Twin-Timescale Radio Resource Management for Ultra-Reliable and Low-Latency Vehicular Networks
11 Twin-Timescale Radio Resource Management forUltra-Reliable and Low-Latency Vehicular Networks
Haojun Yang,
Student Member, IEEE , Kan Zheng,
Senior Member, IEEE ,Long Zhao,
Member, IEEE , and Lajos Hanzo,
Fellow, IEEE
Abstract
To efficiently support safety-related vehicular applications, the ultra-reliable and low-latency com-munication (URLLC) concept has become an indispensable component of vehicular networks (VNETs).Due to the high mobility of VNETs, exchanging near-instantaneous channel state information (CSI) andmaking reliable resource allocation decisions based on such short-term CSI evaluations are not practical.In this paper, we consider the downlink of a vehicle-to-infrastructure (V2I) system conceived for URLLCbased on idealized perfect and realistic imperfect CSI. By exploiting the benefits of the massive MIMOconcept, a two-stage radio resource allocation problem is formulated based on a novel twin-timescaleperspective for avoiding the frequent exchange of near-instantaneous CSI. Specifically, based on theprevalent road-traffic density, Stage 1 is constructed for minimizing the worst-case transmission latencyon a long-term timescale. In Stage 2, the base station allocates the total power at a short-term timescaleaccording to the large-scale fading CSI encountered for minimizing the maximum transmission latencyacross all vehicular users. Then, a primary algorithm and a secondary algorithm are conceived forour V2I URLLC system to find the optimal solution of the twin-timescale resource allocation problem,with special emphasis on the complexity imposed. Finally, our simulation results show that the proposedresource allocation scheme significantly reduces the maximum transmission latency, and it is not sensitiveto the fluctuation of road-traffic density.
Index Terms
Vehicular networks (VNET), ultra-reliable and low-latency communications (URLLC), resourcemanagement, finite blocklength theory, massive MIMO.
Haojun Yang, Kan Zheng and Long Zhao are with the Intelligent Computing and Communications (IC ) Lab, WirelessSignal Processing and Networks (WSPN) Lab, Key Laboratory of Universal Wireless Communications, Ministry of Education,Beijing University of Posts and Telecommunications (BUPT), Beijing, 100876, China (E-mail: [email protected];[email protected]; [email protected] ).Lajos Hanzo is with the School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ,U.K. (E-mail: [email protected] ). a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
Given the rapid development of the Internet of things (IoT), the design of radio access networksis undergoing a paradigm shift from promoting spectral-efficiency and energy-efficiency towardconnecting everything [1]. As a representative application scenario of IoT, vehicular networks(VNETs) support various vehicular applications for reducing the probability of traffic accidentsand for improving both the driving and infotainment experience [2], [3]. Traditional safety-relatedvehicular applications, such as platooning and road safety generally require a transmission latencywithin a few milliseconds and a reliability in terms of error probability down to − (or − ).Furthermore, in order to achieve the ultimate goal of autonomous driving on the road, futureautonomous vehicles need sophisticated sensors to support safety-related applications. Ultra-reliable and low-latency communications (URLLCs) exchanging near-real-time information arealso required to offer assistance to control systems [4], [5]. To this end, it is paramount toconceive URLLC techniques for future VNETs. In the conception of URLLC techniques, radioresource management plays a key role in improving the performance of VNETs, hence ground-breaking research is required.The latency encountered may be classified according to the hierarchical architecture of net-works. The physical (PHY) layer aims for reducing the transmission latency, while the mediaaccess control (MAC) layer aspires to reduce the queueing latency. The theoretical analysis ofPHY layer radio resource management conceived for device-to-device-based vehicle-to-vehiclecommunications can be found in [6], where both the transmission latency and reliability areconsidered simultaneously. However, the study of vehicular mobility was not considered in [6].Moreover, neither the ergodic capacity nor the outage capacity are suitable for characterizingthe tradeoff among the achievable rate, transmission latency and reliability in URLLCs [7]–[9]. Hence, the recent advances in studying an attractive tradeoff among the rate, latency andreliability of multiple-antenna aided channels are illustrated in [10] utilizing finite blocklengththeory. However, there is still a paucity of literature on addressing the URLLC problems of thePHY layer in VNETs under the consideration of vehicular mobility models.With respect to the MAC layer, sophisticated techniques have been used for analyzing andoptimizing the queueing latency, such as Markov decision processes [11], [12], stochastic networkcalculus [13], [14] and queueing theory [15]. Based on the Markov decision process framework, aminimum queueing delay-based radio resource allocation scheme is proposed in [16] for vehicle- to-vehicle communications, where the successful packet reception ratio is considered as thereliability metric. However, it is not practical to inform the base station of the global channelstate information (CSI) including the links which are not connected to the base station. Moreover,the analysis in [16] has not been provided for URLLCs, since it was not based on radical advancesin finite blocklength theory. Relying on finite blocklength theory and stochastic network calculus,a novel probabilistic delay bound is conceived in [17] for low-latency machine-to-machineapplications. Furthermore, a URLLC cross-layer optimization framework is proposed in [18]by jointly considering the transmission latency and queueing latency. However, the advancesmentioned above are not applicable to VNETs due to the lack of vehicular mobility model.In addition to latency, reliability is another important performance metric for VNETs. In theMAC layer, reliability is generally characterized either by the stability of the queue [19] or bythe probability of the queueing latency exceeding the tolerance threshold [20], [15]. As for thePHY layer, typically the latency vs reliability (packet error rate) tradeoff is quantified [21], whereone of the pivotal parameters of ensuring a high reliability is deemed to be the multi-antennadiversity gain. Hence, large-scale MIMO schemes constitute a promising solution of reducinglatency and enhancing reliability in VNETs. On a similar note, massive MIMO schemes canalso be adopted for formulating large-scale fading-based optimization problems by exploitingthe channel hardening phenomenon [22]–[24].Against this background, in this paper we conceive radio resource management for URLLCVNETs based on a realistic vehicular mobility model. To conceive URLLC VNETs, the followingpair of crucial characteristics has to be considered:1) The near-instantaneous CSI rapidly becomes outdated and its more frequent update imposesa high overhead, hence reducing the communication efficiency; and2) Compared to the time-scale of CSI fluctuation on the millisecond level, that of the road-traffic fluctuation is higher (second level). Requiring resource allocation updates on theorder of milliseconds would impose an excessive implementation complexity.In tackling the above-mentioned challenges in the downlink of a massive MIMO vehicle-to-infrastructure (V2I) system conceived for URLLC, our contributions are: • We optimize the transmission latency based on a twin-timescale perspective for reducingthe signaling overhead, where a macroscopic road-traffic model is adopted for characteringthe relationship between the vehicular velocity and road-traffic density. • We first derive the transmission latency based on finite blocklength theory for the matchedfilter (MF) precoder and zero-forcing (ZF) precoder both under perfect and imperfect CSI. • Then, a two-stage radio resource allocation problem is formulated for our V2I URLLCsystem. In particular, based on the long-term timescale of road-traffic density, Stage 1 aimsfor optimizing the worst-case transmission latency by appropriately setting the system’sbandwidth. For Stage 2, the base station (BS) allocates the total power based on the large-scale fading CSI for minimizing the maximum transmission latency guaranteeing fairnessamong all vehicular users (VUEs). Finally, the proposed primary and secondary radioresource allocation algorithms are invoked for optimally solving the above two problems ata reasonable complexity. • Our simulation results show that the proposed resource allocation scheme is capable ofsignificantly reducing the maximum transmission latency, while flexibly accommodating theroad-traffic fluctuation. Hence, the radio resources can be allocated at a larger time interval,which means the proposed algorithms can get the effect of semi-persistent scheduling.Additionally, the ZF precoder is shown to constitute a compelling choice for our V2I URLLCsystem.The remainder of this paper is organized as follows. First of all, finite blocklength theory isintroduced in Section II. Section III describes the V2I URLLC system model, while Section IVformulates the twin-timescale resource allocation problem. Then, the proposed resource allocationalgorithms are discussed in Section V. Finally, Section VI illustrates the simulation results, whileour conclusions are offered in Section VII.
Notations:
Uppercase boldface letters and lowercase boldface letters denote matrices andvectors, respectively, while diag { a } is a diagonal square matrix whose main diagonal is formedby vector a . Furthermore, ( · ) T , ( · ) * , ( · ) H and ( · ) − represent the transpose, conjugate, conjugatetranspose and pseudo-inverse of a matrix/vector, respectively, while (cid:107)·(cid:107) denotes the Euclidiannorm, and [ · ] kk denotes the k -th diagonal element of a square matrix. Finally, E ( · ) representsthe mathematical expectation, while CN ( µ, σ ) is the complex Gaussian distribution with mean µ and real/imaginary component variance σ / . II. P
RELIMINARIES : F
INITE B LOCKLENGTH T HEORY
The Shannon formula quantifies the error-free capacity at which information can be transmittedover a band-limited channel in the presence of noise and interferences, C ( γ ) = E [ B log (1 + γ )] , (1)where B is the bandwidth and γ is the signal-to-interference-plus-noise ratio (SINR). Thiscapacity can only be approached at the cost of excessive coding latency and complexity, i.e., C ( γ ) = lim (cid:15) → C (cid:15) ( γ, (cid:15) ) = lim (cid:15) → lim n →∞ R ( γ, n, (cid:15) ) , (2)where C (cid:15) ( γ, (cid:15) ) is the so-called outage capacity at the error probability (cid:15) . For the operationalwireless systems, both the ergodic and outage capacity are reasonable performance metrics,because the packet size is typically large.However, the assumption of large packet size does not meet the requirements of URLLCs.Thus, a more refined analysis of R ( γ, n, (cid:15) ) is needed. Fortunately, during the last few years,significant progress has been made for satisfactorily addressing the problem of approximating R ( γ, n, (cid:15) ) [7], [8], [25]: R ( γ, L, (cid:15) ) ≈ E (cid:40) B (cid:34) log (1 + γ ) − (cid:114) VLB Q − ( (cid:15) ) (cid:35)(cid:41) , (3)where Q − ( · ) denotes the inverse of the Gaussian Q -function and V is the so-called channeldispersion. For a complex channel, the channel dispersion is given by [7], [8]: V = (cid:18) − γ ) (cid:19) (log e) . (4)Furthermore, L is the transmission latency , while LB , which is also referred to as the block-length of channel coding, represents the number of transmitted symbols. When LB is highenough, the approximation (3) approaches the ergodic capacity. Compared to the ergodic capacity,the approximation (3) also implies that the rate reduction is proportional to / √ LB , when aimingfor meeting a specific error probability at a given packet size. Finite blocklength theory constitutesa powerful technique of dealing with the URLLC-related optimization problems. In addition tothe contributions mentioned above, some new advances based on (3) analyze and optimize theURLLC performance of 5G and IoT networks [26]–[28]. In general, the one-way latency of the PHY layer entails both the propagation latency of electromagnetic wave and the codinglatency. For example, for a
300 m cell radius, the propagation delay of µ s is typically negligible. Hence, the coding latencydominates the one-way delay of the PHY layer’s transmission latency. Base Station
Road-Traffic
Monitoring Starting Point Road-Traffic
Density d B d R d k r Road-Traffic
Prediction
Fig. 1. Illustration of system model.
III. V2I URLLC S
YSTEM M ODEL
As shown in Fig. 1, for the downlink of the V2I URLLC system, we consider a singleroadside BS and a highway road segment of length d R . The BS is d B meters away from theroad. Furthermore, the BS employs M antennas and simultaneously sends information to K single-antenna aided VUEs (massive MIMO with M (cid:29) K ). The system operates in the time-division duplex (TDD) mode. A. Road-Traffic Model
According to the classic theory, the road-traffic model is divided into macroscopic and mi-croscopic scales. Specially, the macroscopic model describes the average behavior of a certainnumber of vehicles at specific locations and instances, treating the road-traffic similarly to fluiddynamics. By contrast, the microscopic model describes the specific behavior of each individualentity (such as vehicle or pedestrian), hence it is more sophisticated than the macroscopicmodel. The open-source simulation of urban mobility (SUMO) mainly focuses on generating themicroscopic model, namely the trajectory of vehicles. However, our proposed algorithms dependboth on the road-traffic density and on the location information of all VUEs at a certain moment.Based on the macroscopic model, the above two types of information can be predicted andperiodically reported by the traffic center [29], [30]. To this end, we adopt the one-dimensionalmacroscopic model in this paper, which is also known as the Lighthill-Whitham-Richards (LWR) model [31]. Given the road-traffic density ρ and the road-traffic velocity v , the road-traffic flowrate (flux) f is generally given by f = ρv. (5)Numerous models have been proposed for characterizing how the vehicular velocity depends onthe road-traffic density. In this paper, the Underwood model is employed, and its speed-densityfunction can be written as [31]: v ( ρ ) = v F exp (cid:18) − ρρ m (cid:19) , (6)where v F is the free-flowing velocity and ρ m is the maximum density. Moreover, the road-trafficdensity can be used for modeling the number of VUEs, according to K = ρd R . B. Channel Model
All channels experience independent flat block-fading, i.e., they remain constant during aa coherence block, but change independently from one block to another. Let G = D / H =[ g , · · · , g K ] T ∈ C K × M denote the system’s CSI, where the diagonal matrix D = diag { β , · · · , β K } ∈ R K × K and H = [ h , · · · , h K ] T ∈ C K × M represent the large-scale fading and fast fading.The channel vector spanning from the BS to the k -th VUE is given by g k = β / k h k ∈ C M × . Furthermore, imperfect channel estimation is considered [32]–[34], i.e., g k = β / k h k = β / k √ χ k ˆ h k + β / k √ − χ k e k , where ˆ h k , e k and χ k ∈ R are the estimate, error and estimationaccuracy of h k , respectively. If χ k = 1 , g k corresponds to perfect channel estimation. Finally, allrandom variables are independent and identically distributed (i.i.d.) complex Gaussian randomvariables with mean 0 and variance 1, namely ˆ h km , e km i.i.d. ∼ CN (0 , . C. Transceiver Model1) Transmitter:
It is widely exploited that low-complexity linear transmit precoders are capa-ble of attaining asymptotically optimal performance in massive MIMO. Therefore we considerthe MF precoder and ZF precoder in this paper. Let V = [ v , · · · , v K ] ∈ C M × K and v k ∈ C M × denote the precoder matrix and vector, where V = ˆ G H , for MF , ˆ G H (cid:16) ˆ G ˆ G H (cid:17) − , for ZF , (7) and ˆ G = [ˆ g , · · · , ˆ g K ] T with ˆ g k = β / k ˆ h k . The normalized form of V is given by W =[ w , · · · , w K ] ∈ C M × K with w k = v k (cid:107) v k (cid:107) , ∀ k. (8)As a result, given the symbol vector s = [ s , · · · , s K ] T ∈ C K × with E [ ss H ] = I K , the transmittedsignal vector of all VUEs can be written as x = WP / s , (9)where P = diag (cid:8) p M (cid:9) = M diag { p , · · · , p K } ∈ R K × K is the power allocation matrix and thetotal power at the BS is P B .
2) Receiver:
The received signal vector is given by y = Gx + n = GWP / s + n , (10)where n = [ n , · · · , n K ] T ∈ C K × is the i.i.d. additive white Gaussian noise with n k i.i.d. ∼ CN (0 , σ ) .The received signal of the k -th VUE can be written as y k = (cid:114) p k M √ χ k ˆ g T k w k s k + K (cid:88) i =1 ,i (cid:54) = k (cid:114) p i M √ χ k ˆ g T k w i s i + K (cid:88) j =1 (cid:114) p j M (cid:112) − χ k β / k e T k w j s j + n k , ∀ k. (11)Hence, the SINR of the k -th VUE is given by γ k = p k M χ k (cid:12)(cid:12) ˆ g T k w k (cid:12)(cid:12) K (cid:80) i =1 ,i (cid:54) = k p i M χ k (cid:12)(cid:12) ˆ g T k w i (cid:12)(cid:12) + P B M β k (1 − χ k ) + σ = p k M χ k (cid:12)(cid:12)(cid:12)(cid:12) ˆ g T k ˆ g ∗ k (cid:107) ˆ g ∗ k (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) K (cid:80) i =1 ,i (cid:54) = k p i M χ k (cid:12)(cid:12)(cid:12)(cid:12) ˆ g T k ˆ g ∗ i (cid:107) ˆ g ∗ i (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) + P B M β k (1 − χ k ) + σ , for MF , (cid:20)(cid:16) ˆ G ˆ G H (cid:17) − (cid:21) kk p k MP B M β k (1 − χ k ) + σ , for ZF . (12) Road-Traffic Density and Location Information of All VUEsStage 1: Long-Term Timescale
AllocationStage 2: Short-Term Timescale
Allocation
Else Loop Stage 2If
Then Go to Stage 1 t current = t report Predicted and
Periodically
Reported by Traffic Control Center
Fig. 2. Illustration of twin-timescale radio resource management.
Recall from ˆ g k = β / k ˆ h k that the instantaneous γ − k can be rewritten as γ − k = K (cid:80) i =1 ,i (cid:54) = k p i p k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ h T k (cid:13)(cid:13)(cid:13) ˆ h T k (cid:13)(cid:13)(cid:13) ˆ h ∗ i (cid:13)(cid:13)(cid:13) ˆ h ∗ i (cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + P B β k (1 − χ k ) + M σ p k χ k β k (cid:13)(cid:13)(cid:13) ˆ h k (cid:13)(cid:13)(cid:13) , for MF ,P B β k (1 − χ k ) + M σ p k χ k β k (cid:20)(cid:16) ˆ H ˆ H H (cid:17) − (cid:21) kk , for ZF . (13)Based on [35], we have the following distributions, i.e., Ω MFB (cid:44) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ h T k (cid:13)(cid:13)(cid:13) ˆ h T k (cid:13)(cid:13)(cid:13) ˆ h ∗ i (cid:13)(cid:13)(cid:13) ˆ h ∗ i (cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ Beta (1 , M − , ∀ i (cid:54) = k, (14) Ω MFG (cid:44) (cid:13)(cid:13)(cid:13) ˆ h k (cid:13)(cid:13)(cid:13) ∼ Gamma − ( M, , ∀ k, (15) Ω ZFG (cid:44) (cid:20)(cid:16) ˆ H ˆ H H (cid:17) − (cid:21) kk ∼ Gamma − ( M − K + 1 , , ∀ k. (16)Then γ − k can be finally expressed as γ − k = P B − p k p k Ω MFB + P B β k (1 − χ k ) + M σ p k χ k β k Ω MFG , for MF ,P B β k (1 − χ k ) + M σ p k χ k β k Ω ZFG , for ZF . (17)IV. P ROBLEM F ORMULATION OF T WIN -T IMESCALE R ADIO R ESOURCE M ANAGEMENT
In this treatise, we optimize the transmission latency of V2I communications based on the road-traffic density and location information, avoiding the frequent exchange of instantaneous globalCSI. As the speed of mobility gradually increases, the fluctuation of the small-scale channelfading envelope generally becomes more rapid. However, the impact of small-scale fading can be mitigated by exploiting the channel hardening and asymptotic orthogonality of massiveMIMO [22], [35], [23]. As a result, the mobility of vehicles (the change of location information)only impacts the large-scale channel fading. Furthermore, compared to the time-scale of CSIfluctuation at the millisecond scale, that of the road-traffic (including the density and locationinformation) fluctuation is on the scale of say 90 km / h (2.5 cm / ms ). Therefore, consideringthe above facts, we formulate a twin-timescale large-scale fading-based radio resource allocationproblem in order to reduce the signaling overhead imposed. As show in Fig. 2, according to theroad-traffic density, the first stage is constructed for optimizing the worst-case latency by settingthe system’s bandwidth from a long-term timescale. As for the second stage of the short-termtimescale, the BS will allocate the total power based on the large-scale fading CSI (equivalentto location information) to minimize the maximum transmission latency for ensuring fairnessamongst all VUEs. A. Large-Scale Fading-Based Transmission Latency
In high-speed vehicular environments, small-scale channel fading tends to fluctuate rapidly,hence the instantaneous V2I CSI is often outdated. Furthermore, exchanging the instantaneousV2I CSI with the BS is not practical. As a result, large-scale fading-based allocation problemsare generally considered by exploiting the channel hardening and asymptotic orthogonality ofmassive MIMO [36], [22], [35], [23]. The theorem for quantifying the maximum achievable rateis first formulated as follows.
Theorem When the MF or ZF precoder is employed in the downlink of our V2I URLLCsystem, the maximum achievable ergodic rate of the k -th VUE can be approximated ( M (cid:29) )by [7], [8], [25]: R k (Γ k , L k , (cid:15) k ) = B log (1 + Γ k ) − log e (cid:114) BL k Q − ( (cid:15) k ) (cid:113) − (1 + Γ k ) − , ∀ k, (18)where Γ k = M p k P B − p k + ϕ k , for MF ,p k ϕ k , for ZF , (19) and ϕ k = P B β k (1 − χ k ) + M σ χ k β k MM − , for MF ,χ k β k ( M − K ) P B β k (1 − χ k ) + M σ , for ZF . (20) Proof:
See Appendix A.According to Theorem 1, we can obtain the following corollary for illustrating the transmissionlatency.
Corollary Given the desired rate R k (Γ k , L k , (cid:15) k ) and the reliability (cid:15) k , the transmissionlatency of the k -th VUE can be expressed as L k = √ BQ − ( (cid:15) k ) log e (cid:113) − (1 + Γ k ) − B log (1 + Γ k ) − R k (Γ k , L k , (cid:15) k ) , ∀ k. (21)With respect to the reliability, Q − ( (cid:15) ) with (cid:15) ∈ [0 , obeys the following properties, Q − ( (cid:15) ) < , (cid:15) > . , = 0 , (cid:15) = 0 . ,> , (cid:15) < . , (22)and lim (cid:15) → Q − ( (cid:15) ) = + ∞ . Thus, a higher reliability (also known as − (cid:15) ) leads to a higherlatency, and vice versa. B. Stage 1: Long-Term Timescale Allocation
In somewhat simplistic, but plausible terms one could argue that doubling the system band-width allows doubling the bitrate, halving the framelength and hence halving the transmissionlatency. This plausible concept is also adopted by the 3GPP [37] for reducing the transmissionslot-duration. Another influential factor is the coding delay, because the ultimate Shannon ca-pacity limit is only achievable, if the codeword-length tends to infinity. Broadly speaking, thesystem bandwidth can be appropriately adjusted for reducing the transmission latency, subjectto the ubiquitous bandwidth constraints. Hence the system bandwidth determined in Stage 1 canalso be viewed as part of the system design. Based on the long-term timescale of road-trafficdensity, the objective of Stage 1 is to optimize the worst-case transmission latency. Notice thatthe road-traffic density is valid for a limited road segment within about 50 m to 200 m [29],[30]. Furthermore, our system model is based on a highway segment rather than on an urban scenario. The BS is deployed very close to the road segment, and thus generally there are nosignal obstructions. To this end, the impact of shadow fading is ignored in this paper [38], [39].Without loss of generality, the worst situation is encountered in the scenario of assuming a largestdistance from the BS in case of equal power allocation. The path loss at the largest distancefrom the BS is given by β W = θ (cid:34) d B + (cid:18) d R (cid:19) (cid:35) − α , (23)where θ is a constant related both to the antenna gain and to the carrier frequency, while α isthe path loss exponent. Then, with p k = P B K = P Bρd R , the worst-case SINR can be expressed as Γ W = M P P ( ρd R −
1) + P β W (1 − χ th ) + M N χ th β W ρd R MM − , for MF ,P ρd R χ th β W ( M − ρd R ) P β W (1 − χ th ) + M N , for ZF , (24)where P and N are the power spectral density of the transmitted signal and noise, respectively,while χ th is the minimum requested estimation accuracy, namely χ th = min k { χ k } . Therefore,the worst-case transmission latency L W is finally given by L W = √ BQ − ( (cid:15) W ) log e (cid:113) − (1 + Γ W ) − B log (1 + Γ W ) − R W , (25)where (cid:15) W = min k { (cid:15) k } and R W = max k { R k (Γ k , L k , (cid:15) k ) } .On the other hand, with (6), the maximum Doppler frequency is given by f MD = v ( ρ ) λ = f C v F exp ( − ρ/ρ m ) c , (26)where f C is the carrier frequency, and c is the speed of light. Hence, the typical coherence timecan be calculated as [40]: T C = (cid:115) πf MD = (cid:115) c πf C v F [exp ( − ρ/ρ m )] . (27)The design of the existing wireless systems follows a rule that CSI remains constant duringthe coherence time [41], [37]. Thus, motivated by this rule, we assume that the worst-casetransmission latency is lower than the coherence time, which is similar with the design of theLTE-based systems. On the basis of the above derivations, the following optimization problemis formulated. Problem
Let δ denote a very small thresholdconstant. Given the road-traffic density ρ , the long-term timescale bandwidth allocation problemof Stage 1 is formulated as P1: L W ( B, ρ ) (cid:54) δT C ( ρ ) . (28)Typically, δ is equal to for the LTE-based systems [41]. Hence, in our simulations, we let δ be equal to or . C. Stage 2: Short-Term Timescale Allocation
By solving Problem 1, the total power at the BS can be obtained, namely P B = P B ∗ ( ρ ) . InStage 2, the BS allocates the total power based on the short-term timescale large-scale fading CSIof all VUEs for minimizing the maximum transmission latency. Then the following optimizationproblem is formulated. Problem
Given the desired rate R k (Γ k , L k , (cid:15) k ) ,reliability (cid:15) k , total power P B and the large-scale fading CSI of all VUEs, the optimizationobjective of Stage 2 is to minimize the maximum transmission latency of all VUEs ensuringfairness amongst all VUEs, i.e., P2: min p max k { L k } (29a) s . t . K (cid:88) k =1 p k = P B , (29b) p k (cid:62) , ∀ k. (29c)V. T WIN -T IMESCALE R ESOURCE A LLOCATION A LGORITHMS FOR
V2I URLLC S
CYSTEM
In this section, the solutions to the above pair of problems are first studied, and then efficienttwin-timescale allocation algorithms are proposed for our V2I URLLC system.
A. Solution to Problem 1
According to Problem 1, we have √ BQ − ( (cid:15) W ) log e (cid:113) − (1 + Γ W ) − B log (1 + Γ W ) − R W ( γ, L, (cid:15) ) (cid:54) (cid:112) δT C , (30) which shows that B (cid:112) δT C log (1 + Γ W ) − (cid:112) δT C R W ( γ, L, (cid:15) ) − √ BQ − ( (cid:15) W ) log e (cid:113) − (1 + Γ W ) − (cid:62) . (31)Upon defining ∆ = (cid:2) Q − ( (cid:15) W ) (cid:3) log e (cid:2) − (1 + Γ W ) − (cid:3) + 4 δT C log (1 + Γ W ) R W ( γ, L, (cid:15) ) > , (32)the optimal system bandwidth B ∗ can be finally expressed as B ∗ = Q − ( (cid:15) W ) log e (cid:113) − (1 + Γ W ) − + √ ∆2 √ δT C log (1 + Γ W ) , (33)where Γ W can be found in (24). Given the system bandwidth, the total power at the BS is P B = P B ∗ . B. Solution to Problem 21) Problem Transformation:
Problem 2 can be transformed into the following form:
P2: max p min k (cid:110) − (cid:112) L k (cid:111) (34a) s . t . (29b) − (29c) . (34b)
2) Property of Objective Function:
Recall from (21) that −√ L k can be expressed as −√ B ∗ Q − ( (cid:15) k ) log e (cid:113) − (1 + Γ k ) − B ∗ log (1 + Γ k ) − R k (Γ k , L k , (cid:15) k ) (cid:44) f k ( p ) g k ( p ) , ∀ k. (35) Theorem When the MF or ZF precoder is employed in the downlink of our V2I URLLCsystem, −√ L k is a quasi-concave function of p k . Proof:
See Appendix B.Given the definition of quasi-concavity, min k (cid:8) −√ L k (cid:9) is also a quasi-concave function. Bycontrast, even if −√ L k , ∀ k is pseudo-concave, min k (cid:8) −√ L k (cid:9) still cannot be pseudo-concave,since the function min( · ) is not differentiable.
3) Optimal Solution:
Since the objective function of (34) is QC, the Karush-Kuhn-Tucker(KKT) condition cannot be adopted. However, we will show that Problem 2 can still be solvedoptimally by an efficient algorithm.Based on (35), the following auxiliary function F ( η ) is constructed, F ( η ) = max p min k { f k ( p ) − ηg k ( p ) } . (36)Intuitively, F ( η ) is strictly monotonically decreasing and F ( η ) = 0 has a unique root. Lemma For any ˜ p ∈ P , we have F ( η ˜ p ) (cid:62) with η ˜ p = min k { f k (˜ p ) /g k (˜ p ) } , and theequality holds when ˜ p = arg max p min k { f k ( p ) − η ˜ p g k ( p ) } . Proof:
See Appendix C.Based on Lemma 1, the following theorem describing the optimal allocation is formulated.
Theorem For p ∗ ∈ P and η ∗ = min k { f k ( p ∗ ) /g k ( p ∗ ) } , p ∗ is the optimal solution of (34)if and only if p ∗ = arg max p min k { f k ( p ) − η ∗ g k ( p ) } . (37) Proof:
It relies on proving both sufficiency and necessity. Firstly, let p ∗ be the optimalpower allocation of (34), η ∗ = min k (cid:26) f k ( p ∗ ) g k ( p ∗ ) (cid:27) (cid:62) min k (cid:26) f k ( p ) g k ( p ) (cid:27) , ∀ p ∈ P . (38)Therefore, min k { f k ( p ) − η ∗ g k ( p ) } (cid:54) , ∀ p ∈ P , (39) min k { f k ( p ∗ ) − η ∗ g k ( p ∗ ) } = 0 , (40)which yields (37) and F ( η ∗ ) = 0 .Conversely, let p ∗ = arg max p min k { f k ( p ) − η ∗ g k ( p ) } , i.e., min k { f k ( p ) − η ∗ g k ( p ) } (cid:54) min k { f k ( p ∗ ) − η ∗ g k ( p ∗ ) } = F ( η ∗ ) (a) = 0 , ∀ p ∈ P , (41)where (a) follows from Lemma 1. Then, (41) shows that η ∗ (cid:62) min k (cid:26) f k ( p ) g k ( p ) (cid:27) , ∀ p ∈ P , (42) η ∗ = min k (cid:26) f k ( p ∗ ) g k ( p ∗ ) (cid:27) . (43) Thus p ∗ is the optimal power allocation. Corollary Based on the monotonicity of F ( η ) and Theorem 3, for any η , we have F ( η ) < , η > η ∗ , = 0 , η = η ∗ ,> , η < η ∗ , (44)where η ∗ is the optimal objective function value of (34).According to Theorem 3 and Corollary 2, solving Problem 2 is equivalent to finding the uniqueroot of F ( η ) = 0 . C. Twin-Timescale Resource Allocation Algorithms for V2I URLLC System
By taking into account the above solutions, the following primary algorithm and secondaryalgorithm are proposed for the optimal V2I URLLC resource allocation.
1) Primary Algorithm:
As shown in [42], there are two classic techniques of solving fractionaloptimization problems, namely Charnes-Cooper transform and Dinkelbach’s method. In simpleterms, Dinkelbach’s method can be interpreted as being reminiscent of Newton’s method appliedto the convex function F ( η ) [43]. Compared to Charnes-Cooper transform, Dinkelbach’s methodhas the advantage of requiring no extra constraints [42]. Therefore, we opt for Dinkelbach’smethod in this paper. Based on the Dinkelbach’s method [43], a twin-timescale iterative algorithmis put forward for jointly solving Problem 1 and 2, which is described in Algorithm 1. Thefollowing theorem illustrates the convergence and optimality of Algorithm 1. Theorem Algorithm 1 must converge to the optimal power allocation.
Proof:
According to Step 5 in Algorithm 1, we have F j = min k (cid:8) f k ( p j ) − η j g k ( p j ) (cid:9) (a) = min k (cid:8) ( η j +1 − η j ) g k ( p j ) (cid:9) (b) (cid:62) . (45)(a) follows from f k ( p j ) g k ( p j ) (cid:62) η j +1 , ∀ k ⇒ f k ( p j ) (cid:62) η j +1 g k ( p j ) , (46)where the equality holds when k = arg min k (cid:8) f k ( p j ) /g k ( p j ) (cid:9) . (b) follows from Lemma 1. Thisimplies that when the algorithm does not achieve convergence, we have η j +1 > η j . Therefore, Algorithm 1 . Road-traffic-aware twin-timescale resource allocation algorithm for V2I URLLCsystem
Initialization: • Periodically reported road-traffic density ρ . • The location information of all VUEs. • The rate requirement R k (Γ k , L k , (cid:15) k ) and reliability requirement (cid:15) k of each VUE. • Iterative index j = 0 , maximum iterative tolerance ζ P > and initial objective value η < . Stage 1: Long-term timescale resource allocation Calculate the optimal system bandwidth B ∗ based on (33). Calculate the total power at the BS with P B = P B ∗ . Stage 2: Short-term timescale resource allocationloop ( j ) Based on Algorithm 2, solve the power allocation p j of Sub-problem 1 with the given η j . Calculate F j = min k (cid:8) f k ( p j ) − η j g k ( p j ) (cid:9) . if F j (cid:54) ζ P then Update p ∗ = p j and η ∗ = min k (cid:26) f k ( p ∗ ) g k ( p ∗ ) (cid:27) .Set j = j + 1 and break . else continue . end if Update η j +1 = min k (cid:8) f k ( p j ) /g k ( p j ) (cid:9) . Set j = j + 1 .
7) If the current slot is the moment of reporting periodic road-traffic density information,
Then go to Stage 1 and continue ; Else loop Stage 2 . end loop as the iterations proceed, η is gradually increased and F ( η ) is gradually decreased, henceAlgorithm 1 converges.The optimality can be proved by the method of contradiction. Let η ˜ p = min k { f k (˜ p ) /g k (˜ p ) } <η ∗ be an improved, but sub-optimal value. Given the monotonicity and convergence of F ( η ) , Algorithm 2 . Max-min resource allocation algorithm for Sub-problem 1
Initialization: • The location information of all VUEs. • The rate requirement R k (Γ k , L k , (cid:15) k ) and reliability requirement (cid:15) k of each VUE. • Total power P B and initial power allocation p = [ p , · · · , p K ] T with p k = P B /K, ∀ k . • Iterative index i = 0 , maximum iterative tolerance ζ S > and adjustable allocation steplength µ ∈ (0 , P B ) . Power allocation for Sub-problem 1: Calculate initial H = [ h ( p ) , · · · , h K ( p )] T with h k ( p ) = f k ( p ) − η j g k ( p ) . loop ( i ) Set i = i + 1 . Calculate k m i = arg min k { H i − } and k M i = arg max k { H i − } . Update p i with p ik m i = p i − k m i + µ i − , p ik M i = p i − k M i − µ i − and p ik = p i − k ( ∀ k (cid:54) = k m i , k M i ) . Calculate H i with h ik ( p i ) = f k ( p i ) − η j g k ( p i ) . if max k { H i } − min k { H i } (cid:54) ζ S then Update p j = p i and break . elseif h ik m i ( p i ) (cid:54) h i − k m i ( p i − ) or h ik M i ( p i ) (cid:62) h i − k M i ( p i − ) or p ik M i (cid:54) or max k { H i − } − min k { H i − } (cid:54) max k { H i } − min k { H i } then Update µ i = µ i − / , p i = p i − and H i = H i − . else Update µ i = µ i − and continue . end ifend if end loop we have F ( η ˜ p ) > F ( η ∗ ) and F ( η ˜ p ) = 0 . On the other hand, based on Theorem 3, we have F ( η ∗ ) = 0 , which leads to a contradiction.
2) Secondary Algorithm:
As illustrated in Algorithm 1, Step 3 raises the following optimiza-tion problem.
Sub-Problem
Given η j < , the max-min power allocation problem is formulated as SP1: max p min k { f k ( p ) − η j g k ( p ) } (47a) s . t . (29b) − (29c) . (47b)To solve Sub-problem 1, the monotonicity of h k ( p ) = f k ( p ) − η j g k ( p ) is first studied. Theproof of the ZF is similar to that of the MF, hence only the case of the MF is shown here. Thederivative ∂h MF k ( p ) /∂p k = ∂f MF k ( p ) /∂p k − η j ∂g MF k ( p ) /∂p k is given by ∂h MF k ( p ) ∂p k = √ B ∗ (cid:18) Γ MF k p k + (Γ MF k ) M p k (cid:19)(cid:113) − (1 + Γ MF k ) − (1 + Γ MF k ) ln 2 (cid:20) − Q − ( (cid:15) k ) − η j √ B ∗ (cid:113) − (1 + Γ MF k ) − (cid:0) MF k (cid:1) (cid:21) . (48)Since the order of magnitude for the bandwidth is generally of kHz, we can obtain − Q − ( (cid:15) k ) − η j √ B ∗ (cid:113) − (1 + Γ MF k ) − (cid:0) MF k (cid:1) (cid:62) − Q − ( (cid:15) k ) − η j √ B ∗ Γ MF k MF k = − Q − ( (cid:15) k ) (1 + Γ MF k ) − η j √ B ∗ Γ MF k MF k > , (49)which implies that the function h MF k ( p ) = f MF k ( p ) − η j g MF k ( p ) is increasing. The theorem capableof achieving the optimal max-min power allocation is presented as follows [33]. Theorem In order to maximize the minimum objective function value of (47), all VUEsshould have the same objective value, hence we have h ( p ) = h ( p ) = · · · = h K ( p ) (cid:44) H p . (50) Proof: (50) can be obtained by the method of contradiction. Let p ∗ = [ p ∗ , · · · , p ∗ K ] T bethe optimal power allocation of Sub-problem 1, which does not satisfy (50). Without loss ofgenerality, let h ( p ∗ ) = max k { h k ( p ∗ k ) } and h K ( p ∗ K ) = min k { h k ( p ∗ k ) } . As h k ( p ) is monotonicallyincreasing, we should have µ ∈ (0 , p ∗ ) satisfying h ( p ∗ ) > h ( p ∗ − µ ) = h K ( p ∗ K + µ ) > h K ( p ∗ K ) . (51)Hence, a new power allocation p ∗ N = [ p ∗ − µ, · · · , p ∗ K + µ ] T satisfying (47b) is constructed. Inother words, the minimum objective function value has been increased, which contradicts to themax-min criterion. Hence the theorem is proved. In the operational wireless systems, the MIMO-related techniques are generally used at the narrowband (subcarriers), hencethe order of magnitude on basic scheduling resource is of kHz such as the resource block (180 kHz) in the LTE-related systems. Based on Theorem 5, an iterative procedure described by Algorithm 2 is proposed for solvingSub-problem 1. According to lim i →∞ ( | h ik m i ( p i ) − h ik M i ( p i ) | ) = 0 , we can intuitively find thatAlgorithm 2 is capable of converging to the same objective function value H p . Meanwhile, theoptimality can be guaranteed by Theorem 5. In the initialization phase of Algorithm 2, the objec-tive values of all VUEs are calculated for the equal power allocation. As the iterations proceed,Algorithm 2 will accordingly increase the minimum power and reduce the maximum powerevery time until we have max k { H i } − min k { H i } (cid:54) ζ S . To avoid the ping-pong phenomenonbetween k m i and k M i , four judgement conditions are listed in Algorithm 2 to adjust the allocationstep length µ i .
3) Algorithm Complexity:
First of all, the optimal solution of Problem 1 is directly achievedby the quadratic formula. Furthermore, since this optimal solution is independent of the numberof VUEs and it is a closed-form solution, the complexity of solving Problem 1 can be ignored.With respect to solving Problem 2, the convergence and optimality are guaranteed by Theorem 4and 5, while the complexity is determined by Stage 2 of Algorithm 1, and Algorithm 2. Basedon the maximum iterative tolerance ζ P of Algorithm 1, the associated precision is given by Z = log ( ζ − P ) -digit. Thus, the complexity order of Algorithm 1 is O (log Z ) [44], [43]. On theother hand, because | h k ( p k ) − H p | (cid:54) ζ S is a sufficient condition for satisfying the terminationcondition max k { H i } − min k { H i } (cid:54) ζ S , the complexity of Algorithm 2 is on the order of O ( (cid:80) Kk =1 log [ h − k ( H p + ζ S ) − h − k ( H p − ζ S )] − ) [24], [33]. Hence, the total complexity order is O (log Z (cid:80) Kk =1 log [ h − k ( H p + ζ S ) − h − k ( H p − ζ S )] − ) [45].Finally, due to the handover in high mobility VNETs, we discuss the scalability of the proposedalgorithms. Remark
The proposed algorithms depend on the road-trafficdensity and the location information of all VUEs. In general, the traffic control center can useroadside monitoring sensors or real-world traffic datasets to predict and periodically report thevehicular density and location information [29], [30]. For multi-cell scenarios, both of the densityand location information can be shared via wired links between the traffic control center and allthe base stations. Therefore, as long as the density and location information are available, ourproposed algorithms can be readily extended to the multi-cell scenarios. In addition, due to thewired links, the sharing of the density and location information do not bring the extra signaloverhead for radio access networks. TABLE IB
ASIC S IMULATION P ARAMETERS
Parameter Value
BS distance d B m Road length d R m Maximum road-traffic density ρ m v F km / h Estimation accuracy χ k , ∀ k f C GHz
Constant θ − Path loss exponent α N -130 dBm / Hz Transmitted signal power spectral density P -10 dBm / Hz VI. S
IMULATION R ESULTS
A. Simulation Setup
In the simulations, the large-scale fading of the k -th VUE is given by β k = θ [( d k − d R / + d B ] − α/ , where θ is a constant related to the antenna gain and carrier frequency, α is the path lossexponent, and d k is the distance between the k -th VUE and the starting point of the road. Thenoise power is calculated by σ = N B , where N and B are the noise power spectral densityand system bandwidth, respectively. Furthermore, all VUEs are uniformly distributed on theone-way road, namely d k ∼ unif (0 , d R ) . Since the average length of a vehicle is 6.5 m (alreadyplus 50% of additional safety distance), the maximum road-traffic density is set to ρ m = 0 . .In the following simulation results, PCSI and IPCSI represent the perfect and imperfect channelstate information, respectively. Finally, all other basic simulation parameters are listed in Table I. B. Simulation Results for the Maximum Achievable Rate and Transmission Latency1) Tradeoff:
Upon using the equal power allocation (EPA) and MF precoder, Fig. 3 illustratesthe tradeoff of a VUE among the maximum achievable rate, transmission latency and reliabilityfor the case of IPCSI. As shown in Fig. 3, regardless of what the latency and reliability are, theergodic capacity remains a constant. By contrast, Theorem 1 indicates that some rate regions arenot achievable (negative) because of the ultra-high reliability and ultra-low latency. Moreover, -310-2 9-1 8 0.10 7 0.091 0.086 0.072 5 0.063 4 0.050.043 0.032 0.021 0.01 Theorem 1Ergodic capacity
Fig. 3. Tradeoff among maximum achievable rate, transmission latency and reliability in (1) and (18) based on the MF precoder,with ρ = 0 . , M = 300 , B = 200 kHz and P B = 10 dBW . P C S I I P C S I A s y m p t o t i cP C S I I P C S I Z F S i m u l a t i o n Z F T h e o r e m 1
8 d B W
Maximum achievable rate Rk (bps/Hz) T h e n u m b e r o f B S a n t e n n a s M P C S I I P C S I M F S i m u l a t i o n M F T h e o r e m 1
Fig. 4. Maximum achievable rate versus the number of antennas in (18), with ρ = 0 . , B = 200 kHz and L k = 1 ms . the maximum achievable rate given by Theorem 1 is lower than the ergodic capacity, whichconfirms that V2I URLLCs are indeed possible, but only at the cost of a reduced rate. Finally,Fig. 3 shows that the transmission latency can only be optimized within a reasonable range.The results of the ZF are similar to those of the MF, thus they are omitted here due to spacelimitations. P C S I I P C S I A s y m p t o t i c M F S i m u l a t i o n M F T h e o r e m 1 Z F S i m u l a t i o n Z F T h e o r e m 1
Transmission latency Lk (ms) T h e n u m b e r o f B S a n t e n n a s M Fig. 5. Transmission latency versus the number of antennas in (21), with ρ = 0 . , B = 200 kHz and R k = 100 kbps .
2) Tightness:
Upon adopting the EPA schemes, Fig. 4 and 5 illustrate the maximum achiev-able rate and transmission latency versus the number of antennas, respectively. The reliabilitythreshold (cid:15) k of these two figures is set to − . The asymptotic results ( M → ∞ ) are calculatedfrom (18) and (21) with Γ ∞ k = p k χ k β k /σ ( χ k = 1 for PCSI). For the PCSI and IPCSI, Fig. 4 and5 indicate that upon increasing the number of antennas, the maximum achievable rate increases,while the latency of a VUE decreases. The approximate results of Theorem 1 and Corollary 1 areclose to the simulation results, and both of them tend to the asymptotic results as the number ofantennas increases. Because of the channel estimation error, the case of IPCSI is always worsethan the case of PCSI. Furthermore, both the rate and latency of the ZF are better than thoseof the MF in the case of PCSI or IPCSI, due to the higher SINR Γ k . Finally, these two figuressuggest that increasing the total transmitted power improve both the rate and latency. Based onthe tightness of Corollary 1, the simulation results of Problem 1 and 2 will be discussed in thefollowing subsections. C. Simulation Results for Problem 1
Based on the PCSI and IPCSI, Fig. 6 illustrates the optimal system bandwidth versus theroad-traffic density, where the MF, ZF and asymptotic results are calculated by substituting(24) and Γ ∞ W = P χ th β W / ( N ρd R ) into (33). In order to keep the worst-case latency below acertain threshold, Fig. 6 shows that with the road-traffic density increasing, the optimal systembandwidth also increases. This means that the more VUEs are supported, the more bandwidth is (cid:1) = 1 / 2 0 Optimal bandwidth B * ( k Hz)
T r a f f i c d e n s i t y (cid:2)
P C S I I P C S I A s y m p t o t i c M F Z F (cid:1) = 1 / 4 0
Fig. 6. Optimal system bandwidth versus road-traffic density in (33), with M = 300 , R W = 100 kbps and (cid:15) W = 10 − . - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 Optimal bandwidth B * ( k Hz)
M i n i m u m r e l i a b i l i t y (cid:1) W P C S I I P C S I A s y m p t o t i c M F Z F (cid:2) = 0 . 1 (cid:2) = 0 . 0 5
Fig. 7. Optimal system bandwidth versus minimum reliability in (33), with M = 300 , R W = 100 kbps and δ = 1 / . required. In the region of low road-traffic density, the optimal bandwidth of the ZF is lower thanthat of the MF, whereas in the region of high road-traffic density, the required bandwidth of theZF is higher than that of the MF due to the lower SINR. Moreover, the increase of the threshold δ requires more bandwidth. Compared to the case of PCSI, the case of IPCSI requires morebandwidth to satisfy the worst-case latency requirement. Finally, when the number of antennastends to infinity, the bandwidth is only reduced by at most 75 kHz (IPCSI at δ = 1 / with ρ = 0 . ), which implies that the V2I URLLC system does not require an excessive numbersof antennas.Fig. 7 illustrates the optimal system bandwidth versus the reliability for the two road-traffic Objective function value F ( (cid:1) ) T h e n u m b e r o f i t e r a t i o n j O b j e c t i v e v a l u e (a) Convergence of Algorithm 1 in (44) and Theorem 4.
Objective function value of SP1
T h e n u m b e r o f i t e r a t i o n i M a x i m u m o b j e c t i v e v a l u e M i n i m u m o b j e c t i v e v a l u e (b) Convergence of Algorithm 2 in Theorem 5.Fig. 8. Convergence of the two proposed algorithms with ζ P = ζ S = 10 − , η = − × − and µ = P B / (2 ρd R ) . densities considered. As shown in Fig. 7, upon relaxing the minimum reliability requirement, theoptimal system bandwidth decreases, regardless of the CSI. The gap between the MF results andZF results is in line with the trend seen in Fig. 6 under these two types of road-traffic densities.Additionally, by comparing the results of (cid:15) W = 10 − to those of − , the optimal bandwidth isonly increased by at most 75 kHz (IPCSI at ρ = 0 . ), which suggests that the reliability can besignificantly improved by only modestly increasing the bandwidth. D. Simulation Results for Problem 21) Convergence:
Based on a specific channel realization in the simulations, Fig. 8(a) and 8(b)illustrate the convergence of the two proposed algorithms, respectively. Fig. 8(a) indicates that as » » Maximum transmission latency (ms)
T r a f f i c d e n s i t y (cid:1)
P C S I I P C S I M F P r o p o s e d M F E P A Z F P r o p o s e d Z F E P A
Fig. 9. Maximum transmission latency versus road-traffic density solved by Algorithm 1 and 2, with M = 300 , δ = 1 / , R k = 100 kbps and (cid:15) k = 10 − for all VUEs. T h e b e g i n n i n g o f F i g . 8
Maximum transmission latency (ms)
T o t a l B S p o w e r P B ( d B W ) P C S I I P C S I M F P r o p o s e d M F E P A Z F P r o p o s e d Z F E P A
Fig. 10. Maximum transmission latency versus total transmitted power solved by Algorithm 1 and 2, with M = 300 , δ = 1 / , ρ = 0 . , R k = 100 kbps and (cid:15) k = 10 − for all VUEs. the number of iterations increases, the auxiliary function F ( η ) converges to zero in Algorithm 1,as stated by Theorem 4. On the other hand, as shown in Fig. 8(b), as the number of iterationsincreases, the maximum objective function value decreases and the minimum objective functionvalue increases. All objective function values finally converge to a constant, which characterizesthe convergence of Algorithm 2.
2) Transmission Latency:
Fig. 9 illustrates the maximum transmission latency among allVUEs versus the road-traffic density. Observe that when the road-traffic density increases, the maximum transmission latency of both the proposed scheme and of the EPA scheme increaseaccordingly. Upon comparing the MF precoder to the ZF precoder, regardless of having eitherPCSI or IPCSI, the performance of the ZF is better than that of the MF both for the proposedscheme and for the EPA scheme. Fig. 9 also indicates that the proposed scheme performs betterthan the EPA scheme both for the MF precoder and for the ZF precoder in the case of either PCSIor IPCSI. Furthermore, the performance gap between the MF and ZF is always about 0.01 ms .Naturally, the channel estimation error increases latency. Nevertheless, the IPCSI results of theproposed scheme are close to the PCSI results of the EPA scheme both in the case of the MFand the ZF precoder. Finally, the slope of the proposed scheme is lower for both the PCSI andIPCSI, which implies that the proposed scheme is not sensitive to the road-traffic density. Hence,the proposed algorithms can get the effect of semi-persistent scheduling for the V2I URLLCsystem.Given ρ = 0 . and increase the total power based on P B ∗ , Fig. 10 quantifies the maximumtransmission latency among all VUEs versus the total transmitted power. As shown in Fig. 10,the maximum transmission latency decreases for both the proposed scheme and for the EPAscheme, as the total power increases. The performance of the ZF is better than that of the MFboth for the proposed scheme and for the EPA scheme in the case of either PCSI or IPCSI.Additionally, the gap between the ZF relying on IPCSI and the MF using PCSI is relativelysmall for the proposed scheme. Hence, the ZF precoder is the better choice for V2I URLLCs,as evidenced both by Fig. 9 and Fig. 10.VII. C ONCLUSIONS
In order to reduce the signaling overhead, this paper studied the problem of minimizingthe transmission latency in the downlink of the V2I URLLC system based on the road-trafficdensity and large-scale fading CSI. The expression of transmission latency was first derived forthe MF precoder and ZF precoder based on both perfect and imperfect CSI. Then, a two-stageradio resource allocation problem was formulated relying on our twin-timescale perspective.More particularly, the first stage optimized the worst-case transmission latency by adjustingthe system’s bandwidth based on a long-term timescale, while the second stage minimized themaximum transmission latency among all VUEs by judiciously allocating the total power on ashort-term timescale basis. For optimally solving our two-stage problem formulated, a moderate-complexity twin-timescale radio resource allocation algorithm was conceived. Our simulation results showed that the proposed resource allocation scheme significantly reduced the maximumtransmission latency and that the ZF precoder was the better choice for the V2I URLLC systemconsidered. A PPENDIX AP ROOF OF T HEOREM k -th VUE is given by R k ( γ k , L k , (cid:15) k ) ≈ E (cid:40) B (cid:34) log (1 + γ k ) − (cid:114) V k L k B Q − ( (cid:15) k ) (cid:35)(cid:41) = E (cid:20) B log (cid:18) γ − k (cid:19)(cid:21) − E (cid:34)(cid:114) BV k L k Q − ( (cid:15) k ) (cid:35) . (52)Let u k = log (1+1 /γ − k ) denote the spectral efficiency of the k -th VUE, then V k can be rewrittenas V k = (cid:18) − γ k ) (cid:19) (log e) = (cid:0) − − u k (cid:1) (log e) , (53)and we have R k ( γ k , L k , (cid:15) k ) ≈ B E (cid:20) log (cid:18) γ − k (cid:19)(cid:21) − log e (cid:114) BL k Q − ( (cid:15) k ) E (cid:104) √ − − u k (cid:105) . (54)Since the function a ( x ) = log (1 + 1 /x ) , ∀ x > is convex, Jensen’s inequality gives U k (cid:44) E [ u k ] (cid:62) U L k (cid:44) log (cid:18) E [ γ − k ] (cid:19) . (55)Substituting E [Ω MFB ] = M , E [Ω MFG ] = M − and E [Ω ZFG ] = M − K into (17), we obtain U L k = log (cid:18) M p k P B − p k + ϕ k (cid:19) , for MF , log (1 + p k ϕ k ) , for ZF , (56)where ϕ k = P B β k (1 − χ k ) + M σ χ k β k MM − , for MF ,χ k β k ( M − K ) P B β k (1 − χ k ) + M σ , for ZF . (57)Note that for M → ∞ , we have U L k = log (1 + p k χ k β k σ ) . On the other hand, due to the channelhardening phenomenon, we arrive at lim M →∞ U k = log (1 + p k χ k β k σ ) , which indicates that U L k is tight for U k when M is large. Therefore, U L k can be used for approximating the first term of(54), i.e., E (cid:20) log (cid:18) γ − k (cid:19)(cid:21) ≈ U L k , M (cid:29) . (58)Similarly, since the function b ( x ) = √ − − x , ∀ x > is concave, with Jensen’s inequality, E (cid:104) √ − − u k (cid:105) (cid:54) (cid:112) − − E [ u k ] . (59)Based on the channel hardening phenomenon, we arrive at lim M →∞ E (cid:104) √ − − u k (cid:105) = (cid:115) − (cid:18) p k χ k β k σ (cid:19) − = lim M →∞ (cid:112) − − E [ u k ] , (60)which implies that √ − − E [ u k ] is also tight for E (cid:2) √ − − u k (cid:3) when M is large, namely E (cid:2) √ − − u k (cid:3) ≈ √ − − E [ u k ] , M (cid:29) .Recalling that U k ≈ U L k , we finally have E (cid:104) √ − − u k (cid:105) ≈ (cid:112) − − U L k , M (cid:29) . (61)Substituting (58) and (61) into (54) yields (18).A PPENDIX BP ROOF OF T HEOREM f MF k ( p ) = −√ B ∗ Q − ( (cid:15) k ) log e (cid:113) − (1 + Γ MF k ) − , (62) ∂f MF k ( p ) ∂p k = − √ B ∗ Q − ( (cid:15) k ) log e (cid:18) Γ MF k p k + (Γ MF k ) M p k (cid:19)(cid:113) − (1 + Γ MF k ) − (1 + Γ MF k ) , (63) ∂ f MF k ( p ) ∂p k = √ B ∗ Q − ( (cid:15) k ) log e ( M + Γ MF k ) (3 M − MF k )( P B − p k + ϕ k ) (cid:113) − (1 + Γ MF k ) − (1 + Γ MF k ) . (64)Since M (cid:29) ⇒ M − > , then f MF k ( p ) < , ∂f MF k ( p ) /∂p k < and ∂ f MF k ( p ) /∂p k > ,namely f MF k ( p ) is negative, differentiable, decreasing and convex. On the other hand, we have g MF k ( p ) = B ∗ log (cid:0) MF k (cid:1) − R k (Γ k , L k , (cid:15) k ) , (65) ∂g MF k ( p ) ∂p k = B ∗ (cid:18) Γ MF k p k + (Γ MF k ) M p k (cid:19) (1 + Γ MF k ) ln 2 , (66) ∂ g MF k ( p ) ∂p k = B ∗ ( M + Γ MF k ) (2 + Γ MF k − M )( P B − p k + ϕ k ) (1 + Γ MF k ) ln 2 . (67)Since M (cid:29) ⇒ MF k − M < , then g MF k ( p ) > , ∂g MF k ( p ) /∂p k > and ∂ g MF k ( p ) /∂p k < ,namely g MF k ( p ) is non-negative, differentiable, increasing and concave. Lemma Let
C ⊆ R n be a convex set and r : C → R . Then, r is QC if and only if itssuperlevel set S t = { x ∈ C : r ( x ) (cid:62) t } is convex for all t ∈ R . Proof:
The omitted proof can be found in [46], [43].According to Lemma 2, the superlevel set S t = { p ∈ C : f MF k ( p ) /g MF k ( p ) (cid:62) t } is taken intoaccount. S t is the empty set for all t > when f MF k ( p ) < and g MF k ( p ) > , thus only t (cid:54) hasto be considered. The equivalent form of S t is given by S t = { p ∈ C : f MF k ( p ) − tg MF k ( p ) (cid:62) } .Let h MF k ( p ) = f MF k ( p ) − tg MF k ( p ) , then ∂ h MF k ( p ) /∂p k = ∂ f MF k ( p ) /∂p k − t∂ g MF k ( p ) /∂p k isgiven by ∂ h MF k ( p ) ∂p k = − √ B ∗ ( M + Γ MF k )( P B − p k + ϕ k ) (cid:113) − (1 + Γ MF k ) − (1 + Γ MF k ) ln 2 × (cid:20) Q − ( (cid:15) k ) (cid:0) − M + 2 − Γ MF k (cid:1) − t √ B ∗ (cid:113) − (1 + Γ MF k ) − (cid:0) MF k (cid:1) (cid:0) M − − Γ MF k (cid:1)(cid:21) . (68)Because M (cid:29) and the order of magnitude on bandwidth is generally of kHz, we obtain Q − ( (cid:15) k ) (cid:0) − M + 2 − Γ MF k (cid:1) − t √ B ∗ (cid:113) − (1 + Γ MF k ) − (cid:0) MF k (cid:1) (cid:0) M − − Γ MF k (cid:1) (cid:62) Q − ( (cid:15) k ) (cid:0) − M + 2 − Γ MF k (cid:1) − t √ B ∗ Γ MF k MF k (cid:0) M − − Γ MF k (cid:1) > , (69)which states that the function h MF k ( p ) = f MF k ( p ) − tg MF k ( p ) is concave, namely the superlevelset S t is convex, and −√ L k is a QC function of p k . A PPENDIX CP ROOF OF L EMMA F ( η ˜ p ) = max p min k (cid:26) f k ( p ) − min k (cid:26) f k (˜ p ) g k (˜ p ) (cid:27) g k ( p ) (cid:27) , ∀ p ∈ P (a) (cid:62) min k (cid:26) f k (˜ p ) − min k (cid:26) f k (˜ p ) g k (˜ p ) (cid:27) g k (˜ p ) (cid:27) (b) = f n (˜ p ) − f n (˜ p ) g n (˜ p ) g n (˜ p ) = 0 , (70)where the equality of (a) holds when ˜ p = arg max p min k { f k ( p ) − η ˜ p g k ( p ) } , and (b) follows from f n (˜ p ) g n (˜ p ) < f k (˜ p ) g k (˜ p ) , ∀ k (cid:54) = n, n = arg min k (cid:26) f k (˜ p ) g k (˜ p ) (cid:27) , (71) ⇒ f k (˜ p ) − f n (˜ p ) g n (˜ p ) g k (˜ p ) > . (72)R EFERENCES [1] T. Qiu, N. Chen, K. Li, M. Atiquzzaman, and W. Zhao, “How can heterogeneous Internet of things build our future: Asurvey,”
IEEE Commun. Surveys Tuts. , vol. 20, no. 3, pp. 2011–2027, Thirdquarter 2018.[2] K. Zheng, Q. Zheng, P. Chatzimisios, W. Xiang, and Y. Zhou, “Heterogeneous vehicular networking: A survey onarchitecture, challenges, and solutions,”
IEEE Commun. Surveys Tuts. , vol. 17, no. 4, pp. 2377–2396, Fourthquarter 2015.[3] Z. MacHardy, A. Khan, K. Obana, and S. Iwashina, “V2X access technologies: Regulation, research, and remainingchallenges,”
IEEE Commun. Surveys Tuts. , vol. 20, no. 3, pp. 1858–1877, Thirdquarter 2018.[4] K. Zheng, Q. Zheng, H. Yang, L. Zhao, L. Hou, and P. Chatzimisios, “Reliable and efficient autonomous driving: Theneed for heterogeneous vehicular networks,”
IEEE Commun. Mag. , vol. 53, no. 12, pp. 72–79, Dec. 2015.[5] J. Wang, J. Liu, and N. Kato, “Networking and communications in autonomous driving: A survey,”
IEEE Commun. SurveysTuts. , vol. 21, no. 2, pp. 1243–1274, Secondquarter 2019.[6] W. Sun, E. G. Ström, F. Brännström, K. C. Sou, and Y. Sui, “Radio resource management for D2D-based V2Vcommunication,”
IEEE Trans. Veh. Technol. , vol. 65, no. 8, pp. 6636–6650, Aug. 2016.[7] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,”
IEEE Trans. Inf. Theory ,vol. 56, no. 5, pp. 2307–2359, May 2010.[8] M. Hayashi, “Information spectrum approach to second-order coding rate in channel coding,”
IEEE Trans. Inf. Theory ,vol. 55, no. 11, pp. 4947–4966, Nov. 2009.[9] G. Durisi, T. Koch, and P. Popovski, “Toward massive, ultrareliable, and low-latency wireless communication with shortpackets,”
Proc. IEEE , vol. 104, no. 9, pp. 1711–1726, Sep. 2016.[10] W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy, “Quasi-static multiple-antenna fading channels at finite blocklength,”
IEEE Trans. Inf. Theory , vol. 60, no. 7, pp. 4232–4265, Jul. 2014.[11] Q. Zheng, K. Zheng, H. Zhang, and V. C. M. Leung, “Delay-optimal virtualized radio resource scheduling in software-defined vehicular networks via stochastic learning,”
IEEE Trans. Veh. Technol. , vol. 65, no. 10, pp. 7857–7867, Oct.2016. [12] K. Zheng, H. Meng, P. Chatzimisios, L. Lei, and X. Shen, “An SMDP-based resource allocation in vehicular cloudcomputing systems,” IEEE Trans. Ind. Electron. , vol. 62, no. 12, pp. 7920–7928, Dec. 2015.[13] G. Yang, M. Xiao, and H. V. Poor, “Low-latency millimeter-wave communications: Traffic dispersion or networkdensification?”
IEEE Trans. Commun. , vol. 66, no. 8, pp. 3526–3539, Aug. 2018.[14] H. Forssell, R. Thobaben, H. Al-Zubaidy, and J. Gross, “Physical layer authentication in mission-critical mtc networks: Asecurity and delay performance analysis,”
IEEE J. Sel. Areas Commun. , vol. 37, no. 4, pp. 795–808, Apr. 2019.[15] J. Choi, “An effective capacity based approach to multi-channel low-latency wireless communications,”
IEEE Trans.Commun. , vol. 67, no. 3, pp. 2476–2486, Mar. 2019.[16] H. Yang, L. Zhao, L. Lei, and K. Zheng, “A two-stage allocation scheme for delay-sensitive services in dense vehicularnetworks,” in
Proc. IEEE International Conference on Communications Workshops (ICC Workshops) , Paris, France, May2017, pp. 1358–1363.[17] S. Sebastian, G. James, and A.-Z. Hussein, “Delay analysis for wireless fading channels with finite blocklength channelcoding,” in
Proc. ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems(MSWiM) , Cancun, Mexico, Nov. 2015, pp. 13–22.[18] C. She, C. Yang, and T. Q. S. Quek, “Cross-layer optimization for ultra-reliable and low-latency radio access networks,”
IEEE Trans. Wireless Commun. , vol. 17, no. 1, pp. 127–141, Jan. 2018.[19] Y. Cui, V. K. N. Lau, R. Wang, H. Huang, and S. Zhang, “A survey on delay-aware resource control for wireless systems–large deviation theory, stochastic Lyapunov drift, and distributed stochastic learning,”
IEEE Trans. Inf. Theory , vol. 58,no. 3, pp. 1677–1701, Mar. 2012.[20] J. Mei, K. Zheng, L. Zhao, Y. Teng, and X. Wang, “A latency and reliability guaranteed resource allocation scheme forLTE V2V communication systems,”
IEEE Trans. Wireless Commun. , vol. 17, no. 6, pp. 3850–3860, Jun. 2018.[21] N. A. Johansson, Y. . E. Wang, E. Eriksson, and M. Hessler, “Radio access for ultra-reliable and low-latency 5Gcommunications,” in
Proc. IEEE International Conference on Communications Workshops (ICC Workshops) , London,UK, Jun. 2015, pp. 1184–1189.[22] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,”
IEEE Trans. WirelessCommun. , vol. 9, no. 11, pp. 3590–3600, Nov. 2010.[23] K. Zheng, L. Zhao, J. Mei, B. Shao, W. Xiang, and L. Hanzo, “Survey of large-scale MIMO systems,”
IEEE Commun.Surveys Tuts. , vol. 17, no. 3, pp. 1738–1760, Thirdquarter 2015.[24] L. Zhao and X. Wang, “Massive MIMO downlink for wireless information and energy transfer with energy harvestingreceivers,”
IEEE Trans. Commun. , vol. 67, no. 5, pp. 3309–3322, May 2019.[25] C. She, C. Yang, and T. Q. S. Quek, “Radio resource management for ultra-reliable and low-latency communications,”
IEEE Commun. Mag. , vol. 55, no. 6, pp. 72–78, Jun. 2017.[26] Y. Hu, Y. Zhu, M. C. Gursoy, and A. Schmeink, “SWIPT-enabled relaying in IoT networks operating with finite blocklengthcodes,”
IEEE J. Sel. Areas Commun. , vol. 37, no. 1, pp. 74–88, Jan. 2019.[27] C. Sun, C. She, C. Yang, T. Q. S. Quek, Y. Li, and B. Vucetic, “Optimizing resource allocation in the short blocklengthregime for ultra-reliable and low-latency communications,”
IEEE Trans. Wireless Commun. , vol. 18, no. 1, pp. 402–415,Jan. 2019.[28] L. Zhou, A. Wolf, and M. Motani, “On lossy multi-connectivity: Finite blocklength performance and second-orderasymptotics,”
IEEE J. Sel. Areas Commun. , vol. 37, no. 4, pp. 735–748, Apr. 2019.[29] Y. Lv, Y. Duan, W. Kang, Z. Li, and F. Wang, “Traffic flow prediction with big data: A deep learning approach,”
IEEETrans. Intell. Transp. Syst. , vol. 16, no. 2, pp. 865–873, Apr. 2015. [30] N. G. Polson and V. O. Sokolov, “Deep learning for short-term traffic flow prediction,” Transp. Res. Part C Emerg. Technol. ,vol. 79, pp. 1–17, Jun. 2017.[31] P. Kachroo and K. M. A. Özbay, “Traffic flow theory,” in
Feedback Control Theory for Dynamic Traffic Assignment .Cham: Springer, 2018, pp. 57–87.[32] S. Kashyap, E. Björnson, and E. G. Larsson, “On the feasibility of wireless energy transfer using massive antenna arrays,”
IEEE Trans. Wireless Commun. , vol. 15, no. 5, pp. 3466–3480, May 2016.[33] L. Zhao, T. Riihonen, W. Xiang, Y. Kuang, and K. Zheng, “Resource optimization of wireless information and energysupply control systems with massive MIMO,”
IEEE Commun. Lett. , vol. 21, no. 12, pp. 2734–2737, Dec. 2017.[34] J. Zhang, L. Dai, X. Li, Y. Liu, and L. Hanzo, “On low-resolution ADCs in practical 5G millimeter-wave massive MIMOsystems,”
IEEE Commun. Mag. , vol. 56, no. 7, pp. 205–211, Jul. 2018.[35] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO systems,”
IEEE Trans. Commun. , vol. 61, no. 4, pp. 1436–1449, Apr. 2013.[36] X. Liu, Y. Li, L. Xiao, and J. Wang, “Performance analysis and power control for multi-antenna V2V underlay massiveMIMO,”
IEEE Trans. Wireless Commun. , vol. 17, no. 7, pp. 4374–4387, Jul. 2018.[37] “NR physical channels and modulation,” 3GPP, Tech. Rep. TS 38.211 V15.6.0, Jun. 2019.[38] F. Abbas, P. Fan, and Z. Khan, “A novel low-latency V2V resource allocation scheme based on cellular V2Xcommunications,”
IEEE Trans. Intell. Transp. Syst. , vol. 20, no. 6, pp. 2185–2197, Jun. 2019.[39] Y. Park, T. Kim, and D. Hong, “Resource size control for reliability improvement in cellular-based V2V communication,”
IEEE Trans. Veh. Technol. , vol. 68, no. 1, pp. 379–392, Jan. 2019.[40] B. H. Fleury, “An uncertainty relation for WSS processes and its application to WSSUS systems,”
IEEE Commun. Mag. ,vol. 44, no. 12, pp. 1632–1634, Dec. 1996.[41] E. Dahlman, S. Parkvall, and J. Skold, . Cambridge, Massachusetts:Academic Press, 2013.[42] K. Shen and W. Yu, “Fractional programming for communication systems-Part I: Power control and beamforming,”
IEEETrans. Signal Process. , vol. 66, no. 10, pp. 2616–2630, May 2018.[43] A. Zappone and E. Jorswieck, “Energy efficiency in wireless networks via fractional programming theory,”
Foundationsand Trends in Communications and Information Theory , vol. 11, no. 3-4, pp. 185–396, Jun. 2015.[44] J. M. Borwein and P. B. Borwein,
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity .Hoboken, New Jersey: John Wiley & Sons, 1987.[45] K. T. K. Cheung, S. Yang, and L. Hanzo, “Achieving maximum energy-efficiency in multi-relay OFDMA cellular networks:A fractional programming approach,”
IEEE Trans. Commun. , vol. 61, no. 7, pp. 2746–2757, Jul. 2013.[46] S. Boyd and L. Vandenberghe,