Effective Energy Efficiency of Ultra-reliable Low Latency Communication
Mohammad Shehab, Hirley Alves, Eduard A. Jorswieck, Endrit Dosti, Matti Latva-aho
11 Effective Energy Efficiency of Ultra-reliable LowLatency Communication
Mohammad Shehab, Hirley Alves, Eduard A. Jorswieck, Endrit Dosti, and Matti Latva-aho
Abstract —Effective Capacity defines the maximum communi-cation rate subject to a specific delay constraint, while effectiveenergy efficiency (EEE) indicates the ratio between effectivecapacity and power consumption. We analyze the EEE of ultra-reliable networks operating in the finite blocklength regime. Weobtain a closed form approximation for the EEE in quasi-staticNakagami- 𝑚 (and Rayleigh as sub-case) fading channels as afunction of power, error probability, and latency. Furthermore,we characterize the QoS constrained EEE maximization problemfor different power consumption models, which shows a signifi-cant difference between finite and infinite blocklength coding withrespect to EEE and optimal power allocation strategy. As assertedin the literature, achieving ultra-reliability using one transmissionconsumes huge amount of power, which is not applicable forenergy limited IoT devices. In this context, accounting for emptybuffer probability in machine type communication (MTC) andextending the maximum delay tolerance jointly enhances the EEEand allows for adaptive retransmission of faulty packets. Ouranalysis reveals that obtaining the optimum error probabilityfor each transmission by minimizing the non-empty bufferprobability approaches EEE optimality, while being analyticallytractable via Dinkelbach’s algorithm. Furthermore, the resultsillustrate the power saving and the significant EEE gain attainedby applying adaptive retransmission protocols, while sacrificinga limited increase in latency. Index Terms —Effective energy efficiency, finite blocklength,URLLC, IoT, optimal power allocation.
I. I
NTRODUCTION
The new generations of mobile communication are expectedto support a multitude of smart devices interconnected viamachine type networks, enabling the Internet of Things (IoT).Energy efficient transmission while guaranteeing quality-of-service (QoS) is an ultimate goal in the design of futurewireless networks. QoS constraints ranging from low latencyin the order of few milliseconds and packet loss rate ( ă ´ ) are key requirements for Ultra-Reliable Low LatencyCommunication (URLLC) [1]. In order to boost throughputand reliability while guaranteeing low latency, it becomescrucial to investigate and optimize the resources that are
M. Shehab, H. Alves, Endrit Dosti, and M. Latva-aho are with Centrefor Wireless Communications (CWC), University of Oulu, Finland. Email:firstname.lastname@oulu.fi. Eduard A. Jorswieck is with the Departmentof Information Theory and Communication Systems, Technische UniverstätBraunschweig, Germany, Email: [email protected]. E. Dosti is with theDepartment of Signal Processing and Acoustics, Aalto University, Finland.Email: endrit.dosti@aalto.fiThis work is partially supported by Academy of Finland 6Genesis Flagship(Grant no. 318927), Aka Project EE-IoT (Grant no. 319008). The work ofE. Jorswieck is partly funded by the German Research Foundation (DFG,Deutsche Forschungsgemeinschaft) as part of Germany’s Excellence Strategy- EXC 2050/1 - Project ID 390696704 - Cluster of Excellence "Centre forTactile Internet with Human-in-the-Loop" (CeTI) of Technische UniversitätDresden. allocated for transmission. In most cases, URLLC devices suchremote sensors have limited energy resources which dictatescareful planning of throughput maximization with wise energyconsumption models [2]. Furthermore, besides possible healthrisks of electromagnetic radiation in over populated areassuch as city centers [3], the information and communicationtechnology industry is projected to contribute to 6 % of globalCO emission in 2020 [4]. This urges the invention of lowpower consumption, green communication schemes, yet ableto perform with QoS guarantees.In order to satisfy extremely low latency in real timeapplications and emerging technologies such as e-health, in-dustrial IoT, and autonomous vehicles, an attractive solu-tion is communication with short messages [5]. Therein, thelengths of the packets to be communicated are short, but theirimportance is extremely high. When the packets are shortand delay requirements are stringent, performance metrics,such as Shannon capacity or outage capacity, provide a poorbenchmark, and therefore, fundamentally new approaches areneeded [6], [7]. In this context, the maximum achievable rateof finite blocklength packets was defined in [8] as a functionof blocklength and error probability.As envisaged by [9], the design of URLLC focuses on thetail distributions of reliability and latency instead of averagemetrics. Here arises the challenge of how to incorporateenergy efficiency with the data rates, delay, and reliabilityrequirements imposed by the International TelecommunicationUnion (ITU) and for MTC towards 6G. In this sense, metricssuch as effective capacity (EC) and effective energy efficiency(EEE) are meant to capture tail statistical delay requirementsin parallel with transmission throughput. A. Related Work
The effective capacity metric was first introduced in [10]to guarantee statistical QoS requirements by capturing thephysical and link layers aspects. It maps the maximum arrivalrate that can be supported by a network with a maximum delaybound of 𝛿 and a delay outage probability. Unlike the theDelay-Sensitive Area Spectral Efficiency metric which onlyaccounts for the transmission delay [11], the EC accounts forthe statistical QoS aspect in terms of delay outage probabilityand the maximum delay bound. In [6], Gursoy characterizedthe EC in bits per channel use (bpcu) for short packets in quasi-static fading channels where the channel coefficients remainconstant for the whole time spanning one packet transmission.Moreover, in [12], Gursoy et al. extended their analysis tomultiple users but without considering the power consumption a r X i v : . [ c s . I T ] J a n and energy efficiency aspects. Meanwhile, the per-node EC inmassive MTC networks was studied in [13] proposing threemethods to alleviate interference namely power control, grace-ful degradation of delay constraint and the hybrid method.Effective energy efficiency is defined as the ratio betweeneffective capacity and the total power consumption. In thissense, EEE captures the interplay between power, delay, andreliability and thus, fits well for dealing with the inherentenergy-limited and bursty traffic scenarios in MTC charac-terized by URLLC. The maximization of EEE is of greatimportance for green IoT, where the goal is to maximize thethroughput for each consumed unit of power. The EEE can beused as a measure of how efficient an IoT system is in terms ofpower consumption and energy saving. This is a very usefulmetric in the study and design of remote IoT devices thatrun on installed batteries with limited energy supply and arerequired to communicate with URLLC requirements. We referto the novel factors that affect energy efficiency in MTC, whichare strict delay and error constraints, bursty traffic, emptybuffer probability, and communication on finite blocklengthpackets. In [14], the empty buffer probability (EBP) model wasconsidered as an EEE booster for long packets transmission.The energy efficiency gap which results from utilizing finiteblocklength packets and the optimum power allocation in thiscase were characterized in [15] for Rayleigh fading channel.The trade-off between EEE and EC was studied in [16], wherethe authors suggested an algorithm to maximize the EC subjectto EEE constraint. However, the probability of transmissionerror that appears in finite blocklength communication dueto imperfect coding was not considered. The authors of [17]showed that the relation between EEE and delay in wirelesssystems is not always a trade-off. They concluded that a linearrelation between service rate and power consumption leads toan EEE-delay non-trade-off region.On the other hand, it has been well-established that achiev-ing ultra reliability requires the utilization of diversity schemessuch as ARQ retransmission protocols. The utilization ofthis family of protocols has been embraced by several upto date systems such as 5G NR [18]. In this context, theauthors of [19] discussed the EC of ARQ schemes for matrix-exponential distributed fading channels. They suggested thatexploiting spatial diversity as the case in MIMO would reducethe sensitivity of EC to variations in the delay exponent 𝜃 .However, only the work in [20] studied the finite blocklengtheffect in ARQ Assisted URLLLC but without accountingpower consumption as in the EEE metric. B. Contributions
In this work, we build upon [15] and propose a finite block-length model for the EEE in quasi-static Nakagami- 𝑚 fadingassuming a linear power consumption model. We characterizethe optimum power allocation strategy that maximizes theEEE. We account for the EBP and in power consumptionmodel and prove that this model is valid for short packets.Our analysis indicates that considering EBP with short packetsallows for a more precise characterization of the EC and EEE.Results show that higher EC and EEE are obtained under EBP in contrast to the full buffer scenario. Besides, we illustrate theEEE gap between infinite and the finite blocklength models,and highlight the effect of network congestion on the EEEperformance. We analyse how the optimum power allocationis affected by limiting the packet length and the performancegap that appears accordingly for different types of fading. Inaddition, we evaluate the trade-off between the EEE and thedelay tolerance.A key contribution of this work is that we exploit thenon-EBP model by incorporating retransmission of faultypackets in the instants when the buffer is empty, whichrenders an EBP-ARQ scheme. We evaluate the EEE of theproposed scheme and also compare this case to the basicEBP model. The results show that this scenario grants asignificant improvement in the EEE and reduction in thepower consumption. Our analysis characterizes the optimumtransmission error probability for each transmission so thatthe global EEE is maximized and the power consumptionis minimized. The average latency is also analyzed for thisscenario where the results indicate a very limited increase inlatency as a cost of the high gain in EEE.The contributions of this work are summarized as follows: ‚ We derive closed form approximation for the EEE underquasi-static fading. ‚ We characterize the optimum power allocation that max-imizes the EEE for linear and EBP power consumptionmodels and highlight the trade-off between EEE andlatency. ‚ We show the performance and optimal power alloca-tion gap that results from utilizing short packets whencompared to finite blocklength. Unlike [21], where wemaximize effective capacity via optimal power allocationunder Rayleigh fading while neglecting energy consump-tion of the devices, herein we analyze the optimal powerallocation for different Nakagami- 𝑚 fading setups. ‚ We present a basic framework for applying ARQ byconsidering retransmission of faulty packets in the EBPmodel. A buffer aware strategy is adopted to allow forrate adaption when the buffer is empty in the timeslot following the current transmission. The proposedframework allows for a significant rate gain and renders asignificant boost in the EEE, while maintaining a limitedincrease in average latency. ‚ We solve the optimization problem for optimum errorallocation at each transmission with a target reliabilityconstraint via low complexity Dinkelbach’s algorithm.The solution shows that minimizing the non-EBP jointlyreduces the power consumption and maximizes the EEE.
C. Outline
The rest of the paper is organized as follows: in Section II,we introduce the system model and clarify the relation betweenEC and EEE. Next, Section III presents the EEE analysisin Nakagami- 𝑚 quasi-static fading and characterizes optimum It is worth noticing that in [15] we evaluated only numerically the optimalpower allocation that maximizes the EEE under Rayleigh fading and withoutany assumptions on the EBP.
TABLE II
MPORTANT ABBREVIATIONS AND SYMBOLS .bpcu bits per channel usemax maximizeNBP non-empty buffer probabilitys.t subject to 𝑚 fading parameter 𝑛 blocklength 𝑃 𝑡 power consumption 𝑝 𝑛𝑏 non-empty buffer probability 𝑟 normalized achievable rate 𝐶 𝑒 effective capacity 𝛿 maximum delay E [ ] expectation of Λ delay outage probability 𝑄 ( 𝑥 ) Gaussian Q-function 𝜃 delay exponent 𝜖 𝑡 target error probability 𝜖 ˚ optimum error probability 𝜂 𝑒𝑒 effective energy efficiency 𝜆 arrival rate 𝜌 average signal-to-noise ratio 𝜁 inverse drain efficiency error and power allocation for the linear power consumptionmodel. We illustrate the EBP model in finite blocklengthtransmission and characterize the EEE maximization with reli-ability, latency, and power consumption constraints in SectionIV. After that, Section V studies the retransmission of faultypackets in the empty buffer instants, while the results arediscussed in Section VI. Finally, Section VII concludes thepaper. To make the paper more tractable, we summarize thekey abbreviations and symbols that will appear throughout thepaper in Table I.II. S YSTEM MODEL AND PRELIMINARIES
We consider a communication scenario in which an energy-limited sensor transmits data to a common aggregator througha quasi-static Nakagami- 𝑚 fading channel. The received vector y P C 𝑛 is y = ℎ x + w , (1)where x P C 𝑛 is the transmitted packet, and the block flat-fading coefficient is denoted by ℎ P C which is assumed to beindependent and identically distributed (i.i.d). This implies that ℎ remains constant over the blocklength 𝑛 , but changes fromblock to block. The blocklength is assumed to be smaller thanthe channel’s coherence time. Lastly, w is the additive complexGaussian noise vector whose entries are of unit variance. Weassume that CSI is available at each node. Note that CSIacquisition at the transmitter in the URLLC setup is feasiblewhenever the channel state remains constant over multiple symbols . In most communication environments, the channelcoherence time is much larger than the URLLC transmissionsin mini-slots of duration 0.1 ms, and thus spans of multipleTTI. This gives the transmitter sufficient time to perfectlyacquire CSI [22]. Recent machine learning tools facilitate thistask specially if the channel coefficients are highly correlatedwithin a short period of time. In this case, the transmitter nodecan exploit a recently received signal from the other node orrequest a training sequence in order to estimate the channel[23]. Additionally, as in [5], we aim to provide a performancebenchmark for energy efficiency of these networks, where theeffect of imperfect CSI is beyond the scope of our work. A. Communication at Finite Blocklength
In finite blocklength transmission, unlike Shannon’s model,short packets are conveyed at rate that depends on the block-length 𝑛 and the packet error probability 𝜖 P [ , ] , which issmall but not vanishing. The normalized achievable rate, in(bpcu), is [6] 𝑟 ( 𝜌 ) « log ( + 𝜌 | ℎ | ) ´ 𝑄 ´ ( 𝜖 ) log ( 𝑒 ) ? 𝑛 d ´ (cid:0) + 𝜌 | ℎ | (cid:1) , (2)where 𝑄 ( ¨ ) = ∫ ? 𝜋 𝑒 ´ 𝑡 d 𝑡 is the Gaussian Q-function, and 𝑄 ´ ( ¨ ) represents its inverse, 𝜌 is the average SNR whichfrankly represents the transmit power in watts since the noise isassumed to be normalized to unity. and | ℎ | is the envelope ofthe channel coefficients. The fading coefficients are presentedby the random variable 𝑍 = | ℎ | , which is gamma distributedwith probability density function given as [24, Eq. (2.21)],[25] 𝑓 𝑍 ( 𝑧 ) = 𝑚 𝑚 𝑧 𝑚 ´ Γ ( 𝑚 ) 𝑒 ´ 𝑚𝑧 , (3)where low values of 𝑚 mark severe fading, high values of 𝑚 mark the presence of line of sight (LOS) and 𝑚 = representsRayleigh fading. B. The relation between Effective Capacity and Effective En-ergy Efficiency
For low latency communication, effective capacity ( 𝐶 𝑒 ) isa powerful metric that characterizes the relation between thecommunication rate and the tail distribution of the packetdelay violation probability [9]. For relatively large delay ofmultiple symbol periods, packet delay violation occurs whena packet delay exceeds a maximum delay bound 𝛿 and theoutage probability is defined as [10] Λ = Pr ( 𝑑𝑒𝑙𝑎𝑦 ě 𝛿 ) « 𝑒 ´ 𝜃 ¨ 𝐶 𝑒 ¨ 𝛿 , (4)where Pr ( ¨ ) denotes the probability of a certain event. Conven-tionally, the tolerance of a system to long delays is measured Channel estimation is a cumbersome task for conventional systems andalso under URLLC constraints. However, estimates can be reliably acquiredvia feedback channel in FDD system, or by exploiting channel reciprocity inTDD in line with channel inversion power control methods. CSI is obtainedat reception if the latency constraint allows additional overhead due to pilotbased transmissions, or alternatively via non-coherent transmissions [1]. by the delay exponent 𝜃 . The system tolerates large delaysfor small values of 𝜃 (i.e., 𝜃 Ñ ), and it becomes stricterdelay-wise for large values of 𝜃 . As an exemplary scenario,consider 5G NR numerology 1 with symbol period of 35.7micro-seconds, effective capacity of 1 bpcu and 𝜃 = . . For adelay outage probability of Λ = ´ (i.e, . reliability,the network can tolerate a maximum delay of 𝛿 = symbolperiods ( « ms) for 𝜃 = . , and 𝛿 = symbol periods( « . ms) when 𝜃 = . . From [6], the EC in bpcu is 𝐶 𝑒 ( 𝜌, 𝜃, 𝜖 ) = ´ ln 𝜓 ( 𝜌, 𝜃, 𝜖 ) 𝑛𝜃 , (5)where 𝜓 ( 𝜌, 𝜃, 𝜖 ) = E 𝑍 (cid:104) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜌 ) (cid:105) , (6)and ln is the natural logarithm, 𝑟 ( 𝜌 ) comes from (2) and E 𝑍 [ ¨ ] denotes expectation over the fading. The above equationsassumes an underlying simple ARQ process and indicate thathigher service rates reduce the amount of data bits storedin the queue, and hence also reduce the delay required totransmit, which boosts the EC. Thus, the EC is a measureof the throughput while statistically guaranteeing the delaytolerance. Remark 1.
Note that, we resort to the more practical conceptof service rate as in [6] rather than the departure rate modelin [10]. The intuition is that the service rate metric is moresuitable for the characterization of the re-invented effectivecapacity (or effective rate) for discrete short packets operatingin the finite blocklength regime. Herein, we consider shortpackets with a packet drop probability of 𝜖 to measure relia-bility. Reliability could be well mapped via service rate ratherthan departure rate which does not account for the droppedpackets with probability 𝜖 . This model allows us to combinethe latency and reliability aspects and accommodate them inthe characterization of QoS constrained energy efficiency. In [6], the effective capacity is studied for single node sce-nario, but never to a closed form expression. It has been proventhat the EC is concave in 𝜖 , and hence has a unique globaloptimum. From (2) and (5), we observe that increasing thetransmission power would definitely raise the EC. However,this comes at the expense of increased power consumption,which is not suitable for energy-limited (battery-operated) IoTdevices. Thus, it is necessary to study the trade-off betweenenhancing EC and power consumption.In this context, effective energy efficiency is defined as theachieved effective capacity per unit of consumed power. TheEEE captures the trade-off between the throughput of the com-munication link, the overall power consumption, and latency.Thus, EEE is a suitable metric to quantify and optimize thethroughput of the communication link per each consumed wattfor energy-limited, low latency IoT. Hence, the optimizationof EEE is of great importance for IoT devices, which areisolated from stationary power sources and are required todeliver packets with low latency in the order of milli-seconds[26]. In what follows, we study the EEE for short packetcommunication. III. E FFECTIVE E NERGY E FFICIENCY UNDER LINEARPOWER CONSUMPTION MODEL
In our analysis, we resort to a linear power consumptionmodel defined as [27] 𝑃 𝑡 ( 𝜌 ) = 𝜁 𝜌 + 𝑃 𝑐 , (7)with 𝜁 ě being the inverse drain efficiency of the transmitamplifier and 𝑃 𝑐 the hardware power dissipated in circuitin watts. The linear power consumption model is a well-established and accepted model that has been widely used inmany studies related to energy efficiency of wireless systemssuch as [14], [27]–[29]. This model captures the linear increaseof the power consumption as a function of the transmit powerand the inverse drain efficiency as well as the idle circuit powerconsumption. Furthermore, it facilitates the analysis that aimsat providing a performance benchmark for the EEE in thecontext of short packet and low latency communication of IoTdevices.For this model, the EEE is given by 𝜂 𝑒𝑒 = ´ 𝑛𝜃 ln (cid:0) E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 (cid:3) (cid:1) 𝜁 𝜌 + 𝑃 𝑐 . (8)This scenario assumes an always full buffer and does notaccount for EBP. In [6], a stochastic model for EC was studied,but never to a closed form expression. Herein, we present atight approximation for the EC and hence, the EEE. Lemma 1.
The effective capacity in Nakagami- 𝑚 quasi-staticfading is approximated by 𝐶 𝑒 ( 𝜌, 𝜃, 𝜖 ) « ´ 𝑛𝜃 ln (cid:20) 𝜖 + ( ´ 𝜖 ) 𝑚 𝑚 Γ ( 𝑚 ) ¨ ∑︁ 𝑛 = 𝛽 𝑛 𝑛 ! ∫ ( + 𝜌𝑧 ) 𝛼 𝛾 𝑛 𝑧 𝑚 ´ 𝑒 ´ 𝑚𝑧 𝑑𝑧 (cid:35) , (9) where 𝛼 = ´ 𝜃𝑛 ln 2 , 𝛽 = 𝜃 ? 𝑛𝑄 ´ ( 𝜖 ) log 𝑒 , and 𝛾 = b ´ ( + 𝜌𝑧 ) .Proof. Please refer to Appendix A (cid:3)
Remark 2.
It is quite straightforward to conclude that EEE isan increasing function of the fading parameter 𝑚 . That is, theEEE becomes worse when the fading becomes more severe (i.e, 𝑚 Ñ ). Hence, starting from here, we focus our analysis onquasi-static Rayleigh fading to provide a benchmark of theEC and EEE in the proposed scenarios, where the resultscan also be extended to any type of Nakagami- 𝑚 fading.However, Lemma 1 facilitates the following derivation of theEEE in Rayleigh fading and later, comparing the optimumpower allocation in each fading scenario. Theorem 1.
For a Rayleigh quasi-static fading channel withblocklength 𝑛 , the EEE of the linear power consumption modelis approximated as 𝜂 𝑒𝑒 ( 𝜌, 𝜃, 𝜖 ) « ´ ln [ 𝜖 + ( ´ 𝜖 ) J ] 𝑛𝜃 ( 𝜁 𝜌 + 𝑃 𝑐 ) , (10) where J = 𝑒 𝜌 𝜌 𝛼 (cid:18) 𝛽 + 𝛽 + (cid:19) Γ (cid:18) 𝛼 + , 𝜌 (cid:19) ´ (cid:18) 𝛽 + 𝛽 (cid:19) Γ (cid:16) 𝛼 ´ , 𝜌 (cid:17) 𝜌 , (11) where Γ ( ¨ , ¨ ) is the upper incomplete gamma function [30].Proof. Please refer to Appendix B. (cid:3)
Remark 3.
It was shown in [15] that both EC and EEE areconcave functions of the error probability 𝜖 , and the optimumvalue of 𝜖 that maximizes them in this case is given by 𝜖 ˚ ( 𝜌, 𝛼, 𝛽 ) « arg min ď 𝜖 ď 𝜖 + ( ´ 𝜖 ) J . (12)In what follows, we study the behaviour of EEE as afunction of transmit SNR 𝜌 . Theorem 2.
The EEE function in Theorem 1 is a quasi-concave function of the transmit SNR.Proof.
Please refer to Appendix C (cid:3)
Fig. 1 illustrates the EEE in Rayleigh block fading channelfor different delay exponents, while applying the expectationin (8) and Theorem 1. The network parameters are 𝑛 = symbol periods and 𝜖 = ´ . The figure proves the accuracyof Theorem 1 as a tight approximation for the EEE, which isalso well established for the EC in [21]. However and unlikethe EC which is an asymptotically increasing function of theSNR, the figure shows the convexity of the upper contour ofthe EEE in the transmit power and that the approximation inTheorem 1 captures this quasi-concavity precisely as statedin Theorem 2. Note that the EEE declines when the delayconstraint becomes more strict. Meanwhile, it is also observedthat the optimum transmit power shifts to a higher value forless strict delay constraints. Finally, we plot the EEE also forsmaller packets with length of 𝑛 = symbol periods. As wecan observe, the EEE is higher for smaller packet size as thedelay is minimized which boosts the EC. The approximationholds tightly for this setup as well. -20 -10 0 10 20 30 40 SNR (dB)
EEE ( bp c u / w a tt ) expectationapproximation n=50=0.001=0.01=0.1 Fig. 1. EEE vs SNR in quasi-static fading for 𝑚 = , 𝑛 = , 𝜖 = ´ , 𝑃 𝑐 = . , 𝜁 = . , 𝜆 = and different delay exponents 𝜃 . The quasi-concavity of the EEE in the SNR aids to char-acterize the optimum power allocation which maximizes theEEE in the linear power consumption model.
Theorem 3.
The optimum power allocation for maximizingthe EEE is 𝜌 ˚ , which is the solution of 𝜂 𝑒𝑒 ( 𝜌 ˚ ) = ´ 𝑛𝜃 (cid:18) J ( 𝜌 ˚ )J ( 𝜌 ˚ ) (cid:19) , (13) where 𝜅 = 𝛽 + 𝛽 + and 𝜅 = 𝛽 + 𝛽 , and J = 𝑒 𝜌 𝜌 𝛼 (cid:18) 𝜅 Γ (cid:18) 𝛼 + , 𝜌 (cid:19) ´ 𝜅 𝜌 Γ (cid:18) 𝛼 ´ , 𝜌 (cid:19)(cid:19) , (14) J ( 𝜌 ˚ ) = B J B 𝜌 = ´ 𝜌 (cid:34)(cid:18) + 𝛼𝜌 (cid:19) J + ( ´ 𝜅 ) 𝑒 ´ 𝜌 𝜌 𝛼 ´ 𝜅 𝜌 Γ (cid:18) 𝛼 ´ , 𝜌 (cid:19)(cid:35) . (15) Proof.
Based on the quasi-concavity of the EEE functionwhich was proven in Theorem 2, we differentiate (10) withrespect to 𝜌 and equate to zero as follows B 𝜂 𝑒𝑒 B 𝜌 = ´ ( ´ 𝜖 ) J ( 𝜁 𝜌 + 𝑃 𝑐 ) 𝜖 +( ´ 𝜖 ) J ´ 𝜁 ln ( 𝜖 + ( ´ 𝜖 )J ) 𝑛𝜃 ( 𝜁 𝜌 + 𝑃 𝑐 ) « ´ J ( 𝜁 𝜌 + 𝑃 𝑐 ) 𝑛𝜃 J ( 𝜁 𝜌 + 𝑃 𝑐 ) ´ 𝜁 ln ( 𝜖 +( ´ 𝜖 ) J) 𝑛𝜃 ( 𝜁 𝜌 + 𝑃 𝑐 ) ( 𝜁 𝜌 + 𝑃 𝑐 ) = , (16)where the above approximation is valid since J is much largerthan 1, and the reliability constraint 𝜖 is very small (i.e, J ąą 𝜖 ). Then, to differentiate J , we apply the derivative of theupper incomplete gamma function [31], which yields B J B 𝜌 = ´ 𝜌 (cid:34) J + 𝛼𝜌 J ´ 𝜅 𝑒 ´ 𝜌 𝜌 𝛼 ´ 𝜅 𝜌 Γ (cid:18) 𝛼 ´ , 𝜌 (cid:19) + 𝑒 ´ 𝜌 𝜌 𝛼 (cid:35) , (17)and after algebraic manipulation we obtain (15), which con-cludes the proof. (cid:3) Despite the fact that we were able to find the partialderivative of J , a closed form solution for (13) does notexist. For this purpose, we can compute a point-wise numericalsolution or utilize Matlab root-finding functions, e.g., fzero ina similar way to [16].IV. E MPTY BUFFER PROBABILITY MODEL
Previously, we assumed that the buffer is always full whichpractically is not always the case. In real scenarios, therewould be instants in which a certain IoT device becomes idleand therefore has no data to transmit. Thus, we need to accountfor the case when the buffer is empty. Accordingly, we applythe model considered in [14] to networks operating in thefinite blocklength regime with non-vanishing probability oferror 𝜖 . After accounting for EBP, the transmission probability 𝑃 𝑛𝑏 is equal to ( ´ the probability of empty buffer) and thetransmission process appears in Fig. 2. 𝑛 . . .Transmission with error 𝜖 (𝑃 𝑛𝑏 ) No transmission (1 − 𝑃 𝑛𝑏 ) Fig. 2. Transmission with empty buffer probability in quasi-static channelwith blocklength 𝑛 . For an average arrival rate of 𝜆 and a stable queue, thepower consumption becomes 𝑃 𝑡 ( 𝜌 ) = 𝑃 𝑛𝑏 𝜁 𝑝 + 𝑃 𝑐 = 𝜆 E [ 𝑟 ] 𝜁 𝜌 + 𝑃 𝑐 , (18)with 𝑃 𝑛𝑏 = 𝜆 E [ 𝑟 ] denoting non-empty buffer probability (NBP),which is bounded between 0 and 1. The EEE with EBP is 𝜂 𝑒𝑒 = ´ 𝑛𝜃 ln [ 𝜖 + ( ´ 𝜖 ) J ] 𝜆 E [ 𝑟 ] 𝜁 𝜌 + 𝑃 𝑐 , (19)where the numerator represents the effective capacity in thefinite blocklength regime as defined in Theorem 1.Note that similar to the fluid model , the NBP indicates theaverage asymptotic probability of transmission over relativelylong time, where 𝑃 𝑛𝑏 = means that the average servicedamount of data per packet time is equal to the average amountof data arrival per packet time. This indicates that, in average,transmission always occurs. A. Verifying the effective energy efficiency model with emptybuffer probability in finite blocklength
According to [28], an energy efficiency function must benon-negative, must be zero when the transmit power is zero,and must tend to zero as the transmit power tends to infinity.It was shown in [14] that this power the EBP model fulfills isvalid for Shannon model. In the following Lemma, we verifythat this EBP power consumption model is valid as well forshort packets transmission.
Lemma 2.
The EEE in (19) is zero for 𝜌 = and tends to 0when 𝜌 Ñ 8 .Proof.
Please refer to Appendix D. (cid:3)
B. Effective energy efficiency maximization with buffer con-straints
We investigate the EEE maximization with EC, delay, andpower constraints. EC should be larger than the arrival rate 𝜆 to guarantee a stable queue, while the transmission SNR The reason behind the choice a fluid model is that these fluid modelsare motivated as approximations to discrete queueing systems. Fluid flowqueues have been well accepted as a useful mathematical tool for modelingand have long been used to evaluate the performance of telecommunicationand computer systems. In particular, we apply the fluid model to characterizethe asymptotic delay probability, and to approximate the NBP by arrival ratedivided by average service rate. The exact (non-asymptotic) analysis of delayoutages and empty buffers is outside the scope of this work. 𝜌 is bounded by 𝜌 𝑚𝑎𝑥 . Thus, the optimization problem isformulated as max 𝜌 ě ,𝜃 ě 𝜂 𝑒𝑒 = ´ 𝑛𝜃 ln [ 𝜖 + ( ´ 𝜖 ) J ] 𝑃 𝑛𝑏 𝜁 𝜌 + 𝑃 𝑐 ,𝑠.𝑡 𝐶 𝑒 ( 𝜌, 𝜃, 𝜖 ) ě 𝜆, 𝑃 𝑛𝑏 𝑒 ´ 𝜃𝜆𝛿 ď Λ 𝜌 ď 𝜌 𝑚𝑎𝑥 , 𝜖 ď 𝜖 𝑡 , ď 𝑃 𝑛𝑏 ď . (20)Note that the reliability constraint here, which is the errorprobability constraint on the first transmission, is importantto improve QoS in URLLC. Meanwhile, the delay outageprobability constraint does not necessarily guarantee reliabilityif the 𝜖 is not imposed.For the full buffer model, we set 𝑃 𝑛𝑏 to 1. We performa line search for 𝜌 in the interval [ , 𝜌 𝑚𝑎𝑥 ] . The optimumerror probability is min [ 𝜖 ˚ , 𝜖 𝑡 ] where 𝜖 ˚ is obtained fromRemark 3. When analyzing the empty buffer scenario, we set 𝑃 𝑛𝑏 = 𝜆 E [ 𝑟 ] . Here, Λ is the maximum allowed delay outageprobability. In all cases, the optimal value of 𝜃 can be obtainedfrom the second constraint at equality as 𝜃 ˚ ( 𝜌 ) = 𝜆𝛿 ln 𝑃 𝑛𝑏 Λ . (21)V. A DAPTIVE R ETRANSMISSION SCENARIO
As shown in [32], achieving ultra-reliability using onetransmission consumes a huge amount of power, which is notapplicable for energy limited IoT devices. Herein, we presenta basic framework for applying ARQ by considering adaptiverate retransmission of faulty packets when the buffer is emptyin the EBP model in order to achieve ultra-reliability. In thisframework, at a certain time instant 𝑡 , we assume a buffer-aware transmission as in [33], [34]; this allows the transmitterto have a prior knowledge of whether the buffer will be emptyor there is a packet that needs to be delivered at the next timeslot 𝑡 + . Then the following is applied:1) If a packet arrives at time slot 𝑡 and there is also a packetarrival at 𝑡 + , normal transmission occurs at 𝑡 with rate 𝑟 ( 𝜖 ) .2) If a packet arrives at time slot 𝑡 and there is no packetarrival at 𝑡 + , we transmit the with rate 𝑟 ( 𝜖 ą 𝜖 ) andapply ARQ at 𝑡 + .However, the non-EBP in this case will be slightly changedto 𝑃 𝑛𝑏 due to the rate variations. The first case occurs whenthe buffer is full at 𝑡 + . Assuming i.i.d arrivals, the probabilityof occurrence of case number 1 is the same as the non-emptybuffer probability 𝑃 𝑛𝑏 .In case 2, we apply the type-I ARQ protocol. In type-IARQ protocol, the node is allowed to retransmit its packet if itreceives a NACK feedback from the receiver, which indicatesthat the packet is not successfully decoded. We assume amaximum of only 1 retransmission in order to satisfy the strin-gent delay requirements in URLLC. Note that, the transmitterperforms the first transmission with an error probability of 𝜖 and the second transmission with an error probability of 𝜖 such that the aggregate error probability satisfies the reliabilityconstraint (i.e, 𝜖 𝜖 ď 𝜖 ). Thus, both 𝜖 and 𝜖 are higher than 𝜖 . The fact that the transmission rate is an increasing functionof the error probability implies that both 𝑟 ( 𝜖 ) and 𝑟 ( 𝜖 ) are higher than 𝑟 ( 𝜖 ) , which reflects the rate gains of this model.In other words, we obtain a significant rate gain in the likelyevent of successful first transmission. Moreover, the aggregatepacket error probability when applying ARQ would be 𝜖 whichmaintains the reliability level. This can be performed by rateadaption as in [35], where sensors generate data in the formof bits that can be relocated from one packet to the other. Thisrate adjustment occurs when the buffer is empty in the nexttime instant 𝑡 + with probability ´ 𝑃 𝑛𝑏 and results in arise in the instantaneous rate with symbols resizing. Generallylet 𝑟 = E [ 𝑟 ( 𝜖 )] , 𝑟 = E [ 𝑟 ( 𝜖 )] and 𝑟 = E [ 𝑟 ( 𝜖 )] . Then theaverage rate becomes E [ 𝑟 ] = 𝑝 𝑛𝑏 𝑟 + ( ´ 𝑝 𝑛𝑏 ) (cid:104) ( ´ 𝜖 ) 𝑟 + 𝜖 𝑟 (cid:105) = 𝑝 𝑛𝑏 ( 𝑟 ´ 𝜅 ) + 𝜅, (22)where 𝜅 = ( ´ 𝜖 ) 𝑟 + 𝜖 𝑟 . Note that the rate is dividedby 2 in case of retransmission because the time durationof transmitting 𝑛 symbols is approximately the double asexpressed in the last term of (22). Due to the rate variation,the modified non-EBP should satisfy 𝑝 𝑛𝑏 = 𝜆 E [ 𝑟 ] = 𝜆𝑝 𝑛𝑏 ( 𝑟 ´ 𝜅 ) + 𝜅 . (23)Solving (23) for 𝑝 𝑛𝑏 , we obtain 𝑝 𝑛𝑏 = ´ 𝜅 + a 𝜅 + ( 𝑟 𝑜 ´ 𝜅 ) 𝜆 ( 𝑟 𝑜 ´ 𝜅 ) ď , (24)and the EC in this case is given by 𝐶 𝑒 = ´ 𝑛𝜃 ln (cid:16) E 𝑍 (cid:104) 𝑝 𝑛𝑏 ( 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 )+( ´ 𝑝 𝑛𝑏 ) (cid:16) ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 + 𝜖 ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃 𝑟 + 𝜖 (cid:17)(cid:105) (cid:17) . (25)To clarify (25), we mention that the transmission occurswith rate 𝑟 when the buffer is not empty in the next time slot.This is indicated by the first term in equation (25) and occurswith probability 𝑝 𝑛𝑏 . When the buffer is empty in the nextslot with probability ( ´ 𝑝 𝑛𝑏 ) , one transmission occurs rate 𝑟 if there is no error where the no error probability is ´ 𝜖 .However, the rate is divided by 2 to become 𝑟 / in case ofretransmission when an error occurs in the first transmissiononly with probability 𝜖 ( ´ 𝜖 ) . While the rate is considered tobe zero when both transmissions fail with probability 𝜖 𝜖 = 𝜖 .In order to map the impact of the second transmission on theoverall effective capacity, we calculate the average of the rateof the first transmission which is zero in case of error and thesecond transmission which is 𝑟 ( 𝜖 ) . Since the transmissionoccurs in the duration of 2 time slots, the rate is virtuallydivided by 2 as indicated in the second term of (25).Since the channel coefficients change from one transmissionto the other, we need to vary the transmit power in order tocompensate for the channel coefficient variation so that theproduct 𝜌 | ℎ | is the same for both transmissions. Althoughthe transmit power varies, the average transmit power E [ 𝜌 ] is the same for both transmissions and independent from 𝑝 𝑛𝑏 or any other parameter. Thus, the consumed power for thisscenario is a probabilistic function of the transmit power of one or two transmissions which after manipulations is allowedto be expressed as 𝑃 𝑡 = (cid:104) 𝑝 𝑛𝑏 + 𝑝 𝑛𝑏 ( ´ 𝑝 𝑛𝑏 ) ( + 𝜖 ) (cid:105) 𝜁 𝜌 + 𝑃 𝑐 , (26)Defined by the quotient of EC to the transmit power, the EEEof this scenario is formulated as 𝜂 𝑒𝑒 = 𝐶 𝑒 (cid:2) 𝑝 𝑛𝑏 + 𝑝 𝑛𝑏 ( ´ 𝑝 𝑛𝑏 ) ( + 𝜖 ) (cid:3) 𝜁 𝜌 + 𝑃 𝑐 . (27)Note that the optimum error probability of each transmissionin the EBP model with two ARQ transmissions is not simplythe square root of the aggregate target error probability 𝜖 = 𝜖 𝑡 .Therefore, we define the optimization problem to determinethe optimum error probability of the first transmission thatmaximizes the EEE subject to a target reliability constraint as max 𝜂 𝑒𝑒 ( 𝜖 , 𝜖 ) (28) 𝑠.𝑡 ă 𝜖 𝜖 ď 𝜖 𝑡 An interesting analysis is to determine the asymptotic be-haviour of the EEE as the delay constraint 𝜃 Ñ 8 or 0.This corresponds to extremely strict or no delay constraint,respectively. It is straight forward to conclude that the EEEtends to zero for extremely stringent delay constraint (i.e, when 𝜃 Ñ 8 ). Herein, we derive the upper bound of the EEE forrelaxed latency constraint as 𝜃 Ñ . Theorem 4.
The EEE of the proposed retransmission scenariois upper bounded by lim 𝜃 Ñ 𝜂 𝑒𝑒 = 𝑝 𝑛𝑏 ( ´ 𝜖 ) 𝑟 𝑜 + ( ´ 𝑝 𝑛𝑏 ) (cid:2) ( ´ 𝜖 ) 𝑟 + 𝜖 ( ´ 𝜖 ) 𝑟 (cid:3)(cid:2) 𝑝 𝑛𝑏 + 𝑝 𝑛𝑏 ( ´ 𝑝 𝑛𝑏 )( + 𝜖 ) (cid:3) 𝜁 𝜌 + 𝑃 𝑐 , (29) and lower bounded by zero.Proof. Please refer to Appendix E. (cid:3)
Remark 4.
As the system approaches ultra-reliability (i.e, 𝜖 Ñ ), the EC of one transmission converges to 𝑟 ,while the EC of the proposed EBP-ARQ scehme convergesto 𝑝 𝑛𝑏 𝑟 𝑜 + ( ´ 𝑝 𝑛𝑏 ) 𝑟 . Hence, the EC is raised to 𝑟 for ( ´ 𝑝 𝑛𝑏 ) portion of the time which indicates the gain in theEC of the proposed EBP retransmission scheme.A. Power saving Returning to (26), which represents the average powerconsumption, we study the effect of varying the non-EBP 𝑝 𝑛𝑏 on the power consumption by obtaining the first derivative of(26) as B 𝑃 𝑡 B 𝑝 𝑛𝑏 = (cid:104) ´ 𝜖 𝑝 𝑛𝑏 + ( + 𝜖 ) (cid:105) 𝜁 𝜌, (30)which is strictly non-negative for all possible values of 𝑝 𝑛𝑏 and 𝜖 (i.e, ď 𝑝 𝑛𝑏 , 𝜖 ď ). Thus, the power consumption 𝑝 𝑡 isstill an increasing function of the non-empty buffer probability 𝑝 𝑛𝑏 . Hence, minimizing the non-empty buffer probability alsoreduces the transmit power which leads to a longer battery lifefor remote sensors that are located far from energy sources. Theorem 5.
The non-EBP 𝑝 𝑛𝑏 in (24) is a pseudo-convexfunction in 𝜖 and therefore, the minimization of 𝑝 𝑛𝑏 is afractional program.Proof. Please refer to Appendix F. (cid:3)
Furthermore, being a psuedo convex fractional program anddue to the analytical intractability, it is easier to find the globalminimum of 𝑝 𝑛𝑏 using well known optimization algorithmssuch as Dinkelbach’s algorithm [28] to minimize 𝑝 𝑛𝑏 andhence, the total transmit power. Later the results section showsthat this optimal solution for minimizing the transmit poweralso highly approaches optimality for maximizing the EEEin this case. Moreover, it is more efficient and numericallytractable to minimize the transmit power instead of the EEEfunction given in (28). Hence, the problem becomes min 𝑝 𝑛𝑏 ( 𝜖 , 𝜖 ) (31) 𝑠.𝑡 ă 𝜖 𝜖 ď 𝜖 𝑡 . Note that the target is to minimize the transmit power. Thus,it is straight forward to conclude that the reliability constraintis optimally achieved at equality since more power is neededto achieve lower error and higher reliability. As proven inTheorem 5, the problem in (31) is a pseudo-convex fractionalprogram. Therefore, the global optimum exists, and can befound by utilizing the Dinkelbach’s algorithm as introduced inSection 3.2 in [28]. It is a parametric algorithm of which thebasic idea is to tackle a pseudo-convex problem by solving asequence of easier problems which are guaranteed to convergeto the global optimum. The minimization procedure is depictedin Algorithm 1.
Algorithm 1:
Minimization of 𝑝 𝑛𝑏 Input : 𝐹 𝜎 ą 𝛿 ą ; 𝑛 = ; 𝜎 = ; Output: 𝜖 ˚ while 𝐹 𝜎 𝑛 ą 𝛿 do 𝜖 ˚ = arg min { ´ 𝜅 ( 𝜖 ) + a 𝜅 ( 𝜖 ) + ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 )) 𝜆 ´ 𝜎 𝑛 ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 ))} ; 𝐹 𝜎 𝑛 = ´ 𝜅 ( 𝜖 ˚ ) + b 𝜅 ( 𝜖 ˚ ) + ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 ˚ )) 𝜆 ´ 𝜎 𝑛 ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 ˚ )) ; 𝜎 𝑛 + = ´ 𝜅 ( 𝜖 ˚ )+ b 𝜅 ( 𝜖 ˚ ) + ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 ˚ )) 𝜆 ( 𝑟 𝑜 ´ 𝜅 ( 𝜖 ˚ )) ; 𝑛 = 𝑛 + ; end The intuition behind the algorithm is as follows. It startsfrom some arbitrary estimate of 𝜖 ˚ and analyzes the level setsof the original problem, which are evidently convex. Then, asthe algorithm progresses, it iteratively corrects the estimate of 𝜖 ˚ and checks if the stopping criterion is satisfied, i.e. a 𝛿 -suboptimal solution has been obtained. If the tolerance marginhas not yet been satisfied, then the algorithm continues scan-ning through the level sets of the function until convergence. Taking into account the energy consumption and computational complexityfor the original online optimization algorithm, it might even turn out thatthe suboptimal one proposed here leads to an overall better effective energyefficiency and lower latency.
Notice that the worst-case computational complexity of thealgorithm is dominated by step 2, which can be solved usinginterior point methods. As a consequence, the convergence ratein the sub-problem sequence is super-linear [36], [37].
B. Average latency
Herein, we analyze the average extra packet delay induceddue to retransmissions when applying the proposed EBP-ARQwith empty buffer instants. Let 𝛿 be the delay per packetin the single transmission scenario and 𝛿 be the total delaywhen two transmissions occur. Then the expected delay whenapplying ARQ with two retransmissions would be 𝜏 = 𝛿 (cid:16) 𝑃 𝑛𝑏 + ( ´ 𝑃 𝑛𝑏 )( ´ 𝜖 ) (cid:17) + 𝛿 ( ´ 𝑃 𝑛𝑏 ) 𝜖 . (32)In case of error in the first transmission, the 1 bit NACK feed-back could be transmitted in a span of « symbols accordingthe Physical Uplink Control Channel (PUCCH) Format 1 in5G NR [38]. Assuming the same transmission rate for theNACK packet, the extra delay due to the NACK packet wouldbe Δ = 𝑛 𝛿 . This occurs during before the second transmissionand is very small compared to the one packet transmissiontime. Thus, we can state that 𝛿 = 𝛿 + Δ = (cid:16) + 𝑛 (cid:17) 𝛿 . Hence,the normalized delay with respect to one transmission time 𝛿 can be written as 𝜏 𝑛 = 𝑃 𝑛𝑏 + ( ´ 𝑃 𝑛𝑏 )( ´ 𝜖 ) + (cid:18) + 𝑛 (cid:19) ( ´ 𝑃 𝑛𝑏 ) 𝜖 , (33)where 𝜏 𝑛 ě . Herein, (33) provides an indication of the QoSwhen applying ARQ wiht EBP for boosting the EEE. Note thatwe mainitain the same QoS constraints 𝜃 snd 𝜖 throughout thewhole analysis.VI. R ESULTS AND DISCUSSION
In this section, we present numerical results to illustratethe behaviour of the EEE function and the trade off betweenthe EEE, power allocation and latency for Shannon’s modeland finite blocklength in different transmission scenarios. Wecompare our results to the infinite blocklength case to show theperformance gap that results from applying the short packetinformation theoretic approach which is more suitable fordelay constrained analysis and compare this gap to the longpackets ideal case. Firstly, Fig. 3 illustrates the EEE of shortpacket transmission as a function of the delay exponent 𝜃 inquasi-static Rayleigh fading for 𝑛 = , 𝜌 = dB, errorprobability 𝜖 = ´ , and different circuit powers 𝑃 𝑐 . Thefigure highlights the energy efficiency gap between long packettransmission which is analyzed via Shannon capacity modeland the finite blocklength model. The EEE of short packets isless than the infinite blocklength Shannon’s model by about20% in this case. Moreover, the figure shows that the EEEdeclines when the delay exponent becomes more strict andwhen the consumed power in circuitry is higher.In Fig. 4, we elucidate the EEE for different transmissionprobabilities (i.e, when the buffer is not empty). The figureshows the EEE gap between infinite and finite blocklengthmodels. Again, it is noted that higher circuit power signif-icantly deteriorates the EEE. It is observed that the EEE -3 -2 -1 Delay exponent E ff e c t i v e E ne r g y E ff i c i en cy ( bp c u / w a tt s ) ShannonFinite blocklength P c = 0 dB P c = - 6 dB Fig. 3. Effective energy efficiency as a function of the delay exponent 𝜃 inquasi-static Rayleigh fading for 𝑛 = , 𝜌 =3 dB, error probability 𝜖 = ´ ,and different circuit powers 𝑃 𝑐 . Non-Empty Buffer Probability P nb E ff e c t i v e E ne r g y E ff i c i en cy ( bp c u / w a tt s ) ShannonFinite blocklength P c = - 6 dBP c = 0 dB Fig. 4. Effective energy efficiency vs as a function of the non-EBP 𝑃 𝑛𝑏 in quasi-static Rayleigh fading for 𝑛 = , SNR=3 dB, error probability 𝜖 = ´ , 𝜃 = . and different circuit powers 𝑃 𝑐 . monotonically decreases with the increase of arrival rate (oralternatively Non-EBP) which indicates higher congestion inthe network. However, this effect becomes marginal whenthe circuit power is higher as the circuit power becomes adominant factor in the calculation of EEE. Thus, for 𝑃 𝑐 = dB, the EEE is nearly constant as a function of the arrival rate.Therefore, careful studying of EEE for different source arrivalrates is crucial for low circuit power.For the following simulations, we fix the network parame-ters as follows: Λ = (cid:8) ´ , ´ (cid:9) , 𝑃 𝑐 = . 𝑊, 𝜁 = . , 𝜆 = , 𝛿 = symbol periods, and 𝑛 = symbol periods,unless stated otherwise. In Fig. 5, we evaluate the EEE as afunction of error probability 𝜖 in case of EBP and compareit to the case where the buffer is always full while fixing -4 -3 -2 -1 Error probability ee ( bp c u / w a tt ) EBP, P out_ delay =10 -2 Full buffer, P out_ delay =10 -2 EBP, P out_ delay =10 -3 Full buffer, P out_ delay =10 -3 Shannon's gapShannon's gap
Fig. 5. EEE vs 𝜖 with and without empty-buffer probability for Λ = ´ , ´ , 𝑃 𝑐 = . , 𝜁 = . , 𝜆 = , 𝛿 = , and 𝑛 = . the transmit power at 𝜌 = dB. We observe that the EEEis concave in 𝜖 as stated in Remark 3. It is obvious thatconsidering the probability of empty buffer reflects a gainin the EEE over the full buffer model, while decreasing thedelay outage probability reduces the EEE. Moreover, the figuredepicts that Shannon’s model considered in [14] overestimatesthe EEE by more than when compared to the finiteblocklength model.Fig. 6 depicts the maximum achieved EEE obtained from(20) for different delay limits 𝛿 , where 𝜌 𝑚𝑎𝑥 = dB (variabletransmission power), and 𝜖 𝑡 = ´ . We observe that the EEEincreases when extending the delay bound 𝛿 and relaxing thedelay outage probability Λ . This implies that networks whichcan tolerate longer packet transmission delay are more energyefficient. From another perspective, it is clear that the sporadicnon-EBP transmission scenario allows for a better modellingof the power consumption in MTC. This reflects that full bufferis the worst case, where we assume that all power will beconsumed. Meanwhile, the Non-EBP models the fraction oftime that is actually used for transmission of packets accordingto the queue congestion, which interprets the gain of thismodel compared to always full buffer. Furthermore, the figureverifies the inaccuracy of Shannon’s model when computingthe EEE for relatively small packets where the inaccuracy gapreaches more than in higher delay region.In order to present an insight about how EBP wouldaffect the performance of multi-user network, we consider asimple exemplary setup where 2 users transmit short packetsto a common BS. The BS applies successive interferencecancellation where User 1 is the primary user assumed to beultra reliable with 𝜖 𝑢 = ´ , and therefore decoded last whileit transmits with higher power 6 dB. Meanwhile, User 2 is thesecondary which has lower priority, where it transmits withlow transmit power of 0 dB, and reliability of 𝜖 𝑢 = . . Fig.7 depicts that the NBP of User 1 does not only affect the EEEof User 1, but also affects both the EEE and NBP of User 2.Taking a close look at Fig. 7, we observe that adjusting
100 200 300 400 500 600 700 800 900 1000
Delay limit (symbol periods) ee ( bp c u / w a tt ) EBP, =10 -2 (Shannon)EBP, =10 -2 =10 -2 Full buffer, =10 -2 =10 -2 EBP, =10 -3 (Shannon)EBP, =10 -3 Full buffer, =10 -3 Fig. 6. EEE vs 𝛿 with and without empty buffer probability for Λ = ´ , ´ , 𝑃 𝑐 = . 𝑊 , 𝜁 = . , 𝜆 = , 𝑛 = , and 𝜖 𝑡 = ´ . Pnb User 1 ee P nb U s e r Fig. 7. Multi-user performance evaluation with EBP for 𝜖 𝑢 = ´ , 𝜖 𝑢 = . , 𝜌 = dB, 𝜌 = dB, 𝜃 = . , 𝑃 𝑐 = . , 𝜁 = . , 𝛿 = , 𝜆 = . and 𝑛 = . the arrival rate of User 1 to a lower level reduces its NBPprobability and improves the EEE of both users. Meanwhile,the NBP probability of User 2 increases when the buffer ofUser 1 is more busy. This happens because User 2 suffersfrom excess interference from User 1, which forces User 2into reducing its transmission rate. Hence, packets accumulatein the buffer of User 2 which in turn becomes more congested.In Fig. 8, we plot the optimum power allocation for max-imizing the EEE as a function of the maximum delay 𝛿 incase of EBP and always full buffer where 𝜌 𝑚𝑎𝑥 = dB. Thetarget error outage probability is fixed at 𝜖 = ´ . The plotshows that the optimal power allocation is significantly higherwhen the delay outage probability Λ is lower and when EBPis considered. The figure also depicts that Shannon’s modeldoes not render an accurate power allocation to maximize theEEE; in fact, it underestimates the optimum power allocationwhen compared to the finite blocklength model. The power gap
100 200 300 400 500 600 700 800 900 1000
Delay limit (symbol periods) O p t i m a l po w e r a ll o c a t i on ( d B ) Rayleigh fading, EBP, =10 -2 Ricean fading, EBP, =10 -3 Rayleigh fading, EBP, =10 -3 Severe fading, EBP, =10 -3 Rayleigh fading, Full buffer, =10 -3 Rayleigh fading, Shannon, EBP, =10 -3 Power gap
Fig. 8. Optimal power allocation vs 𝛿 with and without empty-bufferprobability for Λ = ´ , ´ , 𝑃 𝑐 = . , 𝜁 = . , 𝜆 = , 𝜌 𝑚𝑎𝑥 = dB and 𝜖 = ´ . ranges from to dB as shown in the figure. Thus, we canexploit the extra power allocation that results from consideringempty buffer and applying the finite blocklength model inorder to efficiently boost the EC. It is also observed that theoptimal power allocation increases when the delay tolerancebecomes higher. The intuition behind this is that when thenetwork tolerates higher delays, it allows for improving thethroughput by allocating higher power without wasting thenetwork resources. This improvement occurs in the sameway when considering empty buffer probability and whenincreasing the line of sight (e.g. Ricean fading where 𝑚 ą ).In Fig. 9, we illustrate the EEE gain of the EBP retrans-mission scenario. The system parameters are 𝜌 = dB, 𝜖 = ´ , 𝜆 = . . The figure shows that our proposed EBPscheme with ARQ enhances the EEE when compared to theclassical EBP and full buffer models. In fact, the EEE of EBPmodel with ARQ is more than double of the normal EBP casewhen the delay constraint is very strict and the delay exponentapproaches high values at 𝜃 ą . . Thus, the EEE gain ismore relevant for delay stringent networks. In this case, theEEE is upper bounded by 1.07 (bpcu/watt) as obtained fromTheorem 4. The plot also compares different error allocationstrategies for the first and second transmission rounds. It isobvious that the equal error allocation is not optimal enoughto maximize the EEE. However, the minimum transmit powerstrategy highly approaches EEE optimality.In Fig. 10, we depict the total power consumption accom-panied by each scenario as function of the arrival rate. Thenetwork parameters are 𝜌 = dB, 𝑛 = , 𝜖 = ´ , 𝜃 = . ,where 𝜆 is varied this time as shown on the figure axis.Although this effect is marginal for the full buffer model,the figure depicts that higher arrival rates consume morepower as there are more packets to transmit. However, thepower consumption is lower for the EBP model. Despiteretransmissions which consume high power, the EBP modelwith retransmissions is the most power saving scheme, sincethe packets are transmitted with higher error probabilities for Delay exponent -2 -1 ee ( bp c u / j ou l e ) optimum error allocationminimum power consumptionequal error allocationEBP with no retransmissionFull buffer ARQ
Fig. 9. Effective energy efficiency 𝜂 𝑒𝑒 for different delay exponents 𝜃 , where 𝜌 = dB, 𝑛 = , 𝜖 = ´ , 𝑃 𝑐 = . , 𝜁 = . , 𝜆 = . . Arrival rate -2-101234567 P o w e r c on s u m p t i on P t ( d B ) Full bufferEBPEBP with retransmission
Fig. 10. Power consumption 𝑃 𝑡 for different arrival rates 𝜆 , where 𝜌 = dB, 𝑛 = , 𝜖 = ´ , 𝑃 𝑐 = . , 𝜁 = . , 𝜃 = . . each single transmission which boosts the service rate andreduces the traffic congestion at the buffer and hence, thepower consumption of the whole network.Finally, Fig. 11 illustrates the normalized delay 𝜏 𝑛 as afunction of arrival rate 𝜆 for the EBP-ARQ scheme with twotransmissions. The figure shows that the normalized delay 𝜏 𝑛 diminishes as the network traffic becomes higher. This isbecause, when the queue becomes more congested, there arefewer chances for the buffer to become empty and hence, theopportunity for a second transmission disappears. Hence, thedelay becomes only one transmission delay which is lowerthan the delay in case of two transmissions. It is noted thatthe delay becomes worse for lower reliability requirement asthe error is also relaxed in the first transmission which leads tohigher probability of occurrence for the second transmissionand longer delay.Moreover, for the same reliability constraint, boosting the Arrival rate N o r m a li z ed l a t en cy n t =10 -5 , =6 dB t =10 -9 , =10 dB t =10 -9 , =6 dB Fig. 11. Normalized delay 𝜏 𝑛 for different arrival rates 𝜆 , where 𝑛 = , 𝑃 𝑐 = . , 𝜁 = . , 𝜃 = . . transmit power does not reduce latency. This is because for thishigh power, if the network becomes more congested, it slightlyaffects the non-EBP which maintains its low value and allowsfor second transmission which causes longer delay. The delayin its worst case is still only 3% higher than the delay of onetransmission scheme. Hence, we obtain significantly higherEEE with only limited increase in the delay, while maintainingreliability at the same level.VII. C ONCLUSIONS
In this work, we presented a detailed analysis of theEEE for ultra reliable delay constrained networks in thefinite blocklength regime. For Nakagami- 𝑚 quasi-static fadingchannels, we proposed an approximation for the EC. Thenwe characterized the EEE maximizers in terms of optimumerror probability and power allocation for the Rayleigh fadingcase. The results revealed that Shannon’s model overestimatesthe EEE and underestimates the optimum power allocationwhen compared to the finite blocklength model. Further resultsindicated that allowing for larger delays significantly boostsEEE. We showed that the advantage of considering non-empty buffer probability and flexible transmission power istwofold since it significantly improves the EEE of networksoperating in the finite blocklength regime and allows forretransmission of faulty packets with a significant boost inthe EEE, reliability, and limited delay extension. For theEBP retransmission scenario, we derived the upper boundof the EEE and provided a low complexity solution for theoptimization problem of maximizing the EEE and minimizingthe power consumption. The solution showed that EBP modelwith retransmission is an ultra-reliable power saving schemewhich improves the energy efficiency with limited increasein latency. Better performance and higher EEE gain couldbe achieved by applying Chase Combining (CC-HARQ) andIncremental redundancy (IR-HARQ) protocols [39]. This is left as possible extension for this work along with the analysisof EEE in multi-user networks as in Fig. 7.A PPENDIX
APROOF OF L
EMMA 𝐶 𝑒 ( 𝜌, 𝜃, 𝜖 ) = ´ 𝑛𝜃 ln (cid:18) 𝑚 𝑚 Γ ( 𝑚 ) ∫ (cid:16) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝜃𝑛𝑟 (cid:17) 𝑧 𝑚 ´ 𝑒 ´ 𝑚𝑧 𝑑𝑧 (cid:19) . (34)From (2), we have 𝑒 ´ 𝜃𝑛𝑟 = 𝑒 ´ 𝜃𝑛 log ( + 𝜌𝑧 ) 𝑒 𝜃 c 𝑛 ( ´ ( + 𝜌𝑧 ) ) 𝑄 ´ ( 𝜖 ) log 𝑒 , (35)where 𝑒 ´ 𝜃𝑛 log ( + 𝜌𝑧 ) = ( + 𝜌𝑧 ) 𝛼 (36) 𝑒 𝜃 c 𝑛 ( ´ ( + 𝜌𝑧 ) ) 𝑄 ´ ( 𝜖 ) log 𝑒 = 𝑒 𝛽𝛾 . (37)We resort to the Taylor expansion to obtain 𝑒 𝑐𝑥 = (cid:205) 𝑛 = ( 𝑐𝑥 ) 𝑛 𝑛 ! .It follows from (35),(36) and (37) that the expression in (34)can be written as 𝐸𝐶 ( 𝜌 𝑖 , 𝜃, 𝜖 ) = ´ 𝑛𝜃 ln (cid:20)∫ 𝜖 𝑚 𝑚 Γ ( 𝑚 ) 𝑧 𝑚 ´ 𝑒 ´ 𝑚𝑧 𝑑𝑧 +( ´ 𝜖 ) ∫ 𝑚 𝑚 Γ ( 𝑚 ) ( + 𝜌 𝑖 𝑧 ) 𝛼 ∑︁ 𝑛 = ( 𝛽𝛾 ) 𝑛 𝑛 ! 𝑧 𝑚 ´ 𝑒 ´ 𝑚𝑧 𝑑𝑧 (cid:35) . (38)The infinite series in (38) can be truncated to a finite sum ofterms and we evaluate the accuracy of the expression notingthat the accuracy increases with the number of terms. But, itis noticed that when testing for different system parameters( 𝑁 , 𝜌 , 𝜃 , 𝑛 ), the accuracy for expanding 1 term is 92.7 % , 2terms is 99 % and 99.9 % for 3 terms only. Henceforth, in ouranalysis, 3 terms will be enough and (38) reduces to (9).A PPENDIX
BPROOF OF T
HEOREM 𝑓 𝑍 ( 𝑧 ) = 𝑒 ´ 𝑧 .This corresponds to a Nakagami- 𝑚 fading parameter 𝑚 = .Applying the second order Taylor expansion to obtain 𝑒 𝛽𝛾 = + ( 𝛽𝛾 ) + ( 𝛽𝛾 ) , it follows from Theorem 1 that for Rayleighdistributed channels 𝜓 ( 𝜌, 𝜃, 𝜖 ) = 𝜖 + ( ´ 𝜖 ) (cid:20)∫ ( + 𝜌𝑧 ) 𝛼 𝑒 ´ 𝑧 d 𝑧 + 𝛽 ∫ ( + 𝜌𝑧 ) 𝛼 𝛾𝑒 ´ 𝑧 d 𝑧 + 𝛽 ∫ ( + 𝜌𝑧 ) 𝛼 𝛾 𝑒 ´ 𝑧 d 𝑧 (cid:21) . (39)The first integral can be written as 𝑒 𝜌 𝜌 𝛼 Γ (cid:16) 𝛼 + , 𝜌 (cid:17) By applying Laurent’s expansion for 𝛾 [40], we obtain 𝛾 « ´ ( + 𝜌𝑧 ) . Hence, the second and third inte-grals can be written as 𝑒 𝜌 𝛽𝜌 𝛼 (cid:32) Γ (cid:16) 𝛼 + , 𝜌 (cid:17) ´ Γ (cid:16) 𝛼 ´ , 𝜌 (cid:17) 𝜌 (cid:33) , and 𝑒 𝜌 𝛽 𝜌 𝛼 (cid:32) Γ (cid:16) 𝛼 + , 𝜌 (cid:17) ´ Γ (cid:16) 𝛼 ´ , 𝜌 (cid:17) 𝜌 (cid:33) , respectively leadingto (11). A PPENDIX
CPROOF OF T
HEOREM 𝜙 = 𝑄 ´ ( 𝜖 ) log ( 𝑒 ) ? 𝑛 and note that 𝜙 should be a strictly positiveparameter. Moreover, 𝜙 is less than unity for practical values of 𝑛 and 𝜖 , where 𝑛 ě and 𝜖 ď . , which in fact are guaranteedfor URLLC operation where n>100 and 𝜖 ă ´ [1]. Thisdictates that the denominator of 𝜙 is higher than its numeratorsince ? 𝑛 will be large enough to exceed 𝑄 ´ ( 𝜖 ) log ( 𝑒 ) . Thenfrom (2), we have B 𝑟 B 𝜌 = 𝑧 ( + 𝜌𝑧 ) log 2 ´ 𝜙𝑧 ( + 𝜌𝑧 ) b ´ ( + 𝜌𝑧 ) , (40) B 𝑟 B 𝜌 = 𝜙𝑧 ( + 𝜌𝑧 ) b ´ ( + 𝜌𝑧 ) + 𝜙𝑧 ( + 𝜌𝑧 ) (cid:16) ´ ( + 𝜌𝑧 ) (cid:17) ´ 𝑧 ( + 𝜌𝑧 ) log 2 , (41)which is dominated by the negative term, since the other termsare multiplied by 𝜙 (cid:47) and raised to a high power in thedenominator, and thus vanish faster. This firmly holds for non-extremely low SNR (i.e, ě ´ dB) regions. Following asimilar procedure as in [16] based on [41], we can concludethat the EEE in the finite blocklength regime is also a quasi-concave function of power and strictly concave in its uppercontour. A PPENDIX
DPROOF OF L
EMMA 𝜌 = , the achievable rate 𝑟 = and the numerator of(8) becomes 0. Applying L’Hopital’s rule for the denominator,we have lim 𝜌 Ñ 𝜌 E [ 𝑟 ] = lim 𝜌 Ñ E (cid:104) 𝑧 (cid:16) ( + 𝜌𝑧 ) ln 2 ´ 𝑄 ´ ( 𝜖 ) log ( 𝑒 ) ? 𝑛 ( + 𝜌𝑧 ) 𝛾 (cid:17)(cid:105) = . (42)Thus the denominator of (8) equals to 𝑃 𝑐 yielding 0 for theEEE.For the second condition, the numerator of (8) is upperbounded by ´ ln 𝜖𝑛𝜃 , while L’Hopital’s rule for the denominator,we obtain lim 𝜌 Ñ8 E (cid:104) 𝑧 (cid:16) ( + 𝜌𝑧 ) ln 2 ´ 𝑄 ´ ( 𝜖 ) log ( 𝑒 ) ? 𝑛 ( + 𝜌𝑧 ) 𝛾 (cid:17)(cid:105) = . (43)Thus, the denominator of (8) tends to infinity which nullsthe EEE. Hence, (19) holds as well under finite blocklengthregime, which concludes the proof. A PPENDIX
EPROOF OF T
HEOREM 𝜃 ,so the problem is to define the limits of the EC. First, wedefine the limit of the EC for one transmission scenario andextremely strict delay constraint, where 𝜃 Ñ 8 as lim 𝜃 Ñ8 𝐶 𝑒 ( 𝜖 ) = lim 𝜃 Ñ8 ´ 𝑛𝜃 ln (cid:16) E 𝑍 (cid:104) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:105) (cid:17) = , (44)which means that the EC vanishes as the delay constraintbecomes infinitely strict and consequently, the EEE vanishestoo. A similar procedure shows the same zero lower boundfor the retransmission scenario. Next, we define the limit ofthe EC for loose delay constraint, where 𝜃 Ñ as follows lim 𝜃 Ñ 𝐶 𝑒 ( 𝜖 ) = lim 𝜃 Ñ ´ 𝑓 ( 𝜃 ) 𝑔 ( 𝜃 ) = lim 𝜃 Ñ ´ B 𝑓 ( 𝜃 ) B 𝜃 B 𝑔 ( 𝜃 ) B 𝜃 = lim 𝜃 Ñ ´ B 𝑓 ( 𝜃 ) B 𝜃 𝑛 , (45)which follows from L’Hopital rule, where 𝑓 ( 𝜃 ) = ln (cid:0) E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) (cid:1) and 𝑔 ( 𝜃 ) = 𝑛𝜃 . Differentiating 𝑓 ( 𝜃 ) with respect to 𝜃 , we get B 𝑓 ( 𝜃 ) B 𝜃 = BB 𝜃 (cid:8) E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) (cid:9) E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) = BB 𝜃 ∫ (cid:0) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 (cid:1) 𝑒 ´ 𝑧 𝑑𝑧 E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) . (46)Applying Leibniz’s rule, we obtain B 𝑓 ( 𝜃 ) B 𝜃 = ∫ BB 𝜃 (cid:0) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 (cid:1) 𝑒 ´ 𝑧 𝑑𝑧 E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) = ´ 𝑛 ( ´ 𝜖 ) E 𝑍 (cid:2) 𝑟𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) E 𝑍 (cid:2) 𝜖 + ( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) (cid:3) . (47)Plugging back into (45), we reach lim 𝜃 Ñ 𝐶 𝑒 ( 𝜖 ) = lim 𝜃 Ñ ´ ´ 𝑛 ( ´ 𝜖 ) E 𝑍 [ 𝑟𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) ] E 𝑍 [ 𝜖 +( ´ 𝜖 ) 𝑒 ´ 𝑛𝜃𝑟 ( 𝜖 ) ] 𝑛 = ( ´ 𝜖 ) E 𝑍 [ 𝑟 ( 𝜖 )] = ( ´ 𝜖 ) 𝑟 , (48)which represents the upper bound throughput of the finiteblocklength transmission when no delay constraint is imposed.Following the same procedure, we can deduce that the EC ofthe EBP ARQ scenario is given by the numerator of Theorem4. A PPENDIX
FPROOF OF T
HEOREM 𝑝 𝑛𝑏 in 𝜖 goes asfollows. At the first glance, it is possible to prove that 𝜅 is aconcave function in 𝜖 . First, let 𝑟 ( 𝜖 ) « log ( + 𝜌 | ℎ | ) ´ 𝜇𝑄 ´ ( 𝜖 ) , (49) where 𝜇 = log ( 𝑒 ) ? 𝑛 c ´ ( + 𝜌 | ℎ | ) . Applying the derivatives ofthe inverse Q-function from [6], we obtain the derivatives ofthe rate expectations with respect to 𝜖 as B 𝑟 B 𝜖 = E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) , (50) B 𝑟 B 𝜖 = ´ E [ 𝜇 ] 𝜋𝑄 ´ ( 𝜖 ) 𝑒 ( 𝑄 ´ ( 𝜖 ) ) , (51) B 𝑟 B 𝜖 = ´ 𝜖𝜖 E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) , (52) B 𝑟 B 𝜖 = 𝜖𝜖 E [ 𝜇 ] 𝜋𝑄 ´ ( 𝜖 ) 𝑒 ( 𝑄 ´ ( 𝜖 ) ) + E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) 𝜖𝜖 = ´ 𝜖𝜖 B 𝑟 B 𝜖 + E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) 𝜖𝜖 . (53)Then, we obtain the second derivative of 𝜅 w.r.t 𝜖 as follows B 𝜅 B 𝜖 = ´ 𝑟 + B 𝑟 B 𝜖 ( ´ 𝜖 ) + (cid:18) 𝜖 B 𝑟 B 𝜖 + 𝑟 (cid:19) (54) B 𝜅 B 𝜖 = ´ B 𝑟 B 𝜖 + B 𝑟 B 𝜖 ( ´ 𝜖 ) + (cid:32) B 𝑟 B 𝜖 + 𝜖 B 𝑟 B 𝜖 (cid:33) = ´ B 𝑟 B 𝜖 + B 𝑟 B 𝜖 ( ´ 𝜖 ) + (cid:40) (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) ´ 𝜖𝜖 E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) ´ 𝜖𝜖 B 𝑟 B 𝜖 + (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) 𝜖 E [ 𝜇 ] ? 𝜋𝑒 ( 𝑄 ´ ( 𝜖 ) ) 𝜖𝜖 (cid:41) = ´ B 𝑟 B 𝜖 + B 𝑟 B 𝜖 ( ´ 𝜖 ´ 𝜖 𝜖 ) . (55)Note that practically, the term 𝜖 𝜖 is close to zero since 𝜖 ąą 𝜖 , and hence, ( ´ 𝜖 ) ą 𝜖 𝜖 . Since the derivativesin (50) and (51) are strictly negative, we can deduce that thesecond derivative of 𝜅 in (55) renders a strictly negative value.Therefore, 𝜅 is strictly concave in 𝜖 . Back to equation (24),the denominator of the non-empty buffer probability is a lineardecreasing function of 𝜅 , which indicates that the denominatoris a convex function in 𝜖 . The numerator can be proven tobe a convex decreasing function of 𝑇 which means that thenumerator is a convex function in 𝜖 . First we obtain thederivatives of the numerator of 𝑝 𝑛𝑏 with respect to 𝑇 as follows B 𝑛𝑢𝑚 (cid:8) 𝑝 𝑛𝑏 (cid:9) B 𝜅 = 𝜅 ´ 𝜆 a 𝜅 + 𝜆 ( 𝑟 𝑜 ´ 𝜅 ) ´ , (56) B 𝑛𝑢𝑚 (cid:8) 𝑝 𝑛𝑏 (cid:9) B 𝜅 = 𝜆 ( 𝑟 𝑜 ´ 𝜆 ) (cid:0) 𝜅 + 𝜆 ( 𝑟 𝑜 ´ 𝜅 ) (cid:1) , (57)which gives a strictly positive value and hence, the numeratorof of 𝑝 𝑛𝑏 is convex in 𝜅 . Since the numerator of 𝑝 𝑛𝑏 is a non-increasing convex function in 𝜅 and 𝜅 is a concave function of 𝜖 , it follows from [36] that it is a convex function in 𝜖 , whilethe numerator is also a negative function. Hence and accordingto Proposition 2.9 in [28], minimizing 𝑝 𝑛𝑏 is a psuedo-convexfractional program. R EFERENCES[1] P. Popovski, C. Stefanovic, J. J. Nielsen, E. de Carvalho, M. Angjelichi-noski, K. F. Trillingsgaard, and A. Bana, “Wireless Access in Ultra-Reliable Low-Latency Communication (URLLC),”
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