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

Transcription

1 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

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3 To my family, friends and Paul.

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5 Contents 1 Introduction Green communications Background Communication system Sources of energy consumption Renewable energies and energy harvesting nodes State of the art On the energy-efficiency of channel codes Energy-efficient packet scheduling and rate control Energy-efficiency in the spatial domain Summary of the thesis Efficient data transmission for an energy harvesting node with battery capacity constraint Introduction Problem formulation Properties of the optimal solution Problem insights Problem visualization Constraints mapping into the data domain for a given epoch Maximum and minimum rates Optimal data departure curve construction Finish transmission at a constant rate Get rate and length of the next epoch Algorithm optimality Conclusion Efficient data transmission for an energy harvesting node with battery capacity and QoS constraints 23 i

6 3.1 Introduction Problem formulation Properties of the optimal solution Problem insights Problem visualization Constraints mapping into the data domain for a given epoch Optimal data departure curve construction Algorithm optimality Conclusion Throughput maximization for a multi-antenna energy harvesting node Introduction System model Throughput maximization problem Results Conclusions Conclusions and future work Conclusions Future work A Proofs of Chapter 2 47 A.1 The overflow problem A.2 Proof of Lemma A.3 Proof of Lemma A.4 Proof of Lemma A.5 Proof of Lemma A.6 Proof of Theorem B Proofs of Chapter 3 55 B.1 Proof of Lemma B.2 Proof of Theorem ii

7 List of Figures 1.1 General scheme of a communication system 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 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 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 A.1 Graphical representation of the problem presented in Appendix A iii

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9 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 AGAUR 1. 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 v

10 AWGN BER CSI ICT ISO LT MAC MAP MFSK MIMO ML OFDM Additive White Gaussian Noise Bit Error Rate Channel State Information Information and Communications Technology International Standards Organization Luby Transform Medium Access Control Maximum a Posteriori Multiple Frequency-Shift Keying Multiple-Input Multiple-Output Maximum Likelihood Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access OSI QoS SISO SNR SVD WER WSNs Open System Interconnection Quality of Service Single-Input Single-Output Signal to Noise Ratio Singular Value Decomposition Word Error Rate Wireless Sensor Networks vi

11 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.

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13 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

14 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 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

15 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 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, P T, i.e., ( R = W log γp ) T W N o (bits/second), (1.1) where γ is the channel gain, W is the channel bandwidth and N 0 the noise spectral density. Moreover, we can express the rate as a function of the bit duration time, T b, as R = 1/T b [3]. 3

16 From the previous equation be obtain the transmission power consumption as P T = ( 2 R W 1 ) W N0 γ 1 (W atts). (1.2) Hence, the energy consumed in transmission per bit is E T = P T T b = ( 2 R W 1 ) W N 0 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, T b. RF circuitry energy consumption Transmission energy is one of the most important sources of energy consumption, albeit not the only one. Circuitry energy, E C, 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

17 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 r α, 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 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

18 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 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, d CR, 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 < d CR, while for distances above the critical distance, d > d CR, 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, P C, the transmission power, P T, the decoding power, P Dec, as well as some energy consumption to start-up the different nodes. The results show that in general BCH codes are more energy-efficient 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

19 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 U(R) = R P (R) = R P C + P T (R), (1.4) where P (R) is the total power, i.e., the circuitry power, P C, plus the transmission power, P T. Then, the rate that maximizes energy-efficiency is found, i.e., R = arg max R U(R) = arg max R R P C + P T (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

20 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 P T. 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

21 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 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

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23 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

24 E 0 E 1 D 1 E 2 D N-1 E K-1 s 1 u 1 s 2 u N-1 s K-1 t T? D 0 Figure 2.1: Summary of the packetized arrival model 2.2 Problem formulation Let us consider a node with a finite battery capacity, C max, 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 u i seconds the i-th data packet arrives containing D i 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 s j seconds and a total of E j 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 u i data arrival events. Similarly, the time instants s j 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

25 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 0 g(d (τ)) dτ. (2.1) 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 C max. Thus, B(t) must satisfy that 0 B(t) C max, t 0. The overflows of the battery, denoted by O j, 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 s j is O j = max{0, E j C max + B(s j )}. Note that overflows guarantee that the battery level will never be above the battery capacity, B(t) C max. 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 B A (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, O j, produced due to the limited battery capacity C max, i.e., B A (t) = j : s j <t(e j O j ). From beforehand it is easy to see that the optimal solution will minimize the total overflow of the battery, thus, maximizing the accumulated battery. In other words, if we minimize the overflows the node will be able to use more energy. At every time instant it is possible to obtain the energy stored in the battery, B(t), as the difference between the accumulated battery and the energy expenditure: B(t) = B A (t) E(t). (2.2) 13

26 Definition 5 (Minimum Energy Expenditure). The minimum energy expenditure, E min (t), is the smallest amount of energy that the node must have spent at time t such that no overflow of the battery is produced. The minimum energy expenditure can be obtained from the battery capacity, C max, and the accumulated battery B A (t) as E min (t) = B A (t + ) C max. Similarly to the accumulated battery, we define the accumulated data: Definition 6 (Accumulated Data). The accumulated data D A (t) is the sum of data that has arrived at the node during the time interval [0, t), i.e., D A (t) = i : u i <t D i. Our goal is to find the data departure curve, D(t), that minimizes the total transmission time, T, of the N packets of data by fulfilling at every time instant the following conditions: (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. Moreover, given two data departure curves with the same completion time, the one that needs less energy is always preferred. From all that has been said above, the problem can be expressed as follows: min D(t) s.t. T (2.3) E(t) B A (t), D(t) D A (t), D(T ) = D A (T ). 2.3 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 1 (Transmission Without Events). We want to characterize the optimal departure curve in the time interval (t 1, t 2 ) where there are neither energy nor data arrivals. We also consider that the data departure curve at the boundary of the intervals is D(t 1 ) and D(t 2 ), respectively, and that these two points satisfy the data and energy constraints. Lemma 1. In Problem 1, D (t) is a straight line where the slope, or, equivalently, the transmission rate, is constant and equal to r(t) = D(t 2) D(t 1 ) t 2 t 1, t (t 1, t 2 ). 14

27 Proof. See [13]. Corollary 1. Lemma 1 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 s j or u i. From now on, we will denote a period of time where transmission is done with constant rate/power as an epoch. Lemma 2. D (t) satisfies that if the rate changes at a data arrival event, i.e., at t = u i, then D (u i ) = D A (u i ) and there is a rate/power increase, i.e., r(u i ) < r(u + i ). Proof. See Appendix A.2. Lemma 3. D (t) satisfies that if a rate change is produced in an energy arrival event, i.e., at t = s j, that produces overflow, then D (s j ) = D A (s j ), and the rate/power in s + j is zero. Proof. See Appendices A.1 and A.3. From Lemma 3, we have that battery overflows are only allowed when all the available data has been transmitted, as it is summarized in the following Corollary. Corollary 2. In the optimal transmission strategy, battery may only overflow when there is no data to transmit. Lemma 4. D (t) satisfies that, if a rate change is produced in an energy arrival event, i.e. s j, that does not produce overflow of the battery, then one of these two conditions is fulfilled: 1. E (s j ) = B A (s j ) and there is a rate/power increase, i.e., r(s j ) < r(s + j ). 2. E (s j ) = E min (s j ) and there is a rate/power decrease, i.e., r(s j ) > r(s + j ). Proof. See Appendices A.1 and A.4. Lemma 5. The optimal solution must satisfy that, at the instant T at which all the data has been transmitted, the energy expenditure is equal to the accumulated battery, i.e., E (T ) = B A (T ). Proof. Similar to the proof of Lemma 5 in [15]. 15

28 E(t) E 3 B A (t) E () min t I E 2 C max E 1 A B II E 0 s 1 C s s 2 3 t D(t) D 3 D D 2 1 D 0 D A (t) B C u u1 2 A u 3 T? III IV t Figure 2.2: Visualization of the problem presented in (2.3). 2.4 Problem insights Problem visualization The problem formulated in (2.3) can be graphically seen as shown in Figure 2.2, 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 is depicted by the blue solid line in the figure at the bottom. Hence, a solution must lie within regions II and IV, simultaneously, in order to be valid. Three different data departure curves (A, B, and C) and their associated energy expenditure curves are shown. Note that curve A does not satisfy energy causality constraints so it is not a valid solution. The minimum energy expenditure is shown by the red dashed-line. When an overflow is produced, the solution crosses the dashed-line, i.e., curve C, and the accumulated battery must be re-scaled due to the lost energy. 16

29 2.4.2 Constraints mapping into the data domain for a given epoch Given that the optimal solution is known up to a given reference time instant, i.e., t = 0, where E(t) = D(t) = 0, the energy causality constraint and the minimum energy expenditure can be mapped to the data domain by using the inverse of the rate-power function, g 1 ( ). Note that, at s 1, the maximum allowed transmission power is p 1 = E 0 /s 1. Any power above this would require a power change for some time instant smaller than s 1, which is not optimal as shown by Corollary 1. Then, the maximum data that can be transmitted from the reference point to s 1 due to the energy constraint is D BA (s 1 ) = g 1 (p 1 )s 1. 1 The same approach can be done with the rest of the edges of the B A (t) and E min (t) curves. Hence, given a reference point, B A (t) and E min (t) can be mapped to the data domain obtaining D BA (t) and D Emin (t), respectively, as shown in Figure 2.3. Once this is done, 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., D max (t) = min(d A (t), D BA (t)). Similarly, given a reference point, a lower constraint can be found by mapping E min to the data domain, i.e., D Emin (t). Hence, the lower constraint is D min (t) = D Emin (t). 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. In the following, D(t) is said to be feasible between (t 1, t 2 ) if D min (t) D(t) D max (t), t (t 1, t 2 ). It is important to remark that E min (t) is not a constraint of the problem formulated in (2.3). However, if E(t) crosses E min (t), a battery overflow is produced that, from Lemma 3, is known to be suboptimal unless all the available data has been transmitted, i.e., D Emin (s i ) = D A (s i ). Hence, an important contribution of this work is to show that D Emin (t) can be seen as a strict constraint since it has been proved that any solution below D Emin (t) is suboptimal. As we will see in next section, this mapping is used in order to ease the computation of D (t). Once D min (t) is obtained, the problem is similar to the one studied in [13], where their D min (t) was determined by the QoS constraints. However,we have two main differences: (i.) The constraints must be recalculated at every rate/power change since they depend on the chosen energy expenditure. (ii.) D min (t) can be greater than D max (t) Maximum and minimum rates Let R max denote the set that contains the rates obtained from joining the reference point, i.e., D(0) = 0, with the discontinuities from the left of D max (t) and such that the obtained curve is feasible from the reference point until the discontinuity. Similarly, R min contains 1 Here, we have used that the reference point is t = 0 and that E(t) = 0. For any other reference point, the energy should be taken into account, however, the algorithm that we develop in following sections rescales the problem at every iteration moving the reference at t = 0. 17

30 D(t) x g E E ( ) s2 s2 x x x D B A (t) g 1 ( E / s) s x s 1 u1 u2 x s s 2 3 u 3 x D A (t) D Emin ( t) t Figure 2.3: Mapping of the energy causality constraint and minimum energy expenditure to the data domain. the rates obtained from joining the reference point with the discontinuities from the right of D min (t) and such that the obtained curve is feasible in the same interval. An example of this can be seen in Figure 2.4, where the blue and red solid lines stand for feasible rates of the sets R max and R min, respectively. Let R max denote the infimum of the set R max and R min refer to the supremum of the set R min. Let e Rmax and e Rmin denote the time instants from which R max and R min have been obtained. Note that R max is always greater than R min, otherwise, either R max or R min would not be feasible. Then, it is easy to see that all the rates above R max and below R min are suboptimal since would require a rate change to transmit the same amount of data. Hence, the optimal rate lies within the interval [R min, R max ]. 2.5 Optimal data departure curve construction Let us denote M as the number of epochs of the optimal solution, i.e., the number of rate changes. As stated in Corollary 1, the optimal departure curve is a piece-wise linear function with rates r = [r 1, r 2..., r M ]. The proposed algorithm follows an iterative process where at each iteration the duration and rate of an epoch are determined. We will explain the first iteration, i.e., how to obtain r 1, and the rest can be obtained by following the same approach. The developed algorithm is divided in two parts. In the first part, presented in Section 2.5.1, it is checked whether it is possible to transmit all bits by using an even power allocation, where by even power allocation we mean to transmit all the remaining data by using all the available power at constant rate. In such a case, the algorithm ends and 18

31 D(t) Rmin Rmax D max ( t) D min ( t) s 1 u1u s s u t Figure 2.4: Finding rates of the possible points of rate change. the minimum completion time T has been found. Otherwise, at least two more epochs are required and the strategy for obtaining the following epoch is determined. There are two different strategies or modes, either to minimize the total completion time or to minimize the energy expenditure. The mode obtained in the first part of the algorithm is passed as a parameter to the second part, presented in Section 2.5.2, that finds the optimal rate r 1 and duration l 1 of the epoch. Once the first rate is determined, the problem is rescaled. The transmitted bits are subtracted from D A (t) and the expended energy is subtracted from B A (t). Moreover, in case that r 1 produces a battery overflow at time instant l 1, the amount of energy lost due to the overflow is also subtracted from B A (t). Finally, the origin of coordinates is moved to l 1 and the whole procedure is repeated to find the next point r 2 and l 2. When the algorithm ends, the minimum completion time can be found as the sum of all the length of all the epochs, i.e., T = M i=1 l i. In the following subsections, we explain in more detail each of the aforementioned parts Finish transmission at a constant rate The first step, which is presented in Function 1, checks whether it is possible to transmit all bits by using an even power allocation in just one epoch. If it is possible, the algorithm returns the rate and length of the last epoch, otherwise, it returns the strategy or mode that will be used in order to determine the following epoch. The function first checks whether by transmitting at the maximum feasible rate, R max, it is possible to transmit all the remaining data D T ot. In case it is not possible, the function returns the mode minenergy. Otherwise, the function finds the time ˆT 0 required to transmit the remaining data D T ot with the initial battery. Then, it computes the 19

32 Function 1 checkfinish D T ot = D max (u + N ) Remaining data if (getdataincrossing(r max t, D max (t)) < D T ot ) then return mode = minenergy, finish = 0 It is not possible to finish yet. else for i=0:k do E = i j=0 E j ˆT i is obtained by solving g 1 (E/ ˆT i ) = D T ot / ˆT i ˆR = D T ot / ˆT i Even power allocation among all bits if R min ˆR R max then D(t) = ˆRt is feasible in [0, ˆT i ] S = {E i s i (0, ˆT i )} if S = then The algorithm ends and the rate and length of the last epoch are returned. return r = ˆR, l = ˆT i, finish = 1 end if else if ( ˆR > R max ) then return mode = mint, finish = 0 end if end for return mode = minenergy, finish = 0 end if equivalent rate and checks whether transmitting at this rate is feasible, i.e, the following two conditions are fulfilled: (i.) ˆR < Rmax and (ii.) ˆR > Rmin. In case that (i.) is not 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, ˆT 0 ). In case of no arrivals, the algorithm ends and the last epoch has been found. In case there is an energy arrival in (0, ˆT 0 ), the function repeats the whole process but now using the initial battery and the energy of the first arrival, E 1. This process is repeated until (i.) becomes false or a feasible curve is found. 20

33 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 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 at some rate r with r R max. (ii.) The rate obtained from an even power allocation ˆR is not feasible due to ˆR > R max. 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 = R max and l = e Rmax. 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 S Rmax contains all the rates r such that the associated data departure curve crosses the constraint D max (t) first. Whereas the set S Rmin contains all the rates r such that the associated data departure curve crosses the constraint D min (t) first. Then, the rate of the following epoch is determined as the infimum of S Rmax or, equivalently, the supremum of S Rmin, i.e., r = inf( S Rmax ) = sup( S Rmin ) (2.4) and the duration of the epoch can be obtained as the first time instant t x such that, rt x = D max (t x ) or rt x = D min (t x ). Then, the length of the epoch is l = t x 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 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, trans- 21