Statistical Model of Downlink Power Consumption in Cellular CDMA Networks

Abstract

Present work proposes a theoretical statistical model of the downlink power
consumption in cellular CDMA networks. The proposed model employs a simple but
popular propagation model, which breaks down path losses into a distance
dependent and a log-normal shadowing loss term. Based on the aforementioned
path loss formalism, closed-form expressions for the first and the second
moment of power consumption are obtained taking into account conditions placed
by cell selection and handoff algorithms. Numerical results for various radio
propagation environments and cell selection as well as handoff schemes are
provided and discussed.

Full text

1
STATISTICAL MODEL OF DOWNLINK POWER CONSUMPTION IN CELLULAR
CDMA NETWORKS
Stylianos P. Savaidis1 and Nikolaos I. Miridakis 2, 3
1Department of Electronics and 2Department of Computer Engineering, Technological Educational
Institute (TEI) of Piraeus, 250 Thivon & P.Ralli, Aigaleo, Athens–12244, Greece
3Department of Informatics, University of Piraeus, 80 Karaoli & Dimitriou, 185 34 Piraeus, Greece
Abstract—Present work proposes a theoretical statistical model of the downlink power consumption in cellular CDMA
networks. The proposed model employs a simple but popular propagation model, which breaks down path losses into a
distance dependent and a log-normal shadowing loss term. Based on the aforementioned path loss formalism, closed-form
expressions for the first and the second moment of power consumption are obtained taking into account conditions placed by
cell selection and handoff algorithms. Numerical results for various radio propagation environments and cell selection as well
as handoff schemes are provided and discussed.
Index Terms: Cellular CDMA, Downlink, Power Consumption, Soft Handoff
I. INTRODUCTION
Code division multiple access (CDMA) have been adopted by narrowband 2G and wideband 3G cellular wireless
networks, due to its inherent virtue of providing a single frequency reuse pattern. Since the available spectrum is shared
among all active users, the transmission power is the basic radio resource of CDMA based systems. In this context, power
consumption becomes the dominant performance evaluation figure that determines network resource allocation and capacity.
Power consumption depends on the location of the mobile station (MS), traffic parameters and the QoS requirements of
each service, experienced interference level as well as cell selection and handoff settings. Thus, the development of a power
consumption model which takes into account the aforementioned parameters is a prerequisite for efficient deployment of
CDMA networks. Typically, research activities on the area can be classified into those that examine the uplink [1]-[8] and the
ones referring to the downlink direction [1], [3], [9]-[18]. Taking into account the asymmetric nature of data flows, the
downlink is most likely to be the bottleneck point of CDMA networks. In addition, research studies of the uplink have
provided analytical methodologies concluding to closed form expressions [19], which can tackle both hard and soft handoff
connection modes. Typically, the downlink studies conclude to numerical simulations [1], [13], [14], [19], assumptions that
simplify the examined network scenarios [3], [9], [12] or approximations that mainly resolve the complexity of calculations
regarding soft handoff connection modes [9]-[11], [15]-[18]. Thus, modeling of the downlink in CDMA cellular networks is
a rather important but laborious task.
Several research studies, as mentioned before, have developed an analytical methodology for the downlink performance
evaluation but they resort to Monte Carlo simulations, when soft handoff is taken into account [1], [13], [14], [19]. In [3],
[10] and [12] an analytical framework with closed-form expressions has been obtained but these works do not consider the
soft handoff option, which requires particular attention in CDMA networks. Both hard and soft handoff connection modes are

2
analyzed in [9] but the obtained closed-form expressions estimate the minimum downlink capacity. In [11] the complicated
sums of the log-normal interferences that typically appear in soft handoff connection mode have been approximated by a log-
normal distribution, which concludes to closed form expressions regarding the downlink capacity. Apart from the
aforementioned approximation, the capacity evaluation in [11] simplifies the impact of soft handoff assuming that
interference contributed by the soft handoff users is double, when compared with the hard handoff users. In [15] a rather
efficient calculation methodology is introduced, which can estimate downlink capacity and outage probability considering
both Active Set (AS) size and soft handoff option. The proposed methodology provides general analytical expressions but it
demonstrates a rather high computational load, whereas the capacity calculations are possible using approximations
according to the Central Limit Theory. A soft handoff scheme aiming to minimize power consumption and increase
connection reliability is introduced in [16]. The proposed model in [16] approximates the sums of log-normally distributed
random variables appearing in the various expressions as a single log-normal variable. Closed form expressions for the
average power consumption are provided in [17] but still the numerical implementation requires a Monte Carlo simulation
under soft handoff conditions and balanced power allocation for the involved Base Stations (BSs). In [18] an alternative
calculation methodology is introduced in order to derive closed form expressions of the capacity at a certain outage
probability. Nevertheless, the former expressions were obtained using an approximation of the energy per bit to interference
ratio introducing a “macrodiversity non-orthogonality factor” and Gamma approximations of the interferences and signals in
soft handoff conditions.
According to the above mentioned description, the development of a theoretical statistical model that facilitates
performance evaluation of the downlink in cellular CDMA networks becomes quite laborious, especially when soft handoff
is considered. Approximating assumptions or numerical simulations are typically employed in order to overcome the
complexity of analysis. The presence of sums of log-normally distributed random variables in the various expressions is the
major obstacle regarding the derivation of closed-form analytical expressions. The present work proposes an alternative
approach in order to overcome this kind of complexity and conclude to closed form expressions. In particular, a Taylor series
expansion of the aforementioned complicated expressions is employed, which next makes possible a straightforward
calculation of power consumption moments. In fact present work demonstrates the calculation procedure for the first two
moments of power consumption, although in principle affords calculation of higher order moments. The proposed calculation
scheme can integrate several realistic conditions including a best BS selection condition, the impact of a soft handoff
threshold as well as AS size. Overall, the present work provides a theoretical statistical model, which attempts to balance
efficiently between the assumptions that oversimplify the examined network scenarios, the inaccuracies of the potential
approximations and the physical insight that a closed form expression may provide.
Section II, describes the radio propagation model and the downlink power consumption formulas for hard handoff
(HHO), 2-way and 3-way soft handoff (SHO) connection modes. Section III describes the conditions placed by cell selection
and handoff schemes. In Section IV, the calculation details for the first and second moments of the downlink power
consumption are discussed in details. Section V includes numerical results and verification tests regarding the proposed
calculation scheme. Finally, section VI summarizes the main conclusions and discusses potential extensions of current work.

3
II. DOWNLINK POWER CONSUMPTION
The adopted radio propagation model assumes that fast fading can be compensated by special reception techniques, e.g.
rake receiver, thus it can be considered as a pure large scale path loss model. In particular, path losses are solely determined
by a path loss factor, which determines the distance based losses, and a shadowing loss component, which demonstrates a
log-normal behavior. Thus, the power received from a transmitting BS can be determined by the following expression:
( )
T
PrrP
10
10,
ζα
ζ
−
=
(1)
where r denotes the distance between MS and BS and
T
P
the BS’s total transmitted power; α is the path loss factor and ζ
denotes the shadowing losses as a zero-mean Gaussian distributed random variable with standard deviation σ. The shadowing
loss random variable for a certain BS, i.e. BSi, can be further analyzed into two components, namely ii ba
ξξζ
+= [13],
[14]. The aξ component denotes a part of shadowing that is common for all BSs and it represents the environment near and
around the MS, whereas bξi denotes shadowing effects that depend on the environment near and around BS. The constants a
and b, fulfill the relationship a2+b2=1, whereas ξi are considered as independent zero-mean Gaussian distributed random
variables with standard deviation σ [12]-[14].
The network scenario under investigation considers center feed cells of hexagonal shape and equal size. The interference
and downlink power consumption analysis assumes an MS, which camps in cell 1 with two tiers of neighboring cells around
it, as Fig. 1 depicts. Intra-cell interference calculations require only the knowledge of the distance r1 between the serving BS1
and the MS. However, for inter-cell interference calculations, both distance r1 and angular position θ1, as Fig. 2 shows,
should be considered. The distance r1 between MS and BS1 varies from zero to 2/)cos(3
1
θ
R, whereas the angular
coordinate θ1 varies from 0o to 360°. Due to the hexagonal symmetry, throughout the remaining analysis only angular
positions θ1=0o ~ 30o, will be examined.
Power control function should under ideal conditions regulate downlink power consumption in order to lock energy per
bit to interference value to the target value [Eb/Io]t required by each service. Thus, by calculating interference level and
assuming a perfect power control scheme, downlink power consumption for HHO, 2-way and 3-way SHO connection modes
can be estimated as follows:
A) Hard Handoff Scenario
When a HHO connection mode is assumed, all downlink transmissions to other MSs within the cell as well as in neighbor
cells are considered interference. In principle, the proposed model can tackle network scenarios with unequal traffic loads per
cell and thus different transmit power level TiiT
PP
δ
=
, per base station. However, in order to simplify model’s analysis, we
assume equal total downlink transmission levels PT in each cell (i.e.
)1=
i
δ
. In this respect, power consumption for a single
connection in cell 1 can be calculated as follows [13]-[14]:
( ) ( )
1
1
/10
1 s1 19 /10
/10
11Τ
2
10 P
1-u 10 10 P
i
b
otTs i
i
Er
W
I vR r PP r
ζ
α
ζ
ζ
αα
−
−−
=

= ⇒

 −+
∑

4
TT
ii itTts PPXCPXCP 1
19
1,0
1)(
βξ
=








==⇒ ∑≠=
(2)
where Ct=vR[Eb/Io]t/W,
uX −= 1
0
denotes intra-cell interference,
10)(
,1
1
10
ξξ
−
=i
b
ii
CX
denotes inter-cell
interference and
( )
α
ii rrC 1,1 =
. Vector
),...,(
191
ξξξ
=
denotes the uncorrelated shadowing random variables of BSis,
v is the activity factor which applies to the service under examination, R is the service data rate, W is the chip rate, u denotes
the orthogonality between the various transmissions and β1 is the fraction of PT allocated for a single link. For the sake of
simplicity, in equation (2) and throughout equations (3) and (4), we assume that
( )
TsT PPP ≅− 1
as far as it concerns
intracell interference calculations.
B) 2-way Soft Handoff Scenario
If we assume a maximal ratio combination capability (MRC) and a balanced power allocation scheme
(
kssksPPP
1,1
==
) among BSs in cell 1 and cell k, which participate in the 2-way SHO connection, then power
consumption for a single connection in cell 1 is calculated as follows [9], [13]-[14]:
( )







+
+
=






+






=






∑
=
−−
−
19
2
10/
10/
1
10/
1
1
1010
10
1
1
iΤ
i
T
s1
k
o
b
o
b
t
o
b
PrPru-1
Pr
vR
W
I
E
I
E
I
E
i
ζ
α
ζ
α
ζ
α
( )
=







+=⇒







+
+
∑≠=
−−
−
Ttks
kii Τ
i
T
k
sk
kP
X
CP
PrPru-1
Pr
ik
k
)(
1
)(
1
1010
10 1,
19
,1
10/10/
10/
ξΥξ
ζ
α
ζ
α
ζ
α
TkT
kii i
ii it
PPYXC
1
1
19
,0
1
19
1,0
β
=


















+








=
−
≠=
−
≠= ∑∑
(3)
where similar to eq. (2),
uY −= 1
0
,
10)(
,
10
ki
b
ik
i
CY
ξξ
−
=
and
( )
α
ikik rrC =
,; β1k is the fraction of PT allocated
by each BS, which participates to the 2-way SHO connection.
C) 3-way Soft Handoff Scenario
If we assume MRC reception conditions and balanced power allocation scheme (
klsslsksPPPP 1,1 ===
) between
BSs in cell 1, cell k and cell l, which participate in the 3-way SHO connection, then power consumption for a single
connection in cell 1 is calculated as in the previous cases [9], [13]-[14]:

5
( )







+
+
=






+






+






=






∑
=
−−
−
19
2
10/
10/
1
10/
1
1
1010
10
1
1
iΤ
i
T
s1
l
o
b
k
o
b
o
b
t
o
b
PrPru-1
Pr
vR
W
I
E
I
E
I
E
I
E
i
ζ
α
ζ
α
ζ
α
( ) ( )
⇒







+
+
+
+
∑∑
≠=
−−
−
≠=
−−
−
19
,1
10/10/
10/
19
,1
10/10/
10/
1010
10
1010
10
lii Τ
i
T
l
sl
l
kii Τ
i
T
k
sk
k
PrPru-1
Pr
PrPru-1
Pr
il
l
ik
k
ζ
α
ζ
α
ζ
α
ζ
α
ζ
α
ζ
α
TklT
lii i
kii i
ii itTtkls
PPZYXCP
X
CP
1
1
19
,1
1
19
,0
1
19
1,0
1,
)(
1
)(
1
)(
1
β
ξΖξΥξ
=


















+








+








=







++=⇒
−
≠=
−
≠=
−
≠=
∑∑∑
(4)
where similar to eq. (2)-(3),
uZ −= 1
0
,
10)(
,
10 li
b
ili
CZ
ξξ
−
=
and
( )
α
ilil rrC =
,
. β1kl is the fraction of PT
allocated by each BS, which participates to the 3-way SHO connection.
At this point it should be mentioned that in principle, the proposed model can tackle both balanced and unbalanced power
allocation schemes by defining different weights on
1s
P
,sk
P and
sl
P
. Nevertheless, for simplicity reasons, in our analysis
we assume equal weights on
1s
P
sk
P and
sl
P
, yet without loss of generality.
III. CELL SELECTION AND HANDOFF SCHEMES
Cell selection and handover schemes influence the network performance [20], [21] and thus current section will examine
the conditions that are imposed in our calculations by the aforementioned schemes. If cell 1 is the camping cell and assuming
a best BS selection condition, then the transmission of cell-1 will be the best among the candidate cells i (=2, 3,…, 19), i.e.
ξi≤ξ1-R1,i (R1,i =10log(C1,i)/b). The former condition describes an ideal cell selection scenario and a perfect power control
scheme. The addition of a hysteresis threshold cst=10log(CST)/b can account for possible cell selection and power control
imperfections e.g. ξi≤ξ1-R1,i+cst.
Apart from the above described conditions, the handoff algorithm is placing additional ones. The handoff scheme
considered here is one that accepts a maximum number of simultaneous physical connections equal to the AS size. In
addition, the algorithm places a SHO threshold in order to accept a BS to join the AS. If SHO is not an option, i.e. AS=1, then
the handoff condition is identical to the cell selection one. However, if AS>1, then the HHO scenario implies that the signal
strength of all monitored BSs should not exceed the SHO threshold. The latter statement is expressed as ξi≤ξ1-R1,i-sht,
(sht=10log(SHT)/b). Concluding with the HHO mode, the following conditions apply:
1,
,11
>−−≤ ASshtR
ii
ξξ
(5)
1,
,11
=+−≤ AScstR
ii
ξξ
(6)

6
If 2-way SHO conditions apply, then two simultaneous connections with BS1 and BSk occur. If AS>1, then BSk’s signal is
the strongest signal among the monitored ones and exceeds SHO threshold. After some straightforward calculations, the
former statements can be described as follows:
1,
,11,11 >∈+−≤≤−− ASkcstRshtRkkk
ξξξ
(7)
)2(,
,
=∉−≤ ASiR
ikki
ξξ
(8)
)3(,
,11
=∉−−≤ ASishtR
ii
ξξ
(9)
Finally, when AS=3 a 3-way SHO scenario applies and a single logical network link include physical links with three BSs,
e.g. BS1, BSk and BSl. BSk’s and BSl’s signal are the strongest signals among the monitored ones and both exceed the SHO
threshold. Assuming that BSl’s signal is the weakest among the AS participants, then all other monitored signals should be
weaker than BSl’s signal. After some straightforward calculations the former statements can be expressed as follows:
ASlkcstRshtRlklklk ∈+−≤≤−− )(,
)(,11)()(,11
ξξξ
(10)
ASlandkR
lkkl
∈−≤ ,
,
ξξ
(11)
ASiR illi ∉−≤ ,
,
ξξ
(12)
Concluding, it is worthwhile to mention that no restrictions are placed in non monitored cells, which de facto do not
participate to handoff process. In order to simplify the analysis throughout the remaining analysis all cells in both tiers will be
considered as monitored.
IV. DOWNLINK POWER CONSUMPTION STATISTICS
Three handoff schemes are considered in this section, i.e. AS=1, 2 and 3. In all following calculations, the random
shadowing loss values ξi are restricted by the cell selection and handoff conditions discussed in the previous section. If AS=m
and
m
Ω
is the subset of random ξi values, which allow MS to camp in cell 1, then
m
Ω
can be expressed as
∑∑
∪∪=
lk
m
kl
k
m
k
mm
,11
1
ΩΩΩΩ
. Subsets
m
1
Ω
,
m
k1
Ω
and
m
kl1
Ω
include all ξi values, which conform to HHO, 2-way
SHO and 3-way SHO conditions, respectively. The conditions for each subset are established with eqs. (5)-(6), (7)-(9) and
(10)-(12) of section III. Apparently 1
1k
Ω
=∅ and
1
1kl
Ω
=
2
1kl
Ω
=∅.
The above discussed subsets correspond to all possible connection modes that may occur in the cell under investigation
i.e. cell 1. If the downlink transmitted power for a single user in cell 1 is Ps=βPT, then the actual point of interest in our
calculations is the fraction β of the total transmitted power. The first and the second moment of β can be obtained as
)(
]|[)(]|[)(]|[)(
]|[ ,1
1
11
1
1
1
1
1
m
lk
m
kl
kl
m
kl
k
m
k
k
m
k
mm
m
P
EPEPEP
E
Ω
ΩξβΩΩξβΩΩξβΩ
Ωξβ
∑∑
∈+∈+∈
=∈
(13)
11
22 2
11 1 1 1 1 1
,
2
( )[ | ] ( )[ | ] ( )[ | ]
[| ] ()
k kl
m mm m m m
k k kl kl
k kl
m
m
PE PE P E
EP
βξ βξ β ξ
βξ
Ω ∈Ω + Ω ∈Ω + Ω ∈Ω
∈Ω = Ω
∑∑
(14)

7
where
( ) ( ) ( ) ( )
∑∑
++=
lk
m
kl
k
m
k
mm
PPPP
,11
1
ΩΩΩΩ
.
Since the shadowing random variables ξi are independent their joint pdf is
( )
∏∏ ==
==
19
1
2
19
1
2
)()(
22
nn i
i
i
e
ff
σπ
ξξ
σξ
ξξ
(15)
and thus
)(
1
m
P
Ω
can be calculated as follows:
1
19
21
11
19
2
)(
1
1
)0
),(()()()()(
1
1
1
ξξξξξξΩ
ξ
ξ
ξξ
daAfdffP nn
n
a
n
m
n
n






=










=∏
∫
∏∫∫ =
∞
+
∞−
=∞−
∞+
∞−
(16)
where we define function A(x,y) as
[ ]










−+= 210
)10ln(
2
5.05.0200)10ln(exp),( 2222 b
y
x
erfbyyxA
σ
σ
σ
(17)
and )( 1
ξ
n
a
is the upper limit of inequality (5) or (6), when m >1 or m=1, respectively. In a similar manner
)(
1
m
k
P
Ω
is
obtained by the following expressions:
1
19
,2
)(
)(
1
1)0),(()()()(
1
1
1
ξξξξξΩ
κ
ξ
ξξξ
ddaAffP knn kn
a
bk
m
k
k
k
k


















=∏
∫∫ ≠=
∞+
∞−
(18)
where ak(ξ1) and bk(ξ1) is the upper and the lower limit of inequality (7), respectively. If m=2 then an(ξk) is the upper limit
of inequality (8), otherwise an(ξk)(=an(ξ1)) is the upper limit of equation (9). In addition, if m=2 the integration over ξk can
be only evaluated numerically, whereas for m=3 the integration over ξk is evaluated analytically as [A(bk(ξ1),0)-A(ak(ξ1),0)].
With a similar manipulation
)(
1
m
kl
P
Ω
is obtained by the following expression:
1
)(
)(
19
,,2
)(
)(
1
1)0),(()()()()(
1
1
1
ξξξξξξξΩ
ξ
ξξ
ξ
ξξξ
dddaA
fffP kl
a
blknn lnnl
a
bk
m
kl
kl
kl
l
k
k
k


























=
∫∏
∫∫
≠=
∞+
∞−
(19)
where ak(ξ1) and bk(ξ1) is the upper and the lower limit of eq. (10), al(ξk) and bl(ξ1) is the upper and the lower limit of eqs.
(11) and (10), respectively, whereas an(ξl) is the upper limit of eq. (12).
A) HHO Calculations
According to equation (2) the first and the second moments of β1 can be obtained as follows:
[ ]
[ ]
[ ]










+== ∑ ∑∑∑
≠
=≠≠
=
≠
=
≠
=
19
1
0
19
,1
0
19
1
0
222
1
19
1
0
1
][,][
j
jjii
iij
i
ii
t
i
iit
XXEXECEXECE
ββ
(20)
where for i≠0

8
[ ]
1111
10
1
,1
)()1),(()(10
)( 1
1
ξξΠξξ
Ω
ξ
ξ
daAf
P
C
XE
ii
b
m
i
i
∫
+∞
∞−
−
=
(21)
[ ]
1111
5
1
2
2
)()2),(()(10
)(
1
1
,
1
ξξΠξξ
Ω
ξ
ξ
daAf
P
C
XE
ii
b
m
i
i∫
∞+
∞−
−
=
(22)
[ ]
11111
5
1
,1,1
)()1),(()1),(()(10
)(
1
1
ξξΠξξξ
Ω
ξ
ξ
daAaAf
P
CC
XXE
ji,ji
b
m
ji
ji
∫
+∞
∞−
−
=
(23)
and
∏
≠=
=19
,2 11 )0),(()(
inn ni
aA
ξξΠ
,
∏
≠=
=
19
,,2 11
)0),(()(
jinn nji,
aA
ξξΠ
. In addition, E[
0
X
]=(1-u)/
)(
1
m
P
Ω
, E[
2
0
X
]=(1-
u)2/
)(
1
m
P
Ω
, E[X0Xj]=(1-u) E[Xj] with E[Xj] given by (21) if we replace j with i. The integration limits of the above
expressions are the same with the ones appearing in eq. (16).
B) 2-way SHO Calculations
According to eq. (3) the first and the second moment of β1,k, , can not be evaluated by employing the straightforward
semi-analytical approach of subsection IV.A. In order to overcome this constraint, β1k is approximated by a Taylor expansion
in the neighborhood of E[X(ξ)] and E[Y(ξ)]. Next, by omitting Taylor series terms higher than the second order we conclude
to (see Appendix I):
( )
( )
( ) ( )
[ ]
[ ]










+
−−





−+





−
−
+
=∈ 3
22
2
2
2
2
1
1
2
]|[ YX
YXYXXYXYYYXX
YX
YX
CE t
m
k
k
Ωξβ
(24)
( ) ( ) ( )
[ ]


















+
−
∈
+
+
+
++
=∈ YX
YX
C
E
YX
YX
YX
XYYXYXXY
CE
t
m
k
k
t
m
k
k
]|[
2
2
]|[
1
1
4
2
2
4
2
4
2
1
2
1
Ωξβ
Ωξβ
(25)
where
X
,
Y
,
2
X
, 2
Y
and
XY
correspond to ]|[
1
m
k
XE
Ωξ
∈,
]|[
1
m
k
YE
Ωξ
∈
, ]|[
1
2m
k
XE
Ωξ
∈,
]|[
1
2m
k
YE
Ωξ
∈ and
]|[ 1
m
k
XYE
Ωξ
∈
, respectively. Following a calculation scheme as in section IV.A, the above
mentioned E[.] terms can be expressed as a summation of all possible combinations of E[
i
X
], E[2
i
X], E[
ji XX
], E[i
Y
],
E[
2
i
Y], E[
jiYY
] and E[
ji
YX
] . Also, each E[.] term can be expressed in an integral closed form expression, where the
various integration limits are identical to the ones appearing in eq. (18). The E[Xi] expression is obtained as follows:
[ ]
1
11
11
1
()
i,k
()
1, 10 11 ()
110 kk
()
()((),1) () ,
10 ( )
() 10 ( ) ( ) ,
k
k
k
kk
k
k
a
k ik k k
bb
i
imab
k
kk
b
f Aa d i k
C
EX f d
Pf d ik
ξ
ξ
ξξ
ξξξ
ξ
ξ
ξ ξ ξξ
ξξ
ξ ξξ
+∞ −
−∞
Π≠


= ×
ΩΠ=


∫
∫∫
(26)

9
If m=3 and i≠k (i=k) the kth integral in eq. (26) can be evaluated as [A(bk(ξ1),0)-A(ak(ξ1),0)] ([A(bk(ξ1),1)-A(ak(ξ1),1)]). The
E[Yi] calculations are similar to eq. (26) with one difference, namely, the term
10
1
10
ξ
b−
is transferred to the kth integral as
10
10 k
b
ξ
−
[ ]
1
1
1
1
1
1
1
()
10
1 1 i,k
()
,
()
110 10
1 1 kk
()
() 10 ()((),1) () , 1
() 10 ( ) 10 ( ) ( ) , 1
kk
k
k
kk
k
k
ab
k ik k k
b
ki
imab
b
k
kk
b
f d f Aa d i
C
EY Pf d f di
ξξ
ξξ
ξ
ξξ
ξ
ξξ
ξ
ξξ ξ ξ ξ ξ
ξξ ξ ξ ξ
+∞ −
−∞
+∞ −
−∞
Π≠


= ×
ΩΠ=


∫∫
∫∫
(27)
If m=3 the kth integral in eq. (27) can be evaluated as [A(bk(ξ1),-1)-A(ak(ξ1),-1)]. Similar to the HHO case E[X0]= E[Y0]=(1-
u)/
)(
1
m
k
P
Ω
.
The E[
2
i
X
] and E[
2
i
Y
] expressions can be obtained from eqs. (26) and (27), respectively, if we substitute
10
1
10
ξ
b±
and 10
10 k
b
ξ
± with
5
1
10
ξ
b±
and
5
10
k
b
ξ
±
, respectively, A(ai(ξk),1) with A(ai(ξk),2) and [A(bk(ξ1),±1)-A(ak(ξ1), ±1)]
with [A(bk(ξ1), ±2)-A(ak(ξ1), ±2)]. Similar to the HHO case E[
2
0
X
]= E[
2
0
Y
]=(1-u)2/
)(
1
m
k
P
Ω
.
The terms E[XiXj] are described by the following integral expression:
1
11
11
1
()
i,j,k
()
1, 1, 511 ()
110 ( ) i,j,k
()
()((),1)((),1) () ,,
10 ( )
() 10 ()( (),1) () ,()
k
k
k
kk
k
k
a
k ik jk k k
bb
ij
ij mab
k
k ij k k k
b
f Aa Aa d i j k
CC
E XX f d
Pf Aa d i j k
ξ
ξ
ξξ
ξξξ
ξ
ξ
ξ ξ ξ ξξ
ξξ
ξ ξ ξξ
+∞ −
−∞
Π≠



= ×

ΩΠ=


∫
∫∫
(28)
where
∏
≠=
=19
,,,2 )0),(()( kjinn knkkj,i, aA
ξξΠ
. If j=3 and i, j≠k (i or j=k) the kth integral in eq. (28) can be evaluated as
[A(bk(ξ1),0)-A(ak(ξ1),0)] ([A(bk(ξ1),1)-A(ak(ξ1),1)]).
The expression for the E[YiYj] term is given by the following equation:
1
1
1
1
1
1
1
()
5
1 1 i,j,k
()
,,
()
110 5
1 1 ( ) i,j,k
()
() 10 ()((),1)((),1) () ,, 1
() 10 () 10 ()( (),1) () ,()1
kk
k
k
kk
k
k
ab
k ik jk k k
b
ki k j
ij mab
b
k
k ij k k k
b
f d f Aa Aa d i j
CC
E YY Pf d f Aa d i j
ξξ
ξξ
ξ
ξξ
ξ
ξξ
ξ
ξξ ξ ξ ξ ξ ξ
ξξ ξ ξ ξ ξ
+∞ −
−∞
+∞ −
−∞
Π≠



= ×
 ΩΠ=


∫∫
∫∫
(29)
If m=3 the kth integral in eq. (29) can be evaluated as [A(bk(ξ1),-2)-A(ak(ξ1),-2)].
Concluding the 2-way SHO subsection the E[XiYj] term is expressed below:

10
1
1
1
1
1
1
1
1
()
10 10
1 1 i,j,k
()
()
10 1 1 i,j,k
()
1, ,
1
,
10 () 10 ()((),1)((),1) () , 1
10 () ()((),1) () , ,
()
kk
k
k
k
k
k
ab
b
k ik jk k k
b
a
b
k jk k k
b
i kj
ij m
k
ik
f d f Aa Aa d j
f d f Aa d i k j
CC
E XY P
ξξ
ξ
ξξ
ξ
ξ
ξ
ξξ
ξ
ξξ ξ ξ ξ ξ ξ
ξξ ξ ξ ξ ξ
+∞ −−
−∞
+∞ −
−∞
≠

Π
≠

Π=≠

= ×
 Ω
∫∫
∫∫
{ }
{ }
{ }
1
1
1
1
1
1
()
10
1 1 i,j,k
()
()
1 1 i,j,k
()
1
() 10 ()((),1) () , , 1
() () () , , 1
kk
k
k
k
k
k
ab
k ik k k
b
a
k kk
b
f d f Aa d i k j
f d f d i kj
ξξ
ξξ
ξ
ξ
ξξ
ξ
ξξ ξ ξ ξ ξ
ξξ ξ ξ ξ
+∞ −
−∞
+∞
−∞








Π ≠=



Π==


∫∫
∫∫
(30)
If i=j≠1 and k, A(ai(ξk),1)A(aj(ξk),1) product should be replaced by A(ai(ξk),2). Finally, if m=3 and i≠k (i=k) the kth integral
in eq. (30) is evaluated as [A(bk(ξ1),-1)-A(ak(ξ1),-1)] ([A(bk(ξ1),0)-A(ak(ξ1),0)]).
C) 3-way SHO Calculations
As it was discussed in the 2-way SHO case the first and the second moment of β1kl, can be approximated through a Taylor
expansion of eq. (4). If we omit Taylor series terms higher than the second order the following expressions can be derived for
the first and the second moment of β1kl (see Appendix II):
( )
[ ]
[ ]
−





++
+





−
−
++
=∈
3
2
2
2
1
1
))((2
]|[ ZYZXYX
ZYZYXX
ZYZXYX
ZYX
CE
t
m
kl
kl
Ωξβ
( )
[ ]
[ ]
( )
[ ]
[ ] [ ]
[ ]
+
++
−
+
++
+





−
−
++
+





−
−
3
3
3
2
2
2
3
2
2
2
)(2
))((2))((2
ZYZXYX
ZYXYXXY
ZYZXYX
YXYXZZ
ZYZXYX
ZXZXYY
[ ]
[ ]
[ ]
[ ]





++
−
+
++
−
+
3
3
3
3
)(2)(2
ZYZXYX
XZYZYYZ
ZYZXYX
YZXZXXZ
(31)
( )
[ ]
( )
[ ]
+





++





−
+
++





−
+






++
=∈ 4
4
2
2
4
4
2
2
2
2
1
2)()(
]|[
1
ZYZXYX
ZXYY
ZYZXYX
ZYXX
ZYZXYX
ZYX
CE t
m
kl
kl
Ωξβ
( )
[ ] [ ]
[ ] [ ]
[ ]
+
++
−
+
++
−
+
++





−
+
4
42
4
42
4
4
2
2
)()(2)()(2
)(
ZYZXYX
YZXZXXZ
ZYZXYX
ZYXYXXY
ZYZXYX
YXZZ
[ ]
[ ]













++
−
∈
++
+
++
−
+ZYZXYX
ZYX
C
E
ZYZXYX
ZYX
ZYZXYX
XZYZYYZ
t
m
kl
kl
]|[
2
)()(2
1
1
4
42
Ωξβ
(32)

11
where
X
,
Y
,
Z
,
2
X
,
2
Y
, 2
Z
,
XY
,
XZ
and
YZ
correspond to
]|[
1
m
kl
XE
Ωξ
∈
, ]|[
1
m
kl
YE
Ωξ
∈,
]|[ 1
m
kl
ZE
Ωξ
∈
,
]|[
1
2m
kl
XE
Ωξ
∈
, ]|[
1
2m
kl
YE
Ωξ
∈, ]|[
1
2m
kl
ZE
Ωξ
∈, ]|[
1
m
kl
XYE
Ωξ
∈,
]|[ 1
m
kl
XZE
Ωξ
∈
and
]|[ 1
m
kl
YZE
Ωξ
∈
, respectively.
Following a similar calculation scheme as in previous sections, the above mentioned E[.] terms can be expressed as a
summation of all possible combinations of E[
i
X
], E[
2
i
X
], E[
ji XX
], E[i
Y], E[
2
i
Y
], E[
jiYY
], E[i
Z], E[2
i
Z],
E[
jiZZ
], E[
ji
YX
], E[
ji
ZX
] and E[
ji
ZY
] terms.
In details,
[ ]
i
XE and
[ ]
i
YE terms are given by the following equations:
[ ]
1
1
1
1
1
1
1
1
() ()
i,k,l l
() ()
() ()
1, 10 10
1 1 k,l l
1() ()
()
()
() ()((),1) () , ,
10 ( ) 10 ( ) ( ) ( ) ,
()
()
k lk
kl
k lk
k lk
k
kl
k lk
k
k
k
aa
k k l il l
bb
aa
b
b
i
i kk l l
m
kl bb
a
kk
b
f d f Aa d i kl
C
EX f d f d f d i k
P
fd
ξξ
ξξ
ξξ
ξξ
ξ
ξ
ξ ξξ
ξξ
ξ
ξ
ξ
ξ ξ ξ ξ ξξ
ξξ ξ ξ ξ ξξ
ξξ
+∞ −
−∞
Π≠
= × Π=
Ω
∫∫
∫ ∫∫
()
10 k,l l
()
10 ( ) ( ) ,
lk l
l
lk
ab
ll
b
f d il
ξξ
ξ
ξ
ξ ξξ








Π=

∫∫
(33)
[ ]
1
1
1
1
1
1
1
() ()
10
1 1 i,k,l l
() ()
() ()
,10 10
1 1 k,l l
1() ()
() 10 ( ) ()((),1) () , 1,
10 ( ) 10 ( ) ( ) ( ) ,
()
k lk
k
kl
k lk
k lk
k
kl
k lk
aa
b
k k l il l
bb
aa
b
b
ki
i kk l l
m
kl bb
f d f d f Aa d i l
C
EY f d f d f d i
P
ξξ
ξ
ξ ξξ
ξξ
ξξ
ξ
ξ
ξ ξξ
ξξ
ξξ ξ ξ ξ ξ ξξ
ξξ ξ ξ ξ ξξ
+∞ −
−∞
+∞ −
−∞
Π≠
=× Π=
Ω
∫∫ ∫
∫∫ ∫
1
1
1
() ()
10 10
1 1 k,l l
() ()
1
( ) 10 ( ) 10 ( ) ( ) ,
k lk
kl
kl
k lk
aa
bb
kk l l
bb
f d f d f d il
ξξ
ξξ
ξ ξξ
ξξ
ξξ ξ ξ ξ ξξ
+∞ −
−∞








Π=


∫∫ ∫
(34)
where the various integration limits in eqs. (33), (34) and throughout this subsection are the same as the ones described in eq.
(19). Apparently, the E[Zi] expressions are similar to the ones in eq. (34):
[ ]
1
1
1
1
1
1
1
() ()
10
1 1 i,k,l l
() ()
() ()
,10 10
1 1 k,l l
1() ()
() ( ) 10 ()((),1) () , 1,
10 ( ) ( ) 10 ( ) ( ) ,
()
k lk l
kl
k lk
k lk l
kl
k lk
aa
b
k k l il l
bb
aa
b
b
li
i kk l l
m
kl bb
f d f d f Aa d i k
C
EZ f d f d f d i
P
ξξ
ξ
ξξ ξ
ξξ
ξξ
ξ
ξ
ξξ ξ
ξξ
ξξ ξ ξ ξ ξ ξξ
ξξ ξ ξ ξ ξξ
+∞ −
−∞
+∞ −
−∞
Π≠
=× Π=
Ω
∫∫ ∫
∫ ∫∫
1
1
1
() ()
10 10
1 1 k,l l
() ()
1
( ) 10 ( ) 10 ( ) ( ) ,
k lk
kl
kl
k lk
aa
bb
kk l l
bb
f d f d f d ik
ξξ
ξξ
ξξ ξ
ξξ
ξξ ξ ξ ξ ξξ
+∞ −
−∞








Π=

∫∫ ∫
(35)

12
E[
2
i
X
], E[
2
i
Y
]and E[2
i
Z] expressions can be obtained from eqs. (33)-(35) if we replace i
C,1 with 2
,1 i
C, ik
C,, with
2,ik
C
,
il
C
,
, with
2
,il
C
, )1,( i
aA with
)2,( i
aA
and
10
1
10
ξ
b±
,10
10
k
b
ξ
±,
10
10 l
b
ξ
±
with
5
1
10
ξ
b±
,
5
10
k
b
ξ
±
,
5
10
l
b
ξ
±. As in previous cases, E[
0
X
]=E[
0
Y
]=E[
0
Z
]=(1-u)/
)(
1
m
kl
P
Ω
, E[
2
0
X
]=E[
2
0
Y
]=E[2
0
Z] (1-u)2/
)(
1
m
kl
P
Ω
.
The E[XiXj], E[YiYj] and E[ZiZj] terms can be obtained from eq. (33), (34) and (35) if
10
1
10
ξ
b−
, 10
10 k
b
ξ
− and
10
10 l
b
ξ
−
is replaced by
5
1
10
ξ
b−
,
5
10
k
b
ξ
−
and 5
10 l
b
ξ
−, respectively. In addition, if I and j≠k and l in eq. (33), I
and j≠1 and l in eq. (34) and I and j≠1 and k in eq. (35), A(ai(ξl),1)Πi,k,l(ξl) should be replaced by
∏
≠=
=
19
,,,,2
)0),(()1),(()1),(()(
lkjinn lnljlil
aAaAaAP
ξξξξ
. Furthermore, if i=k or l in eq. (33), i=1 or l in eq. (34) and i=1 or
k in eq. (35), then Πk,l(ξl ) should be replaced by A(aj(ξl),1)Πj,k,l(ξl ). Finally, if j takes the latter I values, the same
expressions still apply if we interchange I with j.
The cross product terms E[XiYj], E[XiZj] and E[YiZj] are expressed below:
[ ]
ll
a
blk
a
bk
b
b
m
kl
jki
ji
dPfdfdf
P
CC
YXE
kl
kl
l
k
k
k
k
ξξξξξξξ
Ω
ξ
ξξ
ξ
ξξ
ξ
ξ
ξ
)()()(10)(10
)(
)(
)(
)(
)(
10
11
10
1
,,1
1
1
1
1
∫∫∫ −
∞+
∞−
−
=
(36)
[ ]
ll
a
bl
b
k
a
bk
b
m
kl
jli
ji
dPfdfdf
P
CC
ZXE kl
kl
l
l
k
k
k
ξξξξξξξ
Ω
ξ
ξξ
ξ
ξ
ξξξ
ξ
)()(10)()(10
)(
)(
)(
10
)(
)(
11
10
1
,,1
1
1
1
1∫∫∫
−
∞+
∞−
−
=
(37)
[ ]
ll
a
bl
b
k
a
bk
b
m
kl
jlik
ji
dPfdfdf
P
CC
ZYE kl
kl
l
l
k
k
k
k
ξξξξξξξ
Ω
ξ
ξξ
ξ
ξ
ξξ
ξ
ξ
)()(10)(10)(
)(
)(
)(
10
)(
)(
10
11
1
,,
1
1
1∫∫∫
−−
∞+
∞−
=
(38)
where we assume i≠k, l, j≠1, l and i≠j in eq. (36), i≠k, l, j≠1, k and i≠j in eq. (37) and i≠1, l, j≠1, k and i≠j in eq. (38). If i=k or
l the
10
10
i
b
ξ
term is transferred to the kth integral (thus 10
10
k
b
ξ
− vanishes in eq. (36)) or to the lth integral (thus
10
10 l
b
ξ
−
vanishes in eq. (37)-(38)). In addition, if i=1 in eq. (38)
10
10
i
b
ξ
term is transferred to the 1st integral. In all
aforementioned cases P(ξl) converts to A(aj(ξl),1)Πj,k,l(ξl ). If j=1 or l the
10
10
j
b
ξ
term is transferred to the 1st integral
(thus
10
1
10
ξ
b−
vanishes in eqs. (36)-(37)) or to the lth integral. In addition, if j=k in eqs. (37)-(38) the
10
10 j
b
ξ
term is
transferred to the kth integral (thus 10
10
k
b
ξ
− vanishes in eq. (38)). In all aforementioned cases P(ξl) converts to
A(ai(ξl),1)Πi,k,l(ξl ). Finally, if i=j≠l in eq. (36), i=j
≠
k in eq. (37) and i=j≠1 in eq. (38) then P(ξl) converts to
A(aj(ξl),2)Πj,k,l(ξl ). Otherwise, if i=j=l in eq. (36), i=j=k in eq. (37) and i=j=1 in eq. (38) then the 5
10 l
b
ξ
,
5
10
k
b
ξ
and
5
1
10
ξ
b
term appears in the lth, kth and 1st integral, respectively, whereas P(ξl) converts to Πk,l (ξl ).

13
V. NUMERICAL RESULTS & DISCUSSION
First, a comparison between the calculations of the proposed theoretical model and the corresponding ones from an
independent numerical simulation will be discussed. The calculations have been performed with respect to the expected value
E[β| ξ ∈ Ωm] (=
β
) and the standard deviation
( )
2
2]|[]|[ mm EE
ΩξβΩξβ
∈−∈ (=
β
σ
) of power
consumption. The under examination scenarios include different MS positions (r,θ), various path loss factors (α) and standard
deviations of shadowing losses (σ), as well as different AS sizes, cell selection thresholds (cst) and SHO thresholds (sht). The
service parameters correspond to a typical voice service in WCDMA UMTS networks: v=0.5, R=12.2 Kbps, W=3.84
Mchips/s and [Eb/Io]t=4.4 dB. Finally, the orthogonality factor is u=0.9.
The numerical simulation model has been configured to generate 100.000 random shadowing samples according to a log-
normal pdf. For each sample the cell selection and the handoff inequalities of Section III are examined, first to decide
whether the sample refers to the cell under examination or not and next to decide which of the three handoff conditions is
fulfilled. According to the latter criterion a power consumption sample is calculated using one of the equations (2)-(4), and
next
β
and
β
σ
is estimated using equations (13) and (14), respectively. In order to facilitate a tabulated comparison
between the numerical results and the corresponding theoretical ones the results from 5 rounds of simulation runs have been
averaged and presented in Tables I, II and III. Each Table refers to a different scenario and proves that theoretical and
numerical estimations converge, which in turn proves the efficiency of the Taylor series approximation.
Next, in order to demonstrate the potential benefits from the adaptation of the proposed theoretical model the power
consumption statistics will be further investigated. The under examination numerical results are illustrated in Figs. 3-8. Figs.
3, 5 and 7 depict
β
for AS=1, 2 and 3, respectively, versus the normalized distance r1/Rmax . Figs 4, 6 and 8 depict
β
σ
for
the former scenarios.
Fig. 3 corresponds to a HHO scenario. According to the illustrated data
β
tends to increase, as expected, when the MS
approaches the cell border. Near BS and up to a distance,
β
increases, when α and σ take higher values. Nevertheless, this is
not valid, when the MS approaches the cell border. Actually, close to the border a hostile propagation environment (i.e. high
α and σ values) results to less power consumption. This behavior can be explained, if we take into the account the possibility
of handoff. Close to the border the MS tends to camp to another cell instead of sustaining the degradation of a hostile
environment. Actually, this is more evident, when the cell selection criterion is more tight, i.e. cst=1 instead of cst=3, and
camping to another cell is encouraged. The comments from Fig. 4 are rather similar to the ones in Fig. 3. The higher (lower)
β
σ
appears, when α and σ take lower (higher) and the cell selection algorithm decision criteria are relatively loose (tight).
According to the aforementioned comments the cell selection imperfections burdens the system, when the propagation
conditions are relatively good and AS=1. In such cases, the MS should be encouraged to camp to a neighbor cell.
Fig. 5 illustrates the expected value of power consumption, when AS=2 and thus a 2-way SHO is also possible. Fig. 5 also
includes results for AS=1 for comparison reasons. According to the illustrated data the highest values of
β
appear, when α
and σ take low values as it was already mentioned in Fig.3. Ιf we compare AS=1 and AS=2 results, it appears that the choice

14
of AS=2 and more than this the encouragement of SHO is beneficial and this is more evident when the MS approaches the cell
border. Actually, when α and σ take low values and the MS moves towards the cell border/corner SHO takes advantage of the
good propagation conditions and allows one neighbor BS to participate instead of being a strong interferer. Fig. 6 illustrates
β
σ
numerical results for the network scenarios examined in Fig. 5. According to the illustrated results, the option and more
than this the encouragement of SHO reduces significantly
β
σ
at least when compared to AS=1 scenarios. Concluding, the
inclusion of a SHO option by setting AS=2, provides significant benefits, in terms of reducing
β
and
β
σ
, even in cases
where the MS is located relatively close to the BS.
Fig. 7 illustrates the expected value of power consumption, when AS=3 and thus a 3-way SHO is also possible. According
to the illustrated data the AS=3 choice gives slightly better results, when is compared with the relevant results of Fig. 5 and
particular with the case of σ=8 dB. However, the encouragement of SHO (sht=3 dB) provides a significant reduction, when
compared with the AS=1 and AS=2 choice and the case of σ=10 dB. Fig. 8 illustrates
β
σ
for the network scenarios
examined in Fig. 7. According to the illustrated results and the comparison with the relevant results in Fig.6, the choice of
AS=3 and the encouragement of SHO provides a significant reduction of
β
σ
and a location insensitive behavior.
Concluding the discussion on the aforementioned results it is worthwhile to mention that as it has been found in similar
research works the resource allocation on CDMA networks strongly depends on the propagation conditions, the MS location
and the various Radio Resource Management (RRM) settings. Thus, an optimized network performance definitely requires a
cross layer approach and prediction models that can incorporate both physical layer and RRM parameters.
VI. CONCLUSION
A theoretical statistical model that provides an estimation of the expected and standard deviation value of power
consumption in the downlink direction has been developed for cellular CDMA networks. The proposed model supports the
aforementioned calculations taking into account cell selection and handoff settings. In this context, present work contributes
to a cross-layer approach, by establishing a theoretical framework, which facilitates performance evaluation and optimization
of CDMA networks under specific radio propagation conditions as well as RRM settings. Current work can be extended with
future studies in several directions. The most challenging future extension is to provide a joint pdf for power consumption
based on the capability to estimate power consumption moments. Furthermore, present work provides estimations on a link
level and thus an extension of the model in order to support performance evaluation on a network level is also another
interesting research direction. A cross layer design approach aiming to develop an optimized soft handoff algorithm, which
will take into account the proposed model’s estimations, is another one possible future research topic. Finally, the under
consideration numerical results are based on several assumptions, which can be easily rearranged. For example, it would be
interesting to produce numerical results by taking into account unbalanced power allocation schemes among the SHO links or
unequal traffic loads per cell.

15
APPENDIX I
In the case of 2-way SHO connections,
k1
β
power consumption metric can be expressed in the form of the following
function:
ΥΧ
ΧΥ
ΥΧ
ΥΧβ
+
=





+= −1
11
),(
(I.1)
Using a Taylor expansion in the neighborhood of E[X(ξ)]=
X
and E[Y(ξ)]=
Y
, where Taylor series terms higher than the
second order are omitted, and next taking the average value of this expression we conclude after a few straightforward
calculations to:
[ ]
[ ]
[ ]
[ ]
[ ]
+
∂
∂
−+
∂
∂
−+= =
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβ
Υ
ΥΥΕΥΧβ
Χ
ΧΧΕΥΧβΥΧβ
),()(),()(),(),( 2
2
2
2
2
2
E
[ ]
[ ]
ΥΥ
ΧΧ
ΥΧβ
Υ
ΥΥΕ
=
=
∂∂
∂
−−+ ),())((2
2
X
XX
(I.2)
where
[ ]
( )
( )
[ ]
( )
( )
[ ]
( )
3
2
3
2
2
2
3
2
2
2
2
),(,
2
),(,
2
),( YX
YX
Y
YX
X
Y
YX
Y
+
=
∂∂
∂
+
−=
∂
∂
+
−=
∂
∂
=
=
=
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβ
Χ
ΥΧβΥΧβ
Χ
(I.3)
By taking the square power of the above mentioned Taylor series expansion and omitting higher order terms, we conclude,
after some manipulation, to the following expression regarding
[ ]
),(
2
ΥΧβ
E:
[ ] [ ]
[ ]
[ ]
[ ]
+








∂
∂
−+








∂
∂
−+= =
=
=
=
2
2
2
222 ),()(),()(),(),(
ΥΥ
ΧΧ
ΥΥ
ΧΧ ΥΧβ
Υ
ΥΥΕΥΧβ
Χ
ΧΧΕΥΧβΥΧβΕ
[ ]
[ ] [ ] [ ]
{ }
),(),(),(2),(),())((2
ΥΧβΥΧβΕΥΧβΥΧβ
Υ
ΥΧβ
Χ
ΧΧΕ
ΥΥ
ΧΧ
ΥΥ
ΧΧ
−+








∂
∂








∂
∂
−−+
=
=
=
=
YY
(I.4)
where
[ ]
( )
( )
[ ]
( )
( )
2
2
2
2
),(,),( YX
X
Y
YX
Y
+
=
∂
∂
+
=
∂
∂
=
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΥΧβ
Χ
(I.5)
APPENDIX II
In the case of 3-way SHO connections,
kl1
β
power consumption metric can be expressed in the form of the following
function:

16
YZXZY
Z
Z
Z++
=





++= −
Χ
ΧΥ
ΥΧ
ΥΧβ
1
111
),,( (II.1)
Using a Taylor expansion as in Appendix II, we conclude after a few straightforward calculations to:
[ ]
[ ]
[ ]
[ ]
[ ]
+
∂
∂
−+
∂
∂
−+=
=
=
=
=
=
=
ZZZZ YY XX ZZZZE
ΥΥ
ΧΧ
ΥΧβ
Υ
ΥΥΕΥΧβ
Χ
ΧΧΕΥΧβΥΧβ
),,()(),,()(),,(),,( 2
2
2
2
2
2
[ ]
[ ]
[ ]
[ ]
+
∂∂
∂
−−+
∂
∂
−+
=
=
=
=
=
=
ZZZZ
Z
X
XXZ
Z
ZZ
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβ
Υ
ΥΥΕΥΧβΕ
),,())((2),,()( 2
2
2
2
[ ]
[ ]
+
∂∂
∂
−−+
=
=
=
ZZ
Z
ZX
ZZXX
ΥΥ
ΧΧ
ΥΧβΕ
),,())((2 2
[ ]
[ ]
ZZ
Z
ZY
ZZYY
=
=
=
∂∂
∂
−−
ΥΥ
ΧΧ
ΥΧβΕ
),,())((2
2
(II.2)
where
[ ]
( ) ( )
( )
[ ]
( ) ( )
( )
,
2
),,(,
2
),,(
3
2
2
2
3
2
2
2
ZYZXYX
ZXZX
Z
Y
ZYZXYX
ZYZY
Z
ZZZZ
++
+
−=
∂
∂
++
+
−=
∂
∂
=
=
=
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΥΧβ
Χ
[ ]
( ) ( )
( )
3
2
2
2
2
),,( ZYZXYX
YXYX
Z
Z
ZZ
++
+
−=
∂
∂
=
=
=
ΥΥ
ΧΧ
ΥΧβ
(II.3)
[ ]
( )
( )
[ ]
( )
( )
3
3
2
3
3
2
2
),,(,
2
),,( ZYZXYX
ZYX
Z
Z
ZYZXYX
ZYX
Z
Y
ZZZZ
++
=
∂∂
∂
++
=
∂∂
∂
=
=
=
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβ
Χ
ΥΧβ
Χ
[ ]
( )
( )
3
3
2
2
),,( ZYZXYX
ZYX
Z
ZY
ZZ
++
=
∂∂
∂
=
=
=
ΥΥ
ΧΧ
ΥΧβ
(II.4)
Using the square power of the Taylor series expansions and omitting higher order terms, we conclude to the following
expression regarding
[ ]
),,(
2ZE
ΥΧβ
:
[ ] [ ]
[ ]
+










∂
∂
−+=
=
=
=
2
222 ),,()(),,(),,(
ZZ
ZZZ
ΥΥ
ΧΧ
ΥΧβ
Χ
ΧΧΕΥΧβΥΧβΕ
[ ]
[ ]
[ ]
[ ]
+










∂
∂
−+










∂
∂
−+
=
=
=
=
=
=
2
2
2
2),,()(),,()(
ZZZZ
Z
Z
ZZZ
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΕΥΧβ
Υ
ΥΥΕ
[ ]
[ ] [ ]
+










∂
∂










∂
∂
−−+
=
=
=
=
=
=
ZZZZ
ZZYY
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβ
Υ
ΥΧβ
Χ
ΧΧΕ
),,(),,())((2

17
[ ]
[ ] [ ]
+










∂
∂










∂
∂
−−+
=
=
=
=
=
=
ZZZZ
Z
Z
ZZZ
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΥΧβ
Χ
ΧΧΕ
),,(),,())((2
[ ]
[ ] [ ]
+










∂
∂










∂
∂
−−+
=
=
=
=
=
=
ZZZZ
Z
Z
Z
Y
ZZYY
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΥΧβΕ
),,(),,())((2
[ ]
{ }
),,(),,(),,(2 ZZZ
ΥΧβΥΧβΕΥΧβ
−+
(II.5)
where
[ ]
( )
( )
[ ]
( )
( )
,),,(,),,(
2
2
2
2
ZYZXYX
ZX
Z
Y
ZYZXYX
ZY
Z
ZZZZ
++
=
∂
∂
++
=
∂
∂
=
=
=
=
=
=
ΥΥ
ΧΧ
ΥΥ
ΧΧ
ΥΧβΥΧβ
Χ
[ ]
( )
( )
2
2
),,( ZYZXYX
YX
Z
ZZZ ++
=
∂
∂
=
=
=
ΥΥ
ΧΧ
ΥΧβ
(II.6)
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Table I. Theoretical vs Numerical Simulation Estimations for AS=1
AS=1
θ=15o, a=3,σ=8,cst=1 r=0.6Rmax r=0.7Rmax r=0.8Rmax r=0.9maxR r=1.0Rmax
Theoretical Model
β
0.0035164 0.0043695 0.0051464 0.0058282 0.0064095
β
σ
0.0032682 0.0037155 0.0040348 0.0042499 0.0043742
Numerical Model
β
0.0035124 0.0043708 0.0051283 0.0057727 0.0064118
β
σ
0.0032512 0.0036875 0.0040272 0.0041949 0.0043510
Table II. Theoretical vs Numerical Simulation Estimations for AS=2
AS=2
θ=30o, a=3,σ=8,cst=1,
sht=3
r=0.6Rmax r=0.7Rmax r=0.8Rmax r=0.9Rmax r=1.0Rmax
Theoretical Model
β
0.0031774 0.0036789 0.0040796 0.0043946 0.0046417
β
σ
0.0017504 0.0018292 0.0018618 0.0018691 0.0018788
Numerical Model
β
0.0031854 0.0036904 0.0040845 0.0044148 0.0046605
β
σ
0.0017671 0.0018521 0.0018910 0.0019045 0.0019193

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Table III. Theoretical vs Numerical Simulation Estimations for AS=3
AS=3
θ=0o, a=4,σ=10,cst=1,
sht=3
r=0.6Rmax r=0.7Rmax r=0.8Rmax r=0.9Rmax r=1.0Rmax
Theoretical Model
β
0.0016295 0.0019904 0.0022905 0.0025476 0.0027399
β
σ
0.0010969 0.0012158 0.0012493 0.0012537 0.0012392
Numerical Model
β
0.0016487 0.0020319 0.0023516 0.0026368 0.0028613
β
σ
0.0011215 0.0012532 0.0013089 0.0013289 0.0013308
Fig. 1. The considered cellular network
Fig. 2. The cell geometry and the spatial coordinates
r, θ.
Fig 3. Expected value of power consumption versus
normalized distance r1/Rmax (AS=1 and θ=15o)
Fig 4. Standard deviation of power consumption versus
normalized distance r1/Rmax (AS=1 and θ=15o)

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Fig 5. Expected value of power consumption versus
normalized distance r1/Rmax (AS=1, 2).
Fig 6. Standard deviation of power consumption
versus normalized distance r1/Rmax (AS=1, 2).
Fig 7. Expected value of power consumption versus
normalized distance r1/Rmax (AS=3).
Fig 8. Standard deviation of power consumption
versus normalized distance r1/Rmax (AS=3).