Calculation of Erlang Capacity for Cellular CDMA Uplink Systems

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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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~

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Calculation of Erlang Capacity for Cellular CDMA Uplink Systems