On the Performance of Power Controlled Random
Multiple Access in Wireless Networks
Arash Behzad and Julan Hsu
CS218 Final Project
Professor Mario Gerla
University of California (UCLA)
Los Angeles, CA 90095-1594
{abehzad, jlhsu}@ee.ucla.edu
Abstract— In this paper, we develop and investigate two novel
schemes to enhance the throughput of the slotted ALOHA
medium access control (MAC) mechanism. We analyze the
sensitivity of the randomized power control algorithms for
slotted ALOHA MAC with respect to the parameters of power
distribution. In particular, we prove that, independent of the
power distribution, (the upper bound for) the aggregate
throughput of the network is proportional to the square of the
coefficient of variation of the power distribution. We show
that this result can be directly utilized in the process of
designing efficient randomized power control algorithms. We
reconfirm the validity of our theoretical results through
simulation experiments, which are partially conducted in
QualNet 3.6 environment.
I.
INTRODUCTION
Although in random multiple access the collision of
packets results in low throughput and energy dissipation
which makes it unacceptable for many applications,
random access in an indispensable approach in wireless
networks due to its ease of implementation and low
overhead. ALOHA is a widely studied and deployed
medium access control (MAC) protocol: almost all
deployed cellular systems use ALOHA (or one of its
variants) as a mean to request network access for mobile
users [4]; ALOHA is commonly used in the control
channel of ad hoc wireless networks [7]; Furthermore,
ALOHA is used as a multiple access protocol in local
wireless communications [6].
ALOHA was first introduced by Abramson [1], where
it was determined that the maximum channel utilization
under “pure ALOHA” is 18 percent of the channel
bandwidth. The slotted variation was then introduced by
Roberts, showing its capacity is doubled (over unslotted
ALOHA), to 36 percent [2].
In classical slotted ALOHA it is assumed that a
transmission is successful if and only if no other user
attempts to use the channel during the same slot ([2],
[3]). This is a reasonable assumption when all the
packets are received under nearly power levels. Clearly,
in the ground radio environment such an assumption is
somewhat pessimistic as the propagation loss and
multipath fading can make the received power of packets
at access point differ by an order of magnitude. Such a
condition allows taking advantage of the capture effect,
the phenomenon that the packet with the strongest power
is successfully received even in the presence of other
interfering packets [5].
In this paper, we develop and investigate two novel
schemes to enhance the throughput of the slotted
ALOHA medium access control (MAC) mechanism
based on the capture effect. In our primary scheme, we
first analyze the sensitivity of the randomized power
control algorithms for slotted ALOHA MAC mechanism
with respect to the parameters of power distribution. In
particular, we prove that independent of the power
distribution, (the upper bound for) the aggregate
throughput of the network is proportional to the square of
the coefficient of variation of the power distribution.
This result can be directly utilized in the process of
designing efficient randomized power control
algorithms, as it is discussed later in the paper.
Furthermore, we provide an analytical bound for the
performance of the class of randomized power control
algorithms (including the optimum algorithm if it exists).
In our second scheme, we present a closed form
formula for the aggregate throughput of the network as a
function of the probability of transmission. We note that
the latter formula is based on the capture effect under the
Physical Interference Model, which takes the specific
coordinates of the nodes in the system into consideration.
Based on the derived closed form, the optimum
probability of transmission for every given
(symmetric/asymmetric) topology can be computed.
The rest of the paper is organized as follows. In
section II, we present the system model and the
underlying assumptions. The mathematical analysis is
elaborated in section III: in part A we provide our
mathematical analysis regarding the optimal performance
of the randomized power control algorithms. We
elaborate our asymptotic results on the optimal
probability of transmission under the capture effect in
part B. Simulation results and conclusions are presented
in section IV and section V, respectively.
II.
SYSTEM MODEL
The nodes of the network are assumed to strive to
communicate with a central entity, which we refer to as
the access point (AP).1 All nodes operate in the same
channel and transmit under a fixed data rate R. Nodes
are equipped with identical half-duplex radios and omnidirectional antennas. During the period of operation
under consideration in this paper, we assume nodes to be
immobile and to be located based on an arbitrary
distribution, unless otherwise is specified. The channel
access protocol is slotted ALOHA. That is, the time axis
is divided into identical synchronized time slots whose
duration is assumed to be equal to the transmission time
of a packet (which is assumed to be fixed) plus some
overhead duration that includes the maximum
propagation delay. Each user can transmit its packet
only at the beginning of a slot. It is assumed that all
nodes always have packets waiting for being transmitted
(i.e., heavy traffic load). Conventionally, a newly
arrived packet is transmitted in the first slot after its
arrival; packets which are not received successfully are
buffered and retransmitted after a random delay.
However, similar to many other studies in the literature
(e.g., [4],[9]), we assume that the newly arrived packets
and packets awaiting transmissions are treated
identically. That is, in every slot, each node transmits a
packet according to a Bernoulli process with parameter
q, 0  q  1 , whether it is a new packet or retransmitted
packet.
Let us denote the thermal noise and the transmission
power from node k to the AP by N and Pk , respectively.
We represent the distance from node k to the AP by dk.
Without loss of generality, throughout this paper we
assume that if there are M simultaneous transmissions in
a slot, then the set of simultaneous transmitters is equal
to {node 1, , node M } . A transmission from node k to
the AP is received successfully if the signal-tointerference and noise ratio (SINR) at the AP is not less
than the minimum required threshold  , i.e.
1
The central node is also sometimes referred to as a base station or
backbone node, according to the underlying application.
Pk / d k 
 , k  1,2,..., M ,
M
N
P /d
i
(1)

i
i 1
ik
where M is the number of simultaneous transmissions in
the underlying slot, and  represents the path loss
exponent. This model is commonly known as the
Physical Interference Model [15]. Similar to many other
papers in the literature, we assume  to be always
greater than one, which is the case in most practical
scenarios. It is straightforward to show that at most one
packet can capture the channel if   1 [7].
III.
MATHEMATICAL ANALYSIS
A. Power Control and Capture Effect
Let consider a symmetric topology, whereby the
distances of all nodes and the AP are identical. We
denote such a distance by d. Every node may adjust its
transmit power in a slot-by-slot fashion. The set of
available transmit power levels (for each node) is subject
to the underlying circuitry design. Such a set might
include a continuous spectrum of power levels (e.g. the
Prism chipset) or only a handful of discrete power levels
(e.g. the Cisco/Aironet cards).
The aggregate throughput of the network (in packets
per slot) can be calculated as the following,
TH  Pr{sin gle transmission}  Pr{capture}
n
  q(1  q) n 1  Pr{capture},
1 
(2)
where n is the number of nodes in the network.
When a node attempts to transmit in a time slot, it
selects its transmit power level randomly and
independently (from the other involving nodes in the
system) based on the probability density function (p.d.f.)
f(P). We refer to such an algorithm and the associated
p.d.f. as the randomized power control algorithm and the
power distribution, respectively ([10], [11], [12], [13]).
To support fairness, we assume that the underlying
power distribution is identical for all the nodes in the
network. Since the probability of transmission (q) is
independent of the power distribution, the latter does not
have any effect on the first term of equation (2).
Consequently, the only effect of the power distribution
on the aggregate throughput is crystallized in terms of
probability of capture. Clearly, from throughput point of
view, it would be optimal to select a power distribution
that maximizes the probability of capture.2
In the following theorem, we address the effect of the
power distribution on the aggregate throughput of the
network. In particular, we demonstrate the impact of the
mean and variance of the power distribution on the
probability of capture.
Pk / d k 
Pr{ node k captures M trans.}  Pr{
 }
M
N
P
r
/d r

r 1
r k
M
 Pr{ Pk  
P
r
Nd  }. (7)
r 1
r k
Therefore,
Theorem 1.Consider a slotted ALOHA medium access
control with a randomized power control algorithm
whose power distribution is f(p). Suppose that the mean
and the variance of the power distribution are equal to 
Pr{node k captures M trans.}
M
 Pr{ Pk  
and  , respectively. Then, independent of the power
distribution, the aggregate throughput of the network is
always bounded as follows:
n 
[ M q
M
 [1   ( M  1)]}
M
 Pr{| Pk  
 P  [1  (M  1)] |  Nd

r
 [1   ( M  1)]}. (8)
r 1
r k
n
TH   q(1  q) n 1
1 
 2

r
r 1
r k
2
n
 P  [1  (M  1)]  Nd
(1  q) n  M
M 2
[1   ( M  1)]M
By considering the Chebyshev’s inequality3 and
noting that
2
[  ( Nd   (1   ( M  1)))]2
].
(3)
M
E[ Pk  
 P ]  [1  (M  1)]
(9)
r
r 1
r k
Proof. The probability of capture can be represented by
conditioning on the number of simultaneous
transmissions in the same slot, i.e.
and
M
Var[ Pk  
 P ]  [1  
r
2
( M  1)] 2
(10 )
r 1
r k
n
Pr{capture} 
 Pr{capture M trans.} Pr{ M trans.}.
(4)
M 2
(since the transmit power
independently), we have
Since at most one node can capture the channel, we
have
 Pr{node k captures M trans.}.
are
selected
Pr{ node k captures M trans.}

M
Pr{capture M trans.} 
levels
(5)
k 1
1   2 ( M  1)
2 . (11)

2
[ Nd  (1   ( M  1))]
By considering relations (4), (6), and (11), we have
Considering relation (5) and due to the topological
symmetry, we have
Pr{capture }
n 
[1   2 ( M  1)]M
[ q M (1  q) n  M
. (12 )
M
[  ( Nd   (1   ( M  1)))]2
M 2 
n
 2
Pr{capture M trans.}  M Pr{node k captures M trans.}, (6)
where node k is an arbitrary node, k {1, , n} .

Considering relations (3) and (12), we conclude that,
independent of the power distribution, the aggregate
We have
3
Let the random variable X have a distribution of probability about
which we assume only there is a finite variance 2 . Then for every
2
Note that the energy optimization is out of the scope of this paper.
k > 0, Pr{| X   | k}  1 / k 2 [14].
throughput of the network is always bounded as the
following,
n
TH   q(1  q) n 1
1 
n
 2
n 
[ M q
M
(1  q) n  M
M 2
[1   2 ( M  1)]M
[  ( Nd   (1   ( M  1)))]2
],
Clearly, the latter result provides a significant insight
into the design of the power distribution of the
randomized power control algorithms.
Corollary 1.2. If the optimum randomized power control
algorithm exists, then it cannot increase the probability
of capture to more than
n
which completes the proof. ■
Theorem 1 indicates that (the upper bound for) the
aggregate throughput is proportional to the variance of
the power distribution. This is intuitively very correct as
we increase the variance of the power distribution
(assuming that the mean of the distribution is fixed),
there is a higher chance for capture. In the extreme, as
the variance of power distribution becomes zero, it
would be impossible to have a capture, which is
consistent with relation (12).
On the other hand, Theorem 1 indicates that (the
upper bound for) the aggregate throughput is inversely
proportional to the mean of the power distribution. This
result is also intuitively plausible as we increase the
mean of the power distribution (assuming that the
variance of the distribution is fixed), we expect to have a
less chance for capture. The latter is due to the fact that
what really matters for capture is the ratio of powers.
We note that in most practical cases, Nd  is
dominated by  (1   ( M  1)) in relation (3).
Therefore, by eliminating the former term in relation (3),
the upper bound is not increased considerably.
Consequently, we have
n
TH   q(1  q) n 1
1 
n
n 

[1   2 ( M  1)]M
 ( )2 [ q M (1  q)n  M
], (13)
 M 2  M 
[  (  ( M  1)  1)]2

which leads to the following corollary:
Corollary 1.1.Independent of the power distribution, (the
upper bound for) the aggregate throughput of the
network is proportional to the square of the coefficient of
variation4 of the power distribution.
min{  2
n 
[ M q
M 2
M
(1  q) n  M
[1   2 ( M  1)]M
[  ( Nd   (1   ( M  1)))]2
Example: Let us consider a wireless LAN consisting of
20 nodes, which their distance from the AP is 20 meter.
Suppose that the underlying MAC is slotted ALOHA
with the probability of transmission q=0.05. The
minimum required SINR is 9.5dB. Every node is
equipped with a wireless card that allows it to transmit
under one of the following power levels: 1mW, 10mW,
and 100mW. Based on Theorem 1, a randomized power
control algorithm that selects each of the transmit power
levels with the same probability (i.e. 1/3) cannot increase
the probability of capture to more than 0.0804. This
implies that the maximum throughput under such a
mechanism cannot be larger than 0.4482 packets per slot.
B. Optimum probability of transmission under the
capture effect
In this section, nodes are assumed to be distributed
based on an arbitrary random pattern. Furthermore, all
transmissions are assumed to be performed under an
identical power level P (i.e., no power control
mechanism is considered). The problem that we address
in this section is to find the optimal probability of
transmission (q), which is assumed to be identical for all
nodes in the network, such that the aggregate throughput
of the network is maximized.
Lemma 2. Let us consider an arbitrary topology with
large number of nodes operating under a slotted ALOHA
medium access control with probability of transmission
q. Based on the Physical Interference Model, the
probability of success for transmission from node k can
be calculated in a closed form as
S k  Q( f k (q)),
where Q(.) is the Q-function, and
4
i.e. the ratio of the standard deviation to mean of the distribution.
,1}. (14 )
(15)
 N / P  1 / d k  
n

r 1
r k
f k (q) 
n
 q(1  q) /d
n
Y /d
q /d r 
r
, q  0, q  1. (16 )
2
Proof. Based on the Physical Interference Model, the
probability of success for transmission from node k to the
AP in an arbitrary slot can be presented as
P / dk 
 },
n
N
Y P / d
r
n
(ir , AP ),
r 1
r k
 q(1  q) /d
2
(ir , AP )], (20 )
r 1
r k
where Q(.) is the Q-function, and
 N / P  1 / d k  
f k (q) 
Relation (17) can be written as
r

/d r  ]   N / P  1 / d k  }.
 q /d
n
 q(1  q) /d

r
, q  0, q  1 .■
2
r
r 1
r k
Theorem 3. Let us consider an arbitrary topology with
large number of nodes operating under a slotted ALOHA
medium access control with probability of transmission
q. Based on the Physical Interference Model, the
aggregate throughput of the network as a function of q
can be calculated in a closed form as follows,
(19 )
r 1
r k
n
r 1
r k
r
1, if node k transmits in the underlying slot
Yk  
. (18 )
0, otherwise
n
 q /d
S k  Q( f k (q)) ,

where Yk ' s, k  1,  , n are independent Bernoulli
random variables with parameter q, defined as follows:
 [Y
r 1
r k
(17 )
r 1
r k
S k  Pr{
n
(ir , AP ) ~ N [
where N[A,B] is a Gaussian distribution with mean A and
variance B.
Consequently, relation (19) can be written as
r
r 1
r k
S k  Pr{

n
TH ( q) 
 qQ( f
k
(q)),
( 21)
r 1
n
 [Y

r
/d r ] is a linear combination of large number
r 1
r k
of i.i.d random variables. Therefore, based on a
generalization of the Central Limit Theorem (i.e. the
Lindeberg-Feller Central Limit Theorem5) we have
where Q(.) is a Q-function, and f k (q) is defined as
before (relation (16)).
Proof. The aggregate throughput of the network (in
packets per slot) as a function of q can be calculated as
the following,
n
TH (q )  E[
X
r
],
(22 )
r 1
5If
the independent random variables Z1 , Z 2 ... satisfy the Lindeberg
condition, then for all a < b,
where X r ' s, r  1,  , n are
variables defined as below:
dependent
Bernoulli
n
[Z  E(Z )]
i
lim P(a 
n 
i
i 1
 b)  (b)  (a),
1, if node r transmits successfully in the underlying slot
Xr  
. (23)
0, otherwise
n
Var (Z )
i
i 1
where  is the normal distribution function. Note that in most
practical cases the Lindeberg condition is satisfied.
Therefore, relation (22) can be written as
n
TH (q) 
 P( X
r 1
r
 1).
(24 )
It can be easily shown that
1
2
P( X r  1)  qS k .
(25) .
Considering relation (24), relation (25), and Lemma
2, we conclude
TH ( q ) 
 qQ( f
k
( q )),
r 1
and the proof is complete. ■
As a consequence of Theorem 3, we can analyze the
aggregate throughput of the network as q converges to 1.
Interestingly, based on relation (21), it can be shown that
lim TH (q)
q 1
n



1, if k : 1 / d (ik , AP)  N / P  q /d (ir , AP)


. (26 )
r 1
r k

0, otherwise

Relation (25) illustrates the fact that if exactly one of
nodes is relatively close to the AP with respect to the
other nodes (Fig. 1), then the aggregate throughput
becomes equal to one; otherwise, throughput becomes
equal to zero, which is intuitively very correct.
IV.
SIMULATION RESULTS
For our simulation experiments, we developed and
incorporated the slotted ALOHA MAC scheme for the
QualNet environment. Our simulation study is partially
based on QualNet 3.6. and the power control algorithm is
not yet incorporated. In the simulation environment, 20
nodes are randomly and uniformly distributed, unless
otherwise is specified. Noise power is set to be equal to 90dBm and the path loss exponent is assumed to be
equal to 4. The minimum required SINR is assumed to
be 10dB. Every node has an infinite reservoir of packets
for transmission (i.e. heavy traffic load) and is assumed
to be immobile during the period of simulation.
In Fig. 2 we illustrate the effect of power control on
the aggregate throughput of the network.
The
randomized power control algorithm selects each of the
two available power levels (i.e., 1mW and 100mW) with
equal probabilities. The alternative scheme always
transmits under 50mW. As expected, even a very simple
6
3
AP
Fig. 1. Illustration of a random multiple access, whereby node 4 is
relatively close to the AP with respect to the other nodes.
power control operation over the two power levels can
improve the
aggregate
throughput somewhat
considerably. When the involving nodes transmit with a
probability equal to one (q=1), still the aggregate
throughput under the randomized power control
algorithm is strictly positive. Furthermore, we observe
that q=0.05 (that is 1/n) does not provide the optimal
probability of transmission. This is due to the fact that
q=0.05 only maximizes the probability of a single
transmission in the system (i.e. the first term in relation
(2)). However, q=0.08 optimizes the sum of the
probability of a single transmission and the probability of
capture (see relation (2)). We note that due to the
capture effect induced by the underlying asymmetric
topology, the aggregate throughput for the slotted
ALOHA (with no power control) under q=0.05 is
significantly larger than 1/e.
In Fig. 3 we examine the fairness of the latter two
algorithms. As it has been illustrated, the minimum
throughput (among all the nodes) under the power
controlled slotted ALOHA is larger than that under the
slotted ALOHA with no power control. Therefore, the
randomized power control operation not only increases
the aggregate throughput of the network, but also
supports fairness (in the context of MaxMin).
No power control (P=50mW)
Throughput (packets/slot)
n
5
4
With power control (P1=1mW;P2=100mW)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.05
0.08
0.3
0.5
0.8
1
Probability of Transmission (q)
Fig. 2. Illustration of the aggregate throughput (in packets per slot) for
slotted ALOHA with no power control and power controlled slotted
ALOHA as a function of probability of transmission.
No power control (P=50mW)
r
With power control (P1=1mW;P2=100mW)
Throughput (packets/slot)
0.05
AP
0.04
R’
R
0.03
0.02
0.01
0
0
5
10
15
20
25
Node ID
Fig. 3. Illustration of the throughput per node (in packets per slot)
under slotted ALOHA (with no power control) and the power
controlled slotted ALOHA.
We compare our theoretical results for calculating the
optimal probability of transmission with q=1/n in Fig. 4.
For this experiment we assume that the distances
between all the nodes and the AP are identical, which
makes q=1/n an optimal choice. Clearly, the theoretical q
converges to q=1/n as n becomes sufficiently large.
In our next experiment, we assume nodes are
randomly and uniformly distributed in an annulus
defined by concentric circles of radius R and R’ (Fig. 5).
Clearly, as r = R - R’ becomes smaller, the LindebergFeller Central Limit Theorem converges to the
conventional Central Limit Theorem (CLT) and the
speed of convergence (as a function of n) becomes faster.
In Fig. 6 we show the sensitivity of our theoretical
results for finding the optimal q with respect to small
number of nodes. We assume 20 nodes are randomly
and uniformly distributed in an
annulus with
parameter r, where R = 250m. For each r we find the
optimal probability of transmission (q*) by running the
simulation in QualNet over different values of q. We
then compare the theoretical q (q_theoretic) derived
according to section III with q*.
Fig. 5. Illustration of an annulus representing the geometric locus of
location of nodes.
We observe that as r decreases the difference between
the theoretical q and the optimal q converges to
approximately zero, even when there are only 20 nodes
in the network. Of course, when r is large, our
asymptotic analysis will not remain valid for only 20
nodes.
V.
CONCLUSIONS
In this paper, we developed and investigated two
novel schemes to enhance the throughput of the slotted
ALOHA medium access control (MAC) mechanism
based on the capture effect. In our primary scheme, we
analyze the sensitivity of the randomized power control
algorithms for slotted ALOHA MAC with respect to the
parameters of power distribution. In particular, we prove
that independent of the power distribution, (the upper
bound for) the aggregate throughput of the network is
proportional to the square of the coefficient of variation
of the power distribution assuming a symmetric
topology. In our second scheme, we present a closed
form formula for the aggregate throughput of the
network as a function of the probability of transmission.
Based on the derived closed form, the optimum
0.5
1
|q*-q_theoretic|
Theoretical q
q=1/n
0.8
0.6
0.4
0.2
0.4
0.3
0.2
0.1
0
0
0
20
40
60
80
Number of nodes (n)
Fig. 4. Illustration of the convergence of the theoretical q and q=1/n
for large values of n.
0
50
100
150
200
250
300
Parameter of annulus (r)
Fig. 6. Illustration of the difference between the optimal q and the
theoretic q when the number of nodes is small.
probability of transmission for every given
(symmetric/asymmetric) topology can be computed.
A more comprehensive simulation study of the latter
schemes with emphasis on their stability and fairness
features is in progress. Moreover, generalization of the
derived results in this paper to mobile multihop wireless
networks is part of our ongoing research.
ACKNOWLEDGMENT
The authors wish to thank Professor Mario Gerla for
discussions and insightful comments leading to
improvement of quality of this paper.
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