Energy-efficient rate scheduling and power allocation ... - UPCommons

Energy-efficient rate scheduling and power allocation for wireless energy harvesting nodes

by

Maria Gregori Casas Master thesis advisor

Dr. Miquel Payaró Llisterri

July 2011

A thesis submitted to the Departament de Teoria del Senyal i Comunicacions of the Universitat Politècnica de Catalunya for the degree Master of Science

Centre Tecnològic de Telecomunicacions de Catalunya Castelldefels, Barcelona

To my family, friends and Paul.

Contents 1 Introduction 1.1 Green communications . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Communication system . . . . . . . . . . . . . . . . 1.2.2 Sources of energy consumption . . . . . . . . . . . 1.2.3 Renewable energies and energy harvesting nodes . . 1.3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 On the energy-efficiency of channel codes . . . . . . 1.3.2 Energy-efficient packet scheduling and rate control . 1.3.3 Energy-efficiency in the spatial domain . . . . . . . 1.4 Summary of the thesis . . . . . . . . . . . . . . . . . . . .

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1 1 2 2 3 5 5 6 6 9 9

2 Efficient data transmission for an energy harvesting node with battery capacity constraint 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Properties of the optimal solution . . . . . . . . . . . . . . . . . . . 2.4 Problem insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Problem visualization . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Constraints mapping into the data domain for a given epoch 2.4.3 Maximum and minimum rates . . . . . . . . . . . . . . . . . 2.5 Optimal data departure curve construction . . . . . . . . . . . . . . 2.5.1 Finish transmission at a constant rate . . . . . . . . . . . . . 2.5.2 Get rate and length of the next epoch . . . . . . . . . . . . . 2.5.3 Algorithm optimality . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 11 12 14 16 16 17 17 18 19 21 21 21

3 Efficient data transmission for an energy harvesting node with battery capacity and QoS constraints

23 i

3.1 3.2 3.3 3.4

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23 23 25 26 26 27 31 31 32

4 Throughput maximization for a multi-antenna energy harvesting node 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Throughput maximization problem . . . . . . . . . . . . . . . . . . . . . . 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 33 36 39 41

5 Conclusions and future work 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46

A Proofs of Chapter 2 A.1 The overflow problem A.2 Proof of Lemma 2 . . A.3 Proof of Lemma 3 . . A.4 Proof of Lemma 4 . . A.5 Proof of Lemma 5 . . A.6 Proof of Theorem 1 .

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47 47 48 49 49 50 50

B Proofs of Chapter 3 B.1 Proof of Lemma 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55

3.5 3.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the optimal solution . . . . . . . . . . . . . . . . . . . Problem insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Problem visualization . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Constraints mapping into the data domain for a given epoch Optimal data departure curve construction . . . . . . . . . . . . . . 3.5.1 Algorithm optimality . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 1.1

General scheme of a communication system. . . . . . . . . . . . . . . . . .

2

2.1 2.2 2.3

Summary of the packetized arrival model . . . . . . . . . . . . . . . . . . . Visualization of the problem presented in (2.3). . . . . . . . . . . . . . . . Mapping of the energy causality constraint and minimum energy expenditure to the data domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding rates of the possible points of rate change. . . . . . . . . . . . . .

12 16

Visualization of the problem presented in (3.1). . . . . . . . . . . . . . . . Mapping of the energy causality constraint and minimum energy expenditure to the data domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mapping of the energy causality constraint and minimum energy expenditure to the data domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

The MIMO communication scheme. . . . . . . . . . . . . . . . . Transmitter block diagram. . . . . . . . . . . . . . . . . . . . . Summary of the time notation. . . . . . . . . . . . . . . . . . . Throughput comparison for N = 40 packets of energy. . . . . . . Throughput comparison when the node does not harvest energy. Throughput comparison for N = 20 packets of energy. . . . . . .

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34 34 36 40 41 42

A.1 Graphical representation of the problem presented in Appendix A.1. . . . .

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2.4 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6

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Acknowledgements I would like to deeply thank the people who, during the several months in which this endeavor lasted, provided me with useful and helpful assistance. First of all, I would like to thank my advisor Miquel Payaró Llisterri for the valuable comments, remarks and contributions to this Master Thesis. This thesis would not have been possible without the financial support of both CTTC and AGAUR1 . I would also like to thank all the PhD students and staff of CTTC for building such a friendly workplace. Last but not least, I will be eternally thankful to my family and friends because without their support, love and help, none of this would have been possible.

1

This thesis was conducted with the support from the Generalitat de Catalunya through the scholarship

2011FI_B 00956

v

AWGN

Additive White Gaussian Noise

BER

Bit Error Rate

CSI

Channel State Information

ICT

Information and Communications Technology

ISO

International Standards Organization

LT

Luby Transform

MAC

Medium Access Control

MAP

Maximum a Posteriori

MFSK

Multiple Frequency-Shift Keying

MIMO

Multiple-Input Multiple-Output

ML

Maximum Likelihood

OFDM

Orthogonal Frequency Division Multiplexing

OFDMA Orthogonal Frequency Division Multiple Access OSI

Open System Interconnection

QoS

Quality of Service

SISO

Single-Input Single-Output

SNR

Signal to Noise Ratio

SVD

Singular Value Decomposition

WER

Word Error Rate

WSNs

Wireless Sensor Networks

vi

Abstract Energy harvesting is increasingly gaining importance as a means to charge battery powered devices such as sensor nodes. Traditional rate scheduling and power allocation strategies for wireless nodes are no longer optimal when these nodes are able to harvest energy over time. Efficient transmission strategies must be node. In this thesis, we have considered that both data and energy arrivals are produced following a packetized model. We have assumed that the node has from beforehand full knowledge of the arrival times and quantities (either bits or Joules) of the packets. This thesis solves three different problems for wireless energy harvesting nodes. First, we find the best rate scheduling strategy for a wireless energy harvesting node with finite battery capacity. Second, we constrain the node to fulfill an additional quality of service constraint and, again, find the optimal rate scheduling strategy. In the third problem, we consider a Multiple-Input Multiple-Output (MIMO) point-to-point transmission. Hence, we consider a multiple antenna energy harvesting node for which we find the precoder that maximizes the data throughput over a certain time window. Finally, we have developed algorithms that compute the optimal solution of each of the aforementioned problems.

Chapter 1 Introduction 1.1

Green communications

Information and Communications Technology (ICT) usage has grown exponentially last years both in terms of number of users and required data rates. In 2007, the ICT sector produced 1.3% of global greenhouse gas emissions [1]. Regarding the electricity use, the ICT community expends around 7% of the global electricity bill [2]. Moreover, the analysts predict an increase of these figures in the upcoming years, i.e., it has been foreseen that the ICT energy expenditure in 2020 will be around 20% of the global energy consumption. The unsustainability of this situation yields the ICT community to focus the attention in the study and design of sustainable data centers, components and communication networks. Traditionally, the design of communication systems follows the Open System Interconnection (OSI) model, a layered architecture where each layer interacts only with the layers that it has directly beneath or above. This model was developed by the International Standards Organization (ISO). At the beginning, it simplified the communication systems design by introducing different levels of abstraction, however, it leads now to suboptimal network designs due to the dependencies between the different layers. For instance, when an energy-efficient wireless communication system is intended, the physical, Medium Access Control (MAC), link, network, and transport layers must be jointly designed. Up until very recently, research efforts have focused on the design of communication systems that meet capacity. Hence, the goal has been to maximize the rate of the network subject to some kind of energy constraint, i.e., the sum of all the power transmitted must be below a certain threshold. This approach has allowed to satisfy the exponentially increasing bit rate demand of the network users. However, users do not only require for higher bit rates, but also more autonomy and mobility. A tremendous increase of the users that work with battery-powered devices has been observed in recent years. Moreover, Gordon Moore predicted more than forty years ago that the number of transistors that can be placed within an integrated circuit doubles every 1

Information Source

Source Encoder

Channel Encoder

Digital Modulator

Channel Output Sequence

Source Decoder

Channel Decoder

Digital Demodulator

Figure 1.1: General scheme of a communication system. two years. Unfortunately, the battery capacity does not follow the same trend but a slower increase has been observed. This makes power consumption one of the major bottlenecks of current handheld devices. There are many reasons why energy-efficient communication systems must be designed. Starting from an ethical point of view, it is clear that there is an urgent need to reduce greenhouse gas emissions. From the users perspective, energy-efficient communications will enlarge the autonomy of battery powered devices. Finally, from the operators point of view, green communication will reduce energy operational costs. Green calls green $!

1.2 1.2.1

Background Communication system

We will start by briefly reviewing some basic concepts of communication systems in order to show the reader in which parts of the system energy-efficient strategies can be developed. Figure 1.1 shows a common representation of a communication system. In the following we give a brief explanation of the function of each of the blocks, for more information see [3]. The first block, i.e., the source encoder, converts the source of information, which can be analog or digital, to a sequence of binary digits. These bits, usually called information sequence, pass through the channel encoder that introduces redundancy on the information in order to reduce errors that could be produced due to noise or interference. Then, the 2

modulator transforms this binary sequence in electrical signals or waveforms. Note that in the modulator it is possible to adjust the transmission rate, the number of bits of information per channel use, and the transmission power. The channel is the physical medium by which the waveforms are sent from the transmitter to the receiver. Many different channels can be considered, i.e., wired or wireless channels. From the channel, it is important to remark that the information can be corrupted in a random manner, which is usually characterized by some probability distribution. For instance, the thermal noise, i.e., the noise introduced by electronic devices, can be modeled with a zero-mean Gaussian distribution. It is said to be Additive White Gaussian Noise (AWGN). Then the corrupted signal arrives to the receiver demodulator that tries to recover the transmitted data symbols. These symbols are passed to the channel decoder that will try to remove the errors that may have been produced by the demodulator by using the redundancy of the information, therefore, recovering the original information sequence. Note that the recovered sequence may contain errors due to different reasons, i.e., noise, interference, distortion, synchronization problems, attenuation, multipath fading, etc. These errors have been traditionally quantified by the Bit Error Rate (BER) of the system, i.e., the probability of an erroneous bit. The BER is a function of the Signal to Noise Ratio (SNR) and can be improved by choosing slow and robust modulations or by applying channel coding strategy. In order to have reliable communication, the BER must be under a certain threshold. Another possibility, which is recently gaining importance, is to quantify the errors in the recovered sequence through the Word Error Rate (WER). Current receivers use iterative decoding techniques to decode the message. Regardless iterative receivers are not optimal, they can perform , in certain situations, close to Maximum Likelihood (ML) or Maximum a Posteriori (MAP) receivers. The performance metric of iterative decoding is, in general, the WER instead of the BER.

1.2.2

Sources of energy consumption

In this section, we briefly summarize the different sources of energy consumption of the transmitter and receiver nodes and also the attenuation of the signal. Transmission energy consumption The well known expression of the Shannon channel capacity for AWGN channels relates the rate of the communication with the transmission power, PT , i.e., (

R = W log2

γPT 1+ W No

)

(bits/second),

(1.1)

where γ is the channel gain, W is the channel bandwidth and N0 the noise spectral density. Moreover, we can express the rate as a function of the bit duration time, Tb , as R = 1/Tb [3]. 3

From the previous equation be obtain the transmission power consumption as (

)

PT = 2 W − 1 W N0 γ −1 R

(W atts).

(1.2)

Hence, the energy consumed in transmission per bit is (

R

) WN 0

ET = PT Tb = 2 W − 1

Rγ

(Joules/bit)

(1.3)

that is monotonically increasing and convex in R. Hence, there exists a clear trade-off between energy and bit-rate. The bigger the required bit-rate, the larger the transmission energy consumption and, due to the concavity of the rate-power function, the more energy is required to obtain a certain increment in the rate. In other words, the transmission energy consumption reduces with an increase on the bit time, Tb . RF circuitry energy consumption Transmission energy is one of the most important sources of energy consumption, albeit not the only one. Circuitry energy, EC , must also be taken into account. Regarding the transmitter node, the main sources of energy consumption are the synthesizer, filters, mixers, digital to analog converter, the power amplifier, etc. As far as the receiver is concerned, we must take into account the energy expended in the low noise amplifier, the different filters, the mixer, intermediate frequency amplifier, analog to digital converter, etc. It is important to remark that in order to model accurately the energy consumption of the circuitry, some consideration must be taken regarding the communication system such as modulation or channel coding strategy, etc. For instance, it is clear to see that the circuitry energy consumption of a multiple antenna node operating in a MIMO channel is higher than the one for a Single-Input Single-Output (SISO) system. As we will see in Section 1.3, in the literature there exist several papers that analyze the energy-efficiency by fixing some of the parameters of the communication system and comparing others, i.e., choosing a fixed modulation and comparing different channel coding strategies. By now, it is important to know that circuitry energy is a source of energy that many times has not been taken into account when doing the communication system design, which has lead to suboptimal designs. Encoding and decoding Source and channel coding and decoding operations are another source of energy consumption. As briefly explained in Section 1.2.1, channel coding (also known as Error Control Coding) introduces redundancy in the information sequences which leads to an smaller BER for the same SNR with respect to the uncoded system. Conversely, the coded scheme requires an smaller SNR than the uncoded scheme to reach a certain BER. This difference in required SNR to achieve a certain BER is known as coding gain. 4

Then, one could think that a possible way to improve energy-efficiency is the following: Given a certain maximum BER, one could reduce the transmitted power by using more complex channel coding schemes, i.e., with high coding gains. But the problem is not so straightforward. By using complex codes, the decoding complexity increases, and hence, more computations are required to recover the original information which in turn may result in a higher energy requirement. This is another trade off in the design of energy-efficient communication systems, which should be carefully studied. Path loss The attenuation that the channel introduces to the transmitted signal is known as path loss, denoted by L. Usually, the path loss is a function of the distance between the transmitter and the receiver, denoted as r. Then, L ∝ rα , where α depends on the nature environment where the communication is taking place and usually, for wireless systems, ranges between 2 and 4. Path loss is not an inherent source of energy consumption for the transmitter and receiver nodes as the sources explained previously. However, it is directly related to the transmission energy. Note that if the path loss is high, then the transmission energy has to be increased in order to fulfill a certain SNR or BER. Hence, the path loss must also be taken into account when designing energy-efficient communication systems.

1.2.3

Renewable energies and energy harvesting nodes

Energy harvesting is the process of collecting natural energy from the environment. Energy harvesting techniques are gaining importance to power autonomous wireless nodes or handheld devices since they allow to span their operational lifetime. Energy can be captured by different means, i.e., solar cells, piezoelectric, thermoelectric, end electrostatic energy generators, among others. As we will see during the thesis, the fact that energy can be harvested over time modifies the optimal behavior of the nodes, for instance, in terms of optimal power allocation strategies since power cannot be used before it is harvested.

1.3

State of the art

Energy consumption of communication networks has always been considered a cost, and hence, engineers and research community have tried to develop designs that consume as little energy as possible. However, energy efficiency has not traditionally been the target of the design but just a constraint. In this section, we summarize the literature that explicitly deals with energy-efficient designs. We have grouped the literature in three different subsections. In Subsection 1.3.1, we present some works that analyze the energy efficiency of different channel coding strategies. Some works that derive the optimal rate control or power allocation strategy are presented in Subsection 1.3.2, both for harvesting 5

and non-harvesting wireless nodes. Finally, works that study energy-efficiency when spatial diversity is used by the transmitter and receiver nodes are introduced in Subsection 1.3.3.

1.3.1

On the energy-efficiency of channel codes

The role of channel coding for energy efficient communications in Wireless Sensor Networks (WSNs) is studied in [4]. Basically, the trade-off between the transmission energy savings due to coding gain and the increase in the decoding energy consumption has been studied. They derive the expression of the critical distance, dCR , between the transmitter and the receiver, at which the decoder’s energy consumption per bit equals the transmission energy savings per bit compared to an uncoded system. In other words, the minimum distance at which using a certain coding scheme is energy-efficient. They find out that in certain situations such as free space communication at low frequencies, uncoded transmission is preferable. In [5], the authors propose a low-complexity and low-energy consumption modulation for dynamic WSNs. This modulation is called Green Modulation/Coding (GMC) and it is based on the use of Luby Transform (LT) codes along with a Multiple Frequency-Shift Keying (MFSK) modulation. It is shown that LT codes achieve an energy-efficiency similar to the uncoded MFSK for distances bellow the critical distance, i.e., d < dCR , while for distances above the critical distance, d > dCR , LT coded MFSK outperforms all the other schemes. This result comes from the flexibility of the code to adjust its rate and, hence, the coding gain, to the channel conditions. A different approach has been followed in [6] where the authors find out the optimal packet size in energy constrained WSNs taking energy efficiency as optimization metric. The main point of packet size optimization is that longer packets introduce less overhead on the network while they suffer from a higher loss rate. They obtain the optimal packet size when there is no channel coding, both with BCH and convolutional codes, by taking into account the circuitry power of the transmitter and the receiver, PC , the transmission power, PT , the decoding power, PDec , as well as some energy consumption to start-up the different nodes. The results show that in general BCH codes are more energy-efficient.

1.3.2

Energy-efficient packet scheduling and rate control

There are a myriad of works in the literature that deal with the problem of packet scheduling and rate control. In this section, a summary of the most relevant contributions in terms of energy efficiency is given, which we have sorted in three groups. First, we present some works that make use of the energy efficiency figure to find the optimal rate scheduling. The second subsection presents some contributions that find the rate scheduling that maximizes the throughput or minimizes the total completion time of a non-harvesting node. Finally, the last subsection considers the same problem for energy-harvesting nodes. 6

Energy efficiency as optimization metric In equation (1.1), it is easy to see that if higher rates are intended, we can either increase the bandwidth or the transmission power. From an energy-efficiency point of view it is preferable to increase bandwidth, however, it is a limited resource. Moreover, each frequency band within the spectrum is affected by different channel gains, hence, link adaption becomes crucial to take advantage of the available resources. The traditional approach has focused on maximizing the sum-rate in all the bands by fulfilling that the sum of all the transmission powers in each of the bands is under a certain threshold. This leads to the well-known waterfilling solution to assign powers to each frequency band. From here it arises the question, is waterfilling the best that we can do in terms of energy efficiency? Of course, the answer is "no". Waterfilling is not focused on maximizing energy efficiency, but, rather, on optimizing the sum rate. Alternatively, some works define the energy-efficiency metric as a function of the rate, R, as R R U (R) = = , (1.4) P (R) PC + PT (R) where P (R) is the total power, i.e., the circuitry power, PC , plus the transmission power, PT . Then, the rate that maximizes energy-efficiency is found, i.e., R⋆ = arg max U (R) = arg max R

R

R . PC + PT (R)

(1.5)

This has been the approach followed in [7], by optimizing the energy-efficiency metric they find the optimal link adaption and resource allocation in order to transmit the maximum amount of data per Joule of energy. Their considered system is the uplink transmission between multiple mobile users and the base station, where Orthogonal Frequency Division Multiple Access (OFDMA) is used as multiple access technique for the case of having flat-fading channels, i.e., channel state remains constant between signaling intervals at which the channel is estimated. This work is extended to frequency selective channels in [8]. Wireless nodes The problem of energy-efficient packet transmission scheduling was formulated in [9]. In this work, the authors propose a packet scheduling strategy that minimize energy consumption while satisfying Quality of Service (QoS) constraints. They exploit the fact that energy consumption can be reduced by reducing the transmission power or, equivalently, the rate. Hence, by increasing the transmission time of the packet. They find the optimal offline schedule, where by offline they mean that the packet arrival instances are known from the beginning. Moreover, the optimal online schedule, where packet arrival times are not known, is obtained through simulation. In subsequent work, their problem is extended for different scenarios: In [10] variable length packets are considered, while [11] considers a fading channels. 7

Zafer et al. analyze the same problem under a different point of view. They propose novel solutions to the problem of rate control in fading channels when having variable QoS by introducing the concept of cumulative curves [12]. This methodology is generalized in [13], where a rate control policy that minimizes the total transmission energy while satisfying any QoS constraint is found. From these two works, we want to emphasize the solution of the B-T problem, i.e., finding the transmission strategy that requires less energy when there are B bits to be transmitted with a maximum delay of T seconds. They prove that the optimal transmission strategy is to transmit data at constant rate r = B/T during the T seconds. The proof follows from the fact that the function that relates the power with the rate is convex. Hence, constant rate transmission saves energy.

Energy harvesting wireless nodes Energy harvesting techniques are gaining importance to power autonomous wireless nodes or handheld devices. The fact that harvesting nodes are able to collect energy introduces a new constraint in the use of energy. Typically, the nodes must fulfill a sum-power constraint in the use of energy, i.e., the sum of the energy expended in the different time slots or frequency carriers has to be under a certain threshold PT . When energy-harvesting nodes are considered, a new constraint appears usually called energy causality constraint that basically deals with the fact that energy can only be used after it is harvested. All of this makes that the optimal rate control strategy is different from the works presented above, which deal with wireless non-harvesting nodes. In [14], the authors consider a wireless energy-harvesting node for which they find the minimum transmission completion time in the following two situations: (i.) The node has all the data to be transmitted available from the beginning and both the arrival time and amount of energy in each of the packets are known. (ii.) The data and energy packets arrive at the node at known instants and with known amounts (bits for data packets and Joules for energy arrivals). In both cases, the channel is assumed to be static. Note that this is an offline approach, since the instances at which data and energy packets arrive are considered to be known. The implementation of an online approach is shown to be complicated and was left for future research. However, the authors of [14] assume that the battery capacity of the node is infinite. In [15], a node with finite battery capacity is considered and the minimum completion time is found, however, considering that all the data is available at the beginning. Both of these works, i.e., [14] and [15], follow the calculus approach introduced by Zafer et al. in [13] that is based on cumulative curves and that has been briefly described in previous section. A different path is taken in [16], where the energy allocation strategy that maximizes throughput is found by means of dynamic programming and convex optimization for a slotted transmission in a time selective fading channel. In a very recent work [17], a new approach is taken for the same problem that leads to a more intuitive solution called directional waterfilling. In [18], we expand the works in [14] and [15], by finding the offline schedule that 8

minimize the total completion time when the node is constrained by the battery capacity. This work is presented in the Chapter 2 of this thesis.

1.3.3

Energy-efficiency in the spatial domain

The problem of energy-efficient transmission has also been studied for multi-antenna nodes. In [19], the authors consider the circuitry energy consumption of the multiple radio frequency front ends, as well as, the transmission energy. Then, they analyze energy consumption versus distance between nodes for MIMO and SISO point-to-point communication. It is observed that there exists a certain critical distance between nodes above which exploiting MIMO is more energy efficient. In [20], an efficient channel access protocol for wireless LANs that is able to dynamically adapt the transmission mode (SISO, MISO, SIMO and MIMO) and the transmission power in order to minimize the total energy consumption is designed for the case of nodes having two antennas.

1.4

Summary of the thesis

The remainder of this thesis is structured as follows. In Chapter 2, we consider a finite battery capacity energy harvesting node that has to transmit a certain amount of data packets, which are received from upper layers in the OSI stack at known time instances and with known amounts of data, by using the initial amount of energy contained in the battery, as well as, the energy packets that the node is able to collect over the time. Then, we find the transmission policy that minimizes the transmission completion time. The work presented in Chapter 2 is later extended in Chapter 3 by considering that the node must fulfill some QoS constraints in the data transmission. In Chapter 4, we find the precoder that maximizes the throughput of a multi-antenna energy harvesting node by taking into account the constraint that the node can only make use of the energy that has been harvested in previous time instants, i.e., energy causality constraints. Finally, the main conclusions of this thesis are summarized in Chapter 5.

9

10

Chapter 2 Efficient data transmission for an energy harvesting node with battery capacity constraint

2.1

Introduction

In this chapter, we consider an energy harvesting node that has a finite battery capacity and where both data and energy arrivals take place following a packetized model at known time instants, which, to the best of our knowledge, has not been studied before. Note that the packetized model for arrivals is accurate enough since both the inter-arrival times and the amount of data or energy in the packets can be done arbitrarily small leading to a continuous model and, hence, has been broadly used in the literature. We solve the problem of minimizing the total completion time under data and energy causality constraints and, also, finite battery capacity constraint. We first define the properties of the optimal transmission policy and afterwards develop an algorithm that iteratively is able to find the optimal solution. As in [13], we formulate the problem by using cumulative curves, which allows an appealing visualization of the solution, however, the battery capacity constraint makes that the cumulative curves depend on the chosen solution, as explained in following sections. 11

E0

E1 D1 E2

DN-1

EK-1

s1

uN-1

sK-1

u1

s2

t

T?

D0 Figure 2.1: Summary of the packetized arrival model

2.2

Problem formulation

Let us consider a node with a finite battery capacity, Cmax , that has to transmit a total of N data packets by using the energy that it harvests over time. We want to find the power allocation/rate scheduling strategy to transmit the data such that the transmission time, T , is minimized. We assume that the instants at which the different data packets arrive to the node and their size are known from beforehand. Hence, it is known that at the time instant ui seconds the i-th data packet arrives containing Di bits, with i = 0 ... N − 1. Moreover, the time instants at which the node harvests energy are also assumed to be known. Hence, the j-th energy packet arrives at the instant sj seconds and a total of Ej Joules are harvested, with j = 0 ... K − 1. Figure 2.1 shows a graphical summary of all the explained above. In this chapter, we call the instants ui data arrival events. Similarly, the time instants sj are named energy arrival events. Moreover, we assume that events cannot be produced in the same time instants, thus, we assume that events are always separated by an infinitesimal amount of time, which is a reasonable assumption in practical scenarios. To describe our model we present the following definitions: Definition 1 (Data Departure Curve). A data departure curve D(t), t ≥ 0, is the total number of bits that have been transmitted by the node in the time interval [0, t]. Definition 2 (Energy Expenditure Curve). An energy expenditure curve E(t), t ≥ 0, is the energy in Joules that has been consumed by the node in the time interval [0, t]. 12

Let us consider a static or slow-fading channel where the power-rate function g(·), i.e., the function that, at any given time instant t, relates the transmitted power, P (t), with the rate, r(t), according to P (t) = g(r(t)). As in [13] and [14], we make the common assumption that the function g(ů) is time-invariant, convex, monotonically increasing, and only depends on r(t). Note that the instantaneous rate, r(t), can be expressed as the derivative with respect to t of the data departure curve, i.e., r(t) = D′ (t). Similarly, the transmitted power, P (t), can be written as the derivative with respect to t of the energy expenditure curve, i.e., P (t) = E ′ (t). Then, the energy expenditure curve can be obtained from the data departure curve as follows: ∫

E(D(t)) =

t

g(D′ (τ )) dτ.

(2.1)

0

Observe that the magnitudes D(t), E(t), r(t), and P (t) are unambiguously related by (2.1) and g(·). Therefore, given the initial states E(0) = 0 and D(0) = 0, the design of the system to be optimized can be described by any of these magnitudes. Definition 3 (Battery). The battery of the node B(t) is the amount of energy that the node has available at a given time instant t. We consider a battery with finite capacity Cmax . Thus, B(t) must satisfy that 0 ≤ B(t) ≤ Cmax , ∀t ≥ 0. The overflows of the battery, denoted by Oj , can only be produced simultaneously with an energy arrival, moreover, they depend on the chosen energy expenditure curve. Then, the energy lost due to overflow at sj is Oj = max{0, Ej − Cmax + B(s− j )}. Note that overflows guarantee that the battery level will never be above the battery capacity, B(t) ≤ Cmax . In the following, we define the accumulated battery, a concept introduced in this work that allows us to characterize the optimal solution when having finite battery capacity constraint. Definition 4 (Accumulated Battery). The accumulated battery BA (t) is the sum of the energy that has been stored in the battery during the time interval [0, t). It can be expressed as the total energy arrivals minus the overflows, Oj , produced due to the limited battery capacity Cmax , i.e., BA (t) =

∑

j : sj Rmax ) then else if (R return mode = minT , f inish = 0 end if end for return mode = minEnergy, f inish = 0 end if

equivalent rate and checks whether transmitting at this rate is feasible, i.e, the following ˆ < Rmax and (ii.) R ˆ > Rmin . In case that (i.) is not two conditions are fulfilled: (i.) R fulfilled, the function returns the mode minT . If (ii.) is not met, it is checked if, by using the following energy arrivals, a feasible curve is obtained. Finally, in case both conditions are fulfilled, it is checked whether any energy arrival has been produced in the time interval (0, Tˆ0 ). In case of no arrivals, the algorithm ends and the last epoch has been found. In case there is an energy arrival in (0, Tˆ0 ), the function repeats the whole process but now using the initial battery and the energy of the first arrival, E1 . This process is repeated until (i.) becomes false or a feasible curve is found. 20

2.5.2

Get rate and length of the next epoch

This part of the algorithm determines the rate and length of the following epoch. It works differently depending on the parameter mode that is obtained from the first part of the algorithm as presented in Section 2.5.1. Minimize the total completion time (mode == minT ): This strategy is used when both of the following conditions are satisfied: (i.) It is possible to finish the transmission ˆ at some rate r with r ≤ Rmax . (ii.) The rate obtained from an even power allocation R ˆ is not feasible due to R > Rmax . Hence, the objective is to find the rate that allows us to finish transmission as soon as possible, without paying attention on saving power, however, without wasting it, either. In such case, the rate and duration of the epoch is r = Rmax and l = eRmax . Minimize the energy expenditure (mode == minEnergy): This strategy is used when it is not possible to finish transmission at any rate and, hence, the objective is to save as much power as possible in order to use it when ending the transmission is feasible. Note that in this situation, the problem of obtaining the following epoch is similar to the problem presented in [13] and, hence, the solution is also similar. The possible data departure curves with constant rate r, i.e., D(t) = rt, are divided in two sets. The first set SRmax contains all the rates r such that the associated data departure curve crosses the constraint Dmax (t) first. Whereas the set SRmin contains all the rates r such that the associated data departure curve crosses the constraint Dmin (t) first. Then, the rate of the following epoch is determined as the infimum of SRmax or, equivalently, the supremum of SRmin , i.e., r = inf( SRmax ) = sup( SRmin ) (2.4) and the duration of the epoch can be obtained as the first time instant tx such that, rtx = Dmax (tx ) or rtx = Dmin (tx ). Then, the length of the epoch is l = tx .

2.5.3

Algorithm optimality

The optimality of the algorithm is summarized in the following theorem and its subsequent proof: Theorem 1. The algorithm presented in this section constructs the optimal data departure curve, D⋆ (t). Proof. See appendix A.6.

2.6

Conclusion

The optimal transmission strategy has been obtained for nodes with finite battery capacity when both the data and energy packets arrive at known time instants. In general, trans21

mitting data at constant rate is the strategy that consumes less energy. However, it has been proved that when a battery overflow is produced transmitting at constant rate is not optimal. The optimal strategy increases the rate before the overflow until either there is no overflow or the data causality constraint is reached. Finally, an algorithm to compute the optimal data transmission strategy has been developed.

22

Chapter 3 Efficient data transmission for an energy harvesting node with battery capacity and QoS constraints 3.1

Introduction

In many real scenarios, the data flow must guarantee certain levels of performance. For instance, the network may require that the data is transmitted with a certain minimum data rate, quality level or with a maximum delay. These requirements, usually referred to as QoS constraints, become crucial in real-time applications that are sensitive to the network delay. QoS constraints also allow to assign different priorities to the packets depending on the type of traffic contained in each packet. In this chapter, we expand the work presented in Chapter 2 for the case where our pointto-point link must satisfy some QoS constraints. We expand the algorithm of the previous chapter to determine the optimal rate scheduling, i.e., the rate schedule that minimizes the transmission completion time while satisfying QoS constraints at every time instant. As we will see during this chapter, this problem may not have always a feasible solution.

3.2

Problem formulation

Along this chapter, we will keep using the concepts that were defined in the previous chapter, namely, data departure curve (D(t)), energy expenditure curve (E(t)), battery (B(t)), accumulated battery (BA (t)), minimum energy expenditure (Emin (t)) and accumulated data 23

(DA (t)). The reader can go back to Definitions 1-6 in Chapter 2 in case some of these concepts are not clear. Moreover, in order to deal with the QoS constraints we will make use of a new concept, i.e., the minimum data departure, which is introduced in the Definition 7. Afterwards, we show some examples on how to express the QoS constraints by means of the minimum data departure curve.

Definition 7 (Minimum Data Departure). The minimum data departure, DQoS (t), is the smallest amount of data that the node must have transmitted at time t such that the QoS constraints are satisfied. Different QoS constraints can be considered, we briefly describe two of the most commonly used [13]: • Deadline Constraint: This constraint considers that the maximum permissible delay for the transmission of a certain data packet, Di , is τi seconds. Then, DQoS (t) is a piecewise constant function that changes at instants qi = ui + τi with an increment of Di . As a specific case, we can consider that the supported delay for all the packets is the same, i.e., τi = τ , ∀i. Then, the minimum data departure is given by DQoS = DA (t − τ ). • Buffer constraint: This constraint considers that the node has a limited data queue of size β. Hence, in order not to loose any incoming data, the minimum data departure must be DQoS (t) = max{DA (t) − β, 0}. Different QoS constraints can be considered by mapping the constraint into an appropriate DQoS (t). For instance, a combination of the two aforementioned constraints can be used. The rest of the chapter considers a general DQoS (t) that is a staircase curve where changes are produced at time instants qi with increments of Qi bits. From now on, the time instants where DQoS (t) changes, i.e., qi , are called quality requirement events. Then, in this chapter, we consider three kind of events, namely, data arrivals, energy arrivals, and quality requirement events. We will loose the restriction that events cannot coincide in time. We do so because depending on the chosen QoS, it is likely that the instants qi are equal to ui , e.g., for the buffer constraint. As before, our goal is to find the data departure curve, D(t), that minimizes the total transmission time, T , of the N data packets. Note that now we have an additional constraint that is, the QoS of the link. Thus, the following conditions must be satisfied: (i.) Energy causality: energy must be harvested before it is used by the node or, which is the same, the battery level in the node must be greater or equal to zero. (ii.) Data causality: it is not possible to transmit more bits than the ones that have arrived to the node. (iii.) Quality of service: At time t, a minimum amount of data DQoS (t) has to be transmitted in order to preserve the link quality of service. Moreover, given two data departure curves with the same completion time, the one that requires less energy is always preferred. From 24

all that has been said above, the problem can be expressed as follows: min T

(3.1)

D(t)

s.t.

3.3

E(t) ≤ BA (t), DQoS (t) ≤ D(t) ≤ DA (t), D(T ) = DA (T ).

Properties of the optimal solution

We start by characterizing the optimal data departure curve, D⋆ (t), and its associated energy expenditure curve, E ⋆ (t):

Problem 2 (Transmission Without Events). We want to characterize the optimal departure curve in the time interval (t1 , t2 ) where there are no changes in DA (t), BA (t), and DQoS (t) . We also consider that the data departure curve at the boundary of the intervals is D(t1 ) and D(t2 ), respectively, and that these two points satisfy the data, energy and QoS constraints.

Lemma 6. In Problem 2, D⋆ (t) is a straight line where the slope, or, equivalently, the transmission rate, is constant and equal to r(t) =

D(t2 )−D(t1 ) , t2 −t1

∀t ∈ (t1 , t2 ).

Proof. See BT-problem in [13].

Corollary 3. Lemma 6 implies that D⋆ (t) is a piece-wise linear function such that its slope is equivalent to the transmission rate, which can only change either at sj , ui or qi . Note that Lemma 5, which states that by the end of the transmission the battery must be empty, is still satisfied. In this chapter, the assumption of having disjoint time events is relaxed, in other words, we allow that in a certain time instant two events are produced, i.e., a data arrival event and a quality requirement event. In the next section, we present some insights of the problem that will allow us to continue modeling the properties of the optimal solution. 25

E (t ) B A (t )

E3 E2

E min (t )

Cmax

E1

E0

A

B

C

t

s 2 s3

s1

D (t ) DA (t ) DQoS (t )

D3 D2 D1 D0

T? C

A

B

u3

u1 u 2

t

Figure 3.1: Visualization of the problem presented in (3.1).

3.4 3.4.1

Problem insights Problem visualization

The problem formulated in (3.1) can be graphically seen as shown in Figure 3.1, where the figures at the top and the bottom show the energy and data domains, respectively. The energy causality constraint is represented by the red solid line at the top figure, whereas data causality and QoS constraints are depicted by the blue and green solid lines in the figure at the bottom, respectively. Hence, a data departure curve, D(t), must lie within the blank region of the data and energy domains in order to be valid. Three different data departure curves (A, B, and C) and their associated energy expenditure curves are shown. The curve A is not valid since it breaks the energy causality constraint. The curve B is valid in spite of having an overflow of the battery at s3 . Finally, the curve C is not valid because it does not satisfy QoS constraints. 26

3.4.2

Constraints mapping into the data domain for a given epoch

As explained in Section 2.4.2, given a reference point, all the edges of BA (t) and Emin (t) can be mapped to the data domain obtaining DBA (t) and DEmin (t), respectively, as shown in Figure 3.2. The value of the mapping is only valid in the points marked with a cross, in the rest of the points we have used a constant value in each interval, i.e., (sj , sj+1 ). We have used this mapping, which we call simplified mapping, in order to reduce the ¯ B (t) and D ¯ E (t), computational complexity of the algorithm. The real mapping curves, D min A are obtained by not only mapping the edges but also all the other time instants. Note that these curves are decreasing functions in each of the intervals and that are bounded in the bottom and top, respectively, by the simplified mapping DBA (t) and DEmin (t). We must be careful when QoS constraints are introduced in the problem because it can happen that ¯ B (t). This is explained in more detail in the the DQoS (t) > DBA (t), however, DQoS (t) < D A rest of the section. We can merge the data and energy constraints in a single constraint that, at every time instant, will be the most restrictive of the two constraints, i.e., Dmax (t) = min{DA (t), DBA (t)}.

(3.2)

Similarly, the lower constraint is Dmin (t) = max{DQoS (t), DEmin (t)}.

(3.3)

Note that every time that there is a rate change, the mapping of the energy constraints to the data domain also change, and hence, will have to be recalculated. Let us define the sets of time instants at which Dmax (t) and Dmin (t) have discontinuities as follows: Zmax = {t | Dmax (t− ) ̸= Dmax (t+ )} (3.4) Zmin = {t | Dmin (t− ) ̸= Dmin (t+ )}

(3.5)

Note that from the definition of Dmin (t) and Dmax (t), it can occur that Dmax (t) < Dmin (t). This happens in the following situations: (i.) The node has to transmit a certain amount of data in order not to have overflow of the battery, however, this data is not still available, i.e, DEmin (t) > DA (t). (ii.) The node has to transmit a certain amount of data in order to satisfy QoS constraints, however, it does not have enough energy to do so, i.e., DQoS (t) > DBA (t). These two situations are completely different and this is reflected in the solution of the problem. Since Emin (t) is not a constraint of the problem (3.1), it is permitted by the problem that D(t) < DEmin (t), however, it has been proved to be suboptimal. Then, the optimal solution transmits at constant rate up to the crossing point of DEmin (t) and DA (t). Then, the rate changes to zero until the following data arrival. In this rate change, DEmin (t) is recalculated because it comes from the energy domain which makes that the optimal solution D⋆ (t) does not cross Dmin (t). The situation in (ii.) is a bit more tricky, for instance, in Figure 3.3, one could think that the problem does not have a solution since, 27

D (t ) DA (t ) DQoS (t ) D B A (t )

x x

D E min ( t )

x x

x A

x

T?

x C

B

u3

u1 u 2

t

Figure 3.2: Mapping of the energy causality constraint and minimum energy expenditure to the data domain. at t = q1 , the data that can be transmitted by using the accumulated battery is smaller than the data required to satisfy QoS. However, this is not true because we have used ¯ B (t). the simplified mapping, DBA (t), instead of the real mapping of each of the points D A ¯ B (q1 ), Then, we can compute the value of the mapping just at the point t = q1 , obtaining D A ¯ B (q1 ), the which is represented as a red cross in Figure 3.3. If we have that DQoS (q1 ) ≤ D A problem has solution, which is modeled in the second point of Lemma 10. Otherwise, the problem does not have a solution, as summarized in the following lemma: Lemma 7. The problem (3.1) does not have solution whenever DQoS (tx ) > tx g −1 (BA (tx )/tx ), for some time tx | tx ∈ (0, T¯). ¯ Proof. Let D(t) be the data departure curve that transmits the maximum amount of data ¯ in the interval (0, tx ). We know that D(t) has constant rate and empties the battery at tx . This rate is g −1 (BA (tx )/tx ) and the maximum data that can be transmitted at tx ¯ x ) = tx g −1 (BA (tx )/tx ). Then, if the QoS requires a higher amount of data to be is D(t transmitted the problem does not have a solution. As we will see in next section, the developed algorithm checks Lemma 7 at every iteration in order to determine whether the problem has a feasible solution or not. As we 28

will see, at every iteration the origin of coordinates is moved to the origin. Then, by T¯ we mean the total completion time minus the reference time of the last iteration. Note at the first iteration T¯ = T . The following lemmas are satisfied whenever the problem has a solution. Due to Corollary 3 we know that D⋆ (t) is constant between events. Then, we can model the behavior of D⋆ (t) at the instances that rate changes are produced. Lemmas 8, 9 and 10 define the behavior of the optimal solution whenever a rate change is produced in a time instant where a single event is produced. Similarly, Lemmas 11 and 12 describe the behavior when a rate change is produced at a time instant where two events are produced. Lemma 8. If a rate change is produced at a certain time instant tx such that tx ∈ Zmax and tx ∈ / Zmin , then D⋆ (tx ) = Dmax (tx ) and the rate increases. Proof. Note that in this situation, Dmax (tx ) > Dmin (tx ) since the function Dmax (t) is nondecreasing and the curve Dmin (t) does not increase in t = tx because tx ∈ / Zmin . Hence, the proof of the lemma is equivalent to the proof of Lemma 2. Lemma 9. If a rate change is produced at a certain time instant tx such that tx ∈ / Zmax , ⋆ tx ∈ Zmin and Dmax (tx ) ≥ Dmin (t+ x ) then D (tx ) = Dmin (tx ) and the rate decreases.

Proof. Since the curves Dmax (tx ) and Dmin (tx ) do not cross, the proof is equivalent to the proof of Lemma 4 point 2. Lemma 10. If a rate change is produced at a certain time instant tx such that tx ∈ / Zmax , tx ∈ Zmin and Dmax (tx ) < Dmin (t+ x ), then one of the following situations occurs: • If tx comes from an energy arrival event, then there is battery overflow, D⋆ (tx ) = Dmax (tx ) and the rate is zero until the next data event. • If tx comes from a quality requirement event, then D⋆ (tx ) = Dmin (t+ x ) and the rate decreases. Proof. See Appendix B.1. Lemma 11. If a rate change is produced at a certain time instant tx such that tx ∈ Zmax ⋆ − + and tx ∈ Zmin and Dmax (t− x ) ≥ Dmin (tx ), then either D (tx ) = Dmax (tx ) and the rate

increases, or D⋆ (tx ) = Dmin (t+ x ) and the rate decreases. 29

D (t ) DA (t )

T?

B

x

A

x

B

DQoS (t )

x x

D B A (t ) D E min ( t )

x

q1 s1 u1

t

Figure 3.3: Mapping of the energy causality constraint and minimum energy expenditure to the data domain. Proof. The proof that, at tx , D⋆ (tx ) is either Dmin (tx ) or Dmax (tx ) can be done by contradiction in the same fashion than in the first part of the proof of Lemma 2. The proof that ⋆ + shows that when D⋆ (tx ) = Dmax (t− x ), the rate increases and when D (tx ) = Dmin (tx ) the

rate decreases follows from contradiction because otherwise a constant rate transmission can be found that expends less energy while transmitting the same amount of data, as done in the Part 2 of the proof of Lemma 2. Lemma 12. If a rate change is produced at a certain time instant tx such that tx ∈ Zmax + ⋆ − and tx ∈ Zmin and Dmax (t− x ) < Dmin (tx ), then D (tx ) = Dmax (tx ) and the rate can either

increase or decrease. Proof. As we said this lemma only applies when the problem has a solution. Due to this, we know that Dmin (tx ) = DEmin (tx ) and that Dmax (tx ) = DA (tx ). Hence, this is the overflow problem with the particularity that a new data arrival is produced at the same time instant than the overflow. Hence, as in the overflow problem (See Appendix A.1) we have that D⋆ (tx ) = Dmax (t− x ), but now we cannot say anything regarding the rate in the following epoch. By using these lemmas, we are able to construct an algorithm that iteratively finds the optimal solution, or concludes that there is no a solution. The construction of this algorithm is presented in the following section. 30

3.5

Optimal data departure curve construction

In this section, we explain the developed algorithm that is able to either construct D⋆ (t) or, alternatively, conclude that the problem (3.1) does not have a solution. The algorithm developed to solve this problem is based on the algorithm presented in the previous chapter to solve problem (2.3) due to the similarities of both problems. Let us denote M as the number of epochs of the optimal solution, i.e., the number of rate changes. As stated in Corollary 3, the optimal departure curve is a piece-wise linear function with rates r = [r1 , r2 . . . , rM ]. The proposed algorithm follows an iterative process where at each iteration the duration and rate of an epoch is determined. We will explain the first iteration, i.e., how to obtain r1 , and the rest can be obtained by following the same approach. The algorithm starts by checking whether the mapping to the data domain satisfies DQoS (qi ) < qi g −1 (BA (qi )/qi ) , ∀ qi . In case this condition is not fulfilled, the problem does not have a solution and the algorithm ends. Otherwise, the algorithm proceeds as the algorithm of problem (2.3). Hence, first it is checked whether it is possible to transmit all the remaining data by using all the available power at constant rate, as explained in Section 2.5.1. In such a case, the algorithm ends and the minimum completion time T has been found. Otherwise, this function returns the mode (minT or minEnergy) to be used to find the rate, r1 , and length, l1 , of the following epoch (see Section 2.5.2). Once the first rate is determined, the problem is rescaled. The transmitted bits are subtracted from DA (t) and DQoS (t). The expended energy is subtracted from BA (t). Moreover, in case that r1 produces a battery overflow at time instant l1 , the amount of energy lost due to the overflow is also subtracted from BA (t). Then, the origin of coordinates is moved to l1 . Finally, the mapping to the data domain is recalculated and Lemma 7 is checked again to determine whether it is still possible to find a solution for the problem. With this, the whole procedure starts again to determine the following epoch. There are two possible reasons for which the algorithm ends: (i.) At some iteration, it is found that DQoS (qi ) > qi g −1 (BA (qi )/qi ), for some qi . As stated in Lemma 7, this implies that the problem does not have a solution. (ii) All the data has been transmitted. Then, the minimum completion time can be found as the sum of all the length of all the epochs, ∑ i.e., T = M i=1 li .

3.5.1

Algorithm optimality

The optimality of the algorithm is summarized in the following theorem and its subsequent proof: Theorem 2. The algorithm presented in this section constructs the optimal data departure curve, D⋆ (t), for the problem (3.1). Proof. See appendix B.2. 31

3.6

Conclusion

The optimal transmission strategy has been obtained for nodes with finite battery capacity that have to fulfill some kind of QoS constraint. We have shown that this problem may not have a solution when the node does not have enough energy to guarantee the QoS constraints. We have developed an algorithm that is able to determine whether the problem has a solution or not and, in case of having a solution, determines the optimal data transmission strategy.

32

Chapter 4 Throughput maximization for a multi-antenna energy harvesting node 4.1

Introduction

In this chapter, we consider a multi-antenna energy harvesting node which transmits its information through a MIMO channel. As in the previous chapters, we assume that energy is harvested following a packetized model. We design the precoder that maximizes the total throughput for this energy harvesting node. The main difference with the traditional approach is that the common sum-power constraint is exchanged by a set of energy causality constraints, which take into account the discrete nature of the energy arrivals. Once the design of the optimal precoder is found, we show its superior performance compared with other schemes that do not exploit the energy harvesting process, such as the broadly known water-filling solution for non-harvesting multi-antenna nodes or uniform power allocation among antennas.

4.2

System model

We consider a point-to-point communication system where the transmitter is a wireless energy harvesting node equipped with nT antennas. Similarly, nR denotes the number of antennas at the receiver. The system model considered in this chapter is graphically shown in Figure 4.1. The nb bits to be transmitted in the n-th channel use are packed 33

xn

Transmitter in

nb

B

yn

Receiver

H

i’n nb

nT x nb

nR x nT

nb x nR

Figure 4.1: The MIMO communication scheme.

in

VH

P1/2

U

ns x nb

ns x ns

nT x ns

Figure 4.2: Transmitter block diagram. in the column vector in . This vector is linearly processed at the transmitter by means of the precoder matrix B, whose dimensions are nT × nb . Thus, the transmitted signal at the n-th channel use, xn , can be expressed as xn = Bin , where [xn ]j represents the transmitted signal through the j-th antenna at the n-th channel use. We define H as the channel matrix, where [H]ij contains the observed channel gain between the j-th antenna of the transmitter and the i-th receiver’s antenna. We consider that, at the receiver, the signal is corrupted by AWGN. The noise vector is defined as wn , which has nR components that represent the observed noise level in each of the receiver’s antenna. Then, the received signal, which is given by yn = Hxn + wn = HBin + wn , is processed at the receiver in order to recover the original bit stream. The bit stream at the output of the receiver is denoted as i′n and may contain errors. In this chapter, we want to carry out the design of the optimal precoder matrix, B, by taking into account the discrete nature of the energy arrivals. As shown in Figure 4.2, B is composed by three basic blocks. The first block, expressed through matrix VH , is in charge of generating the symbols to be transmitted from the bit stream, i.e., in . Hence, it transforms the modulation of bits into symbols. The second step, which is carried out by matrix P1/2 , is the power allocation that assigns a certain power level to each of the symbols. Finally, the last step, denoted by U, is the beamformer that distributes the symbols with the corresponding power among the different antennas. Without loss of generality, we assume that U and V are unitary matrices. Let ns denote the number of 34

symbols. Then, the dimensions of VH are ns × nb . Note that the matrix P1/2 is diagonal with dimensions ns × ns . Finally, the dimensions of U are nT × ns . Thus, the matrix B can be expressed as B = UP1/2 VH with dimensions nT × nb . Then, the transmitted sequence is xn = Bin = UP1/2 VH in with the following covariance matrix Q = E{xn xnH } = E{BBH } = UP1/2 VH VP1/2 UH , where we have applied that E{in iH n } = Ins (the identity matrix of dimension ns ). We will assume that there exists a feedback channel between the receiver and transmitter through which the information of the channel estimations done by the receiver are sent back to the transmitter. Moreover, we also assume that there are no errors in the channel estimation nor in the feedback link, so there is full Channel State Information (CSI) at both the transmitter and the receiver. Moreover, we will consider that the Singular Value Decomposition (SVD) decomposition of the channel is H = VH RH UH H, and the SVD of the channel Gram matrix HH H is HH H = UH DH UH H, where DH = diag(h1 , . . . , hmin(nT ,nR ) ). Note that the hi are the eigenvalues of the channel Gram matrix HH H. We consider the case where the transmitted signal vector xn can be approximated by Gaussian signaling and that the received signal is corrupted by AWGN. Let R0 denote the covariance matrix of the noise, i.e., R0 = σ 2 I. Then, we define the matrix RH as RH = HH R0−1 H =

1 H H H = HH H, σ2

without loss of generality the last step assumes the noise has unitary power. Moreover, we assume that the channel is static. We consider an energy harvesting node that is able to collect energy from the environment. Then, at time si the node is able to collect a packet of energy containing Ei Joules. In the next section, we derive the precoding matrix for this energy harvesting node that maximizes the throughput of transmitted data over a total time T , which is divided in S +1 time slots of length Ls seconds, having energy causality as constraints for the problem. We assume that a total of N energy packets are harvested in this time window, i.e, E1 . . . EN and that the node has an initial amount of energy in the battery of E0 Joules1 . 1

We have assumed that energy arrivals occur at the beginning of a certain time slot. This is a reasonable

assumption since Ls is much smaller than the time between arrivals of energy packets.

35

τ0

E1

τ1

E2

EN-1

….

0

τN-1

EN

τN T

t0 t1

tS

Figure 4.3: Summary of the time notation. In the rest of the chapter, we denote epoch as the period of time between two energy arrivals, i.e., τ1 = [0, s1 ), τ2 = [s1 , s2 ) . . . τn+1 = [sN , T ). Note that there are a total of N + 1 epochs, where the last epoch is the period of time between the last arrival and T . Figure 4.3 shows a summary of the definitions exposed above.

4.3

Throughput maximization problem

We want to find the optimal precoding matrix for each slot, i.e., Bq , for q = 0 . . . S, so that the throughput is maximized while satisfying energy causality constraint, i.e., the energy expended by the node at the end of each time slot, tq , is smaller than the energy harvested up to the beginning of that time slot. max Bq

s.t.

Ls

S ∑

ln ( |InT + Qq RH | )

(4.1)

q=0

Ls

l ∑

Tr(Qq QH q) ≤

S ∑

Ei ,

l = 0 . . . S − 1,

(4.2)

i: si ≤lLs

q=0

Ls

∑

Tr(Qq QH q) =

q=0

N ∑

Ei .

(4.3)

i=0

Observe that, in the optimization problem above we have expressed the dependence on Bq through the set of matrices Qq . We recall that Qq = Bq BH q where Bq denotes the H precoder used in the time slot tq that can be expressed as Bq = UP1/2 q V . Note that since we have assumed a static channel, the dependence on the time slot only appears through the matrix Pq . Let us now consider a set of transformation to simplify the optimization problem in 4.1-4.3. First, we can apply the property of the determinant of a matrix that states that the determinant of the matrix is smaller or equal than the product of the elements in its ∏ diagonal, i.e., |B| ≤ bii , and that equality is achieved when the matrix is diagonal. Hence, if we want to maximize the throughput we want that the term inside the determinant is a 36

diagonal matrix. Let us expand this inner part of the determinant: H 1/2 H InT + Qq RH = InT + UP1/2 q V VPq U

= InT + UPq U

H

U H DH U H H

UH DH UH H.

(4.4) (4.5)

Note that if the beamforming matrix is selected to be equal to the eigenvectors of the channel matrix then the term inside the determinant becomes diagonal and, hence, the determinant is maximized: |InT + Qq RH | ≤ |InT + UPq DH UH H| = |Imin(nT ,nR ) + Pq DH |.

(4.6) (4.7)

Therefore, applying these two properties of determinants we have found that the optimal matrix U of the precoder is equal to the eigenvectors of the channel matrix, i.e., U = UH . Note that since we have considered a static channel this matrix does not vary over time. Then, the problem (4.1)-(4.3) is equivalent to finding the optimal power allocation in each of the eigenmodes of the channel at every time slot, i.e., find Pq,i such that Pq,i = [Pq ]ii , for q = 0 . . . S and i = 0 . . . K − 1 where K is the number of eigenmodes of the channel, K = min{nT , nR }. max Pq,i

s.t.

Ls

S K−1 ∑ ∑

ln ( 1 + Pq,i hi )

(4.8)

q=0 i=0

Ls

l K−1 ∑ ∑

S K−1 ∑ ∑

Ei ,

l = 0 . . . S − 1,

(4.9)

i: si ≤lLs

q=0 i=0

Ls

∑

Pq,i ≤ Pq,i =

q=0 i=0

N ∑

Ei .

(4.10)

i=0

As we mentioned before, an epoch, τj , is the period of time between two energy arrivals. Then, due to the concavity of the objective function it can be shown that the powers of all the time slots within a certain epoch must be equal, i.e., Pq,i = Pqˆ,i ,

∀q, qˆ ∈ τj .

This follows from the fact that, due to concavity of the log function, by equalizing the powers over time we can transmit more data by consuming the same amount of energy, a formal proof can be found in [13]. Hence, now the problem can be simplified. Instead of having to determine the power allocation in each time slot, we just need to determine the power allocation in each epoch. This results is the following equivalent optimization problem: 37

N ∑

max Pj,i

Lj

j=0 l ∑

s.t.

K−1 ∑

ln ( 1 + Pj,i hi )

(4.11)

i=0

Lj

K−1 ∑

l ∑

Pj,i ≤

j=0

i=0

j=0

N ∑

K−1 ∑

N ∑

Lj

j=0

Pj,i =

i=0

Ej ,

l = 0 . . . N − 1,

(4.12) (4.13)

Ei .

i=0

where Lj stands for the length of epoch τj , Pj,i is the power loaded in the channel i at epoch τj . Note that now, instead of having S inequality constraints we just have N (where N is the number of harvested packets). The problem (4.11)-(4.13) is convex, since the objective function is convex and the constraints are affine. The Lagrangian can be written as L =

N ∑ j=0

Lj 

−µ 

K−1 ∑

ln ( 1 + Pj,i hi ) −

N −1 ∑

i=0 N ∑



λl 

l=0

Lj

j=0

K−1 ∑

Pj,i −

i=0

N ∑



l ∑ j=0

Lj

K−1 ∑

Pj,i −

i=0

l ∑



Ej 

j=0

Ej  .

(4.14)

j=0

Taking the derivative with respect Pj,i and equating to zero, we obtain that 

Pj,i

+

1



+

1 1 = −  = Wj −  . ∑N −1 hi hi µ + j λj

(4.15)

This solution looks like waterfilling, however, note that due to energy causality constraints, the water level is 1 Wj = (4.16) ∑N −1 µ + j λj and that it depends on the epoch. From the KKT optimality conditions we have that the optimal solution must fulfill λl ≥ 0 λl

l (∑ j=0

Lj

K−1 ∑ i=0

Pj,i −

l ∑

)

Ej = 0

l = 0 . . . N − 1,

(4.17)

l = 0 . . . N − 1.

(4.18)

j=0

From (4.17) it is easy to prove that the water-level is non-decreasing in time. This comes from the fact that when j increases the summation in the denominator in (4.16) has less terms. Since these terms are greater or equal than zero the denominator is nonincreasing in time which implies that the water-level is non-decreasing. This is summarized in the following property: 38

Property 1. The power allocated in a certain channel is non-decreasing in time. From (4.18) we can get more insights in the solution. There are two possibilities to fulfill (4.18): • Empty Battery: This situation occurs when at the end of a certain epoch, i.e., τl the ∑ ∑K−1 ∑ node has consumed all the energy, i.e., lj=0 Lj i=0 Pj,i − lj=0 Ej = 0. • Energy Flow: This situation occurs when at the end of a certain epoch, i.e., τl the node has not consumed all the energy. This remaining energy will be used in the following epochs. Note that when this happens λl = 0 and the water-level of the following epoch will be equal to the water-level of the current epoch, i.e, Wτl+1 = Wτl . Property 2. A change in the allocated power of a certain channel can only be produced when at the end of the previous epoch the node has consumed all the available energy. The remainder of this chapter deals with the implementation of an efficient algorithm that computes the water-level at each epoch and, with that, determines the power allocated in each of the channels. To do so we define two new concepts: (i.) An epoch interval, i.e., I, is a set of epochs with a constant water level. (ii.) A transition epoch, i.e, Ti , is an epoch at which the water level changes. Note that since at each epoch interval the water level is constant, the optimal power allocation within that epoch interval is to perform the traditional waterfilling where the water-level is found by applying that the energy expended in the slot interval has to be equal to the energy harvested. Consequently, the problem reduces to finding the optimal transition epochs, i.e., T1 , . . . , TM and then applying the traditional waterfilling algorithm to the resulting epoch intervals. To do so, we have implemented a forward-search procedure similar to the one proposed in [16]. We explain how to obtain T1 and the others are found in the same manner: 1. Assume that the first epoch interval contains all the epochs, I = τ0 . . . τN . 2. Perform traditional waterfilling in all the epoch interval. 3. Check whether all the energy causality constraints within the interval are fulfilled. • If they are fulfilled, then the optimal transition epoch, T1 , is the first epoch not included in the interval. • If they are not fulfilled, remove the last epoch from the previous slot interval and go back to step 2. Then, the problem is shifted and this procedure is repeated to determine the following transition epochs until a transition epoch is found that is equal to the total time window under analysis, i.e., T . When this happens, the optimal power allocation has been found for all the time slots and for all the channels.

4.4

Results

In this section we present the results obtained through simulation of the previous algorithm. In order to show the optimality of our algorithm, which we name MIMO staircase 39

Total Throughput for a fixed h = 0.2 0

450 MIMO Staircase Waterfilling Epoch−by−Epoch UPA Epoch−by−Epoch MIMO WF

400

Total Throughput

350 300 250 200 150 100 50

0

0.2

0.4

0.6

0.8

1

h1

Figure 4.4: Throughput comparison for N = 40 packets of energy. waterfilling, we compare the results obtained with two suboptimal strategies: • Epoch-by-Epoch Uniform Power Allocation (UPA): All the energy harvested at the beginning of the epoch is expended in this same epoch and it is distributed with UPA. • Epoch-by-Epoch MIMO Waterfilling: All the energy harvested at the beginning of the epoch is expended in this same epoch and waterfilling is performed among the different channels. We have considered a channel matrix of rank K = 2, where the first eigenvalue of the channel takes value h0 = 0.2 and we make the second eigenvalue, h1 , to vary from 0 to 1. We have considered that the amount of energy packets is N = 40 and that the maximum time window is T = 100 seconds. Energy arrivals are produced at random time instances uniformly distributed in (0, T ) and with random amount of energy. The results are shown in Figure 4.4. As expected, the throughput increases with h1 and the MIMO staircase waterfilling performs better than the two other alternatives. Moreover, we want to remark that the throughput differences between the optimal solution, i.e., MIMO 40

Total Throughput for a fixed h = 0.2 0

40 MIMO Staircase Waterfilling Epoch−by−Epoch UPA Epoch−by−Epoch MIMO WF

35

Total Throughput

30

25

20

15

10

5

0

0.2

0.4

0.6

0.8

1

h1

Figure 4.5: Throughput comparison when the node does not harvest energy. staircase waterfilling, and the other two solutions depend both in the number of energy arrivals and the amount of energy contained in the arrivals. For instance, in Figure 4.5 we show an extreme case where N=0. Note that if there are no energy arrivals our algorithm does not get any gain with respect to the traditional MIMO waterfilling solution and the curves overlap. Finally, Figure 4.6 shows the results for N = 20 and it confirms that the throughput gain of our algorithm increases with the number of energy arrivals. The reason of this gain is that our algorithm saves as much energy as possible by equalizing the power between epochs in comparison with traditional waterfilling. These energy savings are used at the last epochs to speed up the transmission and increase the throughput.

4.5

Conclusions

In this chapter we have presented the optimal precoder for a MIMO energy harvesting node. The fact that the node is able to harvest energy introduces a set of causality constraints when allocating transmission power. We have proved that the optimal precoder diagonalizes the channel, as happens in the optimal precoder for non-harvesting nodes. Furthermore, we have shown that the traditional waterfilling algorithm is not anymore optimal and we have derived the optimal 41

Total Throughput for a fixed h0 = 0.2 300 MIMO Staircase Waterfilling Epoch−by−Epoch UPA Epoch−by−Epoch MIMO WF

Total Throughput

250

200

150

100

50

0

0

0.2

0.4

0.6

0.8

1

h1

Figure 4.6: Throughput comparison for N = 20 packets of energy.

42

power allocation strategy, the MIMO staircase waterfilling, where the water level increases in time as energy is being harvested. Through simulations we have shown a substantial increase in the total throughput with respect to other suboptimal strategies such as Epoch-by-Epoch Uniform Power Allocation and Epoch-by-Epoch MIMO Waterfilling.

43

Chapter 5 Conclusions and future work 5.1

Conclusions

In this Master Thesis we have addressed the problem of energy-efficient communications also referred to as green communications. Traditionally, energy consumption of communication systems has been minimized in order to reduce the network cost, as well as the interference produced to other communication systems. However, energy-efficiency has not been the main aim of the current communication system designs. Recently, a need for green communication systems has appeared, basically, due to the increasing concern regarding the amount of greenhouse gas emissions and the increasing amount of battery powered devices that require an enlargement of its lifetime. Energy harvesting seems an appealing solution to enlarge the autonomy of these devices. In this thesis, we have derived the optimal rate scheduling and power allocation for a wireless energy harvesting node. In Chapter 2, we have found the optimal transmission strategy for a wireless energy harvesting node that has a finite battery capacity. We have considered an offline approach, where the node knows from beforehand the instances at which the data and energy packets arrive to the node, as well as the amounts of bits or joules contained in each packet. We have shown that when the optimal transmission strategy is being used, the battery only overflows when the node does not have more data to be transmitted. Finally, we have developed an algorithm that computes the optimal transmission strategy. In Chapter 3, we have extended the work of Chapter 2 by forcing the node to satisfy a certain QoS constraint. We have proved that the problem may not have a feasible solution, when the node is not able to harvest enough energy to transmit the required data to fulfill the QoS requirements. Finally in Chapter 4, we have considered a multiple-antenna wireless energy harvesting node for which we have derived the throughput maximizing precoder. We have proved that the optimal precoder still diagonalizes the channel. The power allocation among the eigenchannels is given by the MIMO staircase waterfilling where the water level, instead 45

of being constant over time, is a staircase function. Substantial gains are obtained in comparison with classical waterfilling.

5.2

Future work

In this thesis, we have considered that the node has a full knowledge of the instances of arrival of the data and energy packets, as well as the amount of bits and joules contained in these packets, i.e., an offline approach. However, in general, this is not a realistic model since the node cannot have a complete and correct knowledge of this information. Hence, an online approach must be developed where the rate scheduling is performed by only using the knowledge of the successes that have occurred at past time instants. A part from this, the energy consumption model of the node must be more accurate, i.e., the RF circuitry must be taken into account. When this sink of energy is considered, a different solution will be found since it may be preferable to have the node working in sleep mode than transmitting at very low rates. A similar situation happens for a multiple-antenna node, since each of the antennas is a new source of energy consumption. There are still many open challenges to be solved before having operational green communication systems. To this aim, synergy between engineers working in the different layers of the OSI protocol stack is required.

46

Appendix A Proofs of Chapter 2 A.1

The overflow problem

Consider the time interval (t1 , t2 ) where there is an energy arrival at sj ∈ (t1 , t2 ) that produces overflow of the battery, Oj . We want to characterize which of the following solutions, D1 (t) or D2 (t), t ∈ (t1 , t2 ), is more efficient: 1. D1 (t), which is such that D1 (t) = D(t1 ) + (t − t1 )r0 , ∀t ∈ (t1 , t2 ), that is the same as saying that we are transmitting at constant rate, r0 , in the interval (t1 , t2 ). 2. D2 (t), which is such that {

D2 (t) =

D(t1 ) + (t − t1 )r1 if t ∈ (t1 , sj ), D2 (sj ) + (t − sj )r2 if t ∈ (sj , t2 ),

that is the same as saying that we are transmitting at r1 , with r1 = r0 + ϵ1 , in (t1 , sj ) and at r2 , with r2 = r0 − ϵ2 , in (sj , t2 ). Also, ϵ1 and ϵ2 are positive and such that the total transmitted data of both solutions is the same, i.e., D1 (t2 ) = D2 (t2 ). Moreover, ϵ1 must be small enough so that the strategy D2 (t) still produces an overflow of the battery at si . The problem formulated above is graphically presented in Figure A.1, where the blue and red lines represent solutions 1 and 2, respectively. The following Lemma summarizes its solution: Lemma 13. The strategy D2 (t) is more efficient than strategy D1 (t) because the battery level at t2 is higher for D2 (t), i.e., B2 (t2 ) > B1 (t2 ). Remark 1. Note that D2 (t) achieves higher battery level at t2 due to the fact that the reduction in the battery overflow is higher than the energy saved by constant rate transmission. 47

Proof. We denote p0 , p1 , and, p2 the powers obtained by evaluating the rates r0 , r1 , and, r2 , respectively, in the power rate function g(·). Note that p2 < p0 < p1 since r2 < r0 < r1 and that the power-rate function, p(t) = g(r(t)), is convex and monotonically increasing in r(t). This implies that, at sj , the energy expenditure of solution 2, E2 (sj ), is greater than the energy expenditure of solution 1, E1 (sj ). We denote the difference between the energy expenditures of the two solutions as ∆. Then, E2 (sj ) = E1 (sj ) + ∆. Note that solution 2 reduces the overflow of the battery by ∆, therefore, the relation between the accumulated battery is BA2 (t) = BA1 (t) + ∆,

∀ t > sj .

(A.1)

However, solution 1 consumes less energy due to the fact that transmission is done at a constant rate. Let us denote the energy saving of solution 1 with respect solution 2 at time instant t2 as µ. Then, the relation between the two energy expenditure curves at t2 is E2 (t2 ) = E1 (t2 ) + µ.

(A.2)

By subtracting (A.1) evaluated at t2 and (A.2) and using (2.2), the following relation between the battery levels is obtained: B2 (t2 ) − B1 (t2 ) = ∆ − µ. We want to prove that B2 (t2 ) > B1 (t2 ), therefore, that ∆ − µ > 0. Let us first find the expressions for ∆ and µ: ∆ = E2 (sj ) − E1 (sj ) = (sj − t1 )(g(r1 ) − g(r0 )),

(A.3)

µ = E2 (t2 ) − E1 (t2 ) = = (sj − t1 )g(r1 ) + (t2 − sj )g(r2 ) − (t2 − t1 )g(r0 ). Finally, the expression of ∆ − µ is ∆ − µ = (t2 − sj )(g(r0 ) − g(r2 )),

(A.4)

that is greater than zero since g(r0 ) > g(r2 ).

A.2

Proof of Lemma 2

The proof of the Lemma is divided in two parts. We will first show that if a rate change is produced at ui , the data departure curve must be equal to the sum of all the data arrivals 48

before ui . Afterwards, we show that if a rate change is produced, then the rate and the power must increase. Part 1. We start by assuming that a rate change is produced at ui such that D(ui ) < DA (ui ) and that it is optimal. We show by contradiction that this cannot be optimal. Let us denote r− and r+ the rates before and after ui , respectively. Let us consider the time interval (ui − ϵ, ui + ϵ) with ϵ being positive. Note that if we select a sufficiently small ϵ, we can find a straight line with rate r =

D(ui +ϵ)−D(ui −ϵ) , 2ϵ

which satisfies energy and data constraints,

that transmits the same amount of data while having less energy expenditure. Hence, we have proved that if the rate changes in a data arrival event, then D(ui ) = DA (ui ). Part 2. Now we prove that when the rate changes at ui , it must increase. The procedure is the same as the one in the first part of the proof. We start by assuming that a rate decrease is optimal and then, by contradiction we show that it cannot be optimal. As before, we denote r− and r+ the rates before and after ui , respectively, where r− > r+ . We consider the same time interval. In this case, we can also find a straight line whose slope is r =

D(ui +ϵ)−D(ui −ϵ) , 2ϵ

which satisfies energy and data constraints, that transmits

the same amount of data while having less energy expenditure. Hence, we have proved by contradiction that if the rate changes, it must increase.

A.3

Proof of Lemma 3

Note that, since in Lemma 3 it is specified that there is overflow of the battery in sj , this problem is equivalent to the one presented in Appendix A. Hence, the optimal solution is to schedule the maximal rate before the energy arrival, while maintaining energy and data causality constraints. In sj there is a battery overflow which implies that the energy causality constraint is satisfied for any rate. Hence, the maximal rate is given by the rate that meets the data causality constraint at sj , thus, D(sj ) = DA (sj ), and, in order not to break this constraint, the rate at s+ j must be zero.

A.4

Proof of Lemma 4

Since there is no battery overflow, the energy expenditure at sj must be within the interval: Emin (sj ) ≤ E(sj ) ≤ BA (sj ), 49

where the lower bound comes from the constraint of not having an overflow and the upper bound is due to energy causality. The proof of Lemma 4 can be divided in three parts: Part 1. Here we show that a rate change in sj is not optimal unless E(sj ) is equal to BA (sj ) or Emin (sj ). Note that this part of the proof is equivalent to Part 1 of the proof of Lemma 2 and has been skipped for brevity. Part 2. This part of the proof shows that, when there is a rate change and E(sj ) = BA (sj ) then the rate and power must increase. This proof can be done in the same manner than Part 2 of Lemma 2. Part 3. Equivalently, in this point it is shown that if there is a rate change and E(sj ) = Emin (sj ), then the rate and power must decrease. Again, the procedure is the same than Part 2 of Lemma 2 and we just briefly sum the procedure. First, a rate increase is assumed to be optimal and, then, by contradiction it is shown that it cannot be optimal because there exists a straight line that connects the points (si − ϵ, si + ϵ) that satisfies the constraints. For this reason, a rate increase cannot be optimal.

A.5

Proof of Lemma 5

The lemma is proved by contradiction. Assume that the energy expenditure of the optimal solution at T is E(T ) < BA (T ), which implies that the battery of the node at this time instant is greater than zero, i.e., B(T ) > 0. Note that a new solution E ⋆ (t) can be found that increases the power during the epoch right before T in such a way that E ⋆ (T ⋆ ) = BA (T ⋆ ), where T ⋆ is the completion time of the new solution. Note that E ⋆ (t) still satisfies energy and data causality. Since the power-rate function is increasing, the increase in power is mapped to an increase in the rate. Therefore, since the rate of the last epoch is higher, the completion time of the new solution T ⋆ is smaller than T , i.e., T ⋆ < T , and hence, the solution E(t) cannot be optimal.

A.6

Proof of Theorem 1

In order to show that the algorithm computes the optimal data departure curve, we focus on the three different situations that can occur depending on the constraints of the problem: (i.) The algorithm finishes transmission by using an even power allocation among all bits 50

to be transmitted. (ii.) minT strategy is used to obtain the epoch. (iii.) The epoch is computed by using minEnergy mode. In the following, the optimality of these cases is proved by showing that the algorithm-chosen solution satisfies the optimality conditions and that it is unique. Part 1 (Even power allocation). When this situation occurs, the algorithm ends transmission by transmitting at constant rate. Hence, Lemma 1 is satisfied in the interval (0, T ). Lemmas from 2 to 4 do not apply since there are no rate changes. Finally, note that Lemma 5 is also satisfied, since the optimal rate r⋆ is obtained as the maximum rate that allows the transmission of all the data by using all the available energy, moreover, from the properties of the function g(·) this rate is unique. Part 2 (minT mode). We want to demonstrate that the optimal departure curve transmits at Rmax during a period of time of eRmax . Let Tˆ be the total completion time that would ˆ was feasible, hence, Tˆ = DT ot /R. ˆ Similarly, Tmax = be obtained if transmitting at rate R DT ot /Rmax . Note that, since Rmax is a feasible rate, Tmax is an upper bound of the total completion time, T , whereas Tˆ is a lower bound, hence: Tˆ < T < Tmax .

(A.5)

Consider the data departure curve D1 (t) = Rmax t. Note that any other data departure curve, D2 (t), is suboptimal since, in order to satisfy (A.5), D2 (t) will cross D1 (t) for some ty ∈ (0, Tmax ). Hence, at ty , both curves have sent the same amount of data, however, D1 (t) has consumed less energy. Now we must show that, at eRmax , the rate increases. Note that by transmitting at ˆ some energy has been saved. Then, in the following epoch, the Rmax instead of at R, ˆ 2 obtained from an even power available energy per bit is higher and, then, the new rate R ˆ2 > R ˆ > Rmax . Hence, we have allocation among all bits in the following epoch fulfills R proved that at eRmax a rate increase is produced. Part 3 (minEnergy mode). This mode is used when it is not possible to finish transmission at any rate. Note that the algorithm can select three different kind of points, denoted as vi = (Di , li ), for ending the epoch depending on the constraints: 51

• v1 | D1 (l1 ) = Dmax (l1 ) where l1 is either ui or sj . • v2 | D2 (l2 ) = Dmin (l2 ) where l2 is sj . • v3 | D3 (l3 ) = Dmax (l3 ) = Dmin (l3 ) where l3 is sj . Note that Corollary 1 is satisfied for any of the selected points and that Lemma 5 does not apply since transmission cannot be ended, yet. Hence, we have to prove the following three conditions: (i.) If a point such as v1 is selected, there will be a rate increase (Lemma 2 and point 1 of Lemma 4). (ii.) If a point such as v2 is selected, there will be a rate decrease (Lemma 4 point 2). And (iii.), if a point such as v3 is selected, the rate of the following epoch is zero (Lemma 3). Regarding (i.), the rate of the epoch is r1 = D1 /l1 that is the supremum of SRmin . Note that, at the following iteration, the set S2Rmin , where the superindex stands for the new iteration, includes all the rates in the interval (0, r1 ) and the rates contained in the interval (r1 , r1 + ϵ), for some ϵ > 0. Hence, the rate of the following iteration, r2 , satisfies that r2 ≥ r1 , therefore, a rate increase is produced. A similar approach can be done for (ii.), the rate of the epoch is r1 = D2 /l2 that is the infimum of SRmax . At the following iteration the set S2Rmax contains all the rates in (r1 − ϵ, ∞) for some ϵ > 0. Hence, the rate of the following iteration, r2 , satisfies that r2 ≤ r1 and, therefore, there is a rate decrease. Finally, in case (iii.), an overflow of the battery is produced. Note that the solution chosen by the algorithm satisfies that all the available data has been transmitted and this implies that, in order to satisfy data causality constraint, the rate of the following epoch must be zero until the next data arrival.

52

E (t ) p2

p1

µ

Δ

p0

E (t1 ) t1

sj

t2

t

t2

t

D (t ) D(t 2 )

r2

r1

r0

D (t1 ) t1

sj

Figure A.1: Graphical representation of the problem presented in Appendix A.1.

53

54

Appendix B Proofs of Chapter 3 B.1

Proof of Lemma 10

The proof of the first point in the lemma is equivalent to the proof of Lemma 3. For the second point see Figure 3.3 where the tx = q1 . We know that Dmin (tx ) = DQoS (tx ) and that Dmax (tx ) = DBA (tx ), however, note that for simplicity we did not perform the mapping of the curve BA (t) at the time instant t = tx since this point is not a discontinuity of this curve. If we do the mapping of the curve at this time instant, the point marked as a red cross is obtained, we observe that indeed DQoS (tx ) < DBA (tx ) since for this lemma we have assumed that the problem has a solution. Note that if we were not considering the QoS constraints the solution would be given by the curve B as explained in the previous chapter. However, due to the QoS constraint, the rate must be increased in the time interval (0, q1 ) and then decreased in the interval (q1 , s1 ) in order to have the same energy expenditure in s1 . In this situation, proving that D⋆ (tx ) = DQoS (tx ) is equivalent to Part 1 of the proof of Lemma 2 and the procedure to proof that at tx a rate decrease is produced is the same that in Part 3 of the proof of Lemma 4 that can be found in the Appendix A.4.

B.2

Proof of Theorem 2

At each iteration the algorithm checks that the condition DQoS (qi ) < qi g −1 (BA (qi )/qi )

(B.1)

is satisfied ∀ qi . If it is not, Lemma 7 applies, hence, there is no solution for the problem and the algorithm ends. Otherwise, the algorithm must satisfy Lemmas 8- 12 at each rate change and Lemma Lemma 5 by the end of the transmission. The proof that these 55

lemmas are satisfied is equivalent to the proof of the algorithm of the previous chapter, see Appendix A.6.

56

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