMA a s ep 𝑗
𝜇Mean o he p ocess
nex hop on he signal s eng h p edic ion, which akes in o accoun
link s abili y by a dis ance and ime based heo e ical o mula ion,
able o p edic how long a link becomes s able o use by he help o
mobili y. In [12], link li e ime is p edic ed h ough he deploymen o
he Unscen ed Kalman Fil e (UKF), used o model a nonlinea sys em
and o compu e he es ima es o he emaining link li e ime. Au ho s
sugges o apply he UKF ecu si ely, in o de o compu e sys em’s
s a es, using as inpu s pe iodical measu emen s o he dis ance be ween
he wo link’s nodes. The wo k in [20] akes in o accoun he esidual
ene gy concep o p edic ing he li e ime o a link among a couple o
nodes (powe awa e ou ing). In ew wo ds, he au ho s make use o
an op imiza ion p oblem, de ining an objec i e unc ion (maximizing
he li e ime o a chosen pa h) and he associa ed cons ain s, in eg a ed
in o he RREQ/RREP mechanism. The co e o he idea is based on he
indi idual ba e y li e ime p edic ion made by each single node, based
on i s pas ac i i y (using a Simple Mo ing A e age (SMA) p edic o ).
In he nex sec ion, ou p oposal is deeply in oduced and desc ibed.
3. Mobili y Ac i eness in mobile ne wo ks
We s a ou p oposal by de ining he concep o Mobili y Ac i eness
(MA) as a measu e o he unce ain y in a gene ic s a is ical model [21].
This de ini ion is based on he amoun o node mobili y and highe MA
leads o ha de p edic ion ope a ions, wi h lowe accu acy. Table 2
shows he main symbols used in he ma hema ical o mula ion o
explaining ou p oposal.
3.1. Mobili y Ac i eness o mobile nodes: he de ini ion
We associa e a ce ain le el o MA o a node by conside ing i s
geog aphical posi ion, i s way o mo e among di e en a eas o how i
communica es wi h i s neighbo s.
So, i s o all, le us assume ha all nodes in o he sys em a e
mobile (s a ic nodes can be conside ed as a pa icula case o mobile
nodes, wi h MA equals o ze o). A gene ic 2D geog aphical a ea G
can be conside ed as he esul o a pa i ioning ope a ion, able o
subdi ide G(whe e mobile nodes a e mo ing) in o a ini e se o 𝑛𝑥𝑚
squa e/ ec angula sub-a eas 𝑔𝑖𝑗 , such as:
𝐺=𝑔11 ∪𝑔12 ∪⋯∪𝑔1𝑚∪𝑔21 ∪𝑔22 ∪⋯∪𝑔𝑛(𝑚−1) ∪𝑔𝑛𝑚
𝑔11 ∩𝑔12 ∩... ∩𝑔1𝑚∩𝑔21 ∩𝑔22 ∩... ∩𝑔𝑛(𝑚−1) ∩𝑔𝑛𝑚 = ∅.(1)
Fig. 1. An example o pa i ion applied o a geog aphical a ea 𝐺𝑁𝑒𝑤𝑌 𝑜𝑟𝑘 wi h N=12
km, O=22 km, and an a ea o O*N=264 km2; he alues o 𝑚and 𝑛a e 4 and 6
espec i ely, wi h 𝑙𝑥= 3 km and 𝑙𝑦≈ 3.67 km.
which can be ew i en, in compac o m, as:
𝐺=
𝑛
⋃
𝑖=1
𝑚
⋃
𝑗=1
𝑔𝑖𝑗 𝑤𝑖𝑡ℎ
𝑛
⋂
𝑖=1
𝑚
⋂
𝑗=1
𝑔𝑖𝑗 = ∅.(2)
and 𝑛, 𝑚 ∈N+.
The alues o 𝑛and 𝑚can be se o de i ed om he dimensions o
𝐺, assumed o be 𝑁and 𝑂(in me e s), so o each 𝑔𝑖𝑗 ∈𝐺 he ela ions
𝑛=⌈𝑁∕𝑙𝑦⌉and 𝑚=⌈𝑂∕𝑙𝑥⌉a e always alid, wi h 𝑙𝑥and 𝑙𝑦 ep esen ing
he side sizes o he gene ic 𝑔𝑖𝑗 ∈𝐺. We assume ha each sub-a ea has
he same dimensions o he o he ones, as depic ed in he example o
Fig. 1.
Fo simplici y o no a ion, 𝐺can be ep esen ed by i s pa i ion se
𝑔𝑖𝑔 ∈𝐺o sub-a eas as a ma ix (𝑛x𝑚):
𝐺=⎡
⎢
⎢
⎣
𝑔11 𝑔12 ... 𝑔1𝑚
... ... ... ...
𝑔𝑛1𝑔𝑛2... 𝑔𝑛𝑚
⎤
⎥
⎥
⎦
(3)
A his ime, he MA alue o each mobile node should be de ined:
we need o in oduce also an obse a ion ime Window W, du ing
which mobile hos s mo e and de ine hei cu en MA. The idea is
o conside he numbe o isi ed 𝑔𝑖𝑗 ∈𝐺du ing 𝑊and ela ing i
o he de ini ion o MA. To his aim, gi en he se o mobile nodes
𝑀𝑂𝐵 = {𝑣1,…, 𝑣𝑀}, wi h ‖𝑀𝑂𝐵‖=𝑀, hen o he 𝑘 h mobile node
𝑣𝑘, we can de ine he se o a eas isi ed by 𝑣𝑘∈𝑀𝑂𝐵 du ing W:
𝑣𝑊
𝑘= {𝑔𝑘
𝑖𝑗1...𝑔𝑘
𝑖𝑗𝑉𝑘|𝑔𝑘
𝑖𝑗𝑙∈𝐺, 𝑙 = 1..𝑉𝑘},(4)
wi h ‖𝑣𝑊
𝑘‖=𝑉𝑘.
A his poin , he p obabili y o isi ing 𝑔𝑘
𝑖𝑗 by 𝑣𝑘in he cu en W
can be de ined as:
𝑝𝑊
𝑘(𝑔𝑘
𝑖𝑗 ) = ∑𝑉𝑘
𝑙=1 𝑐𝑜𝑢𝑛𝑡(𝑙, 𝑔𝑘
𝑖𝑗𝑙, 𝑣𝑊
𝑘)
𝑉𝑘
,(5)
whe e he a gumen o he summa ion 𝑐𝑜𝑢𝑛𝑡(𝑙, 𝑔𝑘
𝑖𝑗𝑙, 𝑣𝑊
𝑘)coun s how
many imes node 𝑣𝑘 isi ed 𝑔𝑖𝑗 .
A his poin , we apply he undamen al en opy de ini ion gi en
by Shannon in [21], based on a se o symbols and he p obabili ies o
hose symbols o appea in he sequence; so i is easy o see ha :
𝑀𝐴(𝑣𝑊
𝑘)=−
𝑉∗
𝑘
∑
𝑙=1
𝑝𝑊
𝑘(𝑔𝑘
𝑖𝑗𝑙)⋅𝑙𝑛[𝑝𝑊
𝑘(𝑔𝑘
𝑖𝑗𝑙)] (6)
whe e 𝑉∗
𝑘is he numbe o dis inc 𝑔𝑖𝑗 isi ed by 𝑣𝑘. Then, we apply he
de ini ion o MA gi en in (6) o ex ac knowledge abou he e olu ion
o a ne wo k wi h a dynamic opology.
3.2. The Recip ocal Ac i eness: how nodes a e in luenced by each o he
In Wi eless Senso s Ne wo ks (WSNs), Ad-hoc Ne wo ks, MANETs
o Vehicula Ad-hoc NETwo kss (VANETs), du ing ou ing ope a ions
some links may b eak, hen ou e eco e y and main enance p ocedu es
Compu e Ne wo ks 226 (2023) 109689
4
P. Fazio e al.
Fig. 2. The in luence o nodes 𝑣𝑖,𝑣𝑗,𝑣𝑙on 𝑣𝑘’s RA (cu ed a ows), whe e 𝑟𝑖,𝑟𝑗,𝑟𝑘
and 𝑟𝑙a e he co e age adius.
need o be execu ed. Bu , such p ocedu es consume a ious esou ces
such as he ene gy which needs o be p ese ed. To minimize ou e
in e up ions, i is o c ucial impo ance o ind a ou e ha endu es
longe ime. Many wo ks in li e a u e, such as [22–24], emphasize he
impo ance o he link s abili y pa ame e . In he case o dis ibu ed
wi eless ne wo ks, elaying ope a ions a ec he pe o mance o he
whole sys em, so he ou ing me ic should be chosen ca e ully. In his
sense, he MA can p o ide p ecious in o ma ion o cha ac e ize nodes
beha io and hei unce ain y in e ms o eliabili y o e ime.
Ano he key aspec could be ep esen ed, o example, by he e-
lec ion o he MA in o a ou ing able [25]: on he basis o he way
he en ies a e s o ed, i is possible o analyze nodes ou ing s abili y
and, consequen ly, i is possible o in oduce an ageing mechanism o
se he pe iodic signaling in e al (such as Hello messages).
In addi ion, i is easy o see ha , i he ne wo k is dealing wi h
s a ic (o almos s a ic, wi h low mobili y g ade) nodes, he mobili y
con ibu ion o MA is null (MA is equal o ze o o each node), o i
could be e alua ed o a e y la ge 𝑊, gi en ha each node is isi ing
only one a ea in 𝑊, wi h p obabili y equal o 1. So, in his case, he
way o calcula ing MA should be di e en .
In pa icula , we ake in o accoun he numbe o one-hop neighbo
nodes o node 𝑣𝑘, as an index o he local in luence o 𝑣𝑘’s neighbo s
on 𝑣𝑘(local wo ld). As ega ds he implemen a ion o his app oach,
le us imagine ha each node has an associa ed 𝐼𝐷𝑘and i can manage
a sha ed s uc u e (a lis ) in which o each 𝐼𝐷𝑘 he numbe o i s one-
hop neighbo can be inse ed. Neighbo ing in o ma ion can be de i ed,
o example, by he ou ing ope a ions (Hello messages, RREQ/RREP
mechanism, e c.), so we a e conside ing he gene al case. A e he
con e gence ime, each node will know he exac numbe o one-hop
neighbo s o he o he nodes. So, i 𝑣1, ..., 𝑣𝑀a e he conside ed
mobile nodes (o he whole ne wo k) and 𝑛𝑔𝑘is he numbe o one-hop
neighbo nodes o 𝑣𝑘, hen he Recip ocal Ac i eness (RA) con ibu ion
𝑅𝐴𝑠o he 𝑛𝑔𝑘nodes on 𝑣𝑘in he 𝑊pe iod can be exp essed a e he
e-de ini ion o :
𝑝𝑊
𝑘(𝑡) = 𝑛𝑔𝑊
𝑘(𝑡)
∑𝑀
𝑙=1 𝑛𝑔𝑊
𝑙(𝑡)
, 𝑡 ∈𝑊(7)
whe e 𝑡is a ime ins an inside he ange 𝑊( he addi ion o he ime
dependence is needed because in he ime window 𝑊 he numbe o
neighbo s o node 𝑣𝑘can change o e he ime). Then, as om Eq. (6):
𝑅𝐴𝑠(𝑛𝑔𝑊
𝑘)=−∑
𝑡∈𝑊
𝑝𝑊
𝑘(𝑡)⋅𝑙𝑛 [𝑝𝑊
𝑘(𝑡)](8)
ha is o say he in luence o 𝑣𝑘’s neighbo s on 𝑣𝑘in 𝑊(as illus a ed
in Fig. 2). When nodes mo e, assuming an ON-OFF beha io (wi h
ailu es/ e ie als), h ough a beaconing o ou ing signaling each 𝑣𝑘
can ’’sense’’ he absence/p esence o a neighbo . In his case, he ela ed
en y o he sha ed lis is upda ed. I a new node en e s he ne wo k i
will s a he upda e p ocedu e om he beginning.
In gene al, since ou app oach does no conside a speci ic si ua ion
( he e could be ixed nodes wi h a high numbe o neighbo s, o mo ing
Fig. 3. The gene al scheme o an AF applied o RMA p ocess.
Fig. 4. The 𝐺map conside ed o simula ions and i s 10 x 10 pa i ion.
Fig. 5. An example o he end o he ac i eness associa ed o a mobile hos , wi h
𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 o 14 m/s, 𝐿= 50 m and an obse a ion ime window size o 𝑊= 10 s. The
ob ained ends o 𝑀𝐴(𝑣𝑊
𝑘)and 𝑅𝐴(𝑛𝑔𝑊
𝑘)a e shown in wo lines o be e eadabili y.
nodes wi h a low numbe o neighbo s o example), he ac i eness
index should be composed by bo h e ms (mobili y and ecip ocal
ac i enesses); his is he eason why in ou app oach, conside ing (6)
and (7) we de ine he Recip ocal and Mobili y Ac i eness (RMA):
𝑅𝑀𝐴(𝑣𝑊
𝑘) = 𝛾⋅[𝑀𝐴(𝑣𝑊
𝑘)] + (1 − 𝛾)⋅[𝑅𝐴(𝑛𝑔𝑊
𝑘)],(9)
wi h 𝛾∈ [0..1] as smoo hing ac o .
4. Ac i eness P edic ion ia Adap i e Fil e ing
The second idea o his pape elies on he u iliza ion o a p edic i e
app oach, o know in ad ance wha he end o nodes RMA will be.
In he nex subsec ions we desc ibe he made assump ions o he
Compu e Ne wo ks 226 (2023) 109689
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Fig. 6. The end o he RMA (60 samples) associa ed o a mobile node (Eq. (9)), wi h
𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 o 11 m/s, a ea side L =50 m and an obse a ion ime window W =10 s.
Di e en alues o 𝛾ha e been conside ed.
Fig. 7. A e age RMA associa ed o mobile nodes o di e en alues o 𝑊and 𝐿.
p edic i e app oach and how he RMA dynamics du ing nodes mobili y
can be cap u ed, analyzed and p edic ed.
4.1. Adap i e Fil e ing o Dynamic P ocesses
Gi en he pe iodical na u e o he de ined pa ame e s (MA, RA and
RMA) and ou ing p o ocols (gi en hei upda e in e al), we decided
o base ou p oposal on he Adap i e Fil e ing (AF) [26] heo y, since an
AF is able o adap he coe icien s o i s impulse esponse in unc ion
o a gi en op imiza ion algo i hm. In ac , in ou case, he pa icula
ac i eness (MA, RA o RMA) changes in unc ion o 𝑊and he closed
loop (o he il e ) ea s he eedback as an e o signal, o e-de ine
i s ans e unc ion pa ame e s.
Fig. 3 shows he gene al scheme o an AF: he 𝐼𝑅𝑊𝑗is he impulse
esponse a he 𝑗 h obse a ion window 𝑊𝑗,𝑅𝑀𝐴(𝑣𝑊𝑗
𝑘)is he inpu
RMA a 𝑊𝑗; he il e e alua es i s ou pu as he con olu ion i i s
cu en impulse esponse and he mos ecen alues o RMA, hen
he p edic ed ou pu 𝑃 𝑅𝐸𝑅𝑀𝐴 is compa ed wi h he desi ed 𝐷𝐸𝑆𝑅𝑀𝐴
and he di e ence oge he wi h he inpu a e gi en as inpu s o he
Fil e Op imiza ion Algo i hm (FOA), able o ecalcula e and op imize he
impulse esponse weigh s.
We decided o use he Leas Mean Squa e (LMS) [27,28] as FOA, able
o upda e il e weigh s in he ollowing way:
𝑐𝑙,𝑗+1 =𝑐𝑙,𝑗 + 2 ⋅𝑐𝑓 ⋅𝜖𝑗⋅𝑅𝑀𝐴(𝑣𝑊𝑗,𝑙
𝑘),(10)
whe e we conside ed he 𝐾−𝑡ℎ o de Adap i e LMS il e (AF-LMS)
(𝑙= 1..𝐾), 𝑗is he p e ious obse a ion s ep and 𝑊𝑗is i s ela ed
obse a ion window, 𝑐𝑓 is called con e gence ac o and 𝜖𝑗is he
di e ence be ween he p edic ed RMA (𝑃 𝑅𝐸𝑅𝑀𝐴) and desi ed RMA
(𝐷𝐸𝑆𝑅𝑀𝐴) a s ep 𝑗. As ega ds he pa ame e 𝑐𝑓 , i con ols he
speed and accu acy o he algo i hm con e gence: gene ally i is la ge
a he beginning o a apid con e gence and dec eased o minimize
o e shoo ing ac ions (0< 𝑐𝑓 < 1).
4.2. Adap i e Fil e ing as an Au o Reg essi e P ocess
The AF-LMS app oach i s pe ec ly wi h ou scope, since he ac-
i eness is e alua ed pe iodically (le us hink, o example, o he
pe iodic beaconing, o pe iodic ou ing upda es, e c.), gi ing us he
possibili y o assume and conside i as a sequence o ime samples and,
in pa icula , as an Au o Reg essi e (AR) p ocess, whe e he las obse ed
alue depends linea ly on he p e ious K ones. The only emaining
conce n o he p oposed analysis is he de e mina ion o he alue o
𝐾, ha is he o de o he AF-LMS and, hence, o he unde lying AR
p ocess. To his aim, we conside he Au oCo ela ion Func ion (ACF)
and he Pa ial ACF (PACF) [29]. In ac , an index o he co ela ion
be ween wo alues o an 𝐴𝑅(𝐾)p ocess is he ACF. Fo a gene ic
p ocess 𝑋𝑡, 𝑡 = 0,1,2,… he au oco a iance [30,31] a lag 𝐾is de ined
as:
𝛾𝑋
𝑘=𝐶𝑜𝑣(𝑋𝑡, 𝑋𝑡−𝐾) = 𝐸[(𝑋𝑡−𝜇)⋅(𝑋𝑡−𝐾−𝜇)] (11)
whe e 𝜇is he mean o he p ocess, i.e. 𝜇=𝐸[𝑋(𝑡)], and he au oco -
ela ion coe icien a lag 𝐾is:
𝜌𝑋
𝐾=𝛾𝑋
𝐾
𝛾𝑋
0
(12)
whe e he au oco a iance a lag ze o 𝛾𝑋
0is he a iance o he p ocess.
I is clea ha , om he de ini ion, he au oco ela ion coe icien 𝜌𝑋
𝐾
is dimensionless, so independen on he measu emen scale, and i
belongs o he in e al [−1,1]. F om [32], i is known ha he e m
in Eq. (12) is he heo e ical ACF. A lag 𝐾au oco ela ion ep esen s,
in ou case, he ela ion be ween ac i eness alues ha a e 𝐾 ime
pe iods apa . So, he ACF is a way o conside he linea ela ionship
be ween a ime ins an 𝑡and all he p ocess obse a ions a p e ious
imes. In ou wo k, we assume ha he ac i eness dynamics can be
modeled as an 𝐴𝑅(𝐾)p ocess, bu we wan o know which is he
ela ion among 𝑅𝑀𝐴(𝑣𝑊𝑗
𝑘)and 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾
𝑘), wi hou conside ing he
con ibu ions o 𝑅𝑀𝐴(𝑣𝑊𝑗−1
𝑘), ..., 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾+1
𝑘). Clea ly, a lag 1,
PACF(1) is he same as ACF(1). Following he heo y in [33], in o de
o desc ibe he exp ession o he PACF, we ha e o conside 𝐾Yule–
Walke equa ions [33] w i en o he 𝐴𝑅(𝐾)p ocess, and sol e hem
o he 𝐾 a iables 𝜙𝐾1,…, 𝜙𝐾𝐾 . Typically hey a e w i en in a ma ix
o m as ollows (we used he no a ion 𝑅 o he 𝑅𝑀𝐴 p ocess in o de
o ob ain a compac no a ion):
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1𝑅(𝑣𝑊1
𝑘)... 𝑅(𝑣𝑊𝐾−1
𝑘)
𝑅(𝑣𝑊1
𝑘) 1 ... 𝑅(𝑣𝑊𝐾−2
𝑘)
𝑅(𝑣𝑊2
𝑘)𝑅(𝑣𝑊3
𝑘)... 𝑅(𝑣𝑊𝐾−3
𝑘)
... ... ... ...
𝑅(𝑣𝑊𝐾−1
𝑘)𝑅(𝑣𝑊𝐾−2
𝑘)... 1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⋅
⎡
⎢
⎢
⎢
⎢
⎣
𝜙𝐾1
𝜙𝐾2
𝜙𝐾2
...
𝜙𝐾𝐾
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
𝑅(𝑣𝑊1
𝑘)
𝑅(𝑣𝑊2
𝑘)
𝑅(𝑣𝑊3
𝑘)
...
𝑅(𝑣𝑊𝐾
𝑘)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(13)
and he PACF will be ep esen ed by he 𝐾−𝑡ℎ solu ion 𝜙𝐾𝐾 , a unc ion
o lag 𝐾.
Based on he discussion abo e, as shown in nex sec ions, we can
conclude ha using he PACF ins ead o he ACF will lead us o ob ain
a meaning ul alue o 𝐾and, as s a ed in [29,34], he PACF ep esen s
he mos use ul ‘‘ ool’’ o de e mining he o de o an AR model. So, he
ACF and PACF a e s a is ical measu es ha e lec how he obse a ions
o a p ocess e olu ion a e ela ed o each o he . In addi ion, as s a ed
in [32], i is o en use ul o plo hese unc ions agains consecu i e
ime lags. All he g aphical app oaches, o assessing he lag/o de
o an AR model, include looking a he ACF/PACF alues e sus he
lag (co elog am). In an ACF co elog am, as he ones shown in he

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P. Fazio e al.
Fig. 8. RMA end i ing by using linea il e ing.
Fig. 9. PACF end o di e en lags 𝐾and 𝛾 alues (𝑊= 10 s, 𝐿= 50 m,
𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 = 14 m/s).
Fig. 10. Co elog am o he PACF o di e en lags 𝐾and 𝛾 alues (on he X axis),
wi h 𝑊= 10 s, 𝐿= 50 m, 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 = 13.9m/s.
nex sec ion, i he e a e la ge alues wi h a non- andom pa e n, hen
he e is a high p obabili y ha he alues a e co ela ed. In a PACF
co elog am, ins ead, he pa e n is usually andomly de ined, bu la ge
alues o a gi en 𝐾indica e ha i is a possible choice o he lag/o de
o he whole p ocess [32].
In he nex sec ion a ull and deep analysis o hese concep s is
ca ied ou , gi ing o he eade he possibili y o well unde s and how
he ac i eness pa ame e can be analyzed, p edic ed and applied in
mobile ne wo king.
5. Nume ical analysis and esul s
This sec ion is dedica ed o show he main nume ical esul s each-
able by he p oposed AF-LMS app oach. In Table 3 he alues used in
Table 3
The main alues used in nume ical
analysis.
Pa ame e Value
𝑁=𝑂2000 m
𝑁x𝑂4 km2
𝑙𝑥=𝑙𝑦50–200 m
𝑛=𝑚[10, ...,40]
𝑔𝑖𝑗 ∈𝐺100–1600
𝑟𝑘=𝑟50 m
𝛾0.6
𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 11, 14, 20 m/s
he nume ical analysis a e lis ed. Mobili y has been gene a ed on he
basis o he OpenS ee Map co e [15] (which gi es he oppo uni y o
selec and expo a desi ed geog aphical map 𝐺) and Ci y4Roadmaps
(C4R) [14] (able o gene a e mobili y pa e ns by ollowing eal mo e-
men s). A squa e Gwi h N=O=2000 me e s and an a ea o abou 4
km2has been conside ed, ex ac ed om he cen e and pe iphe al o
Rome ci y ( e e o Fig. 4).
Mobili y aces ha e been gene a ed as u ban mobili y, wi h a i-
able a e age speeds (𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑) o 11, 14 and 20 m/s, while he
pa i ioning sub-a eas ha e been conside ed o be squa e, wi h a side
𝑙𝑥=𝑙𝑦=𝐿 anging om 50 m o 200 m (so alues o 𝑛=𝑚∈
[10,…,40] and a o al numbe o sub-a eas 𝑔𝑖𝑗 ∈𝐺going om 100
o 1600). Vehicles a i al imes belong o a Poisson dis ibu ion and
he co e age adius o each node has been conside ed o be 𝑟𝑘=𝑟=
50 m ∀ 𝑣𝑘∈𝑀𝑂𝐵, 𝑘 = 1..𝑀.
An objec -o ien ed Py hon applica ion has been designed, in o de
o c ea e he map, i s pa i ion, mobili y aces and he e alua ion o
Eqs. (6),(8) and (9), aking 𝐺,𝑁,𝑂, and 𝐿as inpu pa ame e s.
Fig. 5 shows he end o 𝑀𝐴(𝑣𝑊
𝑘)and 𝑅𝐴(𝑛𝑔𝑊
𝑘) o a gene ic mobile
node. The shown end is gene al and we e i ied ha i is alid o all
he in ol ed nodes du ing hei ac i e sessions. In o de o gi e an idea
o he gene ic end o he alues o 𝑅𝑀𝐴 as de ined in Eq. (9),Fig. 6
is also shown.
In Fig. 6, he o al numbe o samples has been educed in o de o
make he igu e mo e eadable. I we e e o he a e age end o RMA
(𝛾= 0.6) in unc ion o 𝑊and 𝐿,Fig. 7 can be conside ed.
The i s in e es ing esul o ou analysis shows how he a e age
MA alue is in luenced by he selec ed pa ame e s. In ac , Fig. 7
illus a es he end o 𝑅𝑀𝐴(𝑣𝑊
𝑘)by a ying he 𝑊leng h and he
alue o 𝐿(wi h an 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 o 11 m/s and 𝛾=0.6). Fo bigge sub-
a eas, he ac i eness goes dec easing gi en ha each mobile hos akes
mo e ime o go ou side each a ea ( he cu en one), while o highe
alues o 𝑊RMA inc eases, gi en ha each mobile node is able o
mo e among a highe numbe o loca ions.
A his poin , we ha e o e i y ha an 𝐴𝑅(𝐾)app oach can help
o analyze he dynamics o mobile nodes, gi ing o hem an a-p io i
knowledge o u u e RMA beha io s. As illus a ed in he ollowing,
we p o ided o implemen he LMS adap i e il e in py hon, h ough
he Py hon Adap i e Signal P ocessing lib a y 1.1.1 [35].
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P. Fazio e al.
Fig. 11. Ac i eness samples p edic ion wi h LMS o 𝐾=1, 𝑐𝑓 =0.1, 0.3, 0.5, 0.7.
In pa icula , we in eg a ed he p e iously implemen ed pa se (in
o de o make he mobili y gene a ed by C4R eadable) wi h he
lib a y in [35], in o de o e alua e he ACF and he PACF unc ions
ela ed o he collec ed samples, a e he 𝐺pa i ioning ope a ion,
and assuming ha 𝑅𝑀𝐴(𝑣𝑊
𝑘)is an 𝐴𝑅(𝐾)p ocess. This app oach i s
pe ec ly wi h he main aim o his pape , ha is he possibili y o
p edic u u e samples in eal- ime, such as e y impo an sys em
pa ame e s (weigh s in ne wo k ou ing, he o e all cos on a pa h om
a sou ce o a des ina ion, links s abili y, e c.). In o de o ob ain some
sui able esul s in his di ec ion, we p o ide o apply he ACF/PACF
de ini ions o ind he o de o he RMA p ocess.
Fi s o all, he o de o he adap i e il e needs o be decided. To
his aim, we p o ided o use he 𝑠𝑐𝑖𝑝𝑦.𝑠𝑖𝑔𝑛𝑎𝑙 and 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚 lib a ies in
Py hon, whe e he pyule unc ion is a ailable [33], in o de o ob ain
he PACF ela ed o he lag 𝐾.Fig. 8, o example, shows wo ep esen-
a ions o he end o he o iginal sequence o 20 RMA samples (c oss
poin s) and hei es ima ion by a linea AR il e wi h LMS op imiza ion
(𝐾= 10). On he le side a mo e e iden end o he commi ed e o
is unde lined, while on he igh i can be obse ed how a linea il e
is able o ollow he igh end, wi h a p edic ion e o ( a iance)
𝜎2=0.18. So, in o de o disco e and choose an adequa e alue o he
il e o de 𝐾, we p o ided o e alua e he PACF, conside ed, as de ined
ea lie , as he au oco ela ion be ween 𝑅𝑀𝐴(𝑣𝑊𝑗
𝑘)and 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾
𝑘),
wi hou he linea dependence o 𝑅𝑀𝐴(𝑣𝑊
𝑘)on 𝑅𝑀𝐴(𝑣𝑊𝑗−1
𝑘) h ough
𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾+1
𝑘)[36].
We p o ided o ca y ou he analysis o se e al ac i eness samples
ela ed o di e en nodes in o 𝐺and, consequen ly, he PACF analysis
has been ca ied ou , a ying 𝑊,𝐿,𝛾,𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑: o space issues we
canno show all he ob ained esul s, bu we summa ize hem wi h he
ollowing igu es.
Fig. 9 shows he end o he PACF, aking in o accoun he possible
alues o lags om 𝐾=1 o 𝐾=20. We can obse e and conclude ha
he memo y e ec is no iceable o low alues o 𝐾. In pa icula he
ac i eness e olu ion can be conside ed as an 𝐴𝑅(1) p ocess, since he
unc ion spike occu s o 𝐾=1, independen ly o he chosen 𝛾 alue. Fo
𝐾=2 o 𝐾= 3 he absolu e alue o PACF goes d as ically dec easing,
while o highe alues i app oaches o 0. Clea ly, Fig. 9 is ob ained
o a pa icula combina ion o simula ion pa ame e s, bu i e lec s
he gene al PACF end.
Fig. 10 shows he same alues in a di e en o m (a co elog am):
independen ly om 𝛾,𝐾=1 (bigges and da kes ma ke ) leads o he
highes absolu e alue o PACF, while o dec easing lag alues, PACF
is negligible, showing ha no co ela ion exis s among samples a e
la ge ime pe iods.
Table 4
Pa ame e alues o he simula ion.
Pa ame e Value
Simula ion a ea 1000 ×1000 m2
Numbe o Nodes 10, 30, 50
Numbe o Ne De ices pe node 1
Wi i Phy mode DsssRa e11Mbps
Wi i P opaga ion Delay Cons an Speed P opaga ion Model
Da a T a ic Type UDP Cons an Bi Ra e
Da a T a ic Ra e 512 kbps
Da a T a ic Applica ion OnO Applica ion
Mobili y Model Random WayPoin
Mobili y Model Pause In e al Cons an (0.5 s)
Node speed in mobili y model 10, 30 m/s
To al Simula ion ime 100 s
Fo ha alue o 𝐾 he p edic ion e o could be minimized, gi en
ha he ac i eness e olu ion is no comple ely andom, bu i is egu-
la ed by a hea y co ela ion be ween samples which a e 𝐾s eps away.
So, o he nex esul s, we se 𝐾=1 and applied he LMS algo i hm o
he RMA p ocess in o de o ob ain he op imal coe icien .
Fig. 11 shows he esul s ob ained by conside ing 120 RMA samples,
wi h 𝑊=10s, 𝐿=30 m, 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑=14 m/s. I can be seen how, in
gene al, he LMS is able o e alua e u u e samples wi h high accu acy
and he 𝑐𝑓 pa ame e does no hea ily a ec he o e all e o .
5.1. Me ic applica ion simula ion analysis
To analyze he usage o RA and MA in ou ing p o ocols, we
simula ed ne wo ks wi h andom opologies consis ing o 10, 30 and 50
nodes. We conside ed he impac o node mobili y and he geog aphical
size o sub-a eas ha a e used o calcula e MA alue (𝑔𝑖𝑗 in Eq. (3)).
The simula ions we e pe o med using he NS-3 Simula o o e sion
3.37 [37]. Fo he same pa ame e s o he numbe o nodes, he speed
o mo emen , and he numbe o a ic-gene a ing applica ions, 10
di e en andom scena ios we e gene a ed, which esul ed in o al
1712 simula ions. The BRITE opology gene a o o gene a e andom
opologies since i is suppo ed unde NS-3 and he sou ce code is eely
a ailable [38]. Table 4 lis s he simula ion pa ame e s including pa am-
e e s o WiFi Ne De ices which we e se o p o ide a maximal co e age
a ea o 150 m2and enable mul i-hop communica ion. Pa ame e s no
gi en he e a e de aul pa ame e s o he NS-3 3.37 simula o .
Each simula ion included wo a ic-gene a ed applica ions. Fo
each o he applica ions, a node is andomly selec ed om he (0,(𝑛∕2)−
1) ange o nodes and he sou ce a ic applica ion is ins alled on he
selec ed node. Fo each o he applica ions, a node is andomly selec ed
Compu e Ne wo ks 226 (2023) 109689
8
P. Fazio e al.
om he (𝑛∕2, 𝑛 − 1) ange o nodes and he des ina ion a ic appli-
ca ion is ins alled on he selec ed node. Loca ion-based moni o ing is
implemen ed as dedica ed module in he NS-3 simula o , which pe iodi-
cally analyzes he mo emen o nodes e e y h ee seconds. Based on he
measu ed alues, he MA alue o each node is pe iodically calcula ed.
Also, loca ion-based moni o ing was ex end o p o ide RA alue by
analyzing he ou ing ables o each node in he ne wo k. In pa icula ,
we conside ed he applica ion o he RA and MA in AODV ou ing
p o ocol. AODV is known as a eac i e ou ing p o ocol whe e ou ing
pa hs a e sea ched only when needed by looding he ne wo k [39,40].
The disco e y p ocedu e e mina es when ei he a ou e has been
ound, o no ou e is a ailable a e all ou e pe mu a ions ha e been
checked. Due o looding, an in e media y node may ecei e mul iple
RREQ que ies o ind a pa h o a emo e des ina ion. By de aul , AODV
s o es he i s RREQ while disca ding all subsequen eques s as hey
a e conside ed duplica es. In ou a ian , we conside ed he applica ion
o he RA and MA when analyzing ha b oadcas ed RREQ que ies. Each
ime node ecei es RREQ eques om i s neighbo and he e is al eady
p e iously p ocessed RERQ and s o ed in cache memo y wi h he same
o igin and des ina ion, i will calcula e RA and MA alues using Eq. (9)
o i sel and he neighbo ing node which o wa ded RREQ eques .
Suppose ha he calcula ed RMA alue o neighbo is lowe hen he
calcula ed RMA alue o i sel . Then he ecei ed RREQ eques will
be igno ed. Howe e , in opposi e case, i will be p ocessed and AODV
ou e will be upda ed o e he neighbo ing node which o wa ded
RREQ eques .
Figs. 12 and 13 shows he ob ained esul s. One can no e ha when
pa ame e gamma (𝛾 om Eq. (9)) is se o 0, he alue RMA is based on
RA, and hus, he e is no in luence o he geog aphical size o sub-a eas
ha a e used o calcula e MA alue (𝑔𝑖𝑗 in Eq. (3)). This case is deno ed
wi h blue box-plo s on g aphs ha a e iden ical in sub igu es. Howe e ,
as alue 𝛾inc eases, he RAM alue conside s RA and MA alues. In he
case o a ne wo k wi h a smalle numbe o nodes (i.e. 10), he e a e
no signi ican changes in ob ained esul s. The eason is ha a small
numbe o nodes do no lead o apid changes in he ne wo k om he
aspec o ou ing able en ies and o e all ne wo k dynamics. Howe e ,
when he ne wo k is o med wi h a la ge numbe o nodes, mo e
dynamics lead o signi ican changes in ou ing ables and MA alues.
As he numbe o ne wo k nodes inc eases, mo e mobile nodes a e
ma ked as candida es as messenge nodes be ween di e en mobili y
egions. Thus, he e a e mo e chances o ind a be e AODV ou e.
Simula ions we e pe o med wi h iden ical ne wo k opologies and
andom seeds, gua an eeing he simula ion’s epea abili y. While com-
pa ing esul s om Figs. 12 and 13, one can no e ha he ob ained
PDR alues a e signi ican ly lowe . The eason is ha as he mobili y
o nodes inc eases, he e a e mo e in e up ions o es ablished AODV
ou es. The e is also an inc eased numbe o chances o ind al e na i e
AODV ou es, bu due o high mobili y, hese al e na i e ou es las
only o a sho pe iod o ime. The impac o geog aphical sizes o
sub-a eas is mo e signi ican , and AODVM can o di e en alues o
gamma (𝛾) ou pe o m pu e AODV.
In some cases, ou modi ica ion o AODV esul ed in equal o be e
ou ing (bes exp essed wi h pu ple box-plo s when 𝛾= 1), while in
o he s, i esul ed in deg ada ion. I depends on alues o 𝛾and sizes o
sub-a eas. Howe e , he simula ed scena ios deno e only one example
o using he RMA app oach. I is possible o ind be e scena ios in
which RMA alues will ha e a mo e signi ican impac . In ou example
wi h AODV RREQ eco ds, RMA is conside ed only when he ou e is
in e up ed and needs o be e eshed by p ocessing new RREQ eco ds
( he p ocessing o he i s RREQ eco ds o es ablish he ini ial ou e is
iden ical o AODV and AODVM p o ocols). Such cases a e no equen
(especially o ne wo ks wi h low mobili y and dynamics), and a mo e
signi ican in luence o RMA eco ds is expec ed in p oac i e ou ing
p o ocols, e.g., when p ocessing pe iodic hello messages o conside
he ne wo k s a e (i.e., DSDV ou ing p o ocol [41]). Howe e , he
desc ibed example shows ha he RMA alue can signi ican ly impac
ne wo k pe o mance, e en conside ed h ough applica ion o AODV
RREQ eques s.
Fig. 12. Simula ion esul s compa ing AODV and AODVM ou ing p o ocols o
di e en sizes o ne wo k (numbe o nodes). The mobili y speed o nodes was se
o 10 m/s.
6. Conclusion and u u e wo ks
In his wo k, we p esen ed a s ochas ic analysis o he concep o
mobili y ac i eness in mobile ne wo ks, gi en i s capabili y o in luence
ne wo k dynamics ( ou ing, physical channel, e c.). We p o ided o
de ine i and gi e emphasis o he main ea u es which a e able o
in luence i s alue when mobile nodes mo e inside a geog aphical
a ea. We unde lined he impo ance o conside ing mobili y ac i eness
(di ec o ecip ocal), as well as he possibili y o p edic i , by he use
o an adap i e il e , op imized by he LMS algo i hm o he weigh s
upda e. In addi ion, we disco e ed ha he ac i eness p ocess can
be conside ed o be an o de -1 au o eg essi e p ocess. The ob ained
esul s ha e shown ha he end o mobile ac i eness can be p edic ed
wi h a e y negligible e o and his ea u e can gi e o he ne wo k
a e y impo an eedback on he u u e beha io o mobile nodes,
Compu e Ne wo ks 226 (2023) 109689
9
P. Fazio e al.
Fig. 13. Simula ion esul s compa ing AODV and AODVM ou ing p o ocols o
di e en sizes o ne wo k (numbe o nodes). The mobili y speed o nodes was se
o 30 m/s.
especially on hei s abili y in he nea u u e. We p o ided, also,
o ca y ou a pe o mance compa ison be ween he classical AODV
p o ocol (whose me ic is based on he hop-coun ), and he AODVM
(wi h he RA and MA me ics), in o de o show he bene i s o ou
p oposal.
As u u e ex ensions o he p oposed idea, we plan o conside
also A i icial In elligence (AI)-based p edic i e schemes o ou ing
op imiza ion, in o de o conside and compa e he possible ob ainable
enhancemen s, a he cos o a highe compu a ional complexi y.
CRediT au ho ship con ibu ion s a emen
Peppino Fazio: Concep ualiza ion, In es iga ion, W i ing – o igi-
nal d a , Me hodology, So wa e. Mi alem Mehic: Concep ualiza ion,
W i ing – e iew & edi ing, So wa e. Mi osla Voznak: Visualiza-
ion, Supe ision, Funding acquisi ion. Flo iano De Rango: Resou ces,
In es iga ion, Supe ision. Mau o T opea: So wa e, Da a cu a ion,
Valida ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Da a a ailabili y
Da a will be made a ailable on eques .
Acknowledgmen
The esea ch ecei ed a inancial suppo om he S uden G an
Sys em (SGS) No. SP2018/59 ‘‘Ne wo ks and Communica ion Tech-
nologies o Sma Ci ies’’, VSB - Technical Uni e si y o Os a a, Czech
Republic.
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