Given the rapid development of the Internet of things (IoT), the design of radio access networks is undergoing a paradigm shift from promoting spectral-efficiency and energy-efficiency 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 safety-related 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 safety-related applications. Ultra-reliable and low-latency communications (URLLCs) exchanging near-real-time information are also required to offer assistance to control systems . 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 ground-breaking 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 device-to-device-based vehicle-to-vehicle communications can be found in, where both the transmission latency and reliability are considered simultaneously. However, the study of vehicular mobility was not considered in . 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 multiple-antenna 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, stochastic network calculus  and queueing theory . Based on the Markov decision process framework, a minimum queueing delay-based radio resource allocation scheme is proposed in  for vehicle-to-vehicle 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  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  for low-latency machine-to-machine applications. Furthermore, a URLLC cross-layer optimization framework is proposed in  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  or by the probability of the queueing latency exceeding the tolerance threshold . 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 multi-antenna diversity gain. Hence, large-scale 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 large-scale fading-based 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 near-instantaneous CSI rapidly becomes outdated and its more frequent update imposes a high overhead, hence reducing the communication efficiency; and
Compared to the time-scale of CSI fluctuation on the millisecond level, that of the road-traffic fluctuation is lower. Requiring resource allocation updates on the order 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 reducing the signaling overhead, where a macroscopic road-traffic model is adopted for charactering the relationship between the vehicular velocity and road-traffic density.
We first derive the transmission latency based on finite blocklength theory for the matched filter (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 URLLC system. In particular, based on the long-term timescale of road-traffic density, Stage 1 aims for optimizing the worst-case transmission latency by appropriately setting the system’s bandwidth. 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 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 road-traffic 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 twin-timescale 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.
Uppercase boldface letters and lowercase boldface letters denote matrices and vectors, respectively, whileis a diagonal square matrix whose main diagonal is formed by vector . Furthermore, , , and represent the transpose, conjugate, conjugate transpose and pseudo-inverse 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, while
is the complex Gaussian distribution with mean
and real/imaginary component variance.
Ii Preliminaries: Finite Blocklength Theory
The Shannon-Hartley theorem quantifies the error-free capacity at which information can be transmitted over a band-limited channel in the presence of noise and interferences,
where is the bandwidth and is the signal-to-interference-plus-noise ratio (SINR). This capacity can only be approached at the cost of excessive coding latency and complexity, i.e.,
where is the so-called 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]:
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 URLLC-related 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 single-antenna aided VUEs (massive MIMO with ). The system operates in the time-division duplex (TDD) mode.
Iii-a Road-Traffic Model
According to traffic theory, the road-traffic 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 road-traffic 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 road-traffic on communication, the one-dimensional macroscopic model is adopted here, which is also known as the Lighthill-Whitham-Richards (LWR) model . Given the road-traffic density and the road-traffic velocity , the road-traffic flow rate (flux) is generally given by
Numerous models have been proposed for characterizing how the vehicular velocity depends on the road-traffic density. In this paper, the Underwood model is employed, and its speed-density function can be written as :
where is the free-flowing velocity and is the maximum density. Moreover, the road-traffic density can be used for modeling the number of VUEs, according to .
Iii-B Channel Model
All channels experience independent quasi-static 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 large-scale 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.
Iii-C Transceiver Model
It is widely exploited that low-complexity 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
and with . The normalized form of is given by with
As a result, given the symbol vector with , the transmitted signal vector of all VUEs can be written as
where is the power allocation matrix and the total power at the BS is .
The received signal vector is given by
where is the i.i.d. additive white Gaussian noise with . The received signal of the -th VUE can be written as
Hence, the SINR of the -th VUE is given by
Recall from that the instantaneous can be rewritten as
Based on , we have the following distributions, i.e.,
Then can be finally expressed as
Iv Problem Formulation of Twin-Timescale Radio Resource Management
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 global CSI. A twin-timescale resource allocation problem is formulated in this section for reducing signaling overhead. Specifically, according to the road-traffic density, the first stage is constructed for optimizing the worst-case latency by setting the system’s bandwidth from a long-term timescale. As for the second stage of the short-term timescale, the BS will allocate the total power based on the large-scale fading CSI (equivalent to location information) to minimize the maximum transmission latency for ensuring fairness amongst all VUEs.
Iv-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, because reporting the instantaneous V2I CSI to the BS is not practical. As a result, only a large-scale fading-based allocation problem was considerer in . The theorem for quantifying the maximum achievable rate is first formulated as follows.
See Appendix A.
According to Theorem 1, we can obtain the following corollary for illustrating the transmission latency.
Given the desired rate and the reliability , the transmission latency of the -th VUE can be expressed as
With respect to the reliability, with obeys the following properties,
and . Thus, a higher reliability (also known as ) leads to a higher latency, and vice versa.
Iv-B Stage 1: Long-Term Timescale Allocation
The system bandwidth determined in Stage 1 can be viewed as part of the system design. Based on the long-term timescale road-traffic density, the objective of Stage 1 is to optimize the worst-case transmission latency. Without loss of generality, the worst-case latency is considered in the scenario of the road-edge location (both ends of the road) with equal power allocation. The large-scale fading of the road-edge location is given by
where is a constant related both to the antenna gain and to the carrier frequency, while is the path loss exponent. Then, with , the worst-case SINR can be expressed as
where and are the power spectral density of the transmitted signal and noise, respectively, while is the minimum requested estimation accuracy, namely . Therefore, the worst-case transmission latency is finally given by
where and .
On the other hand, with Equ. (6), the maximum Doppler frequency is given by
where is the carrier frequency, and is the speed of light. Hence, the typical coherence time can be calculated as :
On the basis of the above derivations, the following optimization problem is formulated.
Problem 1 (Long-Term Timescale Bandwidth Allocation)
Let denote a small threshold constant. Given the road-traffic density , the long-term timescale bandwidth allocation problem of Stage 1 is formulated as
Iv-C Stage 2: Short-Term 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 short-term timescale large-scale fading CSI of all VUEs for minimizing the maximum transmission latency. Then the following optimization problem is formulated.
Problem 2 (Short-Term Timescale Power Allocation)
Given the desired rate , reliability , total power and the large-scale 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.,
V Twin-Timescale Resource Allocation Algorithms for V2I URLLC Scystem
In this section, the solutions to the above pair of problems are first studied, and then efficient twin-timescale allocation algorithms are proposed for our V2I URLLC system.
V-a Solution to Problem 1
V-B Solution to Problem 2
V-B1 Problem Transformation
Problem 2 can be transformed into the following form:
V-B2 Property of Objective Function
Recall from Equ. (21) that can be expressed as
When the MF or ZF precoder is employed in the downlink of our V2I URLLC system, is a quasi-concave111Quasi-concave function: Let be a convex set and . Then, for any and , is quasi-concave if . function of .
See Appendix B.
Given the definition of quasi-concavity, is also a quasi-concave function. By contrast, even if is pseudo-concave222Pseudo-concave function: Let be a convex set and . Then, for any , is pseudo-concave if it is differentiable and ., still cannot be pseudo-concave, since the function is not differentiable.
V-B3 Optimal Solution
Since the objective function of Equ. (36) is QC, the Karush-Kuhn-Tucker (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,
Intuitively, is strictly monotonically decreasing and has a unique zero root.
For any , we have with , and the equality holds when .
See Appendix C.
Based on Lemma 1, the following theorem describing the optimal allocation is formulated.
For and , is the optimal solution of Equ. (36) if and only if
It relies on proving both sufficiency and necessity.
Firstly, let be the optimal power allocation,
which yields Equ. (40) and .
Conversely, let , i.e.,
which shows that
Thus is the optimal power allocation.
V-C Twin-Timescale 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.
V-C1 Primary Algorithm
Based on Dinkelbach’s method , a twin-timescale iterative algorithm is put forward for jointly solving Problem 1 and 2, which is described in Algorithm V-C1. The following theorem illustrates the convergence and optimality of Algorithm V-C1.
Algorithm V-C1 can converge to the optimal power allocation.
where (a) follows from
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 V-C1 converges.
The optimality can be proved by the method of contradiction. Let be an improved, but sub-optimal 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.
V-C2 Secondary Algorithm
Sub-Problem 1 (Max-Min Power Allocation)
Given , the max-min power allocation problem is formulated as
To solve Sub-problem 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
Since the order of magnitude for the bandwidth333In the operational wireless systems, the MIMO-related 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 LTE-related systems. is generally of kHz, we can obtain
which implies that the function is increasing. The theorem capable of achieving the optimal max-min power allocation is presented as follows .
In order to maximize the minimum objective function value of Equ. (50), all VUEs should have the same objective value, hence we have