DISTRIBUTED JOINT POWER AND RATE ADAPTATION IN

AD HOC NETWORKS

Fredrick Mzee Awuor, Karim Djouani and Guillaume Noel

Department of Electrical Engineering, French South African Institute of Technology (F’SATI)

Tshwane University of Technology, Private Bag X680, Pretoria, 0001, South Africa

Keywords: Signal-interference plus noise ratio (SINR), Coupled interference network utility maximization (NUM),

Joint power and rate adaptation, Ad hoc networks, Reward, Costing/pricing.

Abstract: Ad hoc networks are dynamic and scalable entities that autonomously adapt to nodes entering the network

(i.e. increasing interference) or exiting the network (i.e. due to energy depletion), poor connectivity among

others. In such networks, nodes exhibit individualistic behaviours where nodes selfishly compete for the

limited network resources (i.e. energy and bandwidth) to maximize their own utilities. This consequently

degrades network performance leading to low data rates, poor power efficiency, loss of connectivity

etcetera. This paper considers a network utility maximization (NUM) strategy based on coupled interference

minimization to adapt the transmission power and data rates in ad hoc networks. The proposed distributive

joint power and rate adaptation (JRPA) algorithm employs costing (and reward) mechanisms to promote

users’ cooperation such that both users’ local and network global optimum is always attained. This is similar

to a super-modular game hence the optimality and convergence of JRPA is analysed using super-modular

game theory. Simulation results show that the proposed algorithm improves network performance since

users’ are compels to transmit at optimal data rates and power levels just enough to sustain the transmission.

1 INTRODUCTION

Preference of wireless networks (WNs) to fixed

networks has incredibly increased in the recent past

due to their cost efficiency and ease of setting-up

and integrating them with other networks. This has

since led to introduction of IEEE standards that

support higher data rate e.g. 802.11a/g. However,

transmitting at higher data rates reduces connectivity

due to decline in communicating range and hence

requires that the transmission power be increased to

sustain transmission.

To attain spectrum efficiency in WNs, resource

sharing and management is critical. Nonetheless,

this is not easily attainable in ad hoc networks due to

dynamic topology and time-variant channel

conditions in such networks hence need for adaptive

approaches.

Though reducing the transmit power allows

multiple simultaneous transmissions, this results to

decrease in SINR performance owing to either weak

received signal strength (RSS) or increased

interference. As a result, transmissions are sustained

at lower data rates. Moreover, such scenarios are

vulnerable to hidden node problems resulting from

high interference range created by the reduced

transmit power. On the converse, transmitting at

high power impedes concurrent transmissions.

Nonetheless, this mitigates hidden terminal

problems and improves SINR thence higher data

rates are achievable. In a nutshell, to attain high data

rates at minimum transmission power in WNs is a

contradictive objective. Huang et al in (Huang and

Letaief, 2005) shows that adapting transmission

parameters (data rate and power) based on link

dynamics can solve the aforementioned objective. In

such a case, the link dynamics can be estimated

based on the RSS, acknowledgment (ACK) history

(Kim and Huh, 2006) or SINR (Olwal et al., 2009,

Grilo and Nunes, 2003, del Prado Pavon and Choi,

2003). However, SINR based schemes has better

performance compared to RSS and ACK since it

responds faster to link variations (Olwal et al.,

2009).

We propose a joint power and rate adaptation

scheme based on NUM problem formulated as a

coupled interference minimization such that nodes

determine their data rates and transmit power based

Mzee Awuor F., Djouani K. and Noel G..

DISTRIBUTED JOINT POWER AND RATE ADAPTATION IN AD HOC NETWORKS.

DOI: 10.5220/0003518100050011

In Proceedings of the International Conference on Wireless Information Networks and Systems (WINSYS-2011), pages 5-11

ISBN: 978-989-8425-73-7

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

on presumed coupled interference at the receiver. In

such a case, users are always aware of channel

conditions as they choose their transmission

parameters. Further, costing (pricing) effect is

imposed on users’ choices to encourage cooperation

and deter selfish behaviors hence both local and

global utility are attainable.

The reminder of this paper is organized as

follows: Section 2 reviews related works; Section 3

gives the system model; JRPA algorithm is

presented in Section 4 while simulation results are

given in Section 5 and finally, conclusion is drawn

in Section 6.

2 RELATED WORK

Most protocols proposed in literature ((Luo et al.,

2010, Hayajneh and Abdallah, 2004, Grilo and

Nunes, 2003) and references therein) considers

power control, rate adaptation or joint rate-power

control in centralized infrastructures WNs where a

centralized station determines and dictates the

power/rate for data transmission in the network.

Such protocols may not be applicable in ad hoc

networks where all stations are at free will to choose

their transmission parameter based on their own

preferences. This may lead to greedy behavior

wherein users adapt their transmission power with

sole objective of achieving individual desired

throughput without considering others users’

interests (Olwal et al., 2009). Such schemes require

much power to sustain a stable SINR in deep fading

environment and causes high interference.

Furthermore, such algorithms tend to diverge in case

of no feasible power allocation due to hard SINR

requirements. However, this divergence problem is

easily solved by adaptive SINR based on coupled

interference at the receiver.

Due to the distributed and heterogeneous nature

of ad hoc network, it is often challenging to design

distributed algorithms that can achieve the global

optimal NUM solution. The difficulty in distributed

algorithm design often lies in the coupling nature of

the NUM problem. NUM problems generally

assume that user’s utilities are uncoupled, i.e., each

utility depends only on local variables (Li Ping et al.,

2009). However, in problems where cooperation or

competition is modeled using the objective function,

each user’s utility depends on both its local variables

and local variables of other users in the network

(Hayajneh and Abdallah, 2004, Wang et al., 2006).

In (Chee Wei et al., 2006, Palomar and Mung,

2006), these NUM problems are formulated as

coupled optimization. Dual decomposition with

significant message passing is used to solve such

coupled NUM problems where the coupling in the

objective function is transferred to coupling in the

constraints. However this requires strict convexity

and exhibits slow convergence. In (Huang, 2005,

Huang et al., 2006), “reverse engineering” with

limited message passing is proposed that solves

coupled NUM problems without need for strict

convexity.

Similar to (Huang, 2005, Huang et al., 2006), our

proposed algorithm considers limited message

passing strategy based on “reverse engineering” to

solve the formulated coupled interference NUM

problem. The proposed JRPA dynamically adjust the

users’ choices of transmission power to curb the

influence of coupled interference. Such dynamic

adjustments exploit the locally observable network

channel conditions and cost charges attached to that

transmit power choice. The users are hence

cognizant of the current link condition while

determining their data rates. Moreover, due to the

ineluctable cooperation, every user’s strategy to

maximize its utility maximizes the utility of other

network users, thus improving global network

performance.

Supermodular game theory is used to show the

existence, convergence and optimality of user’s

utility functions (Saraydar et al., 1999) since in such

games, each player strives to increase its strategy

while increases other players’ strategies as well.

Such a game contains Nash Equilibrium (NE), and

does not necessarily require assumption of convexity

in order to attain NE (Ozdaglar, 2010, Levin, 2003).

3 SYSTEM MODEL

3.1 Problem Formulation

Consider an ad hoc network with N stations where

node

i transmits to node

on a single hop

subjected to path loss, shadowing and multi path

fading dynamics (Olwal et al., 2009). Assume

further that all the nodes in the network are within

the transmission range of their neighbors such that a

node’s transmission interferes with other nodes in

the network. Consider a set of transmission power

levels

and set of data rates

defined as follows:

{

}

min 2 3 max

, , ,...,pppp p=

and

{

}

min 2 3 max

, , ,...,rrrr r=

where

min

and

max

r are the minimum and maximum

data rates while

min

p and

max

are the minimum and

WINSYS 2011 - International Conference on Wireless Information Networks and Systems

maximum transmit power levels possible in the

network. These sets are assumed identical to all

users in the network. The channel gain on link

given by

derived as below:

jiji

pGp=

)

where

is i ’s transmit power and

is received

power at

. Notably,

is not necessarily equal to

since the channel condition is time variant. Half

duplex model is assumed i.e. a user can either

receive or transmit but not both simultaneously.

The objective is to determine

’s power

allocation that maximizes its utility given the

coupled interference perceived at

. Utility function

(())

for user

nN∈

is differentiable, concave

and increasing function of the received SINR

(Saraydar et al., 1999, Huang et al., 2006) and hence

NUM problem based on coupled interference can be

formulated as follows:

max ( ( ))

∈



)

such that

min max

rrrN≤≤ ∀

)

min max

pp N≤≤ ∀

)

where SINR,

()

is given by

ij i

kj k o

kij

Gp n

≠



(5)

where

kj k

kij

≠



is the sum of interference power

at node

due to communication of other users in

the network other than

i .

is the thermal noise,

is the channel gain while

is the transmit power

used by

i to transmit to

3.2 Optimal Power based on Coupled

Interference

Due to existence of mutual interference, network

users have coupled utility function that depends on

both the user’s local decision and decisions of other

users in the network. The global NUM problem can

therefore be formulated from (2) as

{}

()

max

pp P n

∈∀



s.t. (3) and (4)

(6)

To solve the coupled objective function in (6),

(Palomar and Mung, 2006) proposes consistency

pricing which requires significant message passing

to attain optimal decision. Moreover, this approach

requires convexity in (6) but





(. ) in (6) is concave

. Therefore we adopt reverse-engineering

based on KKT conditions (Huang et al., 2006,

Huang, 2005) to solve (6) by localizing the network

objective function and updating users on their

neighbors’ utility choices by means of limited

message passing.

Define

as the power profile of user

in the

network and

−

as the power profile for user 'is

opponents i.e.

()

111

,..., , ,...,

pp p

−

−+

such

that

{

}

;

ppp

−

∈

. Then this utility maximization can

be modeled as a power control game

,{ },{ }

GNpu









where all the players selects

transmit power

p that maximize their utility

whereby

()

represents user 'ispay-off or

reward. User

'isoptimal response is

that

maximizes its utility

u given by

()

ii i i

upp

−

formulated as (7) (Huang, 2005, Huang and Letaief,

2005, Li Ping et al., 2009).





(





)

=

max ( ( , ))

ii i i

upp

−

∈

)

Assuming that

−

is fixed, the reward

(( , ))

iii

upp

−

in (7) is strictly increasing with

In view of a Non Cooperative Game (NCG)

where players select optimal power levels to

maximize their rewards at the expense of others,

then a fixed point p=

defined by (8) would be

the NE.

()

** ' '

ii i i ii i i

upp upp

γγ

−−

≥

(8)

where

pp∈

is any power chosen by any user

other than *p

in view of the fact that each user’s

reward

(( , ))

iii

upp

−

is strictly increasing with

for fixed

−

(Huang, 2005, Huang et al., 2006,

Li Ping et al., 2009).

DISTRIBUTED JOINT POWER AND RATE ADAPTATION IN AD HOC NETWORKS

We seek to improve the NE in (8) by introducing

pricing in users’ choices since pricing discourages

users’ selfish behaviors. In effect, every user strives

to maximize its pay-off or reward

()

in (9) by

minimizing the cost

attached to its transmission

power choice

p .

(, ) ()

ii i ii i

upp f cp

−

=−

(9)

Considering (9) as cost or penalty imposed on

for

generating interference to other network users, user

has to minimize c in (9) to be able to maximize its

utility. Since c depends on channel gain

and

network factor

, surplus function is derived from

(9) as follows:

(; , )

ii i i

Spp

−−

((; ))

ii i i i jij

upp p G

γε

−

≠

−



(10)

Lemma 1 (KKT conditions) (Huang et al., 2006):

For any local optimal

of problem (6), there exist

unique lagrange multipliers

,

∗

,...,μ

,

∗

and

,

∗

,...,μ

,

∗

such that for all nN∈ ,

()

ii kk

iu gu

up u p

γγ

μμ

≠

∂∂

+=−

∂∂



(11

)

where

()

**max

iu i i

−=

()

*max*

ig i i

−=

iu gu

μμ

≥

(12)

The KKT set of problem (6) need to contain all

the solutions that satisfy (11) and (12) hence we

design a distributed algorithm that converges to this

set. Substituting (11) in (6), the KKT condition for

can be expressed as follows

()

** * *

,,,

jj jij iu gu

pp G

μμ

−

≠

∂

=−

∂



(13

)

where

()

(, )

()

jj j j

jj j

upp

−

∂

=−

∂

(14)

In equation (14),

()

−

is the total interference

received by user

given by

iij

≠



. Notably, the

cost function in (14) is always nonnegative and

represents

’s marginal increase in utility per unit

decrease in total interference. The reward is the

product of user’s transmission power

and the

weighted sum of other users’ costs in (10) where

weights equal to the channel gains between

transmitter

and the other users’ receivers. If

the penalty obtruded to other users for generating

interference to user

i defined in (9), then (14) is an

acceptable optimal condition for the problem in

which each user

i chooses a power level

pp∈ to

maximize (i.e. the surplus function in (10)) (Huang,

2005).

At an instance of time

t, network users

announce their cost in reference to (14) and adjust

their transmit power taking into account network

dynamics according to (10). The chosen power is

constrained to (13) and as a result, an optimal

localized distributive power algorithm with costing

constrains is derived. The surplus in (10) and cost

function (14) are formulated as function of the

desired power

and SINR as in (14) and (15)

respectively.

(,)

i ii

S p

−−

min max

min max , ,

() ()

jij

γγ

≠



   



   

   





(15)

(())(())

()

ii i

iiij

up p

ppG

γγ

γβ

∂

(16)

where β is the spreading factor while

()

given by .

3.3 Convergence and Optimality

Lemma 2 (Ozdaglar, 2010): Let

X ⊆ 

and

T ⊂ 

for some k, a partial ordered set with the

usual vector order. Let

:fXT×→ be a twice

continuously differential function. Then, the

following statements are equivalent:

(i) The function

f has increasing differences in (x,t), (ii) For all t’ ≥ t

and

∈

X, we have

(, ) (,)

xt f xt

∂∂

≥

∂∂

and, (iii) For

all

∈

X, t

∈

T, and all i=1,…,k, we have

(,)

fxt

∂

≥

∂∂

() ( )

ii ii

ωω

−

WINSYS 2011 - International Conference on Wireless Information Networks and Systems

Theorem 1: Define

X ⊆ 

as a compact set and T

as some partially ordered set. Assume that the

function

:fT

×→ is upper semicontinuous in x

for all

and has increasing differences in

(x,t).

Define

() argmax ( ,)

tfxt

∈Χ

. Then, we have: for

all

tT∈

, x(t) is nonempty and has a greatest and

least element, denoted by

()

and ()

t respectively

and, for all

t’≥ t, we have

(')

≥

()

and (')

t and

()

From

lemma 2 and theorem 1, every user’s utility

function

(, )

ii i

upp

−

has increasing differences in

(, )

−

given that

()

1, 0

()

ii i

γγ

−

≥∀ ≥

hence the convergence.

Assume

()

,( ),( )

Ipu

is a supermodular game.

Then

()

−

in (7) as a greatest and least element,

denoted by

()

−

and

()

−

, and If

−−

≥

then

(') ( )

ii ii

ββ

−−

≥

and

(') ( )

ii ii

ββ

−− −−

≥

(Levin, 2003).

This implies that each player’s best response is

increasing in the actions of other players. The set of

strategies that survive iterated strict dominance (i.e.

iterated elimination of strictly dominated strategies)

has greatest and least elements

and

, which are

both pure strategy in Nash Equilibrium.

Definition and formulation of supermodular

game theory can be found in (Huang, 2005,

Ozdaglar, 2010, Hayajneh and Abdallah, 2004,

Levin, 2003).

3.4 Rate Adaptation

From the SINR’s of the distributive pricing power

control algorithm above,

best constellation size for

M-QAM modulation is determined that is supported

by the SINR level. From Shannon theory of

communication ((Yu-Chee et al., 2001)) we can

deduce the following:

()

SINR

ln BER



−



−



where BER is the bit error rate while

and

are

modulation type dependent constants. Let

ln( )BER

−

= , then data rate

for transmit power

between the sender i and receiver j is a function of

SINR estimated as

SINR

and hence

() ()

log 1 log

rSINRrSINR

δδ

=+≈=

(17)

where

1SINR



while

is the bandwidth of the

channel used for data transmission. When the signal

level is much higher than the interference level or

when the spreading gain is large then

lies within

(3).

4 JOINT POWER AND RATE

CONTROL ALGORITHM

(JRPA)

The outline of joint power and rate control algorithm

is presented as follows:

1. Initialization Stage: Initialize power

and cost

−

to some non-negative value, and then calculate

from (17).

2. Cost Advertisement and Transmit Power

Adjustment

a. Cost Announcing: interferers

−

update and

advertise their cost

−

according to (16).

Power Updating: based on network cost, user

i updates its transmission power

according to

(15).

Determine data rate according to (17).

Repeat 2 while not end of communication

5 SIMULATION TEST AND

RESULTS

Simulation is performed in MATLAB with 32 nodes

randomly placed in a

20 20mm× field free of

obstacles. It’s assumed that only Tx communicates

with Rx while other network users are actively

interfering. Performance metrics are evaluated for 50

independent runs (transmissions). For all the

simulations, we assume single hop with the

following simulation parameters: path loss model

exponent = 1, AWGN = -96dB, Pmax = 10dB, Pmin

=1dB, Initial cost = 0.1 and utility function,

()

is given by

log( )

. It is further summed that all

transmissions are successful, channel bandwidth =

20MHz and spreading factor,

= 5. We consider 2

scenarios: scenario 1 is a stationary network where

DISTRIBUTED JOINT POWER AND RATE ADAPTATION IN AD HOC NETWORKS

users are static while scenario 2 reflects a mobile

network where users randomly move after every 2

transmissions at a velocity 20kmph. In addition, all

transmissions are assumed to be successful. The

performance of JRPA is compared to IEEE 802.11

and adaptive auto response joint power and rate

control algorithm – LP proposed by (Chevillat et al.,

2005).

Figure 1: Stationary users.

In all the runs, it’s observed that JRPA attains the

highest data rates at minimal transmission power

followed by LP and the legend 802.11 protocols.

The costing mechanism drives the power selection

response in JRPA to the most cost effective option.

At the beginning, transmission power hikes due to

limited information available at Tx about the channel

conditions. As the other network users advertise

their network costs, Tx determines the most feasible

transmission power for the subsequent transmissions

till most optimal transmission power is attained.

This is the NE. LP and 802.11 transmit at higher

power levels and hence achieve higher SINR than

JRPA. Nonetheless, JRPA attains highest data rate.

The improvement on JRPA compared to LP and

802.11 is that JRPA operates at optimal power just

enough to sustain the required transmission and to

decode data packets at the receiver Rx.

Figure 2 shows the performance of JRPA in an

environment where the network users are assumed to

be in random movement. Similar to figure 1,

performance of JRPA in terms of data rates and

power efficiency is relatively better compared to LP

and 802.11. However, low data rates are experienced

due to fast fading channel conditions resulting from

user mobility. The power level that JRPA settles on

is apparently the most optimal power that maximizes

both local and global utility considering the network

dynamics during data transmission. At such power

choices, interference cost function is always

minimized while the reward function (data rate) is

maximized hence improving network performance.

Figure 2: Partial mobility.

6 CONCLUSIONS

This paper proposes distributive algorithm that

jointly adapts transmission powers and data rates in

ad hoc networks by formulating NUM as a coupled

interference minimization problem. The simulation

results have shown that penalizing users’ selfish

behaviors promotes cooperation such that user aim

to optimize both global and local utilities. Future

work may consider cross layering optimization to

incorporate packet routing in the proposed model.

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0 5 10 15 20 25 30 35 40 45 50

SINR - Tx/Rx Stationary

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DISTRIBUTED JOINT POWER AND RATE ADAPTATION IN AD HOC NETWORKS