A General Framework for the Optimization of Energy Harvesting Communication
with Battery Imperfections
Energy harvesting has emerged as a powerful technology for complementing current battery-powered communication systems in order to extend their lifetime. In this paper a general framework is introduced for the optimization of communication systems in which the transmitter is able to harvest energy from its environment. Assuming that the energy arrival process is known non-causally at the transmitter, the structure of the optimal transmission scheme, which maximizes the amount of transmitted data by a given deadline, is identified. Our framework includes models with continuous energy arrival as well as battery constraints. A battery that suffers from energy leakage is studied further, and the optimal transmission scheme is characterized for a constant leakage rate.
A General Framework for the Optimization of Energy Harvesting Communication Systems
with Battery Imperfections
Bertrand Devillers, Deniz Gündüz
00footnotetext: This work is supported in part by EXALTED project (IT-258512) funded by European Unions Seventh Framework Programme (FP7), and by the Spanish Government under project TEC2010-17816 (JUNTOS). Deniz Gündüz is supported by the European Commission’s Marie Curie IRG Fellowship with reference number 256410 under the Seventh Framework Programme.00footnotetext: B. Devillers and D. Gündüz are with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Castelldefels, Barcelona, Spain.00footnotetext: Emails: firstname.lastname@example.org, email@example.com
Battery leakage, battery size constraint, broadcast channel, continuous energy arrival, energy efficient communications, energy harvesting, rechargeable wireless networks, throughput maximization.
Energy efficiency is a key challenge in the sustainable deployment of battery-powered communication systems. Applications such as wireless sensor networks depend critically on the lifetime of individual sensors, whose batteries are limited due to physical constraints as well as cost considerations. Power management is essential in optimizing the energy efficiency of these systems in order to get the most out of the available limited energy in the battery. A complementary approach has recently been made possible by introducing rechargeable batteries that can harvest energy from the environment. Several different technologies have been proposed and implemented for harvesting ambient energy such as solar, radio-frequency, thermoelectric or solar (see [1, 2, 3, 4, 5] and references therein for various examples of energy harvesting technology).
Harvesting energy from the environment is an important alternative to battery-run devices to extend their lifetime. However, it is important to design the system operation based on the energy harvesting process to increase the efficiency. Energy harvesting systems have received a lot of recent attention [6, 7, 8, 9, 10]. Node and system level optimization have been considered from both practical and theoretical perspectives. The previous work that are most relevant to the problems studied in this paper are [11, 12, 13, 14]. In , the problem of transmission time minimization is studied when the data and the energy arrives at the transmitter in packets; and the transmission power is optimized when the data and energy arrival times and amounts are known in advance. In , the amount of transmitted data is maximized for an energy harvesting system under deadline and finite battery capacity constraints. Reference  also shows that the transmission time minimization problem studied in  and the transmitted data maximization problem are duals of each other and their solutions are identical for the same parameters. The problem is extended to the broadcast channel in [13, 14, 15, 16], to the relay channel in , and to the multiple access channel in .
The problem considered in this work is that of maximizing the amount of data that is transmitted within a given deadline constraint under various assumptions regarding the energy harvesting model as well as the battery limitations. Our focus is on the offline optimization of the energy harvesting communication system, that is, we assume that the energy arrival profile is known in advance. We first introduce a general framework for transmitted data maximization by adjusting the transmit power in an energy harvesting system with battery limitations. Our model includes continuous energy harvesting, generalizing the packetized energy arrival model considered in  and . Moreover, different from the previous work, our model also includes the realistic scenario of battery degradation over time by considering a time-varying battery capacity. We show that these constraints can be modeled through cumulative harvested energy and minimum energy curves, which are then used to obtain the optimal transmission policy. The framework introduced for the energy harvesting system optimization is similar to the calculus approach introduced by Zafer and Modiano for energy-efficient data transmission in . We later show that the proposed framework also applies to a broadcast channel with an energy harvesting transmitter.
We then consider a more realistic battery model with energy leakage. Assuming a constant leakage rate, we identify the optimal transmission strategy for the case of a packetized energy arrival model.
The paper is organized as follows. Section II presents the system model. Optimal transmission scheme for a point-to-point system under battery size constraints is derived in Sections III. In Section IV, it is shown that the proposed framework can be used to characterize the optimal transmission scheme in an energy-harvesting broadcast channel. We consider battery leakage in Section V and find the optimal transmission scheme for a linear leakage rate. Finally, conclusions are provided in Section VI.
Ii System Model
We consider a continuous-time model for both the harvested and the transmitted energy, that is, the harvested energy is modeled as a continuous-time process, while the transmitter is assumed to be able to adjust its transmission power, and hence, the transmission rate, instantaneously. This continuous-time model generalizes the discrete-time arrival model considered in  and . A cumulative curve approach is used to described the flow of energy in the system.
Definition II.1 (Harvested Energy Curve)
The harvested energy curve is a right-continuous function of time , , that denotes the amount of energy that has been harvested in the interval .
Definition II.2 (Transmitted Energy Curve)
The transmitted energy curve is a continuous function with bounded right derivative, that denotes the amount of energy that has been used for data transmission in the interval , .
Naturally, we require , i.e., the transmitter cannot use more energy than that has arrived. We also consider a “minimum energy curve” that might model, for example, a battery size constraint.
Definition II.3 (Minimum Energy Curve)
Given an harvested energy curve , a minimum energy curve is a function satisfying , , and denotes the minimum amount of energy that needs to be used by the system until time .
Given the harvested energy curve and the minimum energy curve, a feasible transmitted energy curve should satisfy the conditions , . Among all feasible transmitted energy curves, our goal is to characterize the one that transmits the highest amount of data over a given finite time interval . We consider offline optimization, that is, the harvested and the minimum energy curves are assumed to be known in advance111This is an accurate assumption for systems in which the energy harvesting process can be modeled as a deterministic process. For example, in solar based systems the amount of energy that can be harvested at various times of the day can be modeled quite accurately. In some other systems, harvested energy depends on the operating schedule of the harvesting device rather than the energy source, such as shoe-mounted piezoelectric devices; and the harvested energy curve can be modeled accurately in advance..
We assume that the instantaneous transmission rate relates to the power of transmission at time through a rate function , which is a non-negative strictly concave increasing function of the power with . We note here that many common transmission models, such as the capacity of an additive white Gaussian noise channel, satisfy these conditions . The total transmitted data corresponding to a given curve over the interval is found by
where is the derivative of function at time , and it gives the power of transmission at that instant while is the corresponding transmission rate.
Iii Optimal Transmission Scheme under Battery Size Constraints
In our problem formulation we assume that the transmitter always has data to transmit. Hence, the minimum energy curve can be used to model a constraint on the battery size, forcing the system to use any energy that cannot be stored in the battery for transmission of additional data before it is discarded. For a fixed energy curve and unlimited battery size, the energy that is available in the battery at time instant is given by . However, if the battery size is , we should have . Consequently, the associated minimum energy curve is given by .
We can also consider a time-varying battery capacity , which can model the degradation in the battery capacity over time. This is a common phenomenon in rechargeable batteries used for energy harvesting applications. See Fig. 1(a) for an illustration of the harvested and minimum energy curves for a battery with continuously decreasing capacity.
Now, the optimization problem can be stated as follows.
where specifies the set of all non-decreasing, continuous functions with bounded right derivatives for all and with .
We first present the optimality conditions for the transmitted energy curve. Similar to previous studies, such as , ,  and , our main tool is the Jensen’s inequality given in the following lemma (in the integral form).
[Jensen’s inequality] Let be a non-negative real valued function, and be a concave function on the real line, then
with strict inequality if is strictly concave, , and is not constant over the interval .
Consider the simple setup in which the battery has available energy at time , no further energy is harvested, and the minimum energy curve is given as for and . We will prove for this simple setting that the constant power curve transmits the maximum amount of data over the time interval .
For any transmitted energy curve with non-constant power, by replacing the function in Lemma III.1 with , and letting , and , we obtain
which is equivalent to
Note that is the transmitted data by the constant power scheme. Hence, this proves the fact that the maximum data is transmitted by this scheme. We can express this result in a more general context as in the following theorem.
Let be a feasible transmitted energy curve and be a straight line segment over interval that joins and , . If satisfies for , the transmitted energy curve defined as
The following theorems state, respectively, the uniqueness of the optimal transmitted energy curve and the optimality conditions. Their proofs follow similarly to those of Theorem 2 and Lemmas 2-4 in .
For a strictly concave rate function , if is a feasible transmitted energy curve which does not have any two points that can be joined by a distinct feasible straight line, then is unique and it maximizes the transmitted data.
Let be the optimal energy expenditure curve and be any point at which the power of transmission changes, i.e., the slope of changes. Then, at , intersects either or . If , then the slope change must be positive. If , then the slope change must be negative.
The optimal transmitted energy curve is also the one that has the minimum length, and hence, the same “string visualization” suggested in  can be applied here. The string visualization suggests that, if we tie one end of a string to the origin and connect it to the point tightly while constraining it to lie between and , this string gives us the optimal energy expenditure policy.
A special case of the framework considered here is the one with packetized energy arrivals and without any battery constraint. This is the energy-harvesting dual of the packet arrival problem considered in . As it is shown in , this is equivalent to the problem of transmission time minimization problem studied in . In this problem we have for , and energy packets arrive at times . The algorithm that gives the optimal transmitted energy curve for this problem can be obtained following  and .
Another example that fits into the general structure introduced above is the following. Consider a wireless system with an energy storage unit consisting of batteries. Assume that all the batteries are full initially and a total of energy is available in the system at time , where is the capacity of battery . It is assumed that the batteries in the system have finite lifetime, and they die at certain time instants, , . The problem is to find the maximum amount of data that can be transmitted until the last battery dies, i.e., until . In this problem we have for , and can be obtained as in Fig. 1(b). Note that, since once the battery dies, the energy stored in it is not available for transmission anymore, and since we always have data in the queue to be transmitted, it is always beneficial to use the available energy in a battery before it dies. In this sense, we can consider the time until a battery dies as a deadline constraint on the time the available energy in this battery should be used. The optimal transmitted energy curve can then be found using the string argument as seen in Fig. 1(b).
As an example of continuous energy arrival, we consider here a model of a solar panel harvesting energy during the day. The amount of energy harvested per unit of time changes during the day. While no energy is harvested when there is no sun, the harvested energy is maximized at noon (see ). We model the rate of harvested energy with the function for , and elsewhere, where denotes the time of the day (hours), such that . The unit of energy depends on the solar panel characteristics. The corresponding harvested energy curve is depicted in Fig. 2.
Assume that we want to maximize the amount of data that can be transmitted up to time , i.e., until the panel stops harvesting energy. Based on the above arguments, the optimal transmitted energy curve is identified as follows. First we draw a tangent to the harvested energy curve from the point , and denote its intersection with the curve by . The transmitted energy curve follows the harvested energy curve from the origin up to 222In practice, a continuous adaptation of the transmission rate is unrealistic due to the block structure of channel coding, and the finite number of modulation and coding modes available. However, such practical constraints are out of the scope of this paper.. Afterwards, it follows the straight tangent line, i.e., it uses constant power transmission. Note that, while it is easy to prove the optimality of this strategy using Theorem III.3, the discrete energy arrival models studied in  and  do not apply here.
Iv Optimal Broadcast Scheme with Battery Constraint
In this section, we show that the general approach introduced in Section III can be instrumental in identifying the optimal transmission policy in a broadcast channel (BC) with an energy harvesting transmitter , . Consider the same energy harvesting model at the transmitter as before; however, now there are two receivers in the system, and the transmitter has independent data for each receiver.
The BC problem is studied in  and ; however, the solutions in these papers are elaborated from the basics rederiving the behavior of the optimal transmission policy in the BC scenario. Here, we show that the general approach introduced in previous section for the point-to-point setting can be directly applied to the BC scenario as well. This approach allows to generalize the results in  and  to continuous energy arrivals, and introduce battery constraints in the problem formulation [15, 16].
We consider an additive white Gaussian BC in which the signal received at receiver is given by
where is the channel input of transmitter and is the zero-mean Gaussian noise component with variance . Without loss of generality, we assume that 333The case with reduces to the single receiver problem.. Let denote the total number of bits transmitted to receiver up to time . Our goal is to maximize the weighted sum of transmitted bits by time , for some .
In the broadcast channel setting, the transmitter not only needs to identify the transmitted energy curve , but also has to decide how to allocate the power among the two receivers at each time instant. Accordingly, we denote by and the power allocated to each receiver at time . The optimization problem can be written as follows.
We assume that the rate-power functions are operating on the boundary of the capacity region of the Gaussian BC:
The considered optimization can be decoupled into two maximization problems as follows:
where we define .
First, we consider the maximization problem in between brackets in (12). Defining , we can make the following observations on its solution444The time variable is omitted for conciseness.:
If , no power is allocated to the first receiver, i.e. , independent of the total power.
If , no power is allocated to the second receiver, i.e. , independent of the total power.
When , the optimal power allocation behaves as follows. If the available total power is below , all the total power is allocated to receiver 1, i.e., and . On the other hand, if , then we have and .
Note that, if or , the problem reduces to the point-to-point setting; hence, we assume in the remainder. We can write the outcome of the maximization problem in between brackets in (12) as
Then we can rewrite the optimization problem in (12) in the same form as the point-to-point problem in (III) with a rate function given in (13). We next prove that this rate function is strictly concave.
The rate function in (13) is a strictly concave function of power .
Proof: It is easy to show that is continuous, differentiable, and its derivative is decreasing with ; hence, it is a strictly concave function of .
Now, based on this form of the optimization problem, we can directly use the results of Section III in the broadcast channel setting in order to identify the optimal transmission scheme for an energy harvesting transmitter. Note that as opposed to  and , our solution is valid for continuous energy arrivals as well as transmitters with various battery constraints. Once the optimal total transmit power over time is characterized, the power allocation among the users at each instant can be found using (13).
V Optimal Transmission Scheme with Battery Leakage
In Sections III and IV and references therein, the battery has been considered to be ideal, that is, there was no energy leakage. In this section, we consider the more realistic scenario of a battery that leaks part of the stored energy.
The leakage rate of a battery depends on the type (Li-ion batteries have a lower leakage rate compared to the nickel-based ones), age and usage of the battery as well as the medium temperature. Moreover, even for a fixed type of battery and medium temperature, the leakage rate changes over time; the batteries leak most right after being charged. However, for simplicity, a constant rate leakage model is considered here. If the battery is non-empty at a given time instant, the energy is assumed to leak from the battery at a constant finite rate denoted by . Obviously no leakage occurs if the battery is empty. We use the same cumulative curve approach to model the battery leakage process. Note that the leakage rate can alternatively be interpreted as the constant operation power of the node, that is, the circuit power needed to maintain the node awake.
Definition V.1 (Energy Leakage Curve)
The energy leakage curve is the amount of energy that has leaked from the battery in the time interval , , with . Due to the constant leakage rate assumption, is a continuous, non-decreasing function whose right-derivative is given by
To highlight the effect of leakage, we do not consider any minimum energy curve in this section, i.e., , and we focus only on discrete energy packet arrivals. Defining a maximum energy curve as , the feasibility condition on the transmitted energy curve becomes . We tackle again the problem of characterizing the feasible transmitted energy curve that transmits the most data over a given finite time interval . The corresponding optimization problem can be stated as
Unlike the battery size constraint studied in Section III, the battery leakage phenomenon does not translate into a minimum energy curve, but into a maximum energy curve obtained by removing the total leaked energy from the harvested energy curve. More importantly, the leakage curve is a function of the transmitted energy curve. Consequently, the maximum energy curve inherently depends on the transmitted energy curve, and hence, the solution framework presented in Section III does not directly extend to this setup.
Throughout this section,we consider the discrete energy harvesting process in which the -th energy packet of size arrives at time instant for . Without loss of generality, the first packet is assume to arrive at time (i.e., ). We call this general setup the -packet problem. As before, we assume that the transmitter always has enough data to transmit. Below, we characterize the optimal transmission scheme first for the single-packet problem (i.e., ), and then for the general -packet problem.
V-a The Single-Packet Problem
We consider here the simplified problem consisting of a single energy packet harvested at time . We refer to it as the single-packet problem. The solution of this problem will serve as a building block for the general -packet problem.
First, let us treat the single-packet problem with infinite555Note that, in the case of energy leakage, potential transmit time is finite when the number of harvested energy packets is finite as the available energy decays to zero even if no data is transmitted. deadline constraint (i.e., ), and denote it by S. It is depicted in Fig. 3. Following Section III, it is not hard to show that the optimal transmitted energy curve has to be piecewise linear, and the slope changes occur only if intersects . Consequently, the optimal for the S problem is as shown in Fig. 3: the node transmits at a constant power until the battery runs out of energy. One can see that there is a trade-off in the choice of : while it is more energy efficient to transmit at lower power for a longer period of time, the longer the transmission time, the more energy will be wasted due to leakage. The optimization problem in (15)-(16) becomes
Assuming that is a strictly concave increasing function with , and a finite leakage rate , the function achieves its maximum at a finite , as shown in Appendix A. We denote the corresponding optimal value by . Note that while the total amount of transmitted data is proportional to , is independent of . Summarizing, the optimal transmission strategy for the S problem is to transmit at constant power until the battery is empty. The total amount of transmitted data is .
We next consider the single-packet problem with a fixed transmission deadline , and denote it by S. It is depicted in Fig. 4, and the following notations are defined: (we assume for all , as otherwise the problem is equivalent to the S problem). We denote the point by . Finally, the slope of the line segment from the origin to is denoted by . We have . As before denotes the value that maximizes the function . Note that, as shown in Appendix A, is strictly decreasing for . Hence, building on the solution derived for the S problem, the solution of the S is easily derived:
if , transmit at constant power until the battery is empty.
else, transmit at constant power during the whole interval.
In short, the optimal transmission strategy for the S problem is to transmit at constant power for a time duration (that is, until the battery is empty), and remain silent afterwards. The amount of transmitted data is .
V-B The -Packet Problem
We consider here the general packet problem with finite deadline constraint , denoted as the N problem.
We start with the following lemma which proves that the optimal solution of the N problem can be emulated in the equivalent S problem with . That is, having all energy packets at time is at least as good as having them arrive over time. Let us denote by and the optimal solutions (in terms of total transmitted data) of the N and equivalent S problems, respectively.
The optimal solution of the N problem can be obtained in the equivalent S problem with . That is, we have .
Proof: Consider the optimal curve for the problem, and divide the time interval into sub-intervals: , , , , , . We denote by the duration of the interval, i.e., for , and . From Theorem III.3, we know that the optimal transmitted energy curve is a piecewise linear function, which is composed of constant power periods possibly separated by silent intervals (i.e. horizontal segments) in case the battery runs out of energy. Accordingly, we define the optimal solution of the problem by the sequences and , meaning that the node transmits for time (with ) at power in the interval. The node is silent in the remainder of the interval, i.e., for time .
The data transmitted by this transmission strategy is . The total transmit energy is , while the total energy leakage is . Since the optimal solution should eventually empty the battery, we have
We now argue that this optimal solution can be emulated in the S problem with . Consider the following transmission strategy for the S problem: transmit at constant power equal to for time , followed by for time , and so on, ending with for time . By construction, this strategy transmits the same amount of data as the optimal solution of the N problem. We conclude the proof by showing that this strategy is feasible, that is, for all . Since the node is constantly transmitting during the interval , the curve is constantly decreasing666We assume in the considered time interval, as otherwise the problem can be divided into equivalent subproblems. during this interval at rate , i.e. , for . We have
Having proved the achievability of in the equivalent S problem, the inequality naturally follows.
The counterpart of Lemma V.1 in the other direction does not always hold, that is, the optimal solution of the equivalent S problem cannot always be emulated in the original N problem. A counterexample can indeed easily be constructed, which is depicted in Fig. 5. Part (a) of the figure depicts a 2-packet problem with , , and . In part (b) the equivalent S problem is depicted. Let the optimal transmission power for the S problem be given by . This solution cannot be emulated in the original N problem. In fact, as shown in part (a), the node cannot transmit a constant power during the full time interval as the battery runs out of energy at time , and the node has to remain silent during the time interval .
However, in the following lemma, we provide a sufficient condition for the counterpart of Lemma V.1 to hold. For this, we define as the point on the curve corresponding to the time instant , for , and as the point corresponding to time , as illustrated in Fig. 6.
If the line segment from the origin to the point does not cross the curve at any other point than , then the optimal solution of the S problem with can be obtained in the N problem, and . This sufficient condition is expressed by the following inequalities:
Proof: First note that the set of inequalities in (22) expresses that the line segments from origin to points have slope which are all greater than that of the segment from origin to . This requires that the line segments from the points to the point have slopes respectively, which all are lower than or equal to the slope of the segment from the origin to point :
for . An illustration of an N problem satisfying the conditions of this lemma is given in Fig. 6.
Consider now the S problem with . Remember that the optimal scheme for the S problem requires transmitting at constant power for a duration of , with . We now argue that this solution can be emulated in the N problem. Consider the following transmission strategy for the N problem: transmit at whenever the battery is non-empty, and remain silent otherwise. By construction, this strategy is feasible. Again consider the time intervals between energy arrivals , , , of durations , respectively. We denote by (with ) the time for which the node is transmitting in the interval. The total transmission time is then given by . Moreover, combining the inequalities in (23) with the fact that , we have that for . This ensures that the considered strategy uses up the whole available energy by time , i.e., . Then, by the conservation of energy, the transmit and leakage energies must sum to the total harvested energy:
from which we get that , just like for the optimal solution of the S problem. This transmission strategy thus transmits the same amount of data as the optimal solution of the S problem. Consequently, under the conditions given in the theorem, the inequality holds.
Building on the two previous lemmas, the following theorem can be formulated.
If the inequalities in (22) hold, then:
, that is, the optimal solutions of the N problem and the S problem with are equivalent.
The optimal transmission strategy for the N problem is to transmit at constant power whenever the battery is non-empty, and remain silent otherwise, where the value corresponds to the solution of the equivalent S problem:
The total amount of transmitted data is .
An illustration of the result in Theorem V.3 is provided in Fig. 7 for and . Part (a) of the figure depicts the N problem, while its equivalent S problem is given in part (b). According to Theorem V.3, for both problems the optimal strategy is to transmit at constant power . The only particularity of the N problem is the presence of silent zones in between energy packet arrivals. However, the distribution over time of these silent zones do not affect the total duration of transmission, guaranteeing the equivalence of both solutions in terms of amount of transmitted data.
Now, building on Theorem V.3, we can provide the optimal solution for any N problem. Consider all line segments connecting the origin to points , . Among the segments that do not intersect other than at point , we pick the one with the highest index, i.e., the rightmost end point. We denote this index by . We can now consider the first energy packets only, and solve the corresponding packet problem with deadline , using the equivalence given in Theorem V.3. We then proceed recursively by considering the remaining packet problem separately. This recursive algorithm is described next. It takes as inputs the number of packets , the sizes of energy packets , and the packet interarrival times . It returns as output the set of optimal transmission powers , meaning that the optimal solution of the N problem is to transmit at constant power in the interval as long as the battery is non-empty. The optimality of the algorithm is proved in Appendix B.
: number of energy packets
: amount of energy in each packet
: interarrival times
Find the highest such that
for all .
for all .
If , find the by running
We conclude this section by identifying two special cases of the solution provided here:
The special case of an -packet problem without deadline constraint can be solved by Algorithm V.1 by setting . In this case, the inequalities in (26) hold with , and (27) reduces to for all . Hence, the optimal transmission strategy for the -packet problem without deadline constraint is to transmit at constant power whenever the battery is non-empty, and remain silent otherwise.
We have considered a communication system with an energy harvesting transmitter. Taking into account various constraints on the battery we have optimized the transmission scheme in order to maximize the amount of data transmitted within a given transmission deadline. We have provided a general framework extending the previous work in  and  to the model with continuous energy arrival as well as time-varying battery size constraints. We have also showed that the proposed framework applies to the optimization of energy harvesting broadcast systems. Moreover we have studied the case of a battery suffering from energy leakage, for which the optimal transmission scheme has been characterized for a constant leakage rate.
A Properties of
Remember that is a non-negative strictly concave increasing function, with . We prove here that the function , with , achieves its maximum at a finite , and is strictly decreasing for .
The derivative of is calculated as follows:
1) If , (28) becomes
which is analyzed as follows:
if , both the numerator and the denominator are zero. By l’Hôpital’s rule, we get , which follows from the strict concavity of .
if , the numerator is a strictly negative function. Indeed, the strict concavity of together with the fact that guarantees that for all .
Overall, is thus strictly negative for all . Hence,
finds its maximum at , and is strictly decreasing for .
2) Consider now . The sign of (28) is analyzed by focusing on its numerator only, which is rewritten for clarity as:
We analyze term by term:
The first term is a positive and strictly decreasing function, due to the increasing and strictly concave property of , respectively.
The term in between brackets is equal to zero if , and a strictly negative (as shown above), strictly decreasing function for . Indeed, the strict concavity of guarantees that the derivative of this term is strictly negative for all .
Consequently, overall is a strictly decreasing function of for . More precisely, the lower , the more rapid the decrease of will be. The initial value at is positive and proportional to : . On the other hand, the asymptotic value of as is negative: ,where the inequality follows from the strict concavity of together with the fact that . Between these two extremes, the strict decrease of guarantees that changes sign only once (from positive to negative) at some finite value denoted by , and that it will remain strictly negative for all . Hence, we have that:
has a unique maximum, which is achieved at some finite value of , denoted by . The lower the value of , the lower the value of will be.
is strictly decreasing for .
B Proof of Optimality of Algorithm v.1
Consider now that the inequalities in (26) do not hold for . Then, denote by the highest for which (26) holds. This situation is depicted in Fig. 8. We first argue that the optimal solution is such that it empties the battery before receiving the energy packet, i.e. before . Put differently, the optimal transmitted energy curve should intersect at a time . Assume that the opposite holds, as depicted in Fig. 9. Then, at some time , the slope of the transmitted energy curve has to increase in order to guarantee to empty the battery at time (which is a necessary condition for optimality). However, it is easy to realize (see the dot-dashed curve in Fig. 9) that such strategy is suboptimal since it violates Theorem III.3. Note that the feasibility of the dot-dashed curve in Fig. 9 in ensured by considering the largest rather than any satisfying (26). Now, since the battery has to be emptied before receiving the energy packet, we can optimally decouple the problem. First, the packet problem with deadline is solved independently. This subproblem satisfies the inequalities in (22), such that Theorem V.3 guarantees that its optimal solution is obtained by Algorithm V.1 in (27). Then, proceeding recursively, the algorithm is run for the remaining packet problem which can be considered as a new problem with an empty battery at the origin.
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