senso s
A icle
Ou age Analysis o he Powe Spli ing Based Unde lay
Coope a i e Cogni i e Radio Ne wo ks
Phu T an Tin 1, Van-Duc Phan 2, Tan N. Nguyen 3,* , Lam-Thanh Tu 4, Bui Vu Minh 5, Mi osla Voznak 6
and Peppino Fazio 6,7


Ci a ion: Tin, P.T.; Phan, V.-D.;
Nguyen, T.N.; Tu, L.-T.; Minh, B.V.;
Voznak, M.; Fazio, P. Ou age Analysis
o he Powe Spli ing Based
Unde lay Coope a i e Cogni i e
Radio Ne wo ks. Senso s 2021,21,
7653. h ps://doi.o g/10.3390/
s21227653
Academic Edi o : Pe e Han Joo
Chong
Recei ed: 2 Oc obe 2021
Accep ed: 16 No embe 2021
Published: 18 No embe 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Facul y o Elec onics Technology, Indus ial Uni e si y o Ho Chi Minh Ci y,
Ho Chi Minh Ci y 700000, Vie nam; [email p o ec ed]
2Facul y o Au omobile Technology, Van Lang Uni e si y, Ho Chi Minh Ci y 700000, Vie nam;
[email p o ec ed]
3Communica ion and Signal P ocessing Resea ch G oup, Facul y o Elec ical and Elec onics Enginee ing,
Ton Duc Thang Uni e si y, Ho Chi Minh Ci y 700000, Vie nam
4Ins i u e XLIM, Uni e si y o Poi ie s, 86000 Poi ie s, F ance; [email p o ec ed]
5Facul y o Au omo i e, Mechanical, Elec ical and Elec onic Enginee ing, Nguyen Ta Thanh Uni e si y,
Ho Chi Minh Ci y 700000, Vie nam; [email p o ec ed]
6Facul y o Elec ical Enginee ing and Compu e Science, VSB-Technical Uni e si y o Os a a,
708 00 Os a a, Czech Republic; mi osla [email p o ec ed] (M.V.); [email p o ec ed] (P.F.)
7Depa men o Molecula Sciences and Nanosys ems, Ca’ Fosca i Uni e si y o Venice, Via To ino 155,
30123 Venezia VE, I aly
*Co espondence: [email p o ec ed]
Abs ac :
In he p esen pape , we in es iga e he pe o mance o he simul aneous wi eless in o ma-
ion and powe ans e (SWIPT) based coope a i e cogni i e adio ne wo ks (CCRNs). In pa icula ,
he ou age p obabili y is de i ed in he closed- o m exp essions unde he oppo unis ic pa ial
elay selec ion. Di e en om he con en ional CRNs in which he ansmi powe o he seconda y
ansmi e s coun me ely on he agg ega e in e e ence measu ed on he p ima y ne wo ks, he
ansmi powe o he SWIPT-enabled ansmi e s is also cons ained by he ha es ed ene gy. As a
esul , he ma hema ical amewo k in ol es mo e co ela ed andom a iables and, hus, is o highe
complexi y. Mon e Ca lo simula ions a e gi en o co obo a e he accu acy o he ma hema ical
analysis and o shed ligh on he beha io o he OP wi h espec o se e al impo an pa ame e s,
e.g., he ansmi powe and he numbe o elays. Ou indings illus a e ha inc easing he ansmi
powe and/o he numbe o elays is bene icial o he ou age p obabili y.
Keywo ds:
decode–and– o wa d; ou age p obabili y; elay selec ion; cogni i e adio ne wo k; SWIPT
1. In oduc ion
Cogni i e adio ne wo ks (CRNs) a e conside ed one o he mos e ec i e solu ions
o o e come he sca ci y o he equency spec um [
1
,
2
]. The p incipal idea o CRNs
is o pe mi unlicensed use s o concu en ly ope a e wi h licensed use s while s ic ly
gua an eeing he quali y-o -se ice (QoS) o p ima y use s. To ealize such ne wo ks
wo popula p o ocols a e p oposed in he li e a u e, namely, he o e lay and unde lay
p o ocols [
3
,
4
]. The o me allows seconda y de ices o oppo unis ically occupy he
empo a ily unused spec um in e ms o space, ime, and equency. Rega ding he
unde lay p o ocol, on he o he hand, seconda y use s a e always g an ed pe mission o
access he licensed spec um p o ided ha he agg ega e in e e ence c ea ed by seconda y
ne wo ks measu ed on he p ima y de ices is below he p ede ined h eshold. Compa ed
o he o e lay p o ocol, he unde lay p o ocol is p e e able due o i s high a ailabili y
and, hus, is sui able o u gen se ices, ideo con e ences, ideo gaming, and so o h.
Ne e heless, he cons o his p o ocol a e ha i does no suppo a long ansmission
and/o high-quali y se ices owing o he low ansmi powe o a oid exceeding he
Senso s 2021,21, 7653. h ps://doi.o g/10.3390/s21227653 h ps://www.mdpi.com/jou nal/senso s
Senso s 2021,21, 7653 2 o 17
in e e ence h eshold o he p ima y ne wo ks. As a consequence, in o de o employ he
unde lay CRNs in p ac ice, he combina ion wi h o he echniques is necessa y. Fo una ely,
elaying o coope a i e communica ions is a good complemen o he unde lay CRNs [
5
–
7
].
Mo e p ecisely, coope a i e communica ions ne wo ks a e wi eless ne wo ks whe e one o
se e al elays a e deployed among end-use s o help hem exchange in o ma ion. Wi h
he help o he elay, he ansmission dis ance is d ama ically dec eased, imp o ing he
sys em pe o mance. Addi ionally, elaying echnology has also been p o en o be an
e ec i e way o ex end he co e age a ea. The spec al e iciency and QoS issues can
be add essed in a s aigh o wa d manne by employing he coope a i e cogni i e adio
ne wo ks (CCRNs) [
8
,
9
]. None heless, he e s ill exis s an u gen issue, which is o imp o e
he ene gy e iciency o wi eless ne wo ks. The p oblem escala es se iously in ei he he
In e ne o Things (IoTs), low powe wide a ea ne wo ks (LPWAN) [
10
], o 5G and beyond
ne wo ks owing o he exponen ial g ow h o wi eless-connec ed de ices accompanied by
hei powe -hung y applica ions. In ac , imp o ing ene gy e iciency is one o he highes
p io i ies in wi eless communica ions. Fo una ely, simul aneous wi eless in o ma ion
and powe ans e (SWIPT) has ecen ly been p oposed o deal wi h his issue [
11
–
13
]. In
pa icula , SWIPT ansmi s bo h in o ma ion and ene gy on he same ca ie equency,
hus boos ing bo h he spec al and ene gy e iciency. Consequen ly, in he p esen pape ,
we in es iga e he pe o mance o he SWIPT-based coope a i e cogni i e adio ne wo ks
(CCRNs) wi h he help o mul iple elays. Be o e highligh ing ou no el con ibu ions,
he s a e-o - he-a is gi en as ollows.
The pe o mance o CCRNs ne wo ks was s udied widely in [
14
–
20
]. Speci ically,
he ou age p obabili y (OP) o he cogni i e adio non-o hogonal mul iple access (NOMA)
ne wo ks was de i ed in [
14
]. The au ho s in [
15
], on he o he hand, in es iga ed he
sec ecy pe o mance o he unde lay coope a i e mul ihop CRNs. In pa icula , he sec ecy
ou age p obabili y (SOP) was de i ed in he app oxima ed closed- o m exp ession. Lee e al.
in [
16
] add essed he minimiza ion o he numbe o eedback bi s equi ed in o de o
sa is y he QoS in bo h p ima y and seconda y ne wo ks. The op imal powe alloca ion
o non-o hogonal ampli y–and– o wa d (AF) elaying o unde lay CRNs was p o ided
in [
17
] o maximize he sys em h oughpu . In [
18
], he capaci y o oice o e IP (VoIP) in
CRNs was analyzed and maximized by modelling he VoIP a ic and channel coe icien s
as a Ma ko -modula ed Poisson p ocess. Liu e al. in [
19
] in es iga ed he spec um
sensing p oblem in he ull-duplex coope a i e spec um sensing CRNs. I was no ed ha
spec um sensing is impo an in o e lay CRNs o gua an ee he oppo unis ic access o
seconda y de ices while no in e up ing he p ima y de ice’s ansmission.
Meanwhile, he pe o mance o SWIPT-enabled ne wo ks was in es iga ed
in [21–29]
.
The ene gy-e icien op imiza ion o SWIPT-assis ed elaying ne wo ks was add essed
in [
21
]. Tan e al. in [
22
] de i ed he OP and e godic capaci y o he powe spli ing
(PS) based elaying ne wo ks unde he asymme ic channel, i.e., he Nakagami-
m
and
Rayleigh channels. The e o p obabili y and ou age p obabili y o SWIPT-based NOMA
ne wo ks we e de i ed in [
23
]. In pa icula , he pai wise e o p obabili y was compu ed
in closed- o m exp ession in [
23
]. The asymp o ic amewo k unde a high SNRs egime
was p o ided as well. The wo k in [
24
], di e en ly, de i ed he symbol e o a e (SER) o
SWIPT-enabled elaying ne wo ks, whe e he noncohe en modula ion was employed in
place o he con en ional phase-shi keying (PSK) and/o quad a u e ampli ude modula-
ion (QAM). The au ho s in [
25
] also add essed he noncohe en modula ion. Speci ically,
his wo k i s de i ed he momen s and momen gene a ing unc ion (MGF) o he end- o-
end (e2e) signal- o-noise a ios (SNRs). Based on he MGF, hey hen compu ed he ou age
p obabili y, he amoun o ading, and he sys em h oughpu . The au ho s in [
26
,
27
] deal
wi h he physical laye secu i y (PLS) issue o he simul aneous wi eless in o ma ion and
powe ans e assis ed elaying ne wo ks. Fu he mo e, he pe o mance o SWIPT-based
cellula ne wo ks wi h and wi hou u ilizing millime e wa e (mmWa e) was p o ided
in [28,29].
Senso s 2021,21, 7653 3 o 17
Despi e he ex ensi e s udy o ei he he CCRNs o he SWIPT-aided ne wo ks, he pe -
o mance o SWIPT-enabled unde lay CCRNs is s ill in he in ancy s age. In pa icula ,
he e only a ew wo ks add essing his combina ion [
30
–
33
]. Speci ically, P a hima e al.
in [
30
] s udied he pe o mance o he p ima y ne wo ks wi h he help o he seconda y
use s ha also ac as he elay o p ima y use s. This wo k, howe e , concen a ed on he
pe o mance o p ima y ne wo ks. Mo eo e , i solely conside ed wo elay nodes ins ead
o he gene al scena io. The wo k in [
31
], di e sely, add essed he sec ecy pe o mance
o he SWIPT-assis ed cogni i e elaying ne wo ks. The powe alloca ion and anscei e
design we e in es iga ed in [32].
In his pape , di e en om he abo emen ioned wo ks, we ocus on he pe o mance
o he seconda y ne wo ks as well as he eliabili y o SWIPT-based unde lay coope a i e
cogni i e adio ne wo ks wi h ega d o anscei e design, powe alloca ion, and physical
laye secu i y. In pa icula , he p incipal no el con ibu ions a e summa ized as ollows:
•
We conside a single-inpu single-ou pu (SISO) unde lay coope a i e cogni i e adio
ne wo k wi h he assis ance o mul iple elays. Addi ionally, he ansmi powe o
he elay nodes elies only on he ha es ed ene gy om he ansmi e S. The pa ial
elay selec ion is adop ed o bo h enhance he sys em pe o mance and educe he
complexi y compa ed o he ully elay selec ion.
•
Di e en om he con en ional unde lay CRNs whe e he ansmi powe o he
seconda y ansmi e conside s me ely he in e e ence powe , he ansmi powe
o he conside ed ne wo ks is cons ained by bo h he in e e ence powe and he
ha es ed ene gy. The ma hema ical amewo k, hus, is o highe complexi y owing
o dealing wi h mo e co ela ed andom a iables. None heless, we a e s ill able o
de i e he ou age p obabili y in he closed- o m exp essions.
•
Simula ion esul s a e p esen ed o co obo a e he exac ness o ou analysis and o
iden i y he beha io o OP wi h espec o se e al impo an pa ame e s, namely,
he ansmi powe , he numbe o elays, he powe spli ing a io, and so on. Ou
indings show ha bo h inc easing he ansmi powe and numbe o elays a e
bene icial o he OP. Addi ionally, an op imal alue o he powe spli ing a io exis s
ha minimizes he OP.
The emainde o his pape is o ganized as ollows. The sys em model is gi en in
Sec ion 2. The de i a ion o he OP is p o ided in Sec ion 3. Nume ical esul s a e shown
in Sec ion 4. Sec ion 5concludes he pape .
2. Sys em Model
Le us conside SWIPT-based unde lay cogni i e adio ne wo ks as shown in
Figu e 1.
In pa icula , he seconda y ne wo ks comp ise one sou ce node deno ed by S, one des ina-
ion deno ed by D, and
M
elay nodes deno ed by
Ri
,
i∈{1, . . . , M}
, while he p ima y
ne wo ks a e ep esen ed by a p ima y ecei e deno ed by P. He e, P measu es he agg e-
ga e in e e ence c ea ed by he seconda y ne wo ks on he p ima y ne wo ks.
2.1. Channel Modeling
Conside ing a gene ic ansmission om node X o node Y, he channel coe icien s
deno ed by
hXY
,
X∈{S, Ri}
,
Y∈{Ri, D}
a e ollowed by a Rayleigh dis ibu ion. As a
esul , he channel gain deno ed by
γXY =|hXY|2
is ollowed by an exponen ial dis ibu ion
wi h pa ame e
λXY
whose cumula i e dis ibu ion unc ion (CDF) and p obabili y densi y
unc ion (PDF) a e gi en as ollows [6]:
FX(x) = 1−exp(−λXYx), (1)
X(x) = ∂FX(x)
∂x=λXY exp(−λXYx). (2)
Senso s 2021,21, 7653 4 o 17
He e λXY is also he la ge-scale pa h loss om X o Yand is o mula ed as ollows:
λXY =(dXY)β, (3)
whe e
dXY
is he Euclidean dis ance be ween node
X
and
Y
and
β∈(2, . . . , 6)
is he pa h
loss exponen . Addi ionally, he block ading is aken in o conside a ion in his wo k,
hence he channel coe icien s emain cons an s o he whole ansmission
T
and change
independen ly be ween each ansmission.
Figu e 1. SWIPT-based cogni i e adio elaying ne wo ks.
2.2. PS-Based Relaying Ne wo ks
In his wo k, we adop he powe -spli ing (PS) p o ocol a he elay node. To be mo e
p ecise, he ecei ed powe a R is di ided in o wo sepa a e pa s acco ding o he powe -
spli ing a io
ρ
, 0
<ρ<
1, i.e., one is pu in o he ene gy ha es e and ano he goes o
he in o ma ion decode .
ρ
akes in o accoun all loss in oduced by he ene gy ha es ing
ecei e , e.g., noise in oduced by he ecei ed an enna, loss due o he con e ing RF- o-
DC ci cui , and so on [
34
,
35
]. Addi ionally, o ealize he powe -spli ing p o ocol, each
SWIPT-enabled ecei e needs o be equipped wi h a powe spli e o spli he ecei ed
powe in o wo pa s. The i s pa is sen o he con en ional in o ma ion decoding ci cui ,
and he emaining pa is sen o he ene gy ha es ing ci cui [35,36].
2.3. Oppo unis ic Pa ial Relaying (OPR) P o ocol
In his pape , he oppo unis ic pa ial elaying (OPR) p o ocol is adop ed. In pa -
icula , only he elay
n
deno ed by R
n
, which has he highes channel gain om S o
all elay nodes, is selec ed o help exchange in o ma ion be ween S and D. O he elay
nodes, as a esul , keep silen in o de o sa e ene gy consump ion and a oid c ea ing
co-channel in e e ence.
Rn:γSRn=max
|{z}
m=1,2,...,M
{γSRm}(4)
Compa ed wi h he scena io whe e all elays pa icipa e in he ansmission, ou
adop ed p o ocol is simple since i does no equi e pe ec channel s a e in o ma ion
(CSI) o all nodes o he ne wo ks a he des ina ion and pe ec synch oniza ion among
elays [
37
–
39
]. To be mo e p ecise, he adop ed OPR p o ocol can be employed as ollows.
Each elay is equipped wi h a ime , and he alue o he ime is se in e sely wi h he
channel gain om S o elay. Thus, he bes elay is he one ha ing he smalles ime . When
he ime ends, he bes elay o wa ds he sou ce’s signal o he des ina ion. O he elays
sense he a ailabili y o he medium and keep silen once he medium is occupied.
Senso s 2021,21, 7653 5 o 17
2.4. In o ma ion T ansmission
The whole ansmission akes place in wo phases. In he i s phase, sou ce S b oad-
cas s i s signals o all elay nodes. He e, we assume ha he di ec channel be ween S and
D does no exis due o he long ansmission dis ance and deep ades; hus, des ina ion
D does no ecei e he b oadcas signal om S. Al hough all elays a e ecei ed signals
sen by S, only elay R
n
is selec ed o assis he ansmission om S o D. The c i e ia o
selec ing R
n
is gi en in Sec ion 2.3. A elay R
n
, pa s o he incoming signals a e sen o
he in o ma ion decode o decode he in o ma ion sen by S and a e gi en as
yRn=p1−ρpPShSRnxS+nRn, (5)
whe e
nRn
is he addi i e whi e Gaussian noise (AWGN) a elay R
n
, which ollows a
complex Gaussian dis ibu ion wi h ze o mean and N
0
a iance,
nRn∼ CN (0, N0)
;
xS
is
he ansmi ed signal o S and E
n|xS|2o=
1; E
{•}
is he expec a ion ope a o ; and
PS
is
he ansmi powe o S and is de ined in he sequel. The emaining pa o he incoming
signals om S is pu in o he ene gy ha es ed ecei e . The amoun o ha es ed ene gy
deno ed by ERna e hen o mula ed as
ERn=ηρ(T/2)PS|hSRn|2, (6)
whe e
η
is he ene gy con e sion coe icien [
34
,
40
]; he ac o
T/
2 implies ha he ene gy
ha es ing only akes place in hal o he whole ansmission p ocedu e. A he end o
he i s phase, elay R
n
decodes he in o ma ion sen by S and o wa ds he e-encoded
e sion o he des ina ion D in he second phase. The whole ansmission p ocedu e is
shown in Figu e 2. The ecei ed signals a D is hen o mula ed as
yD=pPRnhRnDxRn+nD, (7)
whe e
nD
is he AWGN noise a D wi h ze o mean and N
0
a iance;
xRn
is he ansmi ed
signals o he elay R
n
wi h E
n|xRn|2o=
1, and
PRn
is he ansmi powe o R
n
and is
de ined in Sec ion 2.5. I is no ed ha he ecei ed signal a
Rn
and D in Equa ions
(5)
and
(7)
is a unc ion o he la ge-scale pa h loss ia he channel coe icien
hSRn
and
hRnD
,
espec i ely. The signal- o-noise a ios a Rnand D a e hen o mula ed as ollows:
γRn=(1−ρ)γSRnPS
N0
,
γD=PRn|hRnD|2
N0
. (8)
Figu e 2.
Ene gy ha es ing (EH) and in o ma ion ansmission (IT) p ocesses. EH akes place
only in he i s hal o he ansmission du a ion, while IT akes place du ing he whole ansmis-
sion du a ion.

Senso s 2021,21, 7653 6 o 17
2.5. T ansmi Powe a Sou ce and Relay Nodes
In he unde lay cogni i e adio ne wo ks, all seconda y ansmi e s, i.e., he sou ce
node S and all elay R, ha e o adjus hei ansmi powe o s ic ly sa is y he in e e ence
powe h eshold deno ed by
IP
(in Wa ) on he p ima y ne wo ks, i.e., he p ima y node P.
As a esul , he ansmi powe o S is hen gi en as ([15], Equa ion (5))
PS=IP
|hSP|2, (9)
Rega ding he ansmi powe o Rn, we ha e
PRn=IP
|hRnP|2, (10)
Addi ionally, he ansmi powe o he elay is also cons ained by he amoun o he
ha es ed ene gy in he i s phase and is o mula ed as ([22], Equa ion (2))
PRn=ERn
T/2 =ηρPS|hSRn|2. (11)
As a consequence, PRncan be ew i en as ollows
PRn=min IP
|hRnP|2,ηρPS|hSRn|2!=minIP
γRnP
,ηρPSγSRn(a)
=IPmin1
γRnP
,ηργSRn
γSP ,(12)
whe e (a)is ob ained by subs i u ing PSin (9).
2.6. End- o-End Signal- o-Noise Ra ios a D
Since he decode and o wa d (DF) p o ocol is employed, he e2e SNRs is hen com-
pu ed as
γe2e =min{γRn,γD}
(a)
=Ψmin(1−ρ)γSRn
γSP
, minγRnD
γRnP
,ηργSRnγRnD
γSP , (13)
whe e Ψ=Ip
N0;(a)is held by subs i u ing PSand PRnin (9) and (12) in o (8).
Th ough di ec inspec ion
(13)
, we obse e ha he e2e SNR o he conside ed sys em
is mo e challenging han o he wo k desc ibed in he li e a u e. Mo e p ecisely, he SNR
is he composi e o wo minimum unc ions ins ead o only one. Addi ionally, he an-
dom a iables inside hese minimum unc ions a e ully co ela ed as well. As a esul ,
he p oposed ma hema ical amewo k is no el and mo e complica ed han o he s.
3. Ou age P obabili y (OP) Analysis
In his sec ion, we in es iga e one o he mos impo an me ics o a wi eless com-
munica ions sys em, namely, he ou age p obabili y which measu es he quali y-o -se ice
o he whole ne wo k. The OP e e s o he p obabili y ha he e2e SNRs a D is be-
low a p ede ined h eshold. Ma hema ically speaking, i is o mula ed as ollows ([
12
],
Equa ion (23)):
OP =P {γe2e <γ h}=P {min(γRn,γD)<γ h},=1−P {γRn≥γ h,γD≥γ h}, (14)
whe e
γ h =
2
2R−
1, and
R
is he a ge ed a e [in bps/Hz]. In o de o compu e OP
in (14), we i s de i e Lemma 1as ollows:
Senso s 2021,21, 7653 7 o 17
Lemma 1.
Gi en
N
independen and iden ically dis ibu ed (i.i.d.) exponen ial andom a iables
(RVs) wi h pa ame e s
Ω
deno ed by
Ym
,
m∈{1, . . . , N}
. The CDF and PDF o he maximal RV
deno ed by Ymax =max
m∈{1,...,N}{Ym}a e gi en as ollows:
FYmax (x)=1+
N
∑
m=1
(−1)mCm
Nexp(−mΩx)(15)
Ymax (x)=Ω
N −1
∑
m=1
(−1)mCm
N −1exp(−(m+1)Ωx)
whe e Ck
N=N!
k!(N −k)!is he binomial coe icien .
P oo . Le us begin wi h he de ini ion o he CDF as ollows:
FYmax (x)=P Ymax =max
m∈{1,...,N}{Ym}<x
(a)
=
N
∏
m=1
FYm(x)(b)
=(1−exp(−Ωx))N(16)
(c)
=1+
N
∑
m=1
(−1)mCm
Nexp(−mΩx),
whe e
(a)
is held owing o he independence p ope y be ween RVs;
(b)
is a ained by
yielding he CDF o
Ym
; and
(c)
is achie ed wi h he help o he binomial heo em. Taking
he i s -o de de i a i e o he CDF wi h espec o x, we a ain he PDF as ollows:
Ymax (x)=∂FYmax (x)
∂x=Ω
N −1
∑
m=1
(−1)mCm
N −1exp(−(m+1)Ωx). (17)
We close he p oo he e.
Nex , he OP in (14) is ew i en as ollows:
OP (a)
=1−P (1−ρ)γSRnΨ
γSP
≥γ h,ΨγRnD
γRnP
≥γ h,ΨηργSRnγRnD
γSP
≥γ h
=1−P X≥γ h
(1−ρ)Ψ,γRnD≥γ hγRnP
Ψ,X≥γ h
ΨηργRnD
| {z }
Ξ
, (18)
whe e
(a)
is a ained by subs i u ing
(13)
in o
(14)
,
X=γSRn
γSP
.
Ξ
in
(18)
can be calcula ed
as ollows:
Ξ=P X≥γ h
(1−ρ)Ψ,γRnD≥γ hγRnP
Ψ,γ h
(1−ρ)Ψ≥γ h
ΨηργRnD
| {z }
Ξ1
+P X≥γ h
ΨηργRnD
,γRnD≥γ hγRnP
Ψ,γ h
(1−ρ)Ψ<γ h
ΨηργRnD
| {z }
Ξ2
. (19)
Senso s 2021,21, 7653 8 o 17
Ξ1in (19) can be spli in o wo p obabili ies, i.e., Ξ11 and Ξ12, as ollows:
Ξ1=P X≥γ h
(1−ρ)Ψ,γRnD≥γ hγRnP
Ψ,γRnD≥(1−ρ)
ηρ 
=P X≥γ h
(1−ρ)Ψ,γRnD≥γ hγRnP
Ψ,γ hγRnP
Ψ≥(1−ρ)
ηρ 
| {z }
Ξ11
(20)
+P X≥γ h
(1−ρ)Ψ,γRnD≥(1−ρ)
ηρ ,γ hγRnP
Ψ<(1−ρ)
ηρ 
| {z }
Ξ12
,
whe e Ξ11 is e alua ed as
Ξ11 =P X≥γ h
(1−ρ)Ψ
| {z }
P1
×P γRnD≥γ hγRnP
Ψ,γ hγRnP
Ψ≥(1−ρ)
ηρ 
| {z }
P2
, (21)
Looking a (21), o compu e Ξ11, we i s compu e P1as ollows:
P1=1−P γSRn
γSP
<γ h
(1−ρ)Ψ=1−
∞
Z0
FγSRnγ hx
(1−ρ)Ψ× γSP (x)dx,
(a)
=
M
∑
k=1
(−1)k+1Ck
M
∞
Z0
λSPexp−xkλSRγ h
(1−ρ)Ψ+λSPdx (22)
=
M
∑
k=1
(−1)k+1Ck
MλSP(1−ρ)Ψ
kλSRγ h +λSP(1−ρ)Ψ,
whe e (a)is achie ed by u ilizing Lemma 1. Nex , P2is calcula ed as
P2=P (1−ρ)Ψ
ηργ h
≤γRnP≤γRnDΨ
γ h =
∞
Z
(1−ρ)Ψ
ηργ h
γRnD(x)dx
xΨ
γ h
Z
(1−ρ)Ψ
ηργ h
γRnP(y)dy
=λRD
∞
Z
(1−ρ)Ψ
ηργ h
exp(−λRDx)exp−(1−ρ)ΨλRP
ηργ h −exp−xΨλRP
γ h dx
=exp−(1−ρ)ΨλRP
ηργ h ∞
Z
(1−ρ)Ψ
ηργ h
λRD exp(−λRDx)dx (23)
−λRD
∞
Z
(1−ρ)Ψ
ηργ h
exp−xλRD +ΨλRP
γ h dx
=exp−(1−ρ)ΨλRP
ηργ h
−(1−ρ)ΨλRD
ηργ h 
−λRD
λRD +ΨλRP
γ h
exp−(1−ρ)Ψ
ηργ h λRD +ΨλRP
γ h ,
Senso s 2021,21, 7653 9 o 17
F om (22) and (23), Ξ11 in (21) is hen compu ed as
Ξ11 =
M
∑
k=1
(−1)k+1Ck
MλSP(1−ρ)Ψ
kλSRγ h +λSP(1−ρ)Ψ
×
exp−ζΨ[λRP +λRD]
γ h −exp−ζΨ
γ h hλRD +ΨλRP
γ h i
1+ΨλRP
γ hλRD 
, (24)
whe e ζ=1−ρ
ηρ . Ha ing ob ained he Ξ11, we now mo e o Ξ12. Le us i s ew i e Ξ12 as
Ξ12 =P X≥γ h
(1−ρ)Ψ,γRnD≥(1−ρ)
ηρ ,γ hγRnP
Ψ<(1−ρ)
ηρ 
=P X≥γ h
(1−ρ)Ψ×P γRnD≥(1−ρ)
ηρ ×P γRnP<Ψ(1−ρ)
ηργ h (25)
=1−FXγ h
(1−ρ)Ψ×1−FγRnD(1−ρ)
ηρ ×FγRnPΨ(1−ρ)
ηργ h ,
(a)
=
M
∑
k=1
(−1)k+1Ck
MλSP(1−ρ)Ψ
kλSRγ h +λSP(1−ρ)Ψexp(−λRDζ)1−exp−λRPΨζ
γ h .
He e,
(a)
is held by employing he esul s o
(22)
. Ha ing
Ξ11
and
Ξ12
in hand,
Ξ1
in (20) is hen compu ed as
Ξ1=
M
∑
k=1
(−1)k+1Ck
MλSP(1−ρ)Ψ
kλSRγ h +λSP(1−ρ)Ψexp−ζΨ[λRP +λRD]
γ h 
−exp−ζΨ
γ h hλRD +ΨλRP
γ h i
1+ΨλRP
γ hλRD +exp(−λRDζ)−exp−ζλRD +λRPΨ
γ h 
. (26)
A e ob aining Ξ1, we now compu e Ξ2in (19) as ollows:
Ξ2=P X≥γ h
ΨηργRnD
,γRnD≥γ hγRnP
Ψ,γ h
(1−ρ)Ψ<γ h
ΨηργRnD
=P X≥γ h
ΨηργRnD
,γ hγRnP
Ψ≤γRnD<ζ
=
ζΨ
γ h
Z0
γRnP(x)dx
ζ
Z
γ hx
Ψ
γRnD(y)dy
∞
Z
γ h
Ψηρy
X(z)dz, (27)
(a)
=
M
∑
k=1
(−1)k+1Ck
M
ζΨ
γ h
Z0
γRnP(x)dx
ζ
Z
γ hx
Ψ
λSPλRDΨηρy
kλSRγ h +λSPΨηρyexp(−λRDy)dy
| {z }
Ξ21
Senso s 2021,21, 7653 16 o 17
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