I Introduction
Intelligent transportation system (ITS) revolutionizes the provisioning of diverse applications associated with driving safety, traffic management, and infotainment [1]. These applications are beyond the farfetched goals of academic, industry, and standardization groups, which aim to streamline the innovative operation of vehicle, facilitate safe and ecofriendly driving, and offer ubiquitous infotainment services for commuting passengers [2].
Typically, potential connectivity disruption as a result of high vehicle speed, timevarying vehicle density, and limited intervehicle contact time, confines vehicletovehicle (V2V) communications to applications and services with short communication range [3]. Fortunately, vehicletoinfrastructure (V2I) communications represent a viable solution to bridge longrange vehicular connectivity by introducing stationary network entities, e. g., road side unit (RSU), to exchange data with vehicles [4]. By leveraging the complementary features of infrastructureless V2V communications and infrastructureassisted V2I communications, the use of hybrid vehicular communication, namely vehicletoeverything (V2X), has been envisioned as a fullfledged solution to capture the connectivity, efficiency, and scalability of vehicular networks [5].
Fifth generation (5G) mobile communications target to support efficient data delivery with bulky data rate, high reliability, and low latency for different V2X services and applications [6]. Specifically, realtime applications, such as collision avoidance and lanechange/merge notification, bring the requirement of low latency. In addition, nonemergency services, such as video streaming, demand high data rate to fulfill capacity burst. Therefore, an efficient data delivery scheme to meet the diverse requirements of V2X communications is expected to be designed properly taking into account the coexistence of low latency and high data rate [7].
Nevertheless, data delivery in V2X networks is particularly challenging, as its performance depends highly on the efficiency of data routing [8]. On the one hand, pure intervehicle data delivery may introduce nonnegligible latency because of frequent disconnection. On the other hand, the limited coverage of each RSU is a major concern of pure interRSU data delivery. Therefore, the design of multihop routing algorithm that minimizes latency as well as maximizes data rate becomes an interesting and challenging topic.
Ia Related Works
Multihop routing for data delivery, particularly for vehicular networks, has been intensively investigated by recent research efforts [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The knowledge of vehicular trajectories plays a key role for optimal data delivery, where the performance of routing algorithm relies heavily on the accuracy of vehicle mobility prediction [9, 10, 11, 12, 13].
It has been envisioned that the assistance of infrastructure facilitates data delivery by latency improvement [14, 15, 16, 17]. However, these works either assumed that the latency of V2V and V2I transmission can be ignored, or limited the latency to be considered in a single hop between vehicles, which is not proper for data delivery in largescale vehicular networks where multihop transmissions are expected. More importantly, models applied to the above works depend on the prerequisite that the size of packets transmitted on V2V and V2I link is small enough, such that the data rate for delivering these packets is omitted. This assumption does not hold for the data delivery of services that rely on abundant data rate to guarantee the enormous requirement of data volume.
When it comes to achieving reasonable data rate for data delivery in V2X networks, a rich body of earlier studies have tackled the problem of how “mobility improves data rate” in vehicular networks [18, 19, 20, 21]. With the exception of some studies that contributed to a limited investigation of latency performance [18], none of the abovementioned works to date has considered the tradeoff between low latency and high data rate of data delivery in V2X networks, which is supposed to be a key enabler in fully exploit the mobility of vehicles complemented by the stability of infrastructures to improve data rate performance while keep latency tolerable.
Storecarryandforward strategy, where data is stored at intermediate nodes along delivery and forwarded at a later time to another intermediate station or the final destination, has been recently recognized as a promising evolution path to improve data delivery efficiency, either in latency or data rate [22, 23, 24, 25]. Nevertheless, except the dropbox functionality that allows data to be stored with some cost, none of the aforementioned studies has considered a realistic model of RSU for providing V2I communications in terms of link establishment and resource allocation. Moreover, the ability of V2X networks to support connections for other devices besides vehicular users (e.g. cellular users), which is one of the key considerations in optimizing the overall system performance in this paper, has not been addressed in any of these works.
IB Contributions
Different from the previous studies [22, 23], where RSU is simply assumed to be a dropbox for data collection and temporary storage, in this paper we consider RSU as a network entity that supports both V2I and cellular communications and serves both vehicular and cellular users. The main contributions of this paper are summarized as follows:

Development of an accurate analytical framework for data delivery in V2X networks: The framework first considers hopwise latency and data rate, which are derived taking into account divergent vehicle mobility patterns and data forwarding behaviors at each hop. Based on these, the expected endtoend (E2E) latency and data rate are obtained by adding the hopwise latency and minimizing the hopwise data rate, respectively.

Formulation of optimization problems that maximize the weighted sum of latency and data rate: Unlike some of the previous works that revolved around the feasibility study of data delivery without explicitly addressing performance optimization ([10, 15, 17]), we obtain mathematical expressions of both hopwise and E2E latency/data rate, based on rigorous derivations, and formulate optimization problems that maximize the weighted sum of latency and data rate considering both global and distributed scenarios, where the weighted sum is optimized in E2E manner and hopwise manner, respectively.

Leveraging the optimization problems and the corresponding solutions to propose multihop routing algorithms: The derived expressions of latency and data rate are transformed into closedform for verifying the convexity of the proposed optimization problems, which are then solved by convex optimization theory. Based on these, multihop routing algorithms to select the optimal route are proposed for both global and distributed data delivery in terms of the weighted sum maximization.

A detailed systemlevel performance evaluation for data delivery: Extensive simulations are conducted under numerous system parameters to demonstrate the efficiency of the proposed algorithms in achieving lower latency and higher data rate compared to classical vehicular routing algorithms. The impact of broadcast scheme, vehicle arrival rate, and backhaul availability on the delivery performance are also analyzed.
The remainder of this paper is organized as follows: Section II presents the system model and Section III addresses the optimization problem formulation. In Section IV, we solve the formulated problems and propose the corresponding routing algorithms. The proposed algorithms are then evaluated by extensive simulations in Section V, followed by a summary concluding this paper in Section VI.
A conference version of this paper has appeared in [26]. The current paper extends the previous work with the development of distributed data delivery optimization, the design of distributed routing algorithm, and the support of wireless backhaul in data delivery. The current paper also includes all derivations, discussions of extensions, and more detailed simulations.
Ii System Model
In this section, we introduce the network, traffic, data forwarding, and radio models considered for finding the best route and optimizing the data delivery.
Iia Network Model
We envision a scalable V2X network for data delivery incorporating both RSUassisted and carryandforward strategies. The network is geographically and equally divided by the coverage of RSUs. Data is generated by a source node and to be carried and forwarded by vehicles to a destination node . In general, and can be either RSU or vehicle. Without loss of generality, we assume that both and are RSUs. The traffic information of all vehicles in the networks are available at RSUs, however not all RSUs are necessarily interconnected^{1}^{1}1The data delivery problem with fully interconnected RSUs can be solved by routing algorithms for fixed network topology, which have been thoroughly studied and are trivial for V2X networks. Nevertheless, in this paper we also consider partially and/or fully interconnected RSUs that enable data forwarding via wireless backhaul between RSUs in Section V.. We further assume routes, denoted as a set , between and . A route could be composed of multiple hops and each hop refers to a road segment within the coverage of corresponding RSU.
IiB Traffic Model
Similar to the previous works [18, 7, 22, 23], we assume that vehicles move at the same speed and stay within each hop for a constant duration ^{2}^{2}2This assumption holds well for highway or rural area, where RSUs are equidistantly deployed and vehicles on each lane move at the same speed with slight deviation. For urban scenario where diverse coverages and speed limits are expected, the networks can be partitioned into blocks and in each block, vehicles are likely to move at the similar speed due to speed limit along the isometric road segment between RSUs, hence the assumption is also valid.
. We further adopt the widely used traffic model where the number of vehicles arrives at a hop and heads to the next hop is Poisson distributed
[18, 7].IiC Data Forwarding Model
Consider one route out of the route set . As there might not exist a single vehicle that moves along all hops of the considered route, the data would need to be forwarded by the vehicle that carries the data, when it no longer heads to , to another vehicle that does. We referred to the vehicle that carries data and moves along a hop of the route as the courier of the hop. If the courier moves towards the next hop of the route when leaving the current hop (the information about whether moving towards the next hop or not can be acquired by e.g. navigation or preconfigured route of autonomous vehicle), the data is carried by the courier to the next hop and consequently the courier of the next hop is still this vehicle. Otherwise, when arriving at the current hop, the courier tries to discover another vehicle that moves towards the next hop, referred to as the candidate of the current hop, to forward the data. The discovery maintains at most a duration (the time for discovery process cannot exceed ), referred to as the global candidate discovery duration, and once succeeds, the courier forwards the data to the candidate via V2V communications within the remaining time it stays in the hop. Otherwise, it sends the data to the RSU that covers the hop via V2I communications within , then the RSU will find a suitable candidate heading to the next hop and forward the data.
IiD Radio Model
For candidate discovery, we further assume that when arriving at a hop, the courier starts to broadcast beacon for the candidate discovery assisted by RSU. Information encapsulated in the beacon could include, e.g., the direction of the next hop. Once the beacon has been received and successfully decoded, with error probability
, the candidate sends feedback to report the successful reception. The feedback is then decoded by the courier, again with error probability , and in case of incorrect decoding or no feedback detected, the courier repeats the discovery trial until a communication link between the courier and the candidate is established within .Iii Problem Formulation
Without loss of generality, in the following sections we focus on a single route, which is referred to as the typical route and randomly picked from the route set , between and . We further denote the number of hops of the typical route as and the th hop of the typical route as hop , respectively. Analysis of other routes can be derived similarly.
Iiia EndtoEnd Latency
As mentioned in Section IIB, the arrival of vehicles at a hop follows a Poisson distribution. Here, we denote the arrival rate of vehicles that arrive at hop and head to hop as . Based on the traffic model and data forwarding model described in Section IIB and Section IIC, respectively, the E2E latency of the typical route is described in the following scenarios. Examples including courier moves to the next stop, successful discovery, and failed discovery are illustrated in hop 1, 2, and 3 of Fig. 1, respectively.
IiiA1 Courier Moves Towards the Next Hop
In this scenario, the courier carries the data to the next hop (hop ), as depicted in hop 1 of Fig. 1. Hence, there is no need for candidate discovery. Denoting the number of wayout directions except Uturn for hop as . Then, the probability of the event “Courier of hop moves towards hop ”, denoted as , satisfies
(1) 
which means when leaving hop , the courier randomly selects a direction with equal probability. Clearly, the hopwise latency of hop for the event “Courier of hop moves towards hop ”, denoted as , is the duration of the courier staying in the hop, namely
(2) 
IiiA2 Courier Succeeds in Candidate Discovery
In case the courier is not heading to the next hop, a candidate discovery for data forwarding is carried out by the courier within the discovery duration , as illustrated in hop 2 of Fig. 1
. When the number of arrivals in a given time interval follows Poisson distribution, interarrival times are known to have the exponential distribution
[27]. Thus, the probability of the event “A candidate heading to hop arrives at hop within on condition that courier of hop does not move to hop ”, denoted as , satisfies(3) 
where represents the time between the courier arrives at hop and a candidate heading to arrives at hop .
When arriving at hop , the courier starts to broadcast beacon to discover a candidate for data forwarding, as described in Section IID. We denote the time of one discovery trial as , which includes beacon broadcasting and feedback receiving time. Then, within the candidate discovery duration , there would be maximally rounds of discovery trials, where represents the floor function. Then, the probability of the event “Courier of hop successfully discovers a candidate”, denoted as , satisfies
(4) 
Similarly to the previous scenario, the hopwise latency of hop for the event “Courier of hop successfully discovers a candidate”, denoted as , satisfies
(5) 
IiiA3 Courier Fails in Candidate Discovery
In this scenario, the courier has to send the data to RSU as no proper candidate can be discovered within , as drawn in hop 3 of Fig. 1. Accordingly, the probability of the event “No candidate towards hop arrives at hop within on condition that courier of hop does not move to hop ”, denoted as , satisfies
(6) 
Then, the probability of the event “Courier of hop fails to discover a candidate”, denoted as , can be derived as
(7) 
As the data is not forwarded to any candidate within , the courier will transmit the data to RSU in the remaining time . Specifically, the courier has to stay in the hop for anyway to move through the hop, which means that except the time for candidate discovery , the remaining time for the courier to transmit the data to RSU is . Afterwards, the RSU will find an appropriate candidate to forward the data and once found, the candidate receives the data from the RSU when moving along the hop within . Therefore, the summarized hopwise latency of hop for the event “Courier of hop fails to discover a candidate”, denoted as , is calculated as
(8) 
Here, represents the time for RSU to find a candidate. Note that the first does not overlap with , as in the remaining time after the courier has failed in candidate discovery, i.e. , the RSU is receiving the data from the courier and not able to start to find a candidate.
Combining these scenarios, the expected hopwise latency for data delivery, denoted as , can be derived as
(9) 
In particular, is also exponentially distributed due to Poisson distribution of vehicle arrivals, and correspondingly we have
(10) 
Finally, the expected E2E latency of the typical route, denoted as , is provided as
(11) 
IiiB EndtoEnd Data Rate
As mentioned in Section I, we consider a realistic model of RSU in terms of providing communication links to both vehicular and cellular users, which is different from the dropbox functionality considered in [22, 23, 24, 25] that only allows data to be stored with some cost. Similar to Section IIIA, three scenarios are addressed here for the analysis of the expected data rate of the typical route.
IiiB1 Courier Moves Towards the Next Hop
In this scenario, RSU is not requested by courier for assisting candidate discovery. Therefore, the RSU exclusively serves cellular users. Denoting the achievable data rate of the services provided for cellular users as , the achievable hopwise data rate of hop for the event “Courier of hop moves towards hop ”, denoted as , is calculated as
(12) 
Note that indicates the overall achievable data rate and can be obtained by e.g. taking average of data rates among all cellular users in the network.
IiiB2 Courier Succeeds in Candidate Discovery
When a communication link has been established between courier and candidate, the carried data is transmitted from the courier to the candidate via V2V communications with data rate , which can be similarly obtained as by e.g. taking average of data rates among all V2V communication links. Denoting the actual candidate discovery time for the courier of hop as , the achievable hopwise data rate of hop for the event “Courier of hop successfully discovers a candidate”, denoted as , can be written as
(13) 
Here, the data rate consists of two parts. The first term of the right hand side of (13) indicates that the V2V communication link maintains a duration of , and the amount of data can be transmitted is calculated as . As the RSU of hop is not aware of the actual candidate discovery time , it preserves for the candidate discovery and correspondingly, the data rate provided by the RSU for cellular users is calculated as .
IiiB3 Courier Fails in Candidate Discovery
In this scenario, the data is first transmitted to RSU after failed candidate discovery within and then forwarded to a proper candidate. The RSU of hop is correspondingly exclusively associated with the courier for candidate discovery and data reception, until the data is successfully forwarded to a candidate. Therefore, the amount of data transmitted from the courier to the RSU via V2I communications is , where indicates the data rate of V2I communications and can be obtained similarly to and . After finding a candidate within and forwarding the received data to the candidate within (the data rate between the candidate and the RSU is also assumed to be ), the RSU is able to serve cellular users with amount of data . In summary, the achievable hopwise data rate of hop for the event “Courier of hop fails to discover a candidate”, denoted as , can be written as
(14) 
Combining these scenarios, the expected hopwise data rate for data delivery, denoted as , can be derived as
(15) 
For a multihop route between and , the achievable data rate is determined by the “weakest” hop in which the lowest rate is achieved. Therefore, the expected E2E data rate of the typical route, denoted as , is calculated as
(16) 
IiiC Global Data Delivery Problem
Based on the analysis in Section IIIA and Section IIIB, it is clear that the global candidate discovery duration plays a vital role in determining the E2E latency and data rate. On the one hand, a larger allows courier a better chance to find a candidate, however leaves less time for data forwarding, which leads to data rate degrade. On the other hand, a decreased brings more time for data forwarding and correspondingly enhances the achievable data rate, while an increased latency is expected due to less opportunity for successful candidate discovery. Therefore, a tradeoff between latency and data rate to optimize the overall performance can be achieved by the adaptation of . The global data delivery optimization problem is formulated as follows:
Problem 1.
(Global weighted sum maximization)
(17) 
Here, is the weight parameter. Distinguished by various use cases and application services, can be flexibly adjusted, e. g., may refer to latencytolerant but ratesensitive use case, while may indicate realtime services which are keen to latency with relative low demand on data rate. Here, the term “global” indicates the universal configuration of the candidate discovery duration, where an identical candidate discovery duration is applied to all hops in the typical route. Note that here and are normalized in , where the motivation of the normalization lies in the fact that data ranges and units of latency and data rate are not directly comparable [28].
IiiD Distributed Data Delivery Problem
In addition to the global data delivery problem, we further propose the distributed data delivery problem, where the weighted sum of latency and data rate is hopwisely maximized, compared to the E2Ewise maximization addressed in Section IIIC. The motivation of the distributed data delivery comes from the fact that the arrival rates , the actual candidate discovery time for RSU , and the actual candidate discovery time for vehicle , are diverse in each hop. Therefore, hoptailored maximization may benefit from a hopspecific configuration of discovery duration, as the hopindividual weighted sum could bring potential gain in increasing data rate while reducing latency.
By replacing in (3), (4), (7), (13), and (14) with , which we refer to as the hopwise candidate discovery duration of hop , the expected hopwise latency and data rate, denoted as and , respectively, satisfy
(18) 
and
(19) 
where the terms with hat in (18) and (19) indicates corresponding expressions in (3)–(14) by replacing with .
Finally, the distributed data delivery optimization problem is formulated as follows:
Problem 2.
(Distributed weighted sum maximization)
(20) 
Here, is the weight parameter. Similar to the global data delivery problem, a tradeoff between latency and data rate to optimize the overall performance can be achieved by the adaptation of .
Iv Data Delivery Optimization and Routing Algorithm Design
In this section, we propose solutions of the data delivery optimization problems formulated in Section III.
Iva Reformation of Problem Formulation
The optimization problem 1 and 2 are formulated in a sophisticated way in Section III, and it is relatively hard to verify the convexity of the optimization problems determined in (17) and (20). In this subsection, the derived latency and data rate are reformed into closedform expressions.
IvA1 Reformation of the EndtoEnd Latency
IvA2 Reformation of the EndtoEnd Data Rate
For simplicity, let , , , , , and . Then, similar to the reformation of the expected hopwise latency, the expected hopwise data rate, which is derived in (15), can be transformed as
(23) 
As the expression of in (23) is a combination of multiplication and summation of hopdependent terms (terms with the subscript ), finding the closedform of the minimum of , namely , is still a very complicated problem. Therefore, we further reformed as
(24) 
where
(25) 
(26) 
and
(27) 
Similarly, , , and represent the expected data rate of the corresponding events and are provided by
(28) 
(29) 
and
(30) 
Now, it is clear that the main task of finding the closedform of the expected E2E data rate is to reform the minimization operators addressed in , , and .
a) Closedform of .
Lemma 1.
Given a set of i.i.d geometrically distributed random variables
with parameter , the probability mass function (PMF) of , denoted as , satisfies(31) 
Proof.
The CDF of the maximum Y can be calculated as:
(32) 
Hence, the PMF of Y can be derived as
(33) 
∎
Considering the scenario of successful candidate discovery in all hops, the E2E data rate is limited by the “weakest” hop, at which the longest time for candidate discovery is consumed. Consequently, the goal of finding turns to find , which is solved by the following theorem.
Theorem 1.
Let and denote the actual number of discovery trials in hop and the maximum of , respectively. Let . Then, the closedform of satisfies
(34) 
where
(35) 
Proof.
can be derived as
(36) 
where . As remains constant, finding turns further to find . Given Lemma 1, the PMF of , denoted as , is calculated as
(37) 
Then , we have
(38) 
∎
b) Closedform of .
The E2E data rate of this scenario can be solved similarly. Specifically, finding turns to find where follows a exponentially distribution with parameter .
Lemma 2.
Given a set of i.i.d exponentially distributed random variables with parameter
, the probability density function (PDF) of
, denoted as , satisfies(39) 
Proof.
The CDF of the maximum Z can be calculated as:
(40) 
Hence, the PDF of Z can be derived as
(41) 
∎
Given Lemma 2, the closedform of the expected E2E data rate is solved by the following theorem.
Theorem 2.
Let and , respectively. Then, the closedform of satisfies
(42) 
where
(43) 
Proof.
can be derived as
(44) 
Given Lemma 2, the PDF of , denoted as , can be written as
(45) 
Then, we have
(46) 
∎
c) Closedform of .
The derivation of the closedform of the expected E2E data rate is a bit tricky as hop latency for this scenario follows a combination of geometric distribution (successful discovery) and exponential distribution (failed discovery). To solve the problem, we first introduce the following lemma:
Lemma 3.
For a random variable X with nonnegative values, the expectation of X, denoted as , satisfies
(47) 
where indicates the CDF of X.
Proof.
Let represents the PDF of X. Then,
(48) 
∎
IvB Optimal Data Delivery
IvB1 Global Data Delivery Optimization
From (22), (34), (35), (42), (43), (50), (51), and (52), it is evident that the global candidate discovery duration is the variable for adapting the E2E latency and data rate, given the typical route with hops, hop duration , decode error rate , vehicle arrival rate , and the data rates , , and . It is shown in Appendix A that the global optimization problem is convex and differentiable, which can be solved by convex optimization theory.
Theorem 3.
Let be the stationary point of , i. e., . Then, is the optimal solution for Problem 1, where
(53) 
Proof.
See Appendix B. ∎
IvB2 Distributed Data Delivery Optimization
Similar to the global data delivery problem, it is evident that the hopwise candidate discovery duration controls the hopwise latency and data rate considering hop duration , decode error rate , vehicle arrival rate , and the data rates , , and . Specifically, the expected hop latency and data rate, which are and addressed in (18) and (19), respectively, can be derived as
(54) 
and
(55) 
where , , , , and .
According to [29], , which is the time of one discovery trial, depends on beam duration, frame length, and error probability, which are independent of the duration . Therefore, the distributed data delivery performance in terms of the weighted sum of the hopwise latency and the hopwise data rate, is maximized by determining the optimal . The convexity of the distributed optimization problem can be verified similarly to the global optimization problem 1 and is omitted here to avoid redundancy.
Ultimately, the optimization problem 2 is solved by the following theorem:
Theorem 4.
Let be the stationary point of , i. e., . Then is the optimal solution for Problem 2, where
(56) 
IvC Routing Algorithm Design
We summarize our routing algorithms to solve both the global data delivery optimization problem 1 and the distributed data delivery optimization problem 2, based on Theorem 3 and Theorem 4, in Algorithm 1 and Algorithm 2, respectively. The algorithms can be e.g. executed at RSUs where vehicles have access to the routing information when entering in the corresponding coverage. In Algorithm 1, the set of routes are planned and created considering all possible routes between and . Routes are iteratively selected from for calculating the weighted sum of latency and data rate until all routes have been traversed, as indicated in line 1. For each route, the weighted sum and the corresponding optimal candidate discovery duration are obtained in line 1 and 1, respectively. The update of the overall maximal weighted sum , the optimal route , and the overall global optimal duration are described in line 1–1. Similar procedures can be found in Algorithm 2.
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