power control algorithms in wireless communication

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Distributed Power Control Algorithms in Wireless Networking

Project Report

EE 359 Wireless Communications

Winter 2009

Jeffrey Mounzer

Stanford University

[email protected]

Advisor: Professor Nick Bambos


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Table of Contents

Abstract .3

1 Introduction ............................................................................................................................. 4

2 Literature Survey ..................................................................................................................... 5

2.1 Wireless Network Model and the Foschini-Miljanic DPC Algorithm ............................... 5

2.2 The Backlog-Driven Distributed Power Control Approach .............................................. 7

2.2.1 Mobile power management for wireless communication networks .................... 7

2.2.2 Power-induced time division on asynchronous channels ..................................... 8

2.2.3 Power-Controlled Multiple Access Schemes for Next-Generation Wireless Packet

Networks ................................................................................................................................ 8

2.2.4 Distributed Backlog-Driven Power Control in Wireless Networking ................... 12

3 Simulation of Distributed Power Control Algorithms............................................................ 13

3.1 Simulation Design ........................................................................................................... 13

3.2 Results ............................................................................................................................ 14

4 Future Work ........................................................................................................................... 17

4.1 Simulation Extensions .................................................................................................... 17

4.2 Proving the Structural Properties of Backlog-Driven DPC ............................................. 17

4.3 Design of Stronger DPC Algorithms................................................................................ 18

5 Conclusion ............................................................................................................................. 18

6 References ............................................................................................................................. 19


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Abstract

In wireless networks, transmitter power control can play an important role in improving a number of key

performance parameters, including energy usage, network capacity, and network reliability. As wireless

ad hoc networks become more prevalent, it is critical that power control algorithms operate in a

distributed fashion, since centralized network controllers are rarely available for such systems. This

project explores the evolution of distributed power control (DPC) algorithms over the last 15 years

through a literature review and simulations, and it examines future research directions in this field. In

particular, we consider the emergence of backlog-aware algorithms for DPC, which exploit the tradeoff

between power and delay to induce cooperation between links in the network. The simulations that

were conducted provide a comprehensive platform for the evaluation of such algorithms, which was

largely lacking in the literature.

Several interesting aspects of backlog-aware algorithms are highlighted by our simulation results. The

most powerful result is a validation of the intuition that backlog-aware algorithms, by relying on

teamwork between links, can provide significant performance gains as compared to competitive

algorithms such as the seminal Foschini-Miljanic approach. These gains are particularly large when the

network is under duress either when the load is too great for the links to clear their queues, or when

the cross-link interference becomes large. In addition to presenting and discussing the simulation

results, this work also describes the next steps in this line of research, including extending the currently-

used simulations, overcoming difficulties in the comparison of different algorithms, mathematically

proving the observed structural properties of backlog-aware DPC techniques, and designing better

algorithms for future systems.


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1 IntroductionTransmitter power control can significantly improve a number of critical performance parameters for

wireless communication networks; it can serve to minimize energy usage, improve network capacity,

and mitigate the effects of cross-link interference [2, 4-5]. For networks with centralized controllers(e.g., a cellular network with a base station), the power control problem is relatively simple. However,

in ad hoc networks which lack a central regulator, power control proves to be much more difficult and

interesting [5]. In general, it is necessary for each transmitter in an ad hoc network to regulate its own

power autonomously (this is called distributed power control, or DPC), since centralized coordination

between transmitters is very difficult and susceptible to such problems as having a single point of

failure. As wireless ad hoc networks become more prevalent (for example, in wireless LAN technologies

[5]), it is important to develop high-performance DPC algorithms to meet the challenges presented by

future wireless systems. This report explores the evolution of DPC algorithms over the last 15 years

through a literature review and simulations, and it examines future research directions in this field.

The starting point for much of the research done in DPC is an algorithm proposed by Foschini and

Miljanic [1], which describes a constant signal-to-interference ratio (SIR) approach. At its core, this

approach is competitive each link attempts to continuously maintain its target SIR by overcoming the

interference presented by all the other links. The later works considered in this report explore a

fundamentally different approach in order to improve upon the Foschini-Miljanic algorithm [2-5]. The

unifying theme of these later efforts is the power versus delay tradeoff [4], in which performance

parameters such as throughput and power consumption are improved by allowing the SIRs of the

various links to fluctuate and letting transmitters delay sending information when the interference is

high. The algorithms that are based on this approach allow the links to cooperate instead of compete.

In particular, as is shown through our simulations, networks designed using these algorithms exhibit a

soft TDMA effect, in which the links coordinate themselves to take turns transmitting at high power,

even though their power control algorithms are completely distributed.

While the papers considered in our survey devote much of their effort to the design of the various DPC

algorithms, the literature is much more sparse with regard to evaluating their performance. This

provided the motivation for our development of a comprehensive simulation platform with which we

can closely examine various performance parameters of DPC algorithms and compare them with each

other. This platform, built in MATLAB, allows a large range of network parameters to be modified,

including the number of transmitters and receivers and their spatial distributions, packet arrival rates,

and channel models.

For this project, three algorithms are simulated, taken from the paper by Bambos and Kandukuri

entitled Power-Controlled Multiple Access Schemes for Next-Generation Wireless Packet Networks

[4]. One of these algorithms is a slight modification of the basic Foschini-Miljanic algorithm, which is

essentially competitive in nature. The other two algorithms, based on backlog-driven DPC, are more

cooperative. Our simulations provide a much more rigorous validation of some of Bambos and


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Kandukuris results than is presented in their paper. Furthermore, our simulations illuminate some

important attributes of backlog-driven DPC that are not directly addressed in their results. Our key

conclusion drawn from the simulations is that backlog-driven DPC algorithms perform significantly

better than the Foschini-Miljanic approach when the network is under duress either when the load

becomes very large or when the cross-link interference is high. These characteristics can be extremely

useful for real ad hoc networks because they can prevent network failures in periods of high congestion.

The simulation platform and associated results provide a stepping stone to future research in this area.

Using the insight gained from our simulations, we can determine which structural properties of backlog-

driven DPC are the most prominent and desirable, and we can move forward with both proving these

properties mathematically and with using them to design the next generation of DPC algorithms. As we

show, there is significant research yet to be done in this field.

The remainder of this report is organized as follows. In Section 2, the evolution of DPC algorithms for

wireless networks is examined through a literature review, which includes the seminal work of Foschini-

Miljanic [1] and a subsequent series of papers that employ backlog-aware DPC approaches. Section 3

describes the simulation that was undertaken as part of this project, including an explanation of the

simulation design and a presentation and discussion of the results. Next, we look at future directions for

this line of research in Section 4, and Section 5 provides some concluding remarks.

2 Literature SurveyIn the literature survey portion of this report, we mainly consider five papers on DPC. Throughout the

survey, our focus will be on the power control algorithms and associated results presented in these

papers, although there are certainly many other interesting aspects that could be discussed. We begin

with a description, based on the concise formulation in [6], of the simple network model that is used

throughout papers in this research area, and then proceed to the seminal work of Foschini and Miljanic

entitled A Simple Distributed Autonomous Power Control Algorithm and its Convergence [1] in Section

2.1. After reviewing this paper, which introduces what we shall refer to as the constant SIR algorithm,

we will proceed in Section 2.2 to briefly discuss two papers by Rulnick and Bambos [2,3] that show the

beginning of the movement toward backlog-aware DPC. Finally, we will take a look (also in Section 2.2)

at works by Bambos and Kandukuri [4] and by Dua and Bambos [5], both of which display the trend

toward more sophisticated approaches to backlog-driven DPC through dynamic programming

techniques.

2.1 Wireless Network Model and the Foschini-Miljanic DPC AlgorithmIn order to establish a standard upon which we can compare various DPC algorithms, we begin this

section with a description of a simple wireless network model. This model or a slight variant is used in

all of the papers that we consider in this report, and it provides the framework for the simulations that


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are conducted. Although it is presented in several places (see, for example, [7] Chapter 14.4), we will

largely follow the exposition provided in [6]. Once this framework is established, we will describe the

operation of the basic Foschini-Miljanic algorithm, which serves as a baseline metric for all of the other

papers we will look at.

Consider a channel in which there are N links, and let Gij be the power gain from the transmitter of linkj

to the receiver of link i. Although the wireless channel typically exhibits path loss, shadowing, and

multipath fading as components of this gain [7], all of the papers that we consider in this report take the

gain to be deterministic (e.g., a function of path loss only). A commonly used measure of link quality of

service is the SIR, which we define as

= = (1)where Pi is the transmit power of link i and

> 0is the thermal noise power at the receiver of link i [6].

Lets also assume that each link has a minimum SIR threshold that must be met in order to meet itsquality of service requirements. We will see that in the Foschini-Miljanic algorithm, the objective is to

simultaneously satisfy all of these thresholds for all links in the network, whereas for the backlog-aware

techniques, this requirement is relaxed to achieve other performance benefits. Lets call the threshold

for link ii. Setting up our basic equation in matrix form, the Foschini-Miljanic objective is to achieve

(I F)P u, with P 0 component-wise, where P = (P1, P2, , PN) is the vector of transmitter powers,

= , , , (2)is a vector of normalized noise and interference powers, and

= for i,j {1,2,,N}, (3)where is the indicator function which equals 1 if ij and 0 if i=j [6].The matrix Fis nonnegative element-wise and irreducible, so by the Perron-Frobenius theorem, the

maximum modulus eigenvalue ofF, F, is real, positive, and simple, and its corresponding eigenvector is

component-wise positive. Furthermore, the existence of a vector P 0 satisfying (I F)P u is

equivalent to F < 1 and also to the fact that (I F)-1exists and is component-wise positive [6]. If any of

these conditions hold, then the power vector

= ( ) (4)is the Pareto optimal solution which satisfies all the minimum SIR thresholds simultaneously [6]. By

Pareto optimal, we mean that it is the solution which minimizes the transmit power of each user

clearly, a desirable situation.


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The landmark result obtained by Foschini and Miljanic can be summarized as follows: as long as there

exists a vector under the conditions described above (equivalently, as long as F < 1), then a simpleiterative DPC algorithm for the links in the network will always result in exponentially fast convergence

to [1]. In their own words, Foschini and Miljanic describe the algorithm as:Each user proceeds to iteratively reset its power level to what it needs to be to have acceptable

performance as if the other users were not going to change their power level. Yet the other

users are following the same algorithm and, therefore, are changing their power levels. [1]

Rather remarkably, as Foschini and Miljanic go on to show, this iterative approach results in

exponentially fast convergence to anytime such a vector exists for the overall system. In equationform, for each link, the algorithm can be written as

( + 1) = () ( ) (5)where we use the discrete-time index kand we recall that is the minimum SIR threshold for link i. It isimportant to observe that each link makes its power decision for the next step autonomously the nextpower level chosen is simply a function of the links individual SIR target, its current power level, and its

own observed SIR. If there does not exist a vector which solves the equation (I F)P u accordingto the requirements above, then this algorithm will result in all of the transmitter powers diverging to

infinity [1].

This simple algorithm, which we shall refer to for convenience as the constant SIR algorithm (since the

goal of the algorithm is to maintain a constant link SIR) provides an often-used baseline metric for the

evaluation of other DPC algorithms. For our purposes in the simulations below, we use a slightly

modified form of this algorithm (described in Section 2.2.4) to compare it with backlog-driven DPCalgorithms.

2.2 The Backlog-Driven Distributed Power Control ApproachNow, we examine the progression from the constant SIR algorithm to more recent research in backlog-

driven DPC by reviewing four papers ranging in publication date from 1997 to 2007. The first two of

these papers [2,3] clearly show the beginnings of the movement toward backlog-aware DPC, while the

later two [4,5] represent more sophisticated attempts to design backlog-driven algorithms.

2.2.1 Mobile power management for wireless communication networksIn [2], Rulnick and Bambos address the problem of conserving energy in wireless communication

networks. The stated objective of this work is to guarantee an average transmission rate (used here as

the primary quality of service metric) while keeping power consumption as low as possible. They set up


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the problem as an optimization problem, seeking a rule which minimizes energy consumption while

maintaining a fixed transmission rate. The problem formulation assumes a single link in stationary,

unresponsive interference. This assumption plays a significant role in determining their solution, but it

is clearly undesirable for wireless ad hoc networks, in which it is highly likely that the interference will be

responsive to the actions of each link (for example, this is the case in the Foschini-Miljanic formulation).

While the authors recognize this, they proceed to use the solution that they obtain for this special case

to suggest a distributed power control algorithm called DPMA (dynamic power management algorithm),

and they test the results of this algorithm in a responsive interference environment. The key aspect of

the DPMA is that the links behave aggressively (i.e., transmit at higher power) during low interference

but back off during high interference. Although DPMA is not backlog-aware, we mention it because this

type of behavior is a general characteristic of the backlog-driven DPC algorithms that we will consider,

and it shows a marked difference in approach from the constant SIR algorithm we discussed above.

Perhaps most significantly, this paper relaxes the requirement for a constant minimum SIR and allows

the SIRs to fluctuate this flexibility is a core element of the later backlog-driven DPC algorithms. It

should be noted that there are hard-delay-constrained applications, such as voice, for which it is

important to maintain a constant minimum SIR, and accordingly approaches which allow the SIR to

fluctuate are perhaps less useful in these cases. However, for many modern and emerging wireless

networks, data traffic is extremely important, and data-driven applications are not as constrained as

voice in terms of delay therefore, the backlog-driven algorithms we examine have a high level of

applicability for real systems.

2.2.2 Power-induced time division on asynchronous channelsTwo years after [2] was published, Rulnick and Bambos authored a paper entitled Power-induced time

division on asynchronous channels [3], which builds on the work in [2] and explicitly introduces the idea

of backlog-sensitive power management. In this work, the authors explore how distributed power

control methods can induce TDMA-like behavior in a wireless network a feature that is prominently

displayed in the algorithms we consider later in this report. This paper captures the essence of the

backlog-driven DPC problem, which can be summed up as: is it possible to induce the links in a wireless

network to cooperate with each other instead of compete against each other through a fully distributed

algorithm, so as to achieve better overall network performance? The authors proceed to design a

backlog-sensitive DPC algorithm, which they call modified DPMA, and although the algorithm does not

perform particularly well, they show that it has promise as a method to induce TDMA-like effects in a

wireless network [3].

2.2.3 Power-Controlled Multiple Access Schemes for Next-Generation Wireless PacketNetworks


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The next paper we consider is [4], which provides the basis for the simulations conducted in this project.

The DPC algorithms presented in this work are the first that we have looked at that can truly be

described as backlog-driven. The authors use a dynamic programming approach to look at thepower

vs. delay dilemma [4], which they describe in the following manner. In a wireless packet communication

network, when a transmitter observes high interference in the channel, it realizes that it will require

high power in order to overcome the interference and successfully transmit a packet to its receiver.

Accordingly, it might consider backing off and waiting to transmit until the interference is lower, while

buffering its incoming traffic basically trading off increased delay for reduced power. However, when

it has backed off, its buffer begins to fill up, putting pressure on the transmitter to be aggressive in order

to rapidly reduce its backlog trading off increased power for reduced delay [4].

The authors method for designing backlog-driven DPC algorithms begins by using dynamic

programming to find an algorithm which minimizes the average overall incurred cost when a single

communication link operates in a channel with extraneously driven random interference (the same basic

idea used in [2], as we described above). The overall cost which they attempt to minimize is composed

of a power cost and a backlog cost, which are increasing functions of the link power and backlog,

respectively. They proceed to solve for the optimal power which minimizes this overall cost for different

functional forms of the probability of a successful transmission within a time slot. Then, they leverage

the observed structural properties of these solutions to design backlog-driven DPC algorithms. Their key

observations are that under low interference, it is effective to transmit and thereby reduce the

backlog/delay cost by incurring a moderate power cost. However, as the interference goes up, the

probability of a successful transmission decreases, and it becomes preferable to incur some delay cost to

avoid an excessive power cost. Interestingly, they observe that this tradeoff process is ubiquitous across

different functional forms of the probability of a successful transmission, s(p,i), which is only constrained

to be increasing in p (power) and decreasing in i (interference). Furthermore, as the backlog increases,

the optimal behavior across different functional forms of s(p,i) is for the link to transmit more

aggressively (higher power) [4].

Although these are similar to the observations as the ones made in [2], they come from a fundamentally

different mathematical approach (dynamic programming) to setting up and solving the problem. The

authors use the observations they derive from the process of obtaining a solution to the extraneous

interference case to propose two different classes of backlog-driven DPC algorithms, which we describe

here.

Consider a responsive interference environment with multiple links, where, for example, if a link

increases its transmit power, the interference observed by all the other links increases. Suppose also

that we operate in slotted time, indexed by k, and that each link can observe only its own packet queue

(current backlog), its last transmitted power level, and the collective interference level that the entire

network induced on it in the last time slot. A packet arrives into each links transmitter queue in each

time slot with probability (for the simulations below, we assume is drawn from a uniform [0,1]

distribution), and packet arrivals are statistically independent of each other. In the kth time slot and for a

particular link, let bk be the transmitter queue backlog, ik be the interference observed at the link


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receiver, and pk be the link transmit power. Finally, let G be the power gain from the links transmitter

to receiver [4]. Then the family of algorithms entitled PCMA-1 algorithms has each link in the network

update its power autonomously according to the following scheme:

= ( ) , 0. By modifying

X(b), , and , an entire family of algorithms can be generated [4].

The second family of algorithms presented in this work, entitled PCMA-2 algorithms, is characterized

by the equation:

= log () , < ( )0, ( )

(7)

where > 0. This family of algorithms is parameterized by X(b) and . A simulation of PCMA-2 for two

interfering links with X(b) = b+4 and = 1 is shown in Figure 1 below. It is clear that the two links are

taking turns using the channel, alternating between periods of high and low transmit power. This can

be thought of as a soft TDMA type of behavior, where the links independently configure themselves to

behave like a TDMA channel.

Figure 1: Example Power Evolution of PCMA-2 with Two Links Showing Soft-TDMA Effect

The authors then proceed to evaluate the performance of these two algorithms in comparison with a

modified version of the constant SIR algorithm, which is, for each link i:

= , > 00, = 0 (8)

3000 3500 4000 4500 5000 5500 6000 6500 7000 750010

20

30

40

50

60

70

Time

Power

PCMA-2 Model with Two Links


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where we recall from above that is the minimum threshold SIR for link i and Rk is the SIR of link i attime k. We note that this algorithm does in fact depend on the backlog, but only in a trivial sense

1.

While the authors put much of their effort into setting up the problem and designing PCMA-1 and 2, the

simulation that they conduct to obtain their performance evaluation is rather simplistic and leaves some

room for doubt about the benefits of PCMA-1 and 2 in comparison to the constant SIR algorithm it isthis fact that was the major motivating factor for the simulations conducted in this project. They use a 4

x 4 square lattice of 16 square cells with the edges wrapped around to create a torus (to avoid boundary

effects), and each square has a single link located at its center. A packet arrives at the transmitter

queue of each link with probability , and a packet is successfully transmitted for link n in each time slot

with probability

( , ) = = (9)where again Rn is the SIR of link n at time k. For this simulation, they use ===1. Note that for the

constant SIR algorithm (when it converges), the success probability of each link will be constant and a

function of the minimum threshold SIR. The constant SIR algorithm target for all links is set to 1.5. The

authors use a simple path loss model for the gains Gij, where

= (10)and rij is the distance from the transmitter of link j to the

receiver of link i (note that for their simulation, the

transmitter and receiver of link i are assumed to be in the

same location, and the gain G ii is taken to be 1) [4]. Finally,

the function X(b) in PCMA-1 and 2 is taken to equal b+4.

The results of this simulation are conveyed through a plot

which shows average backlog vs. average load and is

reproduced to the right as Figure 2. The curve that turns

vertical the furthest to the left is for the constant SIR

algorithm, while the other two curves are for PCMA-1 and 2.

Although this graph is used to claim that there is a 20 percent

improvement in throughput by using PCMA-1 and 2 (since

they can support an average load of .5 before blowing up, as

opposed to the average load of .4 that can be supported by

the constant SIR algorithm), these results actually do not

conclusively show the benefits of their proposed backlog-

driven algorithms, for several reasons. One of the key issues

1Note that this modification is quite practical in the case of data-driven networks, since it is unnecessary to waste

transmit power if there is no data to be sent.

Figure 2: Taken from [4]


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with their claim is that the target SIR for the constant SIR algorithm appears to be strictly a function of

the chosen link spatial distribution2, and their simulation assumes a direct relationship between the

probability of a successful transmission and the link SIR. Therefore, changing the SIR target to a larger

value will directly result in a shift of the constant-SIR curve to the right, potentially erasing any of their

performance gains. Secondly, the transmitter and receiver spatial distribution is highly unrealistic, and it

throws into doubt the broader applicability of the observed performance gains. Furthermore, the

relative power consumption of the algorithms is completely neglected, and this parameter is obviously

one of importance for the evaluation of backlog-driven DPC. Therefore, it was clear after an

examination of their results that there remained significant work to be done to obtain meaningful

comparisons of these algorithms, which led to the simulations created for this project.

2.2.4 Distributed Backlog-Driven Power Control in Wireless NetworkingThe last paper we will briefly consider is [5], published in 2007 by Dua and Bambos. We would mainly

like to highlight the approach taken in this paper, which is very interesting and perhaps indicative of thedirection that future work in this field will take. Once again, dynamic programming is the main

mathematical tool used in their design of a backlog-driven DPC algorithm, but this time, the authors

begin by considering a situation in which there are two interfering links and solving for the optimal

power control solution a marked difference in approach from the previous works, which solved for

optimal solutions in the extraneous interference case. Although the solution obtained is only for two

links because of the complexity involved in using a higher number, this is still a step in the right direction

as far as obtaining a truly optimal backlog-driven DPC algorithm.

Another intriguing aspect of this approach is the attempt to design an Oracle what the authors

conceive of as the optimal solution which would be obtained if there were a centralized controller and

the use of this solution as a benchmark with which to compare the performance of their DPC algorithm,

which they call BDD (backlog-driven distributed power control). They show that the Oracle naturally

induces a load balancing effect, in which a transmitter with a large backlog is assigned a transmit power

to reduce its backlog rapidly, and one with low backlog is assigned a lower transmit power [5].

The third feature of this paper that we would like to call attention to is that their method of deriving

BDD incorporates a quasi-game-theoretic approach, because the transmitter behavior of link 1 is

determined for each of three possible strategies chosen by link 2, roughly categorized as aggressive,

backoff, and static. There is much potential in this line of research, and it will certainly be

interesting to see how it will evolve in the coming years.

2It appears that the authors set the constant SIR target to be near the maximum for which the constant SIR

algorithm will converge, given their chosen spatial distribution of links. However, this still means that this target

SIR, and therefore the average load that can be supported by their simulation of the constant SIR algorithm, is a

function of this particular spatial distribution.


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3 Simulation of Distributed Power Control AlgorithmsIn this section, we discuss the simulation that was conducted for this project. The motivation for this

work has already been alluded to above we are seeking a more comprehensive, accurate, and useful

evaluation of the properties of backlog-driven DPC algorithms. In particular, we are looking to improveupon the simulation done in [4]. With this in mind, a large-scale, robust, and easily configurable

simulation was designed, and the results provide some interesting insights into the advantages of

PCMA-1 and PCMA-2 [4] over the constant SIR algorithm.

3.1 Simulation DesignWe begin by explaining the structure of the simulation that was performed. One of the major design

considerations was that we wanted to be able to simulate a very large and randomly dispersed number

of links (i.e., transmitter/receiver pairs) which can be chosen by the user. Then, transmitters are

distributed uniformly over a two-dimensional playing field, and the receiver associated with each

transmitter is placed at (r,) with

respect to its transmitter, where r

is the distance from the transmitter

chosen randomly from a Gaussian

distribution, and is the angle

chosen from a uniform distribution

on [0,2). We should note that in

order to avoid boundary effects in

our performance calculations, we

actually place many more links in

the playing field than the number

of links we intend to use, and then

choose the user-defined number of

links for performance calculations

from the middle of the playing

field3. A sample of this spatial

distribution is included in Figure 3.

Other important parameters that

can be set by the user include the

average arrival rate of new packets

to be transmitted (arrival

3This is in contrast to [4], in which boundary effects are avoided by turning the playing field into a conceptual

torus. We felt that this method is unnecessarily unrealistic, and it introduces numerous problems, such as

potentially double-counting the interference from a neighboring link.

Figure 3: Transmitters and Receivers in the Two-Dimensional Playing Field.Transmitters are represented by blue circles, and receivers are represented as

orange squares.


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probabilities are drawn from a Bernoulli distribution, as in [4]), the various configurable parameters in

the PCMA-1 and 2 schemes such as X(b), and the constant SIR algorithm target SIR. A version of the

simulation code is available at www.stanford.edu/~jmounzer/359.

3.2 ResultsA number of simulations were run to test PCMA-1 (Equation 6), PCMA-2 (Equation 7), and the modified

constant SIR algorithm (Equation 8) in order to show a variety of different relationships, as will be

described below. Key simulation settings are listed in Table 1. Wherever possible, they were chosen for

consistency with the simulation in [4].

Simulation Parameter Parameter Value or Function

Length of simulation 25,000 time steps, unless otherwise noted

Probability of successful transmission

( , ) =

=

(Equation 9)

Gains Gij = (Equation 10)Interference on link i + Thermal noise .1

X(b) b+4

Number of links considered in performance calculations 75

Total number of links in playing field 200

, , 1

Table 1: Simulation settings

Based on these simulations, we can make several observations regarding the relative performance of

PCMA-1 and PCMA-2. Perhaps the most dramatic and notable performance improvements of PCMA-1

and PCMA-2 over the constant SIR algorithm are observed when the network is under duress. This is

very significant, since real networks are likely to have traffic fluctuations that result in temporary or

long-term congestion. For example, one can easily envision a network experiencing a period of

extremely heavy traffic, during which the average load nears unity. In this type of situation, the

Foschini-Miljanic algorithm has disastrous results either the transmitter powers will all explode if the

target SIR is adapted to attempt to keep up with the load (these simulations showed that it takes

relatively few time steps for this to occur in the tens of time steps) or the backlog will rapidly increase

if the target SIR is not adapted, at a far faster rate than the backlog of PCMA-1 or PCMA-2. In contrast,PCMA-1 and PCMA-2 both exhibit much more robust performance in such stressful situations. By

cooperating with each other, the links are able to keep their powers from exploding while managing

their backlogs more effectively. These effects are clearly shown in the set of figures below. In Figures 4

and 5, we show the analogous plots to those in [4], where we set the constant SIR algorithm to achieve a

steady-state success probability of .7. As expected, we see that in Figure 4, the average backlog in the

constant SIR algorithm when the load is less than .7 is driven to zero, while the average backlog when


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the load is greater than .7 grows rapidly (and goes to infinity as the simulation length goes to infinity).

However, the PCMA-1 and 2 curves both perform better in terms of average backlog than the constant

SIR algorithm as the load becomes very large. We also observe that while the PCMA-2 curve performs

rather poorly in terms of backlog compared to the constant SIR algorithm for average loads between .3

and .8, it uses less power over the same range. As we look at the average power for PCMA-1, we see

that for average loads less than .7, it remains very close to the average power of the constant SIR

algorithm4, but when the load becomes large, the power consumption goes up. However, we see that

the PCMA-1 algorithm is trading off increased power for reduced backlog in this situation; over the same

region, the constant SIR algorithm has allowed the backlog to explode.

The next pair of plots in Figures 6 and 7 provides a slight but illuminative twist on the graph in [4]. This

time, we allow the constant SIR algorithm to adapt its constant SIR target to keep up with the average

load, i.e., we set the target SIR so that a success probability higher than the average load is maintained,

if possible. Here, we clearly see that there is a hard limit with regard to the feasibility of the constant

SIR algorithm. For some average load (in this case, approximately .7), the target SIR will become

unattainable for the links in the network, and as Foschini and Miljanic showed, this results in the link

powers rapidly exploding to infinity. This will be the case as the average load increases regardless of the

network spatial distribution at some point, the network under the constant SIR algorithm will not be

able to support the requested target SIR. We see that PCMA-1 and 2 vastly outperform the constant SIR

algorithm when the load becomes high for this situation. The backlog and power for the constant SIR

algorithm immediately explode once the target cannot be reached, whereas PCMA-1 and 2 exhibit a

smooth, gradual decay in performance.

4The linear growth in average power for the constant SIR algorithm in Figure 5 for average loads less than .7 is

simply a result of the modification made to the Foschini-Miljanic algorithm for these simulations namely, that

there is no power spent when there are no packets waiting to be transmitted. After the average load becomes

higher than the success probability at .7, we see the constant SIR algorithm result in a constant average power, as

we expect, since the links never clear their queues.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

4000

Average Load

AverageBack

log

Average Backlog vs Average Load with Constant SIR Algorithm target SIR of 2.33

PCMA-1

PCMA-2

Constant SIR

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Average Load

AveragePo

wer

Avg Power vs. Avg Load with Constant SIR algorithm target of 2.33

PCMA-1

PCMA-2

Constant SIR

Figure 4 Figure 5


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The last plot we consider is Figure 8, which clearly shows how PCMA-1 and 2 can outperform the

constant SIR algorithm when the spacing between links decreases (and consequently, when the

interference between links goes up). We see

that at a certain link density, the constant SIR

algorithm can no longer converge, and all the

transmit powers explode to infinity. However,

the PCMA-1 and 2 algorithms are able to

prevent their powers from exploding over

much higher densities through their use of

cooperation5. This robustness can be very

valuable in cases of high spatial congestion.

Finally, we observe that the simulations

conducted represent a preliminary attempt at

validating and extending the results in [4]. It

proves to be surprisingly difficult to compare

the algorithms along any particular dimension, such as average power consumption, because of their

different structures. For example, a cursory glance at Figure 5 shows that the functional forms of

average power with respect to average load are all quite different, which makes their comparison highly

nontrivial. Future efforts in this line of research will include finding methods to make these types of

comparisons, and we hope that the process of doing so will lead to new insights regarding the

algorithms behaviors.

5The fluctuations in the PCMA-1 and 2 curves in Figure 9 are due to the fact that the simulation was reset for each

density and the links were randomly redistributed. Future work will include taking averages over many simulations

so that the curves are smoother, but the current plot is enough to illustrate our main point.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Average Load

AveragePower

Average Power vs. Average Load with Adaptive Constant SIR Target

PCMA-1

PCMA-2

Constant SIR

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

12000

Average Load

AverageBacklog

Avg Backlog vs. Avg Load with varying Constant SIR Target for each Average Load

PCMA-1

PCMA-2

Constant SIR

Figure 6 Figure 7

10-3

10-2

10-1

0

50

100

150

200

250

300

Transmitter Density

AverageTransm

itterPowerafter12000timesteps

Average Power vs. Transmitter Spatial Density for Arrival Rate of .7

PCMA-1

PCMA-2

Constant SIR

Figure 8


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4 Future WorkThere is much work yet to be done in this line of research, and we roughly characterize these future

directions into three groups: extending the simulations described above to learn more about the

performance of backlog-driven DPC, proving mathematically the structural properties of backlog-drivenDPC that are observed (such as finding the mathematical reasons for the robust behavior when the

network is under duress), and using the intuition we have gained to design improved DPC algorithms in

the future.

4.1 Simulation ExtensionsThere are a number of ways that the simulations used in this project can be extended to provide a more

complete characterization of the behavior of backlog-driven DPC. For example, it would be very

interesting to employ more realistic transmitter and receiver spatial distributions and configurations tobetter simulate a real wireless environment, as well as to vary the packet arrival process. Furthermore,

employing a stochastic wireless channel model (perhaps including shadowing and multipath fading)

could lead to new insights. Another important area for research would be to make the network itself

more dynamic by letting links enter and leave and by allowing transmitters and receivers to move

around. This would enable us to more accurately characterize properties such as convergence of the

algorithms and the maximum allowable steady-state number of nodes under different algorithms.

Additionally, the PCMA-1 and 2 algorithms are parameterized by a number of different values, which

can be changed to induce dramatically different link behavior. There has been little done as far as

exploring the optimal values and functional forms for these parameters to achieve a desiredperformance gain, and further simulations could look into deepening our knowledge on this front.

Another important direction for extending the simulations is to find better ways to compare the

algorithms with each other. As described in Section 3.2, the different natures of the algorithms make it

highly nontrivial to compare them along any particular dimension (such as power consumption, average

backlog, etc.). There remains a significant amount of work to be done in this regard.

4.2 Proving the Structural Properties of Backlog-Driven DPCThe structural properties of backlog-driven DPC are not yet very well understood, and we hope that by

leveraging the insights gained through extensive simulations, we can begin to characterize these

properties mathematically. For example, it would be interesting to prove that the mathematical

properties of PCMA-1 and 2 naturally result in their gradual performance decay under large average

loads, as we observed in our simulations.


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4.3 Design of Stronger DPC AlgorithmsThe ultimate goal of this entire line of research is to develop optimal DPC algorithms suitable for a wide

range of wireless network environments. As we have seen in our literature review, this is a difficult

problem to solve, and the current state of the art involves developing well-informed heuristics based on

analyzing simplified models of the wireless environment, such as the case of unresponsive, extraneous

interference. However, we have also seen that the techniques employed have become more and more

sophisticated over time, and with a better understanding of the properties of backlog-driven DPC

obtained through simulation and mathematical analysis, we hope to push the state of the art in this field

further.

5 ConclusionIn this report, we have examined the evolution of backlog-driven DPC algorithms over the past 15 years,

beginning with the benchmark, backlog-agnostic Foschini-Miljanic approach and ending with recent

work in the dynamic-programming-based backlog-driven approach. Through our analysis and

simulations, we have shown that there are significant performance gains for wireless networks in which

the links can be induced to cooperate with each other through backlog-driven DPC. Although the full

extent these performance gains is not trivial to characterize because of the different natures of the DPC

algorithms, we have clearly highlighted the fact that when the network is under duress, backlog-driven

DPC can provide enormous gains over the standard constant-SIR benchmark in terms of power

consumption, backlog, and overall reliability. There remains much work to be done to find appropriate

ways to compare DPC algorithms, prove the structural properties of backlog-driven DPC, and create new

DPC algorithms which build on our previous insights to achieve even larger performance improvements.


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6 References[1] G.J. Foschini and Z. Miljanic, "A Simple Distributed Autonomous Power Control Algorithm and Its

Convergence," IEEE Trans. Vehicular Technology, vol. 42, no. 4, pp. 641-646, 1993.

[2] J. Rulnick and N. Bambos, "Mobile power management for wireless communication networks," Proc.IEEE INFOCOM , vol. 3, pp. 3-14, 1997.

[3] J. Rulnick and N. Bambos, "Power-induced time division on asynchronous channels," Wireless

Networks, vol. 5, pp. 71-80, 1999.

[4] N. Bambos and S. Kandukuri, "Power-Controlled Multiple Access Schemes for Next-Generation

Wireless Packet Networks," IEEE Wireless Communications , pp. 58-64, June 2002. s

[5] A. Dua and N. Bambos, "Distributed Backlog-Driven Power Control in Wireless Networking," Proc.

IEEE Workshop on Local & Metropolitan Area Networks, pp. 13-18, June 2007.

[6] N. Bambos, Toward Power-Sensitive Network Architectures in Wireless Communications: Concepts,

Issues, and Design Aspects, IEEE Personal Communications, pp. 50-59, June 1998.

[7] A. Goldsmith, Wireless Communications, Cambridge University Press, New York, 2005.

power control algorithms in wireless communication

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