I Introduction
With the rapid development of wireless communication systems, intelligent transportation systems (ITSs) have been widely studied in recent years. More and more vehicular services can be efficiently supported by the evolving wireless networks [1, 2]. As a typical dense scenario in vehicular networks, urban intersection is studied in this paper. In order to meet various vehicular requirements, there exist two categories of applications in urban environments, namely non and delaysensitive ones [3]. In general, the delaysensitive services are safetyrelated, and mainly focus on the performance metrics about low latency and high reliability, such as cooperative driving and road safety, etc. On the other hand, as for nondelaysensitive services, data rate is a key performance indicator.
Because of the poor deployment of roadside infrastructures, dedicated short range communication (DSRC) systems are paid less attention in current vehicular networks. Instead, long term evolution (LTE) and its beyond are regarded as the most promising solution to meet various vehicletoeverything (V2X) communications. Recently, the 3rd generation partnership project (3GPP) declares that LTEbased V2X services adopt PC5, Uu interface and their hybrid to implement information exchange.
A theoretical analysis about radio resource management for D2Dbased vehicletovehicle (V2V) communication is given in [4], where the scenario of cellular and vehicular users coexistence is studied in detail. However, the characteristics of mobility are not considered in that paper. Although the authors consider traffic model in [5], their optimization objective is just the delay without paying attention to the reliability. Moreover, their research scenario is focused on the highway. Besides the above work, some other problems about vehicular communications are also studied in [6, 7, 8, 9, 10, 11, 12], such resource allocation and performance analysis, etc.
Therefore, motivated by the above facts, this paper focuses on the scenario of urban intersection, and aims to investigate radio resource allocation policy to minimize the latency of delaysensitive services, where the corresponding reliability is considered at the same time. Furthermore, in order to reduce the complexity, a twostage allocation policy is also proposed, where the allocation based on the traffic density information (TDI) is separately considered. Finally, with the aid of traffic flow theory, we develop a delay utility function adopting macroscopic vehicular mobility model in this paper.
The remainder of this paper is organized as follows. In Section II, the system model and some assumptions are introduced. Section III first studies the allocation policy of Stage two based on channel state information (CSI) and queue state information (QSI). Then, Stage one based on TDI, namely intersubregion resource allocation is discussed in Section IV. Finally, Section V illustrates the simulation results and conclusions are drawn in Section VI.
Ii System Model
Iia Scenario Description
As shown in Fig. 1, consider an urban vehicular network with one base station (BS). Assume that each vehicle associating with BS is equipped with one receiving antenna and transmitting antennas. There exist two kinds of services in the network, namely non and delaysensitive V2V services. As for delaysensitive V2V services, LTEbased D2D communication is utilized. On the other hand, nondelaysensitive services can be provided via traditional LTE network. Note that we only pay attention to the uplink (UL) in this paper.
In order to efficiently allocate radio resources in dense urban intersection and reduce the complexity, we propose a twostage allocation policy. As illustrated in Fig. 1, the intersection is divided into four subregions. The first stage is to allocate the resources of each subregion based on the corresponding TDI. Here we assume that different subregions use orthogonal resources. The second stage is about the allocation among intrasubregion. In contrast to that of Stage one, Stage two uses reusable resources.
Assume that the number of non and delaysensitive vehicles in a subregion are and , respectively. Since the broadcast characteristic of delaysensitive services, a number of broadcast links are equivalent to one link for simplicity in this paper. Then the total number of links in the subregion is . Moreover, there are independent resource blocks (RBs) in the subregion. Each link can be allocated at most one RB. Based on the assumption in most existing works [13], the resource allocated to a delaysensitive link can be reused by at most one nondelaysensitive link.
IiB Channel Model
The network is assumed to work in slotted time , and we use slot to denote the time interval . Let denote the CSI matrix from transmitter to receiver on th RB during slot , where is the largescale fading coefficient containing the path loss and shadow, and
is the smallscale fading random variable. Assume that the elements of
are independent and identically distributed (i.i.d) complex Gaussian random variables, namely . Note that represents the receiver is BS. At last, let denote the network CSI at slot .IiC Queue Model
Each vehicle maintains one traffic queue with a finite queue length . Let denote the QSI (the number of bits) of vehicle at the beginning of slot . Hence, the queue dynamic is given by
(1) 
where denotes the traffic arrival at the end of slot , and the traffic departure at slot is given by . We assume that the traffic arrival is independent w.r.t. and i.i.d. over slots obeying a general distribution with mean . Let denotes the network CSI at slot .
IiD Performance Metrics
Each service has its specific communication requirements in vehicular network. Hence, it is necessary to study the performance metrics of different services. Let be the RB allocation at slot , the value of is defined as
(2) 
where and .
IiD1 Delaysensitive Service Metric
As for delaysensitive services, we first focus on the packet reception ratio (PRR) which is defined in [1]. So we have the following definition.
Definition 1 (Packet Reception Ratio)
Let denote the number of the neighborhoods of vehicle at slot , then the PRR is defined as the ratio of successful reception among , i.e.,
(3) 
where is the receiving signaltointerferenceplusnoise ratio (SINR) of vehicle among , is the transmit power of vehicle , and is the power of additive white Gaussian noise. Here successful reception is considered as the fact that SINR is greater than or equal to a threshold . Specially, the average PRR can be calculated by the following formula, i.e.,
(4) 
The PRR is a good proxy for reliability. As for delay, we have the following definition.
Definition 2 (Average Queue Length)
Assume that is a discrete time queue, then the average queue length under a policy is given by
(5) 
Furthermore, if the average queue length , the discrete time queue is strongly stable. A network of queues is stable if all individual queues of the network are stable. Based on the Little’s law, we can also calculate the average delay.
IiD2 Nondelaysensitive Service Metric
With the regard to the nondelaysensitive services, we mainly focus on the data rate. In order to simplify the communication model, the perfect CSI at the receiver and transmitter are assumed. Therefore, the maximum achievable data rate of vehicle at slot is given by
(6) 
where denotes the bandwidth of one RB, and can be calculated as
(7) 
where . Similarly, we can also utilize Equ. (4) to calculate the average data rate .
Iii Intrasubregion Resource Allocation
Iiia Resource Allocation Policy
In general, a resource allocation policy is a mapping function from the system state to the resource allocation actions. A policy is called feasible if the relevant actions satisfy the required constraints. As previously mentioned, our policy of intrasubregion resource allocation satisfies the following constraints, i.e.,
(8)  
(9)  
(10) 
IiiB Problem Formulation
In this paper, our main objective is to minimize the latency of delaysensitive services, while satisfying corresponding reliability requirements and data rate requirements. Thus, we consider the following optimization problem, i.e.,
Problem 1 (Delayoptimal Policy for Intrasubregion Resource Allocation)
Given a set of feasible policies , and assuming and are the minimum data rate of all nondelaysensitive vehicles and reliability requirements of all delaysensitive vehicles, the optimization problem is then formulated as
(11) 
where , , and is the positive weighted factor for each delaysensitive vehicle.
In general, with the regard to a unichain policy
, the induced Markov chain is ergodic and there is a unique steady state distribution
. Hence, we have(12) 
where denotes the average delay.
IiiC Elements of MDP
(14) 
The optimization problem is formulated as an infinite horizon average cost constrainedMarkov decision process (MDP). In general, MDP is characterized by five elements, i.e., system state space, action space, state transition kernel, average cost function and constraint conditions as follows.

System State Space: .

Action Space: , which is a set of unichain feasible policies under the system state .

State Transition Kernel: . Since the property of Markov process, we have
(13) 

Average Cost Function and Constraint Conditions: They are described in detail at Equ. (11).
Because of the constraints in Problem 1, the standard Lagrangian approach is utilized here. Then the constrained MDP can be transformed to the unconstrained MDP, and the Lagrange dual function is also defined as Equ. (IIIC) listed at the top of this page, where , , and are the Lagrange multipliers. Therefore, the average cost function of the corresponding unconstrained MDP can be obtained from Equ. (IIIC). As a rule, the delayoptimal policy can be obtained by solving the Bellman equation [14], we discuss it in the next subsection.
IiiD Optimal Solution of MDP
As previously mentioned, we have converted Problem 1 into the unconstrained MDP, thus it can be solved by Bellman equation expressed as follows.
Lemma 1 (Bellman Equation)
For any given , , and , if there exist a scalar
and a vector
satisfy the Bellman equation for the delayoptimal unconstrained MDP in Equ. (IIIC), namely(15) 
then is the optimal average cost perstage, and the optimal policy for Problem 1 is , which minimizes the R.H.S. of Equ. (1) for any state . Similarly, as for a unichain policy, there is a unique solution to Equ. (1). Therefore, we only consider the unichain feasible policy in this paper.
It is well known that the system state space gradually becomes huge with the increasing number of vehicles. Therefore, in order to reduce the complexity, the reducedstate Bellman equation can be adopted to solve Problem 1 [15], which only takes advantage of the QSI. Then we have the following lemma, i.e.,
Lemma 2 (ReducedState Bellman Equation)
In general, the equation can be given by
(16) 
where , and are conditional potential function, average cost perstage and average transition kernel, respectively.
Iv Intersubregion Resource Allocation
Iva Fundamentals of Traffic Flow Theory
According to the different traffic characteristics, vehicular mobility models are usually classified into two categories, namely macroscopic and microscopic models. Each category of model focuses on different performance indicators. The macroscopic models generally describe the average behavior of many vehicles at specific location and time, treating traffic flow as fluid dynamics. Therefore, vehicular density and mean velocity are considered in the macroscopic models, which raises the traffic flow theory. However, the microscopic models describe the precise behavior of each system entity (i.e., vehicle or driver), hence they are more complicated than the macroscopic models.
In order to allocate wireless resources efficiently among intersubregion, we model the TDI adopting the traffic flow theory. It is well known that there are many macroscopic models, such as Greenshield’s model, Greenberg’s model, Underwood’s model, etc. For the sake of simplicity, we utilize linear Greenshield’s model in this paper. Here we give a brief introduction about Greenshield’s model. In general, there exist two parameters in the Greenshield’s model, namely free flow speed and jam density [16]. The relationship between flow and density is given by
(17) 
IvB Delay Utility Function
In order to reflect the influence of TDI on delaysensitive services, we construct a delay utility function with the help of the Greenshield’s model. As illustrated in Fig. 2, the utility function should satisfy the following properties, i.e.,

When , the flow increases with the increase of density, hence the number of delaysensitive services increases, and the delay requirement gradually increases; and

When , the flow decreases with the increase of density, hence the delay requirement gradually decreases for the same reason; and

Furthermore, no matter how much the value of , the delay requirement is not equal to zero. Meanwhile, the requirement is normalized for the sake of simplicity; and

Let denote the ratio of allocation for subregion . For any , the allocation efficiency increases with the increase of .
In conclusion, the utility function is given by
(18) 
where are constants, which is related to the practical traffic condition. The logarithmic utility function can ensure the fairness, and thus is employed in this paper. In Equ. (18), the first and second terms represent the normalized delay requirement and the allocation efficiency, respectively. Note that the utility function is just a proxy for delay, not the true value.
IvC Problem Formulation
Comparing to the CSI and QSI in Stage two, the TDI in Stage one changes at a longer timescale. Therefore, we can formulate a new problem independent of Problem 1. The main objective of Stage one is to maximize the sum of delay utility defined in Equ. (18) based on the corresponding TDI. The following optimization problem is considered, i.e.,
Problem 2 (Delayoptimal Policy for Intersubregion Resource Allocation)
Given the TDI of four subregions , the utility maximization problem of Stage one is then formulated as
(19) 
IvD Resource Allocation for Stage One
Since the logarithmic function is convex, the compound utility function is convex. We can solve Problem 2 utilizing convex optimization theory [17]. First of all, we write the Lagrange function of Problem 2 as follows.
(20) 
where and are the Lagrange multipliers. Therefore, based on the KarushKuhnTucker (KKT) condition, we get
(21) 
where . Then, the utility maximization resource allocation can be given by
(22) 
where the Lagrange multiplier is determined by equation .
IvE Resource Allocation Algorithm for Urban Vehicular Network
V Simulation Results
In order to evaluate the performance of the proposed allocation algorithm, part of the simulation results are shown in this section. For the purpose of better illustration, some simulation assumptions are summarized in Table as follows.
Parameter  Assumption  
Bandwidth  5 MHz  
2 transmitting antennas  
Average packet size 


Average arrival rate  5:5:30 packets/s  
Queue size  10 packets  
Scheduling slot  1 ms (one slot in LTE)  
TDI update interval  500 ms  
2 
Fig. 3 illustrates that the performances of average delay versus average arrival rate. As can be seen from Fig. 3, the average delay increases with the increasing TDI , where the high and low TDI are generated by Uniform(0,0.5) and Uniform (0.8,1.2), respectively. Moreover, we also find that the optimal policy solved by the original Bellman equation has the best delay performance at the expense of high implementation complexity. As for the proposed algorithm, it acquires an asymptotically optimal performance, but its complexity has a significant decrease, which is very satisfactory. In particular, when packets/s, the proposed algorithm has the approximately equal performance with the optimal one in the case of high TDI.
Vi Conclusion
In order to reduce the allocation complexity in dense urban intersection, this paper proposed a twostage allocation algorithm, where Stage one utilized the TDI of corresponding subregion to maximize the delay utility. While for Stage two, its main optimization objective was to minimize the latency of delaysensitive services, meanwhile satisfying the corresponding reliability requirements and data rate requirements. Finally, comparing to the optimal solution of MDP, simulation results illustrated that the proposed scheme can acquire an asymptotically optimal performance with the reasonable complexity comparing to the optimal one.
Acknowledgment
This work is supported by the National High Technology Research and Development Program of China under Grant 2014AA01A705, the “Research and evaluation on key technologies of 5G mobile wireless transmission”, Ministry of Education–China Mobile Research Foundation under Grant MCM20150101, the National Key Scientific Instrument and Equipment Development Project under under Grant 2013YQ20060706, the National Natural Science Foundation of China under Grant 61331009.
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