M. ELMUSRATI ET AL. 77

fixed power. This solution has many problems and limi-

tations such as the near-far problem and the unnecessary-

ily power consumptions for good channel sensors.

New randomized power allocation strategy was sug-

gested in [10], which proposed the use of uniformly dis-

tributed transmitter power levels to mitigate the near-far

effect in congested systems without any channel feed-

back. That work was based on so called snapshot analy-

sis approach and thus neglected the effects of the channel

fading. The performance analysis of the uniform random

power allocation in Rayleigh fading channel is evaluated

in [11].

In this paper, a framework of the performance analysis

for a general distributed random power allocation is in-

troduced. The paper is organized as follows. In the next

section, a description of the system model is given. For

the logical information flow and for comparison purpose

we introduce the system performance of the fixed power

transmission in Section 3. In Section 4, a general treat-

ment of the performance analysis of random power allo-

cation algorithms is given. New empirical random power

distribution is suggested in Section 5. Simulation results

are shown in Section 6. Finally the paper conclusion is

presented in Section 7.

2. System Model

In this paper we assume multiuser environment with broad-

casting devices (sensor nodes) randomly distributed in

certain region. We refer to the transmitters as terminals

and sometimes as sensors. All terminals send their sig-

nals to one or more access points with CDMA multiple

access method. Because of the lack of the feedback

channels it is not possible to use CSMA/CA or any other

protocols that require receiving capabilities in the sensor

nodes. Multi-hop scenarios are not possible as well be-

cause of natural deafness. Every transmitter has different

spreading code, however we do not assume that they are

perfectly orthogonal at the access point.

We consider dynamic scenario, where the terminals or

the access points may have mobility or the environment

is highly dynamics. The transmitted signals arrive from

sensors to the access point in multi-path manner without

dominant path, in other words we assume Rayleigh chan-

nel. The time slot length is small enough to assume that

second order effects such as shadow fading and distance

based attenuation remain constant during the time dura-

tion of the time slot. Although the mean of the received

signal is constant but the instant value of the received

signal magnitude is random variable with Rayleigh pro-

bability density function . In case of Rayleigh fading whi-

ch is considered here, the link gain, i.e. the fraction be-

tween received power and transmitted power becomes

Exponential distributed random variable.

Time is assumed to be slotted such that slot duration is

approximately the same as the coherence time of the

channel. For instance in some sensor network applica-

tions, the duty cycle of the transmitters is low and thus

the channel state in consecutive time slots allocated to

single transmitter node become independent of each other.

Let G denote the link gain between transmitter and re-

ceiver. In case of frequency-non-selective Rayleigh fad-

ing, it can be shown to follow the Exponential distribu-

tion with parameter 1

where

denotes the ex-

pected channel gain which depends on the distance based

attenuation and shadow fading. Let

1

G

ge

de-

notes the cdf of the link gain G and

G

ge

de-

notes its pdf.

Let I and 2

n

denote the received interference and

noise powers, respectively. Let γ denotes the minimum

required signal to interference and noise ratio (SINR) at

the receiver. When the received SINR is less than γ, we

assume that the receiver cannot decode the transmitted

packet correctly. The requ ired SINR depends on the util-

ized modulation and coding method and is out of the

scope of the paper. The outage probability of sen sor i, i.e .

the probability that a packet error occurs, is given by

2

Pr ii

in

GP

I

(1)

where Gi is the channel gain of sensor i, Pi is the trans-

mitted power from sensor i, and the interference term is

given by

N

i

ji

jj

GP

(2)

Note that in this model interference is treated as noise.

The use of multi-user detection could be taken into ac-

count by scaling down the interference power I by some

factor 01

. However, this is not considered in this

paper. We assume that the sensor in outage whenever its

power at the access point is less than some threshold.

3. Performance Analysis of Fixed Power

Allocation

In this section we analyze the system performance of fi-

xed power transmission strategy. The results of this sec-

tion is well known in the literature [6,11], however it is

given here for the subject integrity and for comparison

purposes. Moreover some intermediate results have been

used in next sections. First we assume fixed average in-

terference power (or single sensor scenario). In order to

simplify our notation, let us define n

2

ii

I

Con-

ditioned on Pi = P the outage probability becomes

Pr, i

iii iG

i

GPPF P

(3)

1

i

i

e

(4)

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