calculation of erlang capacity for cellular cdma uplink systems
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7/25/2019 Calculation of Erlang Capacity for Cellular CDMA Uplink Systems
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Calculation of Erlang Capacity for Cellular CDMA Uplink Systems
Ling Ding and James
S.
Lehnert
School of Electrical and Computer Engineering
Purdue University, West Lafayette, IN 47907-1285,
U.S.A.
E-mail: [email protected], [email protected]
Abstract - Three methods of determining the
Erlang capacity of a cellular CDMA uplink sys-
tem are presented. The first method decouples
the analysis of blocking and outage performance,
and avoids iterative search. Based
on
two ap-
proximations regarding the mobile traffic and the
interference, the second method jointly examines
blocking and outage performance. The third,
more detailed, method considers the mobile traf-
fic characteristics and the interference with mo-
bile traffic fluctuations. The Erlang capacity of
a cellular CDMA uplink system using a power
control scheme called truncated channel inver-
sion is calculated with each
of
the methods. The
methods are also used to compare the Erlang ca-
pacities for different power control schemes.
I. INTRODUCTION
The economic value of cellular wireless systems can
be effectively measured by Erlang capacity, defined as
the maximum load that can be supported with a given
blocking probability. In channelized multiple-access sys-
tems, such
as
TDMA/FDMA, each cell is typically as-
signed
a
fixed number of channels. Therefore, Erlang
capacity of these systems can be easily obtained by
the well-known Erlang-B formula. However, in cellu-
lar CDMA systems, where users all share a common
spectral frequency allocation over time, the notion of
channels per cell is soft, in the sense that
a
new user can
be admitted
as
ong as the signal-to-interference ratio is
adequate for receiver processing
[I].
Consequently, the
blocking performance is coupled with the outage perfor-
mance, and it is difficult to calculate Erlang capacity in
cellular CDMA systems.
Here, we present three methods of determining the
Erlang capacity. T he first method decouples the analy-
sis of blocking and outage performance t o avoid iterative
search. Based
on
two approximations on the mobile traf-
fic and the interference, the second method attempts to
jointly analyze blocking and outage performance. The
third method exactly captures the mobile traffic charac-
teristic and the interference under mobile traffic fluctu-
ation, thereby providing the most accurate results.
The research described in the paper is supported by the
U.S.
Government DARPA Glomo Project
A 0
No.
F383,
AFRL con-
tract number
F30602-97-C-0314.
0-7803-6596-8/00/ 10.00 2000 IEEE
To compare these methods of calculating Erlang ca-
pacity, we consider
a
power-controlled cellular CDMA
system. The power control scheme is based on trun-
cated channel inversion, which was recently proposed in
[2] for data traffic and shown to outperform substan-
tially the traditional channel inversion scheme in terms
of system throughput and power consumption. The Er-
lang capacities of the truncated and traditional power
control schemes are also compared using the presented
methods.
11. SYSTEM DESCRIPTIONF THE
POWER
CONTROLLEDDMA UPLINK
Consider a cellular CDMA system where basestations
are located at centers of the hexagons and serve mobiles
in the system. In this paper, we concentrate on the
uplink, which is from
a
mobile to its basestation.
A data mobile arrives at a cell according to a Poisson
process with rate
A,.
The basestation decides whether
to accept the mobile based on some admission control
policy. In this paper , we consider
a
simple admission
control in which a mobile is admitted provided that the
total number of mobiles in the cell does not exceed a
fixed threshold
K .
Once the mobile has been admit-
ted, it stays in the cell for an exponentially distributed
duration with mean l/pu. The arrival and departure
processes of mobiles are all independent. In this paper,
we do not consider handoffs.
While in the cell,
a
mobile generates and transmits
da ta packets. Data traffic is described by an on-off
model. Specifically, a data source assumes alternate
Lonand LLof f l tates. Durations of being in on and
off
states are exponentially distributed with means
b-l
and U- respectively. At on state , data packets
are generated according to a Poisson process with rate
A . No packets are generated at off state. The size of
each packet
is
independently exponentially distributed
with mean
L,.
In order to satisfy the bit error rate requirement, it is
generally required that exceed
a
certain threshold,
where Eb is the received signal energy per bit and No is
the power density of interference. Let Pt and P, denote
the transmitted and received powers, respectively. It
follows that =
=
q here R is the trans-
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7/25/2019 Calculation of Erlang Capacity for Cellular CDMA Uplink Systems
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mission bit rate and C is the channel gain between the
mobile and its basestation. In a wireless environment,
the channel gain generally consists of three components,
namely, path loss, slow log-normal shadowing, and fast
Rayleigh fading. For the purpose of studying power con-
trol,
it
is commonly assumed [3] that Rayleigh fading is
averaged, and thus it will not be taken into account in
this paper . Hence, the channel gain at time t is mod-
eled by C =
r(t)-~lOO.'c(t),
here r( t ) s the distance
between the mobile and the basestation, q is an order of
power path
loss,
and t(t2 s
a
zer eme an normal random
variable with variance U representing the shadowing ef-
fect.
As the channel gain C changes from time to time,
power control is used to adjust Pt to meet the Eb/No
requirement. Note th at in
a
CDMA system, the signal
from
a
mobile is the interference to all other mobiles.
The idea of power control is to allocate an appropriate
transmission power level for each mobile in the CDMA
system to maintain its requirement without generating
unnecessary interference to others.
The power control scheme employed in the current
voice CDMA system is based on channel inversion [3].
The requirement of constant bit rate for voice mobiles
indicates that P,. has to be controlled
at a
constant level,
normalized to be 1. Therefore, the power control scheme
is to make Pt proportional to the inverse of the channel
gain. The problem of the channel inversion scheme is
that when the channel condition is bad, mobiles have
to increase their transmission power dramatically, thus
causing excessive interference to neighboring cells and
adversely impacting system capacity.
The requirement of constant received power is unnec-
essary for data traffic that does not have a strict delay
constraint. In order t o achieve higher system capacity,
the power control for data traffic can be done in an a da p
tive manner with respect to the channel condition. Now,
let the received power
P,.
be
g
where
g
represents an
adaptive strategy to be used. In
[2],
a truncated chan-
nel inversion power control scheme is examined, where
g is given by
Here, P
is
the power control threshold. Equation (1)
indicates that depending on whether C
2
p, the chan-
nel condition is categorized into good and bad states.
The channel service time of transmitting dat a packets at
good state is exponentially distributed with mean
p - I ,
where
p
=
z
Mobile transmission is suspended when
LP
the channel is in a bad state, thereby reducing interfer-
ence to neighboring cells.
Blocking and outage are two important performance
measures in the above power-controlled
CDMA
system.
Blocking occurs when a n incoming mobile cannot be ad-
mitted. Outage occurs when a mobile admitted in t he
cell cannot maintain the
Eb/No
requirement. Given the
requirements of the blocking probability
(Pb)O
and the
outage probability
P,,t)o,
rlang capacity is defined
as
the maximum load, in terms of Au / p u that the system
can support. In order to determine the Er lang capac-
ity, in the next section we develop an analysis model to
calculate the blocking and outage probabilities.
111. ANALYSIS ODEL
Given the admission control threshold K , the call-
level performance can be model by a Markov chain. The
probability that there are i mobiles in the cell is given
by
for i
=
0,1,2,
...,
K .
The blocking probability is there-
fore equal to p ~ .
Note that in the admission control, K is treated
as the effective system capacity. Recall that in a
TDMA/FDMA system, the system resource is com-
pletely channelized, and
K
is thus simply equal t o the
number of channels. In a CDMA system the concept
of system resource, i.e. capacity, is
soft.
The deter-
mination of K is based on the outage probability that
mobiles can tolerate. In the remainder of this section,
we examine the outage probability.
Recall from Equation 1 ) that the wireless channel
switches between good and bad sta tes as
r-~(t)lOO.lc(t)
crosses level P It has been shown in
[2]
that with a
certain wireless channel propagation model, the dwell
time of the wireless channel staying in either
a
good
or bad condition can be approximately modeled as an
exponentially distributed random variable.
Thus,
the
wireless channel can be modeled as a two-state Markov
chain, where the channel switches between
a
good state
and
a
bad state according to an al ternat ing renewal pro-
cess. Recall tha t the data arrival process
is
a Markov
modulated Poisson process, which is independent of the
service process of the wireless channel. Based on the
memoryless property
of
the two processes, the arrival
and service process models can be integrated together
to develop
a
two-dimensional continuous-time Markov
chain for the entire queueing system of a data mobile
transmitting packets. The Markov chain model is used
to investigate the power activity of data traffic, defined
as a binary random variable Z ( r ) ,where
Z T)
=
1
if
the mobile is transmitting, and
Z T)
=
0,
otherwise.
From the above discussion of the wireless channel model,
it follows that
Z r ) =
X ( r ) Y ( r ) ,where Y ( r ) s
a
bi-
nary random variable representing whether the channel
is good, and X T ) s
a
binary random variable repre-
senting whether the mobile is transmitting given that
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the channel is good. We define conditional power activ-
i t y factor C(r)as the expected value of X r ) . C(r)can
be determined from the stochastic models discussed pre-
viously.
C(r)
s determined by solving the Markov chain.
The details can be found in
[2],
and are omitted here.
We next examine the interference. We focus on an
arbitrary cell 0, The interference received
at
the bases-
tation
0
consists of intra-cell interference
I,
caused by
mobiles belonging to cell 0, and out-cell interference
I,
caused by mobiles belonging to all other cells.
Suppose mobiles are densely and uniformly dis-
tributed in the system. Denote by 7 he user density.
With the radius of each cell normalized to unity, the av-
erage number of mobiles per cell is given by K =
9 7 0
It
follows that
K
I,
=C Z j ) ,
j=1
where
Z j )
s the power activity variable of mobile j. It
can be shown that
E[Ia] =
K G ,
and
(3)
mr[~a I = K [ I n a Cna121 (4)
with na
=
a+bpL
I consists of interference from all neighboring cells.
It follows that
.
+(to
z , ~ o / r i )dA,
where @ E o ,ro/ri) is the indicator function of the
mobile belonging to cell
i
i.e.,
if
ro/ri)QIO c~-cO)/10
1, then @ SO
i ,ro/ri)=
1. Here,
r0,ri
denote the
distance from the mobile to basestations 0 and i , re-
spectively, and
&,
denote the associated shadowing
variables. The cell index i is determined with the small-
est distance principle
[3]
ri
=
minrl.
I f
It can be shown that
(5)
2 / 2
where Q g )
=
s g w k d x and f E ) is the Gaussian
density function with zero mean and variance cr2.
We are now ready to examine the outage probability.
The outage probability
is
defined as
Pout Prob (-)o ,
(
2
where
( )o
is
a
constant depending on the physi-
cal layer communication requirement. Clearly, received
&/No is given by
2
=3 here W denotes total
bandwidth. With the assumption that l
Ie
s Gaus-
sian, the outage probability at cell 0 is given by
IV.
ERLANG
APACITY
Given the above analysis model of the blocking and
outage probabilities, we examine three methods to d e
termine the Erlang capacity in this section.
A . Method I
This first method is very close to the analysis
ap-
proach commonly used in a TDMA/FDMA system.
Specifically, the approach is to first determine the ef-
fective system capacity K given the outage probability
requirement, and then determine the Erlang capacity
given
K
and the blocking probability requirement.
When calculating
K ,
we assume that each cell is
equally loaded with K mobiles, that is, operates at its
capacity.
Hence, for a given PqUt)oequirement, we
determine the maximum K such that PoutI Pout)o.
From K and a given p b ) O requirement, we determine
the maximum
Xu/pu
uch that p~ 5 p b )O , where p~
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is given by Equation 2). This method decouples the
blocking and outage performance, and thus does not
re-
quire any iterative search.
B.
Method
I1
The second method follows an approach as in [l]. n
[l],
o explicit admission control is applied, and incom-
ing mobiles are always admitted. Therefore, there is no
blocking. The only performance measure is the outage
probability. When calculating the outage probability,
two important approximations are made. Firs t, though
random, the number of mobiles is the same in each cell.
Second, the out-cell interference, in terms of both mean
and variance, is equivalent to a fraction of the intra-cell
interference.
For the sake of comparison, admission control is still
employed in the second method. We use iterative search
to determine the Erlang capacity. Specifically, for
a
given Au / p u we determine the minimum
K
such that
p~ 5 (Pa),-,,and given K, further calculate the distri-
bution of the number of mobiles in
a
cell, according to
Equation 2). Then, the outage probability is obtained
using the method described in
[I].
We adjust Au / p u and
repeat the above procedure until the outage probability
requirement is satisfied.
C.Method III
The two methods presented above are not always very
accurate. Specifically, as mobiles arrive at and depart
from cells randomly, the loads in individual cells fluc-
tuate and are not always equal to If The first method
assumes all cells operate at the capacity level, thus over-
estimating the interference and leading to pessimistic
results of the Erlang capacity. The second method at-
tempts to capture the fluctuation
of
mobile numbers.
However, the method is based on the two approxima-
tions, which may not accurately reflect mobile traffic
and interference.
We next present an exact calculation method that
captures both the interference and mobile traffic pre-
cisely. Suppose cells
i
= 1,. .
6
are located in the first
ring and cells i
=
7 , . . . 18 are located in the second
ring. We denote by Ni the number of mobiles in cell i
and make the following observations.
1.
The out-cell interference generated by the mobiles in
the first
or
second ring has the equivalent interference
effect as tha t generated by
E:=,
Ni or
cfil
i mobiles
that are uniformly distributed in the first
or
second ring.
2. The out-cell interference, in terms of mean and vari-
ance, is directly proportional to the number of mobiles
in other cells, given that these mobiles are uniformly
distributed.
The first assumption follows from the distance symmetry
of all first ring cells and all second ring cells. The second
assumption
is
based on Equations 3), (4),
(6),
and
7).
Let us denote the out-cell interferences from the first
ring and the second ring
as Iel
and
l e 2 ,
respectively.
Then, according to the above observations, we have
A 18
where E&, , ,
N2= Ni,
nd Ie ,1 and Ie ,2
are the out-cell interferences from the first ring and the
second ring, respectively. The mean and variance of
le l
or l e , p are readily calculated by Equations
(6)
and
(7)
except tha t the integration area is now only the first ring
or the second ring, as discussed below.
The smallest distance principle, Equation
(5),
basi-
cally sta tes that the mobile belongs to either basestation
i or basestation 0. Thus, the out-cell interference from
a first-ring neighboring cell is only caused by mobiles 10-
cated in that cell and mobiles located in the 60 sector
of cell
0
that
is
closest to tha t cell. The out-cell interfer-
ence from
a
second-ring neighboring cell is only caused
by mobiles located in tha t cell. However, the intra-cell
interference is caused by mobiles located in any cell as
long as they belong to cell 0. Hence, the integration
area for EIIe,l] nd
var[ le , l]
overs cells
0,1,.
. 6, and
the integration area for
E[le,2]
nd va~-[l,,~]overs cells
7 , . ..
18.
Hence, given
No
mobiles in cell 0, NI mobiles
in the first ring, and N mobiles in the second ring, the
conditional outage probability
at
cell 0 is given by
The probability distributions of
Nl
and N2 can be
obtained via their characteristic functions, denoted by
@ 1 z ) and @ 2 ( z ) , respectively. Assume that { N i } are
independent for i
=
1,. . 18. Given A u / p u we deter-
mine the minimum K such that p~