I Introduction
Given the rapid development of the Internet of things (IoT), the design of radio access networks is undergoing a paradigm shift from promoting spectralefficiency and energyefficiency toward connecting everything [1, 2]
. As a representative application scenario of IoT, vehicular networks (VNETs) support various vehicular applications for reducing the probability of traffic accidents and for improving both the driving and infotainment experience
[3, 4]. Traditional safetyrelated vehicular applications, such as platooning and road safety generally require a transmission latency within a few milliseconds and a reliability in terms of error probability down to . Furthermore, in order to achieve the ultimate goal of autonomous driving on the road, future autonomous vehicles need sophisticated sensors to support safetyrelated applications. Ultrareliable and lowlatency communications (URLLCs) exchanging nearrealtime information are also required to offer assistance to control systems [5]. To this end, it is paramount to conceive URLLC techniques for future VNETs. In the conception of URLLC techniques, radio resource management plays a key role in improving the performance of VNETs, hence groundbreaking research is required.The latency encountered may be classified according to the hierarchical architecture of networks. The physical (PHY) layer aims for reducing the air interface latency, while the media access control (MAC) layer aspires to reduce the queueing (scheduling) latency. The theoretical analysis of PHY layer radio resource management conceived for devicetodevicebased vehicletovehicle communications can be found in
[6], where both the transmission latency and reliability are considered simultaneously. However, the study of vehicular mobility was not considered in [6]. Moreover, neither the ergodic capacity nor the outage capacity are suitable for characterizing the tradeoff among the achievable rate, transmission latency and reliability in URLLCs [7, 8]. Hence, the recent advances in studying an attractive tradeoff among the rate, latency and reliability of multipleantenna aided channels are illustrated in [9, 10] utilizing finite blocklength theory. However, there is still a paucity of literature on addressing the URLLC problems of the PHY layer in VNETs under the consideration of vehicular mobility models.With respect to the MAC layer, sophisticated techniques have been used for analyzing and optimizing the queueing latency, such as Markov decision processes
[11], stochastic network calculus [12] and queueing theory [13]. Based on the Markov decision process framework, a minimum queueing delaybased radio resource allocation scheme is proposed in [14] for vehicletovehicle communications, where the successful packet reception ratio is considered as the reliability metric. However, it is not practical to inform the base station of the global channel state information (CSI) including the links which are not connected to the base station. Moreover, the analysis in [14] has not been provided for URLLCs, since it was not based on radical advances in finite blocklength theory. Relying on finite blocklength theory and stochastic network calculus, a novel probabilistic delay bound is conceived in [15] for lowlatency machinetomachine applications. Furthermore, a URLLC crosslayer optimization framework is proposed in [16] by jointly considering the transmission latency and queueing latency. However, the advances mentioned 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 the MAC layer, reliability is generally characterized either by the stability of the queue [17] or by the probability of the queueing latency exceeding the tolerance threshold [18]. As for the PHY layer, typically the latency vs reliability tradeoff is quantified [19, 20], where one of the pivotal parameters of ensuring a high reliability is deemed to be the multiantenna diversity gain. Hence, largescale MIMO schemes constitute a promising solution of reducing latency and enhancing reliability in VNETs. On a similar note, massive MIMO schemes can also be adopted for formulating largescale fadingbased optimization problems by exploiting the channel hardening phenomenon [21, 22].
Against this background, in this paper we conceive radio resource management for URLLC VNETs based on a realistic vehicular mobility model. To conceive URLLC VNETs, the following pair of crucial characteristics has to be considered:

The nearinstantaneous CSI rapidly becomes outdated and its more frequent update imposes a high overhead, hence reducing the communication efficiency; and

Compared to the timescale of CSI fluctuation on the millisecond level, that of the roadtraffic fluctuation is lower. Requiring resource allocation updates on the order of milliseconds would impose an excessive implementation complexity.
In tackling the abovementioned challenges in the downlink of a massive MIMO vehicletoinfrastructure (V2I) system conceived for URLLC, our contributions are:

We optimize the transmission latency based on a twintimescale perspective for reducing the signaling overhead, where a macroscopic roadtraffic model is adopted for charactering the relationship between the vehicular velocity and roadtraffic density.

We first derive the transmission latency based on finite blocklength theory for the matched filter (MF) precoder and zeroforcing (ZF) precoder both under perfect and imperfect CSI.

Then, a twostage radio resource allocation problem is formulated for our V2I URLLC system. In particular, based on the longterm timescale of roadtraffic density, Stage 1 aims for optimizing the worstcase transmission latency by appropriately setting the system’s bandwidth. For Stage 2, the base station (BS) allocates the total power based on the largescale fading CSI for minimizing the maximum transmission latency guaranteeing fairness among all vehicular users (VUEs). Finally, the proposed primary and secondary radio resource allocation algorithms are invoked for optimally solving the above two problems at a reasonable complexity.

Our simulation results show that the proposed resource allocation scheme is capable of significantly reducing the maximum transmission latency, while flexibly accommodating the roadtraffic fluctuation. Additionally, the ZF precoder is shown to constitute a compelling choice for our V2I URLLC system.
The remainder of this paper is organized as follows. First of all, finite blocklength theory is introduced in Section II. Section III describes the V2I URLLC system model, while Section IV formulates the twintimescale resource allocation problem. Then, the proposed resource allocation algorithms are discussed in Section V. Finally, Section VI illustrates the simulation results, while our conclusions are offered in Section VII.
Notations:
Uppercase boldface letters and lowercase boldface letters denote matrices and vectors, respectively, while
is a diagonal square matrix whose main diagonal is formed by vector . Furthermore, , , and represent the transpose, conjugate, conjugate transpose and pseudoinverse of a matrix/vector, respectively, while denotes the Euclidian norm, and denotes the th diagonal element of a square matrix. Finally, represents the mathematical expectation, whileis the complex Gaussian distribution with mean
and real/imaginary component variance
.Ii Preliminaries: Finite Blocklength Theory
The ShannonHartley theorem quantifies the errorfree capacity at which information can be transmitted over a bandlimited channel in the presence of noise and interferences,
(1) 
where is the bandwidth and is the signaltointerferenceplusnoise ratio (SINR). This capacity can only be approached at the cost of excessive coding latency and complexity, i.e.,
(2) 
where is the socalled outage capacity at the error probability . For the operational wireless 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 is needed. Fortunately, during the last few years, significant progress has been made for satisfactorily addressing the problem of approximating [7, 23, 24]:
(3) 
where denotes the inverse of the Gaussian function and is the socalled channel dispersion. For a complex quasistatic fading channel, the channel dispersion is given by [23, 24]:
(4) 
Furthermore, is the transmission latency, while , which is also referred to as the blocklength of channel coding, represents the number of transmitted symbols. When is high enough, the approximation (3) approaches the ergodic capacity. Compared to the ergodic capacity, the approximation (3) also implies that the rate reduction is proportional to , when aiming for meeting a specific error probability at a given packet size. Finite blocklength theory constitutes a powerful technique of dealing with the URLLCrelated optimization problems. In addition to the contributions mentioned above, some new advances based on Equ. (3) analyze and optimize the URLLC performance of 5G and IoT networks [25, 26].
Iii V2I URLLC System Model
As shown in Fig. 1, for the downlink of the V2I URLLC system, we consider a single roadside BS and a road segment of length . The BS is meters away from the road. Furthermore, the BS employs antennas and simultaneously sends information to singleantenna aided VUEs (massive MIMO with ). The system operates in the timedivision duplex (TDD) mode.
Iiia RoadTraffic Model
According to traffic theory, the roadtraffic model is divided into macroscopic and microscopic models. Especially, the macroscopic model describes the average behavior of a certain number of vehicles at specific locations and instances, treating the roadtraffic similarly to fluid dynamics. By contrast, the microscopic model describes the specific behavior of each individual entity (such as vehicle or pedestrian), hence it is more sophisticated than the macroscopic model. To model the effect of roadtraffic on communication, the onedimensional macroscopic model is adopted here, which is also known as the LighthillWhithamRichards (LWR) model [27]. Given the roadtraffic density and the roadtraffic velocity , the roadtraffic flow rate (flux) is generally given by
(5) 
Numerous models have been proposed for characterizing how the vehicular velocity depends on the roadtraffic density. In this paper, the Underwood model is employed, and its speeddensity function can be written as [27]:
(6) 
where is the freeflowing velocity and is the maximum density. Moreover, the roadtraffic density can be used for modeling the number of VUEs, according to .
IiiB Channel Model
All channels experience independent quasistatic flat fading, i.e. they remain constant during a basic time block, but change independently from one block to another. Let denote the system’s CSI, where the diagonal matrix and represent the largescale fading and fast fading. The channel vector spanning from the BS to the th VUE is given by
. Furthermore, imperfect channel estimation is considered
[28, 29], i.e., , where , and are the estimate, error and estimation accuracy of , respectively. If ,corresponds to perfect channel estimation. Finally, all random variables are independent and identically distributed (i.i.d.) complex Gaussian random variables with mean 0 and variance 1, namely
.IiiC Transceiver Model
IiiC1 Transmitter
It is widely exploited that lowcomplexity linear transmit precoders are capable of attaining asymptotically optimal performance in massive MIMO. Therefore we consider the MF precoder and ZF precoder in this paper. Let and denote the precoder matrix and vector, where
(7) 
and with . The normalized form of is given by with
(8) 
As a result, given the symbol vector with , the transmitted signal vector of all VUEs can be written as
(9) 
where is the power allocation matrix and the total power at the BS is .
IiiC2 Receiver
The received signal vector is given by
(10) 
where is the i.i.d. additive white Gaussian noise with . The received signal of the th VUE can be written as
(11) 
Hence, the SINR of the th VUE is given by
(12) 
Recall from that the instantaneous can be rewritten as
(13) 
Based on [30], we have the following distributions, i.e.,
(14)  
(15)  
(16) 
Then can be finally expressed as
(17) 
Iv Problem Formulation of TwinTimescale Radio Resource Management
In this treatise, we optimize the transmission latency of V2I communications based on the roadtraffic density and location information, avoiding the frequent exchange of instantaneous global CSI. A twintimescale resource allocation problem is formulated in this section for reducing signaling overhead. Specifically, according to the roadtraffic density, the first stage is constructed for optimizing the worstcase latency by setting the system’s bandwidth from a longterm timescale. As for the second stage of the shortterm timescale, the BS will allocate the total power based on the largescale fading CSI (equivalent to location information) to minimize the maximum transmission latency for ensuring fairness amongst all VUEs.
Iva LargeScale FadingBased Transmission Latency
In highspeed vehicular environments, smallscale channel fading tends to fluctuate rapidly, hence the instantaneous V2I CSI is often outdated, because reporting the instantaneous V2I CSI to the BS is not practical. As a result, only a largescale fadingbased allocation problem was considerer in [31]. The theorem for quantifying the maximum achievable rate is first formulated as follows.
Theorem 1
Proof:
See Appendix A.
According to Theorem 1, we can obtain the following corollary for illustrating the transmission latency.
Corollary 1
Given the desired rate and the reliability , the transmission latency of the th VUE can be expressed as
(21) 
With respect to the reliability, with obeys the following properties,
(22) 
and . Thus, a higher reliability (also known as ) leads to a higher latency, and vice versa.
IvB Stage 1: LongTerm Timescale Allocation
The system bandwidth determined in Stage 1 can be viewed as part of the system design. Based on the longterm timescale roadtraffic density, the objective of Stage 1 is to optimize the worstcase transmission latency. Without loss of generality, the worstcase latency is considered in the scenario of the roadedge location (both ends of the road) with equal power allocation. The largescale fading of the roadedge location is given by
(23) 
where is a constant related both to the antenna gain and to the carrier frequency, while is the path loss exponent. Then, with , the worstcase SINR can be expressed as
(24) 
where and are the power spectral density of the transmitted signal and noise, respectively, while is the minimum requested estimation accuracy, namely . Therefore, the worstcase transmission latency is finally given by
(25) 
where and .
On the other hand, with Equ. (6), the maximum Doppler frequency is given by
(26) 
where is the carrier frequency, and is the speed of light. Hence, the typical coherence time can be calculated as [32]:
(27) 
On the basis of the above derivations, the following optimization problem is formulated.
Problem 1 (LongTerm Timescale Bandwidth Allocation)
Let denote a small threshold constant. Given the roadtraffic density , the longterm timescale bandwidth allocation problem of Stage 1 is formulated as
(28) 
IvC Stage 2: ShortTerm Timescale Allocation
By solving Problem 1, the total power at the BS can be obtained, namely . In Stage 2, the BS allocates the total power based on the shortterm timescale largescale fading CSI of all VUEs for minimizing the maximum transmission latency. Then the following optimization problem is formulated.
Problem 2 (ShortTerm Timescale Power Allocation)
Given the desired rate , reliability , total power and the largescale fading CSI of all VUEs, the optimization objective of Stage 2 is to minimize the maximum transmission latency of all VUEs ensuring fairness amongst all VUEs, i.e.,
(29)  
s.t.  (30)  
(31) 
V TwinTimescale Resource Allocation Algorithms for V2I URLLC Scystem
In this section, the solutions to the above pair of problems are first studied, and then efficient twintimescale allocation algorithms are proposed for our V2I URLLC system.
Va Solution to Problem 1
VB Solution to Problem 2
VB1 Problem Transformation
Problem 2 can be transformed into the following form:
(36)  
s.t.  (37) 
VB2 Property of Objective Function
Recall from Equ. (21) that can be expressed as
(38) 
Theorem 2
When the MF or ZF precoder is employed in the downlink of our V2I URLLC system, is a quasiconcave^{1}^{1}1Quasiconcave function: Let be a convex set and . Then, for any and , is quasiconcave if . function of .
Proof:
See Appendix B.
Given the definition of quasiconcavity, is also a quasiconcave function. By contrast, even if is pseudoconcave^{2}^{2}2Pseudoconcave function: Let be a convex set and . Then, for any , is pseudoconcave if it is differentiable and ., still cannot be pseudoconcave, since the function is not differentiable.
VB3 Optimal Solution
Since the objective function of Equ. (36) is QC, the KarushKuhnTucker (KKT) condition cannot be adopted. However, we will show that Problem 2 can still be solved optimally by an efficient algorithm.
Based on Equ. (38), the following auxiliary function is constructed,
(39) 
Intuitively, is strictly monotonically decreasing and has a unique zero root.
Lemma 1
For any , we have with , and the equality holds when .
Proof:
See Appendix C.
Based on Lemma 1, the following theorem describing the optimal allocation is formulated.
Theorem 3
For and , is the optimal solution of Equ. (36) if and only if
(40) 
Proof:
It relies on proving both sufficiency and necessity.
Firstly, let be the optimal power allocation,
(41) 
Therefore,
(42)  
(43) 
which yields Equ. (40) and .
Conversely, let , i.e.,
(44) 
which shows that
(45)  
(46) 
Thus is the optimal power allocation.
Corollary 2
VC TwinTimescale Resource Allocation Algorithms for V2I URLLC System
By taking into account the above solutions, the following primary algorithm and secondary algorithm are proposed for the optimal V2I URLLC resource allocation.
VC1 Primary Algorithm
Based on Dinkelbach’s method [33], a twintimescale iterative algorithm is put forward for jointly solving Problem 1 and 2, which is described in Algorithm VC1. The following theorem illustrates the convergence and optimality of Algorithm VC1.
Theorem 4
Algorithm VC1 can converge to the optimal power allocation.
Proof:
According to Step 18) in Algorithm VC1, we have
(48) 
where (a) follows from
(49) 
and (b) follows from Lemma 1. This implies that when the algorithm does not achieve convergence, we have . Therefore, as the iterations proceed, is gradually increased and is gradually decreased, hence Algorithm VC1 converges.
The optimality can be proved by the method of contradiction. Let be an improved, but suboptimal value. Given the monotonicity and convergence of , we have and . On the other hand, based on Theorem 3, we have , which leads to a contradiction.
VC2 Secondary Algorithm
SubProblem 1 (MaxMin Power Allocation)
Given , the maxmin power allocation problem is formulated as
(50)  
s.t.  (51) 
To solve Subproblem 1, the monotonicity of is first studied. The proof of the ZF is similar to that of the MF, hence only the case of the MF is shown here. The derivative is given by
(52) 
Since the order of magnitude for the bandwidth^{3}^{3}3In the operational wireless systems, the MIMOrelated techniques are generally used at the narrowband (subcarriers), hence the order of magnitude on basic scheduling resource is of kHz such as the resource block (180 kHz) in the LTErelated systems. is generally of kHz, we can obtain
(53) 
which implies that the function is increasing. The theorem capable of achieving the optimal maxmin power allocation is presented as follows [28].
Theorem 5
In order to maximize the minimum objective function value of Equ. (50), all VUEs should have the same objective value, hence we have
(54) 