I Introduction
Very recently, energy harvesting (EH) has emerged as a promising technology to achieve a sustained and lowcost operation of lowpower devices such as sensors or tiny actuators. Apart from the conventional EH sources [1] such as solar, wind, kinetic, or other ambient energy source, the radiofrequency (RF) signals are very attractive because they can carry energy and information simultaneously [2], which enables energy constrained nodes to harvest energy and receive information. Notice that RF EH, also known as Wireless Energy Transfer (WET), is envisioned to be of utmost importance in scenarios where replacing or recharging the batteries may be dangerous, as in a toxic environment, or even impossible, as in sensors implanted in human bodies. Those scenarios will be also present in future communication paradigms such as the Internet of Things (IoT) [3], where powering a potentially massive number of devices will be a major challenge.
Wireless Powered Communication Networks (WPCN) is already a welladdressed topic in the literature. Readers can find in [4] an overview of WPCNs including system architecture, WET techniques, and existing applications. As shown in [5] for the case of throughput maximization, WPCN can even achieve a performance comparable to that of conventional (nonWET) networks when intelligent policies are applied. Some of the main strategies that have been considered in the scientific literature are relayassisted [6, 7, 8, 9], Automatic RepeatreQuest (ARQ) [10, 11], and power control [12, 3, 13, 14, 15, 16] mechanisms. Specifically, the work in [6]
deals with RF energy transfer in a threenode cooperative network, and focuses on the optimum switching between EH and data relaying, while the outage probability of a dualhop decodeandforward (DF) relay system utilizing the powersplitting EH protocol is investigated in
[7]. Authors in [8] also consider a cooperative WPCN but this time the source and relay nodes have no embedded energy supply and are wirelessly powered by an access point (AP). The performance of a largescale WPCN with random number of transmitterreceiver pairs and potential relays that are randomly distributed into the network, is characterized in [9] by using stochasticgeometry tools. In the latter two cases authors show the benefits of cooperation in terms of throughput and outage probability, respectively. An investigation on the usage of the timeswitching protocol when hybrid ARQ schemes are used for information transmission is presented in [10], while results demonstrate the improvement of the system performance for high SignaltoNoise Ratio (SNR). Additionally, a novel ARQ protocol for EH receivers is proposed in [11] by allowing the ACK feedback to be adapted based upon the receiver’s EH state. Therein, authors derive optimal reception policies including the sampling, decoding and feedback strategies, for different ARQ protocols and assuming several power consumption sources. Regarding power control, authors in [12] study a WPCN that employs dynamic power splitting. Under a pointtopoint flatfading channel setup, they resort to the optimal power splitting rule at the receiver based on the Channel State Information (CSI) to optimize the rateenergy performance tradeoff. Meanwhile, [13] investigates the optimal energy beamforming and time assignment in WPCN for smart cities, and [14] proposes a lowcomplexity solution, called fixed threshold transmission (FTT) by setting a transmit power threshold to determine whether transmission takes place or not. On the other hand, the energy efficiencyoptimal downlink transmit power for a massive multipleinput multipleoutput (MIMO) base station (BS) using WET to charge singleantenna EH users, is derived in [15]. Therein, authors use a scalable model for the devices’ circuit power consumption and show that it is indeed energy efficient to operate the system in the massive MIMO regime. Finally, a timedivision multiple access (TDMA)based WPCN was shown in [16] to be more spectralefficient than its nonorthogonal multiple access (NOMA) counterpart after optimizing the transmit power and time allocation in both cases. Specifically, the scenario consists of a power beacon (PB) powering wirelessly a set of devices, which in turn use such harvested energy for feeding their circuits and transmitting information to a receiving AP, while all the devices are singleantenna.Aforementioned research works, and many others related, design and/or evaluate strategies to maximize the probability of transmission, the throughput, or the energy efficiency, or to minimize the transmitted power, or the outage probability. However, as enabler of IoT paradigm, WPCNs find a natural application in MachineType Communication (MTC) use cases of coming 5th Generation (5G) of wireless systems, where some stringent requirements in reliability and delay may be present, as is the case of ultrareliable MTC (uMTC) scenarios [17]. Critical connections for industrial automation, smart grids, reliable wireless coordination among vehicles, and others, are some examples of uMTC scenarios [18] with general delay constraints less than 10ms and error probability requirements not greater than . The interplay between reliability/delay makes physical layer design of uMTC very complex and challenging [19], and WPCN systems need to be designed properly to meet such demands.
Ia Related Works
Some recent works [20, 21, 22, 23, 24, 25, 26, 27, 28] deal with those issues by considering short packets (stringent delay constraints) and/or ultrareliability. In [20] we analyze and optimize a singlehop wireless system with energy transfer in the downlink and information transfer in the uplink, under quasistatic Nakagamim fading in uMTC scenarios. The results demonstrate that there is an optimum number of channel uses for both energy and information transfer for a given message length. Cooperative WPCNs are considered in [21, 22, 23], where the impact of an amplifyandforward (AF) [21] and DF [22] relayassisted communication setup without direct link is evaluated, and power consumption sources beyond data transmission power, as well as imperfect CSI and a direct link, are incorporated in [23] for the DF setup. A savethentransmit protocol is investigated in [24] where authors address the scenario in which one or more wireless PBs use WET to charge a node over fading channels, and then it attempts to communicate with another receiver over a noisy channel and using a finite number of channel uses. The system performance is characterized using metrics, such as the energy supply probability at the transmitter, and the achievable rate at the receiver. In [25] we study transmission strategies for scenarios with/without energy accumulation between transmission rounds, where a power control protocol for the energy accumulation scenario is proposed in order to improve the reliability performance under the given delay constraints. On the other hand, subblock energyconstrained codes are investigated in [26], and a sufficient condition on the subblock length to avoid energy outage at the receiver is provided. Meanwhile, authors in [27] propose using concatenation of a nonlinear trellis code (NLTC) with an outer lowdensity paritycheck (LDPC) code taking into account both energy transmission and error rate requirements, and show that the designed codes operate at dB away from the information theoretic limits. Besides, retransmission protocols, in both energy and information transmission phases, are implemented in [28] to reduce the outage probability compared to openloop communication. Nevertheless, none of these works deal with rate allocation strategies for WPCNs under reliability and delay constraints. Notice that in WPCNs with elastic
applications, in the sense that devices can transmit data over a wide range of data rates, the transmission rate is a degree of freedom that should be properly exploited.
IB Contributions and organization of this paper
Herein we aim at filling the above gap in the literature by addressing the rate control problem in such WPCNs with reliability and delay constraints. Specifically, our goal is to maximize the instantaneous message size such that requirements of latency and reliability are met. For a twophase wireless system, where first a WET process powers a lowpower sensor node that uses that energy to transmit its data in the Wireless Information Transmission (WIT) phase, we propose two rate control strategies: i) fixed transmit rate (FTR) and ii) transmit rate with known state of charge (KSC). We compare both with the ideal scheme that requires full CSI at transmitter side (fCSI). The main contributions of this work can be summarized as follows:

We show the advantages of deciding on the transmit rate based on the battery charge, since it allows considerable rate improvements when compared to the fixed rate scheme, achieving a performance close to that of the ideal scheme where full CSI is also available at the transmitter side;

Closedform approximations for the attainable average message size of the proposed schemes are obtained, and also for the optimum WET and WIT blocklength when the fixed transmit rate scheme is used;

Results show the positive impact of a greater number of antennas at the destination, and evidence that the more stringent the reliability constraints, the smaller the message sizes on average, and the smaller the optimum WIT blocklength. The inverse relation between the optimum WIT blocklength and the reliability constraints was proved analytically for the fixed transmit scheme, but numerical results evidence that this characteristic holds also for the other schemes;

We develop the asymptotic formulation for the rate control problem under the proposed schemes. We show how to depart from these expressions to attain nonasymptotic results, while illustrating numerically the gaps of both formulations. We corroborate the appropriateness of using the asymptotic formulation as an approximation of the nonasymptotic finite blocklength results in this scenario, specially at data rates that are not low.
Next, Section II presents the system model and assumptions. Sections III, IV and V discuss the rate control schemes, FTR, KSC and fCSI, respectively, designed to meet the given reliability and delay constraints. Section VI presents the numerical results, while Section VII concludes the paper.
WET channel uses, WIT channel uses,  
Optimum values of and , respectively  
Duration of a channel use  
Path loss between nodes and  
Normalized channel power gain of the link  
Normalized channel power gain of the link between and the th antenna of  
Nakagamim shape factor of link and , respectively  
Number of antennas at  
g  RV denoting the normalized sum of for 
Energy harvested at during the WET phase  
Transmit power of node  
Energy conversion efficiency  
Message size in bits  
Fixed minimum message size  
Transmit rate,  
Maximum allowed error probability  
Average noise power at  
Instantaneous SNR at the th antenna of  
Instantaneous SNR after Maximum Ratio Combining (MRC) at  
Average SNR at normalized by and the quotient  
RV denoting the product  
Error probability for an Additive White Gaussian Noise (AWGN) channel  
Channel thresholds for KSC and fCSI schemes, respectively  
Average error probability when operating above the channel thresholds, e.g., above for KSC or above for fCSI 
Notation:
is a normalized gamma distributed random variable (RV) with shape factor
, Probability Density Function (PDF)
and Cumulative Distribution Function (CDF)
for . is the probability of event , while denotes expectation with respect to RV . is the modified Bessel function of second kind and order , is the exponential integral [29], while is the main branch of the Lambert W function [30], which satisfies for and it is defined in . The Gaussian Qfunction is denoted as and is the supremum operation. Table I summarizes the main symbols used throughout this paper.Ii System Model
Consider the twophase scenario in Fig. 1. First, powers during channel uses in the WET phase. Then, , which may be a sensor node with very limited energy supply, uses the energy obtained in the WET phase to transmit its information to over channel uses in the WIT phase. We do not consider other power consumption sources at , e.g., as sensing operations and/or receive/processing/transmit circuitry functioning as considered in [11, 15, 16, 13, 23]. However, notice that our derivations keep accurate in case that such additional consumption is small compared to the average energy harvested by as assumed implicitly or explicitly in [3, 5, 6, 7, 8, 9, 10, 12, 20, 21, 22, 24, 25, 28, 31]. Otherwise, modeling and including into the analysis other power consumption sources may be strictly necessary, as failing to satisfy the corresponding energy requirements could become the main cause of system failure, as shown for instance in [23] when transmitting short data messages in a cooperative scenario. Meanwhile, notice that in systems operating under stringent reliability and delay constraints, which are the focus of this work, such energyoutage events could be critical and difficult to predict/avoid, hence, deploying a very small battery, but still able of supporting the basic operation of the EH devices, could be required. In such case the harvested energy would support only the transmission operations, while once again our derivations and analysis hold. On the other hand, we assume that both, energy and data transfer, utilize the same frequency spectrum, which guarantees small form factors, but perhaps more important it allows saving spectrum and enables Simultaneous Wireless and Information Transfer (SWIPT) [9, 28] in the link , e.g., for the transmission of control messages from to . We denote by the duration of a channel use, thus, the duration of an entire transmission round is , where . Also, and are single antenna devices while is equipped with antennas. The path loss between and , and between and each antenna of , is denoted as , , respectively.
Nakagamim quasistatic channels are assumed where the fading is considered to be constant and independent and identically distributed over each phase. We consider normalized channel power gains and , where characterizes the link , is the coefficient of the link between and the th antenna of , and are their corresponding Nakagamim LineOfSight (LOS) factors^{1}^{1}1Usually since WET processes require greater LOS than WIT.. Also, the antennas at are sufficiently separated such that are independent, and spatial diversity can be fully achieved.
A typical application scenario could be that in which is a BS who powers a set of nearby low power devices including , while enabling their transmissions to a sink or aggregating node with antennas. Notice that our analyses considering a single device remain valid even in a multiuser setup if the simultaneous transmissions in the WIT phase occur in different frequency subbands^{2}^{2}2In such case, the MRC scheme (and consequent derivations) considered in Subsection IIC remains valid as it can be applied after filtering to each subband to maximize its associated SNR. On the other hand, could also be a dedicated RF source deployed to provide a predictable energy supply to those devices, or an interrogator requesting information from them, for which and . The latter case implies that instead of a singleantenna device, would be equipped with antennas. In that sense we would like to point out that our derivations remain valid in setups where is equipped with antennas in two different scenarios:

when uses Maximum Ratio Transmission (MRT) to serve a single device . In such case, our derivations hold by utilizing and instead of and , respectively;

when uses the CSIfree Switching Antennas (SA) strategy proposed in [32] under which transmits with full power through one antenna at a time such that all antennas are used during a coherence block and channels are uncorrelated. Assuming equaltime allocation for each antenna the performance is equivalent to the one attained under the assumption of a single antenna at but utilizing instead of .
Notice that the SA scheme is particularly suitable for lowpower multiuser setups as it does not require CSI acquisition neither other signaling procedures, and still is capable of providing an fold diversity gain in the energy supplying.
Iia WET Phase
During each WET phase the energy harvested at is given by
(1) 
where is the transmit power of and sufficiently large such that the energy harvested from noise is negligible, while is the energy conversion efficiency. Notice that we are considering the simple linear EH model as in [3, 5, 6, 7, 8, 9, 10, 12, 13, 20, 21, 22, 23, 24, 28] to allow some analytical tractability and facilitate the discussions. Although the specific performance results must vary when utilizing different EH models, the functioning of the proposed rate control schemes does not change, while their relative performance is expected to be similar. In any case, the analysis of the proposed schemes under more evolved/practical EH models, such as the piecewise linear model [25, 32] and/or other nonlinear models [31, 33], is left for a future work.
IiB WIT Phase
After the WET phase, uses the harvested energy to transmit its message using a code, which is defined as the collection of

an encoder , which maps the message into a length codeword satisfying the power constraint
(2) where is given by
(3) assuming that uses all the harvested energy in (1) for transmission;

a decoder satisfying the maximum error probability constraint
(4) where
denotes the channel output induced by the transmitted codeword at the end of each WIT phase, e.g., the received signal vector, and the expectation is taken over the channel realizations
and .
The maximum message size is then
(5) 
Notice that the value of in (5) is largely influenced by the channel conditions as in (4) depends on them. Therefore, we next focus on characterizing the SNR at the destination.
IiC SNR at
The instantaneous SNR at antenna of is
(6) 
where is given in (3), is the average noise power at , and
(7) 
is the average SNR at normalized by the number of antennas, , and the quotient of the WET and WIT blocklengths, . Notice that for a fixed , by increasing such that collects more energy, can increase its transmit power and consequently the SNR at each antenna of improves, but that leads to a decrease in , and therefore an increase in the coding rate which degrades the reliability of the system. This is a well known tradeoff of WPCNs.
Finally, perfect CSI at is assumed in the decoding after the WIT phase^{3}^{3}3CSI acquisition in an energylimited setup is not trivial and including the effect of imperfect CSI and associated costs would demand a more elaborated mathematical analysis that is out of the scope of this work. However, notice that when channels remain constant over multiple transmission rounds, which should be the case in most scenarios when operating with short blocklengths, the cost of CSI acquisition can be negligible [20]., and uses MRC to combine the arriving information at each antenna. Therefore, the overall SNR seen is
(8) 
where . Notice that comes from defining , whose PDF and CDF are given in the following result.
Lemma 1.
The PDF and CDF of are given by
(9)  
(10)  
(11) 
for , where (11) holds only for Rayleigh fading scenarios, e.g., .
Proof.
See Appendix A.
IiD ReliabilityRate Tradeoff
As commented in Subsection IIB, , and are related according to (5). Recently, Polyanskiy et. al [34] have provided an accurate characterization of the tradeoff between these parameters in AWGN channels and for channel uses. By explicitly including the effect of the SNR at the receiver, the tradeoff is given by
(12) 
where is the Shannon capacity and is the channel dispersion, which measures the stochastic variability of the channel relative to a deterministic channel with the same capacity. Now, we can rewrite (12) as
(13) 
which is the maximum error probability when transmitting information bits over a channel with SNR and using complex symbols. Notice that (13) matches the asymptotic outage probability when and/or . Meanwhile, for quasistatic fading channels the average maximum error probability is [35, Eq.(59)]
(14) 
since channel becomes conditionally Gaussian on , and we only require to take expectation over that RV to attain the corresponding average error probability. However, and as it has been shown in [36], the effect of the fading when evaluating (14) makes disappear the impact of the finite blocklength when is not extremely small and there is not a strong LOS component, thus, the asymptotic outage probability, which is the Laplace approximation of (14), is a good match in those cases and is given by
(15) 
IiE Problem Statement
Our goal in this work is finding the maximum instantaneous message size such that for given , thus, fixed delay .^{4}^{4}4Notice that and are the duration of the WIT and WET phases, and in most scenarios these durations must be fixed, e.g., for simpler synchronization and/or timeslotted operation. We constrain to be above some fixed value in each transmission, which represents the minimum possible message size. Such value is determined by the minimum frame size in the physical layer, which includes the mandatory metadata fields in the particular frame structure, hence it would be impossible communicating with a message size smaller than . Notice that in our scenario we do not perform reliability analysis of metadata and payload separately and still assume a fixed . In such case our results hold by assuming that metadata and payload are jointly encoded, which favors the system reliability [18].
We refer to this as the rate control problem since by finding we set the transmission rate as , and we address it for three different schemes, when

adopts a fixed transmit rate (Section III) independently of the randomness of the system parameters, e.g., fading realizations;

uses the information on the remaining battery charge (Section IV), which is intrinsically related to the WET channel realization, to decide on its rate;

uses the full knowledge of all, WET and WIT channel realizations, to decide on its rate (Section V), which is idealistic for a practical implementation and it is used as benchmark.
We assume that is aware of the number of antennas at and the value of in (7). Notice that and in (7) can be acquired, with low frequency, by via a lowrate feedback channel from
, also the remaining terms there can be estimated by knowing the average energy harvested at
, .Finally, we adopt the asymptotic expression of in (15) as an initial guess or lowerbound for gaining in analytical tractability. This is, we start with the asymptotic formulation and then we state the procedure to obtain the nonasymptotic results based on the former. Additionally, in Section VI we investigate the gap in performance between both, asymptotic and nonasymptotic, formulations. In the entire work we focus on the region where .
Iii Fixed Transmit Rate (FTR)
Herein, we analyze a rate allocation scheme which does not rely on any instantaneous information at , hence, adopts a fixed rate . Notice that for being possible to meet the required reliability constraint it is necessary that since is an increasing function of and . The allocated message size comes from solving equation for , which can only be performed numerically when using (14). Thus, we use the approximation in (15) as follows
(16) 
where comes from using (8) and from the definition of the CDF. Now, the maximum message size comes from setting in (16) and isolating , which yields
(17) 
where . In case that the FTR scheme can not satisfy the reliability requirements at the same time that satisfying the message size constraint, thus, the rate allocation is infeasible and no transmission occurs. Notice that using (17) requires finding the inverse of (10), which is not possible in closed form. Following result addresses that issue for the reliability region of our concern.
Theorem 1.
Proof.
See Appendix B.
For setups with we were not able of finding an accurate approximation, but notice that this case corresponds to very specific scenarios and its importance would be mainly theoretical rather than practical.
Returning to (17), notice that works as an equivalent SNR which increases linearly with . Additionally, it increases with and , and following a powerlaw of , inversely proportional to the minimum between and . The latter statement comes from using the definition of along with (18).
As stated in Subsection IIE our aim is to find the maximum instantaneous message size for a given and , while the relevance of fixing and was also highlighted (see Comment 4). However, in more flexible scenarios where choosing and under the delay constraint is allowed, there is an optimum blocklength for WET and WIT phases, which we further investigate next.
Theorem 2.
Taking advantage of (17), the optimum blocklength for WIT and WET phases is approximately given by
(19)  
(20) 
respectively, and the approximation includes taking the nearest integer.
Proof.
See Appendix C.
From (19) notice that increases linearly with , while it is also an increasing function of as shown in Fig. 2. Although (19) is indeterminate for dB we know that , thus, . Since is also an increasing function of (because increases with by definition), then, is an increasing function of . This result is very interesting since it clearly states that the harvested energy, through the WET channel uses, becomes more and more relevant when targeting greater reliabilities than the information data rate, . Thus, it is better improving the harvested energy statistics while transmitting with greater information data rates (since delay is fixed) for meeting more stringent reliability constraints. Finally, notice that does not depend on the fading characteristics when using the asymptotic formulation as so far.
Until this point we have been using the asymptotic approximation (15) in order to attain analytical and insightful results: the message size (17) and the optimum blocklengths (19), (20) for a given delay . Notice that at finite blocklength we can use the asymptotic value provided by (17) as an initial step to solve with (14). The procedure is illustrated in Algorithm 1, while
in Section VI numerical results evaluate the gap between the infinite and finite blocklength formulations and further discussion follows.
In the next section we propose a rate allocation scheme that forces to exploit its battery state of charge information for performance improvements.
Iv Transmit rate with Known State of Charge (KSC)
In the previous section had to rely on a conservative fixed transmit rate since no instantaneous CSI was available. Herein we investigate the optimum variable transmit rate if exploits the information related with the state of charge of its battery, which is strictly related with the CSI of the link . Different from the FTR scheme where the transmit rate is fixed along the transmission rounds, in this case the required transmit rate is computed after concluding each WET phase. In that case the asymptotic outage probability is given by
(21) 
where follows directly after using (8), while comes from isolating g and using the definition of CDF. Notice that this is different than the result in (16) because herein the battery charge is known, thus is not random anymore and the only RV is g. Finally, comes from incorporating as an input parameter in the error function.
Obviously, for some values of , , we cannot find such that . This is because the WET channel was so poor such that for we have . Therefore, the transmission strategy under this scheme is

If , where works as a threshold, transmits with ;

otherwise; adopts a rate which is dependent on the specific such that on average it performs with error probability . This allows compensating the average error probability to meet the reliability requirement.
Thus, when it is required that
(22) 
Also, above scheme implies that
(23) 
and notice that
while that is required to be equal or smaller than the target for feasibility. This same result was attained in the previous section for the FTR scheme. Thus, neither KSC, nor other transmission rate allocation strategy operating with the same values of and , is capable of providing a rate that meets the average reliability constraint when .^{5}^{5}5However larger data rates are provided when the feasibility condition holds, which is illustrated in Section VI.
Now we are able of providing the following result characterizing the rate allocation under the KSC scheme when .
Theorem 3.
Proof.
See Appendix D.
Then, based on (24) the average message size is given by
(25) 
where comes from using the CDF and PDF of , and changing variable . Unfortunately there is not a closedform solution for the integral^{7}^{7}7There are numerous numerical methods that could be used to solve efficiently (25
), for instance, importance sampling by taking samples from an exponential distribution with mean
, or the GaussianChebyshev quadrature method as in [7]. In our specific setup the latter would require some kind of transformation to make the integrating interval to be finite, e.g., using , . unless for the specific case of Rayleigh fading, for which(26) 
where comes with the help of [29, Eq.(4.337.1)] and some algebraic manipulations. Even when attaining a closedform expression in (IV) for the average message size under Rayleigh fading, it becomes intractable finding the optimum value of under this scheme, mainly because , which is the solution of (22), depends also on . However, either evaluating numerically (25) or finding numerically in (25) or (IV) is still useful since the Monte Carlo computation alternative could be practically infeasible for some system parameter values. This is because depends on , and using Monte Carlo to find accurately requires enormous computation resources when is very small, e.g., .
Next, we analyze the procedure for attaining the nonasymptotic results, and explain how to depart from our above results to reach them.
Iva Finite Blocklength formulation
Under the finite blocklength formulation is greater than in the asymptotic case, thus, the value of found after solving (22) can be used as a starting point to solve the following moreelaborate equation
(27) 
which is attained in an analogous way to (22), but this time we require to use (13) with according to (8).^{8}^{8}8Differently to the case when solving (22), here it is not recommended using a Newtonlike method in order to solve (27) since computing the derivative of the function takes more computational resources than the function itself. The fact that the solution of (27) is close to the solution of (22) should be exploited. After finding we can evaluate
(28) 
The procedure for allocating the rate under the finite blocklength formulation is described in Algorithm 2, and in line 3 notice that according to (24), thus we can use an iterative procedure as in the previous section, starting from and decreasing at each iteration until (29) is met.^{9}^{9}9From a practical and feasible perspective, should have tabulated many channel realizations, , in order to decide on its rate based on the minimum of the interval where the current realization lies in.
(29) 
In the next section we analyze the case where, in addition to the battery state of charge information, knows the CSI of the channels in the link and uses also that information for allocating its transmit rate.
V Transmit rate with full CSI at (fCSI)
In this setup knows all the channels and , thus it knows what would be the value of , and it can adjust its transmit rate to meet the required reliability. Therefore, the channel is AWGN and the maximum message size comes from using (12) and is given by^{10}^{10}10
For the FTR and KSC schemes we have departed from the asymptotic analysis since they provide a valid approximation that can be used as a starting point for finding their nonasymptotic equivalent. For the fCSI case this is no longer necessary since (
30) is nonasymptotic and it is already in closedform.(30) 
Obviously the impact of finite blocklength is not negligible here. Notice that for sufficiently long blocklength, , yields , thus, following Shannon, it is possible to transmit with an arbitrary small (not bounded) error probability with such rate.
Similar to the previous section where was known, now will transmit with an adjustable when , and can be found by solving
(31) 
where is given in (10), and
(32) 
with as in (8). Herein we do not discuss any particular method for solving (V). It does not worth our attention since the fCSI method does not seem practically feasible, because the full knowledge of all the channels at is a very strong assumption, particularly in powerlimited networks; and it is only useful to provide benchmark results, therefore we resort to standard numerical solutions available in MatLab, e.g., fsolve, fzero, vpasolve, and Wolfram Mathematica, e.g., FindRoot.
Now, we have concluded the analytical discussions around the three rate allocation schemes and in the following section we provide some numerical performance comparisons.
Vi Numerical Results
In this section, we present numerical results to investigate the performance of the proposed schemes as a function of the system parameters. Unless stated otherwise, results are obtained by using the values shown in Table II. Notice that dB may correspond to a system with the following practical values (similar to those in [9, 8, 20, 25, 23, 4]): , which is feasible according to the stateoftheart in EH circuit design, W, dBm, logdistance path loss model with and , where dB is the average signal power attenuation at a reference distance of 1 meter and accounts for other factors such as the carrier frequency and heights and gains of the antennas, m and m are the lengths of the links and , and is the path loss exponent. While by setting and we model both WIT and WET links with certain LOS as required in practical WPCN setups. Also, by taking into account that the typical message size in uMTC use cases is 32 bytes [37] while the value of must be much smaller we have set bytes. Finally, notice that the overall delay channel uses^{11}^{11}11Since must be small for shortlatency transmissions, both and are small as well (strictly smaller than ) but has usually a stronger impact on the system performance than according to the analysis around Fig. 2, which is also illustrated in [20, 25] for different setups, hence, its value should be set larger than . As shown in Table II we utilize channel uses. corresponds to ms by assuming s as in [20].
Parameter  Value 

bytes (16 bits)  
, (similar to [20, 25])  
channel uses (similar to [25])  
dB ([8, 20, 25, 23, 9, 4])  
s [20] 
Fig. 3 shows the attainable average message size for each scheme as a function of with and . Obviously, an increase in improves the system performance according to (7); in fact, operating with is only possible when dB. Notice that the schemes KSC and fCSI, that take advantage of certain levels of information, allow to attain greater data rates. In that sense, KSC seems attractive since its performance overcomes significantly the one attained by FTR while making use only of the battery charge information, which is practically viable. However, neither KSC nor fCSI expand the region for which is feasible, below dB, and the reasons are highlighted in Section IV. Also, the greater the reliability constraints, the smaller the message sizes on average. Notice that (18) is accurate, and the results coming from the finite blocklength formulation approximate the asymptotic case for relatively large , e.g., for bits.
The average message size as a function of the WIT blocklength, , is shown in Fig. 4 for fixed channel uses while setting and . As shown in Fig. 3, it is not possible operating with when dB and , thus, Fig. 4 shows only the performance for , while for both reliability constraints are possible to satisfy as shown in Fig. 4. From both, Fig. 4 and , we can appreciate the existence of an optimal point, , which was analytically proved for the FTR scheme, and notice that matches (19) even when using the approximation in (18), thus, validating those expressions. As pointed out in Section III for the FTR scheme, is an increasing function of and that is why when for channel uses, while for the value of increases to channel uses. This fact holds also for the KSC and fCSI schemes. Additionally, notice that , and KSC and fCSI seem to benefit more of setups, which is an interesting result.
Fig. 5 shows the attainable average message size for each scheme as a function of for . Notice that has a nearlinear behavior with respect to for all the schemes in the loglog scale, thus, , where and depend on the system parameter values, but as shown in the figure they are strongly influenced by , thus, also by and . For the FTR scheme this is expected also from (18). Stringent reliabilities, in the order of and greater, are available when using four antennas. More on this is discussed in Fig. 6 where we show the positive impact of a greater number of antennas at for . In fact, the system cannot operate under the reliability requirement when , and notice that by increasing beyond , the average message size increases while meeting the required reliability. As , the performance of KSC and fCSI tends to overlap since the resultant WIT channel behaves as an AWGN channel and the randomness in the WIT phase disappears. It is remarkable the performance gain that results from using KSC and fCSI schemes with respect to FTR. For example, for , transmits with bits on average when using FTR, while if the information of the battery state of charge is used (KSC scheme), it can operate with up to bits on average. Additionally, the accuracy of (18) is verified again and its significance is shown to be even greater since for the exact inversion of in (10) was not possible to be performed even numerically. In Fig. 6 it is also shown the probability of those events in which the allocated message size is the minimum possible, e.g., probability of transmitting with , which is desired to be as small as possible. Since the FTR scheme uses a deterministic rate, we only plot the performance of the KSC and fCSI schemes, for which and , respectively.^{12}^{12}12Notice that holds always unless the channel becomes deterministic, e.g., , for which there are only two possibilities: either or . As increases, the chances of transmitting with the minimum message size decreases, and the average message size increases. Notice that the gap between the finite blocklength and the asymptotic formulations is considerable when analyzing the probability of transmitting with minimum message size, specially when increases^{13}^{13}13This is because the resultant WIT channel tends to behave as an AWGN channel for which the impact of the finite blocklength is unquestionable as shown in [34]., but it does not have a practical impact as shown when evaluating the average message size.
Figs. 36 corroborate the appropriateness of using the asymptotic results as starting points when getting their finite blocklength counterparts. Finally, Fig. 7 shows the probability that transmits with the minimum message size as a function of that size, for and . As the minimum message size increases, the chances of transmitting messages with that size increases. Differently from the FTR scheme, will never reach the unity when using the KSC^{14}^{14}14Unless for the particular in (22). and fCSI schemes, thus, .
Vii Conclusion
In this paper, we proposed two rate control strategies in order to comply with the reliability and delay constraints of an uMTC WPCN, where multiple antennas are available at the information receiving side. We show the performance gains, with respect to a fixed rate transmission scheme, when the instantaneous battery charge information is used to adapt the transmit rate, while we have also analyzed the ideal scheme requiring full CSI at transmitter side as a benchmark. The greater the reliability constraints, the smaller the message sizes on average, and the smaller the optimum WIT blocklength. We attain a closedform expression for the optimum blocklengths when the allocated transmit rate is fixed, and we show that incorporating more information to the rate control schemes, e.g., variable rate schemes, increases the optimum WIT blocklength. Results also show the positive impact of a greater number of antennas, and corroborate the appropriateness of our procedures when using the asymptotic formulation as an approximation of the real finite blocklength results.
Appendix A Proof of Lemma 1
We proceed from the PDF of the product of two independent random variables as follows
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