# The Optimal Packets Scheduling for Buffer-Aid Energy Harvesting RSUs in Cooperative Vehicle Infrastructure System

Authors: Ying Shi; Chi Feng; Tong Wang; Qingjiang Yang; Yiping Chen
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this paper, we investigate the optimal packet scheduling for RSU-to-vehicle downlink communication in Cooperative Vehicle Infrastructure System (CVIS). The RSU is powered by a capacity-limited battery storing harvested energy with reliable energy source as backup, which meets requirement of renewable energy and ensures sustainability of communication system. We aim to find the optimal packet scheduling policy that minimizes the data queuing delay under a given the reliable power constraint and downlink availability rate. According to probabilistic scheduling policy, we adopt data buffer queue and harvesting energy queue as the state-space, and determine the transition action by the joint state of packet arrivals and channel state to establish a two-dimensional Markov chain. Based on this, the optimization problem is formulated. By solving the problem and analysing its solution, we derive the optimal transmission parameters and the optimal scheduling policy which turns out to be threshold-based. Simulations are demonstrated to verify the accuracy of the theoretical derivation results.


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## Body

## 1. Introduction

### 1.1. Motivations

Internet of Vehicles (IoV) is the application of Internet of Things (IoT) in urban traffic, which promotes the development of urban intelligent transportation system (ITS). At present, the Cooperative Vehicle Infrastructure System (CVIS) has become one of the most promising and fastest growing research in ITS. In CVIS, accurate and timely information interaction between vehicles and infrastructure can contribute to solve intelligent information management and decision-making problems, such as traffic congestion and road safety problems [1, 2]. Data transmission is completed in the way of “store-carry-forward” through passing vehicles within roadside units (RSUs) coverage. Therefore, the RSUs are not only important network access equipment with providing high-speed message forwarding services for vehicles over high bandwidth links, but also gateway nodes of the surrounding environmental monitoring sensor network, which undertakes the function of transmitting monitoring data to roadside units connected with the Internet [3, 4]. Nevertheless, the high-speed and high bandwidth forwarding services require high power consumption [5]. In addition, the RSUs are powered by the electrical grid instead of no-renewable energy, which costs too much, and increased usage of such energy resources will lead to increase carbon dioxide emission. Thereby, renewable and harvested energy resources, such as solar energy, wind energy and RF energy, are generally considered as substitute for conventional grid power [6].However, harvested energy arrives randomly and sporadically due to environmental influence, such as weather and geographic position, etc. To avoid excessive energy and save for future use, battery is used to save the collected energy, but the capacity is usually limited and inconsistent availability [7]. It is possible that RSUs powered by harvested energy alone may not guarantee the quality of service. Thus, the use of conventional grid power cannot be eliminated. When renewable energy sources are not available, using grid energy as a substitute power supply can minimize grid consumption, which will help in reducing energy cost and harmful emission. Hence, there is a hybrid energy supplies formed by harvesting energy and reliable power (for example, grid power) to ensure quality of service and the communication sustainability [8, 9]. In fact, the RSU is mainly powered by harvested energy and the grid is used as a backup.In the above scenario, guaranteeing data timeliness under the constraint of reliable power consumption is a most important metrics when dealing with the packet scheduling optimization of each RSU. Nevertheless, this is not easy. First, in order to achieve efficient data dissemination, there is usually the cooperation of hybrid infrastructure-to-vehicle (I2V) and vehicle-to-vehicle (V2V) communication in CVIS, which makes the data conflicts in the RSU coverage area, causing the interference in the establishment of data communication links, and directly affecting the data transmission timeliness [10, 11]. Second, the RSU is encouraged to use the free and renewable energy whenever available, but the waiting time could be undesirably long if it only relies on the harvested energy. Third, even if a reliable energy is available when renewable energy is unavailable, power consumption limit makes it impossible to use indefinitely, and it will increase the waiting time. To address the above issues, it is necessary to jointly research data transmission delay-minimal and power-constrained trade-off problem to ensure the RSU-to-vehicle downlink data communication timeliness with using grid power as little as possible.


### 1.2. Related Works

In recent years, the use of energy harvesting RSUs in vehicular environment has gradually received the attention in the research literature. In [12], the design of a solar energy harvesting circuit RSUs is presented, and the usability of energy harvesting systems in vehicular networks is investigated in [13]. In [14], the authors use a smart scheduling to reduce RSU energy costs in green vehicular roadside infrastructure, and the authors of [8] assumed that RSUs are powered by both grid source and renewable energy, and proposed an efficient sleep scheduling method to an overall decrease of energy cost. Several works have been carried out recently for optimal scheduling in RSU-to-vehicle communication. The main contribution of [15] is to investigate the problem of scheduling the downlink communication from renewable energy-powered RSUs toward vehicles, with the objective of maximizing the number of served vehicles. For the purpose of optimizing the RSU’s downlink traffic scheduling, a protocol for energy-efficient adaptive scheduling using reinforcement learning is proposed, focusing on guaranteeing the operation of the vehicular network [16]. Regarding to energy consumption and time delay minimal in the CVIS, packet scheduling optimization strategy for energy-delay trade-off in self-powered RSUs is proposed [17, 18]. And a rudimentary online scheduling algorithm for estimating the energy consumption, average packet delay and required battery capacity is proposed [19].In the process of RSU-to-vehicle communication, it involves data arrival, queuing behaviour, power allocation and data transmission in RSUs, corresponding to the network layer, the data link layer and physical layer, respectively. The cross-layer has been proved as one of the most efficient solutions for wireless communications involving multiple layers [20–23]. The cross-layer method was firstly studied in [20], where the scheduling policy is studied for transmission under a time-varying channel with delay constraint. In [21], the sensing-throughput trade-off crosses physical layer and MAC layer in multi-channel cognitive radio networks. In [22], the authors achieved energy efficient and delay trade-off using cross-layer stochastic optimization approach. In addition, it is well applied to green vehicular networks [23].


### 1.3. Contributions

In this work, we address an optimal resource scheduling for the RSU powered by both grid-powered and energy harvesting in RSU-to-vehicle downlink communication. The objective is to minimize the packet queuing delay given the available power constraint with considering the connectivity probability between RSU and vehicle. To analyse the proposed scheme, we adopt the cross-layer framework by combining the data packet arrival in the network layer, queuing state in the MAC layer, and transmission over the wireless channel as well as the energy harvesting behaviour in the physical layer. Then, we formulate a probability scheduling, considering the joint state of random energy arrivals, random data packet arrivals and time-varying channel states. Based on the scheduling, we draw up a two-dimensional Markov chain by using both data queue and harvested energy queue as state variables, and formulate an optimal scheduling problem to describe the delay minimal and reliable power consumption trade-off. With the solution to the optimization problem, the optimal scheduling policy can be revealed as a threshold-based policy. At the end, our theoretical analysis is verified by simulations.


## 1.1. Motivations

Internet of Vehicles (IoV) is the application of Internet of Things (IoT) in urban traffic, which promotes the development of urban intelligent transportation system (ITS). At present, the Cooperative Vehicle Infrastructure System (CVIS) has become one of the most promising and fastest growing research in ITS. In CVIS, accurate and timely information interaction between vehicles and infrastructure can contribute to solve intelligent information management and decision-making problems, such as traffic congestion and road safety problems [1, 2]. Data transmission is completed in the way of “store-carry-forward” through passing vehicles within roadside units (RSUs) coverage. Therefore, the RSUs are not only important network access equipment with providing high-speed message forwarding services for vehicles over high bandwidth links, but also gateway nodes of the surrounding environmental monitoring sensor network, which undertakes the function of transmitting monitoring data to roadside units connected with the Internet [3, 4]. Nevertheless, the high-speed and high bandwidth forwarding services require high power consumption [5]. In addition, the RSUs are powered by the electrical grid instead of no-renewable energy, which costs too much, and increased usage of such energy resources will lead to increase carbon dioxide emission. Thereby, renewable and harvested energy resources, such as solar energy, wind energy and RF energy, are generally considered as substitute for conventional grid power [6].However, harvested energy arrives randomly and sporadically due to environmental influence, such as weather and geographic position, etc. To avoid excessive energy and save for future use, battery is used to save the collected energy, but the capacity is usually limited and inconsistent availability [7]. It is possible that RSUs powered by harvested energy alone may not guarantee the quality of service. Thus, the use of conventional grid power cannot be eliminated. When renewable energy sources are not available, using grid energy as a substitute power supply can minimize grid consumption, which will help in reducing energy cost and harmful emission. Hence, there is a hybrid energy supplies formed by harvesting energy and reliable power (for example, grid power) to ensure quality of service and the communication sustainability [8, 9]. In fact, the RSU is mainly powered by harvested energy and the grid is used as a backup.In the above scenario, guaranteeing data timeliness under the constraint of reliable power consumption is a most important metrics when dealing with the packet scheduling optimization of each RSU. Nevertheless, this is not easy. First, in order to achieve efficient data dissemination, there is usually the cooperation of hybrid infrastructure-to-vehicle (I2V) and vehicle-to-vehicle (V2V) communication in CVIS, which makes the data conflicts in the RSU coverage area, causing the interference in the establishment of data communication links, and directly affecting the data transmission timeliness [10, 11]. Second, the RSU is encouraged to use the free and renewable energy whenever available, but the waiting time could be undesirably long if it only relies on the harvested energy. Third, even if a reliable energy is available when renewable energy is unavailable, power consumption limit makes it impossible to use indefinitely, and it will increase the waiting time. To address the above issues, it is necessary to jointly research data transmission delay-minimal and power-constrained trade-off problem to ensure the RSU-to-vehicle downlink data communication timeliness with using grid power as little as possible.


## 1.2. Related Works

In recent years, the use of energy harvesting RSUs in vehicular environment has gradually received the attention in the research literature. In [12], the design of a solar energy harvesting circuit RSUs is presented, and the usability of energy harvesting systems in vehicular networks is investigated in [13]. In [14], the authors use a smart scheduling to reduce RSU energy costs in green vehicular roadside infrastructure, and the authors of [8] assumed that RSUs are powered by both grid source and renewable energy, and proposed an efficient sleep scheduling method to an overall decrease of energy cost. Several works have been carried out recently for optimal scheduling in RSU-to-vehicle communication. The main contribution of [15] is to investigate the problem of scheduling the downlink communication from renewable energy-powered RSUs toward vehicles, with the objective of maximizing the number of served vehicles. For the purpose of optimizing the RSU’s downlink traffic scheduling, a protocol for energy-efficient adaptive scheduling using reinforcement learning is proposed, focusing on guaranteeing the operation of the vehicular network [16]. Regarding to energy consumption and time delay minimal in the CVIS, packet scheduling optimization strategy for energy-delay trade-off in self-powered RSUs is proposed [17, 18]. And a rudimentary online scheduling algorithm for estimating the energy consumption, average packet delay and required battery capacity is proposed [19].In the process of RSU-to-vehicle communication, it involves data arrival, queuing behaviour, power allocation and data transmission in RSUs, corresponding to the network layer, the data link layer and physical layer, respectively. The cross-layer has been proved as one of the most efficient solutions for wireless communications involving multiple layers [20–23]. The cross-layer method was firstly studied in [20], where the scheduling policy is studied for transmission under a time-varying channel with delay constraint. In [21], the sensing-throughput trade-off crosses physical layer and MAC layer in multi-channel cognitive radio networks. In [22], the authors achieved energy efficient and delay trade-off using cross-layer stochastic optimization approach. In addition, it is well applied to green vehicular networks [23].


## 1.3. Contributions

In this work, we address an optimal resource scheduling for the RSU powered by both grid-powered and energy harvesting in RSU-to-vehicle downlink communication. The objective is to minimize the packet queuing delay given the available power constraint with considering the connectivity probability between RSU and vehicle. To analyse the proposed scheme, we adopt the cross-layer framework by combining the data packet arrival in the network layer, queuing state in the MAC layer, and transmission over the wireless channel as well as the energy harvesting behaviour in the physical layer. Then, we formulate a probability scheduling, considering the joint state of random energy arrivals, random data packet arrivals and time-varying channel states. Based on the scheduling, we draw up a two-dimensional Markov chain by using both data queue and harvested energy queue as state variables, and formulate an optimal scheduling problem to describe the delay minimal and reliable power consumption trade-off. With the solution to the optimization problem, the optimal scheduling policy can be revealed as a threshold-based policy. At the end, our theoretical analysis is verified by simulations.


## 2. System Model

We consider a packet transmission scenario between RSUs relayed by vehicle is shown in Figure1. For sake of reducing environmental pollution, the RSU is powered by two sources of energy namely, a fixed power line using the conventional grid energy and Photo Voltaic cells to utilize the solar energy, but grid power is only used to ensure that the RSU is always operational when necessary [8]. The collected data (such as surrounding traffic, monitoring information) is stored in the data buffer of RSU, waiting to be sent to another RSU with data aggregation centre in form of packet queuing through mobile vehicles transmission. To improve the connectivity probability of vehicular communication, vehicle may complete the whole data transmission through a wireless multi-hop path with V2V communication [24]. However, in this paper, we only focus on the downlink communication from RSU-to-vehicle. Considering the discrete time-slotted system, in a certain time slot, if RSU does not establish a downlink communication link with passing vehicles, the data packet will wait in the buffer. However, if the downlink link is available, RSU determines whether to send packets to passing vehicles and the corresponding power consumption according to the packet scheduling strategy.Figure 1
Schematic diagram of RSU-to-vehicle scene.In Figure2, we formulate a cross-layer system model. At the RSU, the data packets (such as monitoring data) generated by the upper layers arrive at the MAC layer and are stored at a data buffer. A battery storing the free and renewable energy harvested from the environment is represented in a virtual queue. If a downlink link has been established in the slot, the RSU will combine the data buffer status, energy queue status and new packet arrival to determine whether to send data packet to the vehicle according to the packet scheduling strategy, so as to control the reliable energy consumption and queuing delay of data.Figure 2
A cross-layer system model.


### 2.1. Packet Arrival Model

In discrete-time slotted system, assuming that packets are generated randomly, following the Bernoulli process, we can define that the data packets randomly arrive with rateηd. Each data arrival contains kd∈ℕ+ packets. Let d[n] denotes the number of packets arriving at the RSU at the beginning of the n-th timeslot. The mass probability function of d[n] is given by
(1)Prdn=kd=ηdPrdn=0=η¯d,where ηd∈0,1 and η¯d=1−ηd. At the RSU, a buffer is adopted to store the packets that cannot be sent immediately, and its capacity is expressed by Qd. Let vdn and qdn note the number of packets transmitted in the n-th slot and the queue length at the end of n-th timeslot, respectively. Thus, the update of the data packet queue can be obtained as
(2)qdn=minqdn−1+dn,Qd−vdnIn Figure2, the RSU can harvest energy from the environment. The energy storage and consumption processes of the battery can be described as a queuing system. In particular, the harvested energy packets arrive randomly following a certain distribution, and leave the queue when the scheduler spends a few energy packets for data packet transmission. There also exists an upper limit Qe for the virtual queue corresponding to the storage capacity of the battery. This means that the battery can store at most Qe energy packets, each of which contains a certain amount of energy. qenis used to measure the length of the energy queue, i.e., the remaining energy in the battery. We use en describe the process of energy harvesting. Suppose that a certain amount of energy is harvested each time. Accordingly, ke∈ℕ+ energy packets arrive at the battery with probability ηe. Therefore, the mass probability function of en is given by
(3)Pren=ke=ηePren=0=η¯e,where ηe∈0,1means the energy arrival rate, and η¯e=1−ηe. Let ven notes number of energy packets consumed in n-th timeslot. Similarly, the update of the energy queue is characterized as
(4)qen=minqen−1+en,Qe−ven


### 2.2. Transmission Channel Model

In Figure1, all the data packets in the RSU will be transmitted to the vehicle through wireless channel, which is supported by dedicated short range communication (DSRC) protocol. However, in the CVIS, the vehicle can only establish communication connection with RSU within its coverage. When the vehicle drives out the coverage of RSU, the connection will be disconnected. Therefore, there are two communication channel states of “on/off” between RSU and vehicle, which represent the working state and offline state between vehicle and RSU. Moreover, within the coverage aera of RSU, there are not only RSU-to-vehicle downlink transmission channels, but also vehicle-to-vehicle transmission channels, which may interfere with each other. It cannot be guaranteed that the downlink communication channels between RSU and vehicle are available in each time slot. So, in this paper, we use “on” and “off” to describe the channel state, which is denoted by hn in the n-th timeslot. The mass distribution function of the state is given by
(5)Prhn=′on′=αPrhn=′off′=α¯,where α represents the probability that the downlink channel remains working, namely “on,” while α¯ represents the probability that the downlink channel remains offline, namely “off,” and α+α¯=1 in n-th timeslot.


### 2.3. Probabilistic Scheduling Policy

For convenience of analyse, the RSU is supposed to successfully deliver one data packet with one energy packet from the energy queue or with grid power in watts when the channel state is “on.” Usually, the scheduler prefers to use the harvested energy because it is free and renewable, which is also desirable. However, due to the uncertainty of external environment, it may have to wait a long time to harvest sufficient power for next packet transmission, which is intolerable for latency-sensitive application. In this situation, reliable power source should be adopted instead. However, if the RSU relies too much on the reliable power source, the communication cost will increase inevitably, which is inconsistent with the original intention. Thus, there exists a trade-off between the queuing delay and the transmission power drawn from the reliable energy source. In this work, the scheduler aims to find an optimal scheduling strategy to minimize the average queuing delay constrained by the grid power.Letvn=vdn,ven note the actions taken in the n-th slot, namely, transmitting vdndata packets with ven energy packets. The transmission channel state plays a key role, because it be possible to transmit data packets only when hn=′on′. Specifically, when the harvested energy is enough for one transmission, one data packet should always be transmitted to decrease the buffer occupation. Otherwise, the scheduler must decide whether to transmit with a certain grid power in current timeslot. Simultaneously, we define the joint state vector sn=dn,en,hn and the queue state vector qn=qdn,qen. Based on the packet arrival and queuing states, the scheduling policy can be described below in three cases.Case 1.
whenqn−1=i,0,i≥0,
(6)vn=1,1w.p.1sn=⋅,ke,′on′1,0w.p.gisn=kd,0,′on′1,0w.p.fisn=0,0,′on′0,0otherwisewhere “w.p.” is short for “with probability of” and the symbol “·” in the expression of sn means no matter how many packets (data packet or energy packet) arrive in this slot. Let gi∈0,1 and fi∈0,1 be the probabilities of packet transmission with or without packet arrivals in this time slot, respectively. Accordingly, it is also possible for the RSU to remain silent with probabilities 1−gi and 1−fi, respectively. In this case, there are data packets waiting in the data buffer but no renewable energy packet can be used for transmitting in RSU. Taking the first term in (6) for example, when the channel is “on,” regardless of the data packet arrival state, one packet will be delivered vdn at the cost of one energy packet ven if ke energy packets being harvested. As for the second term, there is no energy packet newly harvested. Thus, the scheduler decides to transmit one data packet vdn=1 with probability gii≥0 using paid reliable power ven=1, otherwise, remain silent vdn=0 with probability 1−gi. Similarly, when no data packets newly arrive, the RSU would transmit one packet using the paid power with probability fii>0 or remain silent vdn=0 with probability 1−fi, respectively. Whether the packets are reached or not, only if the channel state is offline, then vn=0,0.Case 2.
whenqn−1=i,j,i,j>0,
(7)vn=1,1w.p.1sn=⋅,⋅,′on′0,0w.p.1sn=⋅,⋅,′off′
In this case, both data buffer and energy queue are not empty. Then, no matter whether the new data and harvested energy packets arrive inn-th slot, a data packet will be delivered with an energy packet if the downlink channel is available.Case 3.
whenqn−1=0,j,j>0,
(8)vn=1,1w.p.1sn=kd,⋅,′on′0,0w.p.1sn=0,⋅,′on′0,0w.p.1sn=⋅,⋅,′off′
In this case, there are enough harvested energy storing at the battery but the data queue is zero. Then, one energy packet is spent for delivering one data packet newly arriving. And if no packets arrive atn-th slot, or the channel is offline, there is no energy consumption.


### 2.4. Markov Chain Model

Based on the probabilistic scheduling policy, we know that the transmission decision is made only in each time slot based on the queuing stateqn−1 and the joint statesn, Hence, the queuing system can be characterized by a two-dimensional Markov chain with qn being the state variables i,ji=0,⋯,Qd;j=0,⋯,Qe. For simplicity, we firstly define the following four joint probability parameters about packets arrival, shown as
(9)ρ0=Prdn=0,en=ke=η¯dηeρ1=Prdn=0,en=0=η¯dη¯eρ2=Prdn=kd,en=0=ηdη¯eρ3=Prdn=kd,en=ke=ηdηeAccording to the scheduling scheme defined by the formulas (6)-(8), we focus on deriving the one-step state transition probability Prqnqn−1 between states, which are listed in Table 1 in detail. In Table 1, the j+ke∨Qe denotes minj+ke,Qe, and the state transition probabilities locating below dashed line are the special case what i =0 or j=0,Qe−1,Qe. Except for the special states, the rest state transition probabilities Prqni,j are exhibited above the dashed line. In particular, the case j=Qe requires special processing, because the battery is full and the newly harvested energy must be discarded. Besides, the case j=Qe−1 also discard the remaining packets when the packet number of newly harvested energy exceeds 1.Table 1
The one-step transition probabilitiesPrqnqn−1.
q[n-1]q[n]Prqnqn−1q[n-1]q[n]Prqnqn−1(i, j)(i-1, (j+ke)∨Qe-1)ρ0α(i, j)(i, j+ke)ρ0α¯(i, j)(i-1, j-1)ρ1α(i, j)(i, j)ρ1α¯(i, j)(i+kd -1, j-1)ρ2α(i, j)(i+kd, j)ρ2α¯(i, j)(i+kd -1, (j+ke)∨Qe-1)ρ3α(i, j)(i+kd, j+ke)ρ3α¯(i, 0)(i+kd -1, 0)ρ2αgi(i, 0)(i+kd, 0)ρ2α1−gi+ρ2α¯(i, 0)(i-1, 0)ρ1αfi(i, 0)(i, 0)ρ1α1−fi+ρ1α¯(0,j)(0, (j+ke)∨Qe)ρ0(0,j)(0,j)ρ1(i, Qe)(i-1, Qe-1)η¯dα(i, Qe)(i+kd-1, Qe-1)ηdα(i, Qe)(i, Qe)η¯dα¯(i, Qe)(i+kd, Qe)ηdα¯(i, Qe-1)(i, Qe)ρ0α¯(0,Qe)(0,Qe)η¯d(0,Qe-1)(0,Qe)ρ0(0, 0)(0, 0)ρ1Based on Table1, we construct a 2D Markov chain model shown in Figure 3, where solid lines present the fixed state transitions while all dotted lines indicate state transitions that vary with different kd and ke. To check the transition probability given in Table 1, assume kd=ke=2in Figure 3. Let m,n∈0,1,2,3, we modify the definition of variable α as, αm=ρmα, α¯m=ρmα¯ in Figure 3. Moreover, the scheduler makes a decision of transmission with a probability only under the situation that there are no harvested energy packets can be used, that is the state transition between states (i,0). We define four variables λ1,i=ρ2αgi, λ1,i′=ρ2α1−gi+ρ2α¯, μ1,i=ρ1αfi, μ1,i′=ρ1α1−fi+ρ1α¯ on the first column in Figure 3.Figure 3
The two-dimensional Markov chain model.For ease of discussion, we give the simplified Markov chain which is shown in Figure4, where the transition probabilities are obtained from the substitution of kd=1, ke=1 in Figure 3. For example, the transition from i,j to i+kd−1,j in Figure 3 becomes that from i,j to i,j in Figure 4, for i,j>0. For this state transition, one is that both data and energy packet arrive and the transmission link is available so that a data is delivered with the corresponding probability of α3, the other is neither data nor energy packet arrives and the transmission link is unavailable, so that no backlogged data packet is transmitted with the corresponding probability α¯1. Therefore, a new notation is needed for the transition from i,j to i,j, which α¯1+α3, namely, α′, for i,j>0. The transition between these states in Figure 4 can easily derived based on the dynamic change of the data buffer queue and the virtual energy queue as given in (2) and (4).Figure 4
The two-dimensional Markov chain model.Denote byπi, πj and πi,j the steady-state probabilities of the data queue state, the energy queue state and the joint queue state, respectively. The relationship between the steady-state probabilities of the 2D Markov chain can be represented in the following theorem.Theorem 1.
Defineμ~1,i=ρ1αfi+α0, then the local balance equations of the Markov chain is obtained as
(10)μ~1,iπi,0+α0∑j=1Qeπi,j+α1∑j=1Qeπi,j=πi−1,0λ1,i−1′+α¯2∑j=1Qeπi−1,j+α¯3∑j=0Qeπi−1,j
The expression of steady-state probabilities of the data queueπi can be derived as
(11)πi=∑j=0Qeπi,jWith the definition ofπi, α0+α1=η¯dα, and α¯2+α¯3=ηdα¯, we can then reformulate (10), as following
(12)η¯dαπi−ηdα¯πi−1=λ1,i−1′−α¯2πi−1,0+η¯dα−μ~1,iπi,0LetΛ denote the transition probability matrix, we know that the steady-state probability πi,jshould meet constraint Λπ=π, where vector π is defined as π0,0,π0,1⋯π0,Qe,π1,0,π1,1⋯π1,Qe,⋯,πQd,0,πQd,1⋯πQd,Qe. Also, πi,j should meet the normalization constraint, namely, ∑i=0Qd∑j=0Qeπi,j=1.


## 2.1. Packet Arrival Model

In discrete-time slotted system, assuming that packets are generated randomly, following the Bernoulli process, we can define that the data packets randomly arrive with rateηd. Each data arrival contains kd∈ℕ+ packets. Let d[n] denotes the number of packets arriving at the RSU at the beginning of the n-th timeslot. The mass probability function of d[n] is given by
(1)Prdn=kd=ηdPrdn=0=η¯d,where ηd∈0,1 and η¯d=1−ηd. At the RSU, a buffer is adopted to store the packets that cannot be sent immediately, and its capacity is expressed by Qd. Let vdn and qdn note the number of packets transmitted in the n-th slot and the queue length at the end of n-th timeslot, respectively. Thus, the update of the data packet queue can be obtained as
(2)qdn=minqdn−1+dn,Qd−vdnIn Figure2, the RSU can harvest energy from the environment. The energy storage and consumption processes of the battery can be described as a queuing system. In particular, the harvested energy packets arrive randomly following a certain distribution, and leave the queue when the scheduler spends a few energy packets for data packet transmission. There also exists an upper limit Qe for the virtual queue corresponding to the storage capacity of the battery. This means that the battery can store at most Qe energy packets, each of which contains a certain amount of energy. qenis used to measure the length of the energy queue, i.e., the remaining energy in the battery. We use en describe the process of energy harvesting. Suppose that a certain amount of energy is harvested each time. Accordingly, ke∈ℕ+ energy packets arrive at the battery with probability ηe. Therefore, the mass probability function of en is given by
(3)Pren=ke=ηePren=0=η¯e,where ηe∈0,1means the energy arrival rate, and η¯e=1−ηe. Let ven notes number of energy packets consumed in n-th timeslot. Similarly, the update of the energy queue is characterized as
(4)qen=minqen−1+en,Qe−ven


## 2.2. Transmission Channel Model

In Figure1, all the data packets in the RSU will be transmitted to the vehicle through wireless channel, which is supported by dedicated short range communication (DSRC) protocol. However, in the CVIS, the vehicle can only establish communication connection with RSU within its coverage. When the vehicle drives out the coverage of RSU, the connection will be disconnected. Therefore, there are two communication channel states of “on/off” between RSU and vehicle, which represent the working state and offline state between vehicle and RSU. Moreover, within the coverage aera of RSU, there are not only RSU-to-vehicle downlink transmission channels, but also vehicle-to-vehicle transmission channels, which may interfere with each other. It cannot be guaranteed that the downlink communication channels between RSU and vehicle are available in each time slot. So, in this paper, we use “on” and “off” to describe the channel state, which is denoted by hn in the n-th timeslot. The mass distribution function of the state is given by
(5)Prhn=′on′=αPrhn=′off′=α¯,where α represents the probability that the downlink channel remains working, namely “on,” while α¯ represents the probability that the downlink channel remains offline, namely “off,” and α+α¯=1 in n-th timeslot.


## 2.3. Probabilistic Scheduling Policy

For convenience of analyse, the RSU is supposed to successfully deliver one data packet with one energy packet from the energy queue or with grid power in watts when the channel state is “on.” Usually, the scheduler prefers to use the harvested energy because it is free and renewable, which is also desirable. However, due to the uncertainty of external environment, it may have to wait a long time to harvest sufficient power for next packet transmission, which is intolerable for latency-sensitive application. In this situation, reliable power source should be adopted instead. However, if the RSU relies too much on the reliable power source, the communication cost will increase inevitably, which is inconsistent with the original intention. Thus, there exists a trade-off between the queuing delay and the transmission power drawn from the reliable energy source. In this work, the scheduler aims to find an optimal scheduling strategy to minimize the average queuing delay constrained by the grid power.Letvn=vdn,ven note the actions taken in the n-th slot, namely, transmitting vdndata packets with ven energy packets. The transmission channel state plays a key role, because it be possible to transmit data packets only when hn=′on′. Specifically, when the harvested energy is enough for one transmission, one data packet should always be transmitted to decrease the buffer occupation. Otherwise, the scheduler must decide whether to transmit with a certain grid power in current timeslot. Simultaneously, we define the joint state vector sn=dn,en,hn and the queue state vector qn=qdn,qen. Based on the packet arrival and queuing states, the scheduling policy can be described below in three cases.Case 1.
whenqn−1=i,0,i≥0,
(6)vn=1,1w.p.1sn=⋅,ke,′on′1,0w.p.gisn=kd,0,′on′1,0w.p.fisn=0,0,′on′0,0otherwisewhere “w.p.” is short for “with probability of” and the symbol “·” in the expression of sn means no matter how many packets (data packet or energy packet) arrive in this slot. Let gi∈0,1 and fi∈0,1 be the probabilities of packet transmission with or without packet arrivals in this time slot, respectively. Accordingly, it is also possible for the RSU to remain silent with probabilities 1−gi and 1−fi, respectively. In this case, there are data packets waiting in the data buffer but no renewable energy packet can be used for transmitting in RSU. Taking the first term in (6) for example, when the channel is “on,” regardless of the data packet arrival state, one packet will be delivered vdn at the cost of one energy packet ven if ke energy packets being harvested. As for the second term, there is no energy packet newly harvested. Thus, the scheduler decides to transmit one data packet vdn=1 with probability gii≥0 using paid reliable power ven=1, otherwise, remain silent vdn=0 with probability 1−gi. Similarly, when no data packets newly arrive, the RSU would transmit one packet using the paid power with probability fii>0 or remain silent vdn=0 with probability 1−fi, respectively. Whether the packets are reached or not, only if the channel state is offline, then vn=0,0.Case 2.
whenqn−1=i,j,i,j>0,
(7)vn=1,1w.p.1sn=⋅,⋅,′on′0,0w.p.1sn=⋅,⋅,′off′
In this case, both data buffer and energy queue are not empty. Then, no matter whether the new data and harvested energy packets arrive inn-th slot, a data packet will be delivered with an energy packet if the downlink channel is available.Case 3.
whenqn−1=0,j,j>0,
(8)vn=1,1w.p.1sn=kd,⋅,′on′0,0w.p.1sn=0,⋅,′on′0,0w.p.1sn=⋅,⋅,′off′
In this case, there are enough harvested energy storing at the battery but the data queue is zero. Then, one energy packet is spent for delivering one data packet newly arriving. And if no packets arrive atn-th slot, or the channel is offline, there is no energy consumption.


## 2.4. Markov Chain Model

Based on the probabilistic scheduling policy, we know that the transmission decision is made only in each time slot based on the queuing stateqn−1 and the joint statesn, Hence, the queuing system can be characterized by a two-dimensional Markov chain with qn being the state variables i,ji=0,⋯,Qd;j=0,⋯,Qe. For simplicity, we firstly define the following four joint probability parameters about packets arrival, shown as
(9)ρ0=Prdn=0,en=ke=η¯dηeρ1=Prdn=0,en=0=η¯dη¯eρ2=Prdn=kd,en=0=ηdη¯eρ3=Prdn=kd,en=ke=ηdηeAccording to the scheduling scheme defined by the formulas (6)-(8), we focus on deriving the one-step state transition probability Prqnqn−1 between states, which are listed in Table 1 in detail. In Table 1, the j+ke∨Qe denotes minj+ke,Qe, and the state transition probabilities locating below dashed line are the special case what i =0 or j=0,Qe−1,Qe. Except for the special states, the rest state transition probabilities Prqni,j are exhibited above the dashed line. In particular, the case j=Qe requires special processing, because the battery is full and the newly harvested energy must be discarded. Besides, the case j=Qe−1 also discard the remaining packets when the packet number of newly harvested energy exceeds 1.Table 1
The one-step transition probabilitiesPrqnqn−1.
q[n-1]q[n]Prqnqn−1q[n-1]q[n]Prqnqn−1(i, j)(i-1, (j+ke)∨Qe-1)ρ0α(i, j)(i, j+ke)ρ0α¯(i, j)(i-1, j-1)ρ1α(i, j)(i, j)ρ1α¯(i, j)(i+kd -1, j-1)ρ2α(i, j)(i+kd, j)ρ2α¯(i, j)(i+kd -1, (j+ke)∨Qe-1)ρ3α(i, j)(i+kd, j+ke)ρ3α¯(i, 0)(i+kd -1, 0)ρ2αgi(i, 0)(i+kd, 0)ρ2α1−gi+ρ2α¯(i, 0)(i-1, 0)ρ1αfi(i, 0)(i, 0)ρ1α1−fi+ρ1α¯(0,j)(0, (j+ke)∨Qe)ρ0(0,j)(0,j)ρ1(i, Qe)(i-1, Qe-1)η¯dα(i, Qe)(i+kd-1, Qe-1)ηdα(i, Qe)(i, Qe)η¯dα¯(i, Qe)(i+kd, Qe)ηdα¯(i, Qe-1)(i, Qe)ρ0α¯(0,Qe)(0,Qe)η¯d(0,Qe-1)(0,Qe)ρ0(0, 0)(0, 0)ρ1Based on Table1, we construct a 2D Markov chain model shown in Figure 3, where solid lines present the fixed state transitions while all dotted lines indicate state transitions that vary with different kd and ke. To check the transition probability given in Table 1, assume kd=ke=2in Figure 3. Let m,n∈0,1,2,3, we modify the definition of variable α as, αm=ρmα, α¯m=ρmα¯ in Figure 3. Moreover, the scheduler makes a decision of transmission with a probability only under the situation that there are no harvested energy packets can be used, that is the state transition between states (i,0). We define four variables λ1,i=ρ2αgi, λ1,i′=ρ2α1−gi+ρ2α¯, μ1,i=ρ1αfi, μ1,i′=ρ1α1−fi+ρ1α¯ on the first column in Figure 3.Figure 3
The two-dimensional Markov chain model.For ease of discussion, we give the simplified Markov chain which is shown in Figure4, where the transition probabilities are obtained from the substitution of kd=1, ke=1 in Figure 3. For example, the transition from i,j to i+kd−1,j in Figure 3 becomes that from i,j to i,j in Figure 4, for i,j>0. For this state transition, one is that both data and energy packet arrive and the transmission link is available so that a data is delivered with the corresponding probability of α3, the other is neither data nor energy packet arrives and the transmission link is unavailable, so that no backlogged data packet is transmitted with the corresponding probability α¯1. Therefore, a new notation is needed for the transition from i,j to i,j, which α¯1+α3, namely, α′, for i,j>0. The transition between these states in Figure 4 can easily derived based on the dynamic change of the data buffer queue and the virtual energy queue as given in (2) and (4).Figure 4
The two-dimensional Markov chain model.Denote byπi, πj and πi,j the steady-state probabilities of the data queue state, the energy queue state and the joint queue state, respectively. The relationship between the steady-state probabilities of the 2D Markov chain can be represented in the following theorem.Theorem 1.
Defineμ~1,i=ρ1αfi+α0, then the local balance equations of the Markov chain is obtained as
(10)μ~1,iπi,0+α0∑j=1Qeπi,j+α1∑j=1Qeπi,j=πi−1,0λ1,i−1′+α¯2∑j=1Qeπi−1,j+α¯3∑j=0Qeπi−1,j
The expression of steady-state probabilities of the data queueπi can be derived as
(11)πi=∑j=0Qeπi,jWith the definition ofπi, α0+α1=η¯dα, and α¯2+α¯3=ηdα¯, we can then reformulate (10), as following
(12)η¯dαπi−ηdα¯πi−1=λ1,i−1′−α¯2πi−1,0+η¯dα−μ~1,iπi,0LetΛ denote the transition probability matrix, we know that the steady-state probability πi,jshould meet constraint Λπ=π, where vector π is defined as π0,0,π0,1⋯π0,Qe,π1,0,π1,1⋯π1,Qe,⋯,πQd,0,πQd,1⋯πQd,Qe. Also, πi,j should meet the normalization constraint, namely, ∑i=0Qd∑j=0Qeπi,j=1.


## 3. Establishment of Optimization Problem

At the above section, we have illustrated the fundamental trade-off between queuing delay and transmission power paid at cost in the RSU. It is of great significance to derive the average queuing delay and the power consumption mathematically. In this section, we will discuss the delay and power metrics based on the properties of two-dimensional Markov chain model in Figure4.


### 3.1. The Delay and Power Metrics

Since the average queuing delay is associated with the occupancy rate of the data queue, combined with the steady-state probabilities of the data queue, the average occupancy rate of the data queue can be obtained as∑i=1Qdiπi. The average packet arrival rate is equal to kdηd according to (1). Thus, the average queuing delay is derived based on Little’s Law [25] as
(13)Dav=1kdηd∑i=1QdiπiNext, we discuss the analytical expressions of the average power consumption. Notice that only when no harvested energy but data packets waiting for transmission, the reliable energy source will be possibly used if the transmission channel is available. Thus, the power consumption from grid-powered only depends on two probabilitiesλ1,i and μ1,i.Theorem 2.
The average power consumptionPav drawn from the reliable energy source is obtained as
(14)Pav=∑i=0Qdξiπi,0−∑i=0Qdζiπi,1,where
(15)ξ0=ρ0+α¯3Qd+ρ2ξi=ηeα¯Qd−i+ρ2−α0ζ0=α2Qdζi=η¯eαQd−i+α1for i∈0,1,⋯,Qd.Proof.
As discussed above, the normalized average power consumption from grid-powered can be obtained as(16)Pav=∑i=0Qdπi,0ρ2αgi−∑i=1Qdπi,0ρ1αfi
By insertingρ2αgi=ρ2−λ1,i−1′ and ρ1αfi=μ~1,i−α0 into (16) and extracting Pav, we arrive at
(17)Pav=ρ2∑i=0Qdπi,0−α0∑i=1Qdπi,0+∑i=1Qdπi,0μ~1,i−πi−1,0λ1,i−1′.
From the Markov chain shown in Figure4, the local balance equation at state (i,0) can be expressed as
(18)πi+1,0μ~1,i+1−πi,0λ1,i′=ρ0+α¯3πi,0−α2πi,1−α1πi+1,1i=0μ~1,i+ηeα¯πi,0−α2πi,1−πi−1,0λ1,i−1′−α1πi+1,1i≥1
For the partition function, we rewrite the second section and obtain(19)πi+1,0μ~1,i+1−πi,0λ1,i′=πi,0μ~1,i−πi−1,0λ1,i−1′+ηeα¯πi,0−α2πi,1−α1πi+1,1for all i≥1. Through recursion of (19) and by inserting π1,0μ~1,1−π0,0λ1,0′ given by (18), we can further get
(20)πi+1,0μ~1,i+1−πi,0λ1,i′=ηeα¯∑m=1iπm,0−α2∑m=0iπm,1−α1∑m=1iπm,1+ρ0+α¯3π0,0
By substituting (20) into (17), we can express the average power Pav as
(21)Pav=ρ0+α¯3Qd+ρ2π0,0−α2Qdπ0,1+∑i=1Qdηeα¯Qd−i+ρ2−α0πi,0−∑i=1Qdη¯eαQd−i+α1πi,1which perfectly corresponds to the (14) and (15).Therefore, by eliminating the dependence ofPav on the parameters gi and fi, the Pav have converted into a linear function of the steady-state probabilities πi,j. Since the analytical expressions of both average queuing delay and average power consumption are determined by steady-state probability, we can achieve the excepted scheduling strategy by optimizing πi,j, which is debated in Section 4.


### 3.2. Delay Minimization Problem under Power Constrained

We can find that the average delay is a linear combination of the steady-state probabilities in (13), so it can be minimized by restricting the steady-state probabilities. In addition, the average power consumption is also a linear function of the steady-state probabilities and coefficients are related to the transition probabilities in (21). Furthermore, the relationship between power consumption limitation and delay optimization can be established through the steady-state probabilities. Meanwhile, some transition probabilities can be expressed by the transmission parameters gi and fi, so we can get several constraints.Due togi∈0,1 and fi∈0,1, the transition probabilities λ1,i′=ρ2α1−gi+ρ2α¯ and μ~1,i=ρ1αfi+α0 should satisfy the following inequalities
(22)ρ2α¯≤λ1,i′≤ρ2(23)α0≤μ~1,i≤η¯dαIn (22), we get the minimum of λ1,i′ called ρ2α¯ when gi=1 and the maximum called ρ2 whengi=0. However, we get the minimum of μ~1,i called α0 when fi=0 and the maximum called η¯dα when fi=1 in (23). Consequently, recalling the local balance equation (12), we can obtain the following the steady-state probabilities constraint
(24)γπi−1≤πi≤γπi−1+η¯eπi,0+δη¯eπi−1,0where δ=ηd/η¯d and γ=α¯ηd/αη¯d. Let Pmax denotes the upper bound of allowable power from reliable energy source. Based on the above constraints and the scheduling probability as the optimization variable, the average delay minimum problem under given power restriction is established as follows
(25)minπi,jDav=1kdηd∑i=1Qdiπias.t.∑i=0Qdξiπi,0−∑i=0Qdζiπi,1≤Pmaxγπi−1≤πi≤γπi−1+η¯eπi,0+δη¯eπi−1,0πi,j≥0∑i=0Qd∑j=0Qeπi,j=1Λπ−π=0bcdefIn problem (25), the objective and the first constraint are exactly the average queuing delay and the paid power consumption. Constraint (25.c) represents the mapping from the scheduling probabilities to the steady-state probabilities. Constraints (25.d)- (25.f) indicate the properties of the Markov chain. Problem (25) is a linear programming (LP) problem with the steady-state probabilities being the variables. With the optimal solutionπi,j∗, we can obtain the optimal scheduling probabilities gi∗ and fi∗ as shown in the sequel.


## 3.1. The Delay and Power Metrics

Since the average queuing delay is associated with the occupancy rate of the data queue, combined with the steady-state probabilities of the data queue, the average occupancy rate of the data queue can be obtained as∑i=1Qdiπi. The average packet arrival rate is equal to kdηd according to (1). Thus, the average queuing delay is derived based on Little’s Law [25] as
(13)Dav=1kdηd∑i=1QdiπiNext, we discuss the analytical expressions of the average power consumption. Notice that only when no harvested energy but data packets waiting for transmission, the reliable energy source will be possibly used if the transmission channel is available. Thus, the power consumption from grid-powered only depends on two probabilitiesλ1,i and μ1,i.Theorem 2.
The average power consumptionPav drawn from the reliable energy source is obtained as
(14)Pav=∑i=0Qdξiπi,0−∑i=0Qdζiπi,1,where
(15)ξ0=ρ0+α¯3Qd+ρ2ξi=ηeα¯Qd−i+ρ2−α0ζ0=α2Qdζi=η¯eαQd−i+α1for i∈0,1,⋯,Qd.Proof.
As discussed above, the normalized average power consumption from grid-powered can be obtained as(16)Pav=∑i=0Qdπi,0ρ2αgi−∑i=1Qdπi,0ρ1αfi
By insertingρ2αgi=ρ2−λ1,i−1′ and ρ1αfi=μ~1,i−α0 into (16) and extracting Pav, we arrive at
(17)Pav=ρ2∑i=0Qdπi,0−α0∑i=1Qdπi,0+∑i=1Qdπi,0μ~1,i−πi−1,0λ1,i−1′.
From the Markov chain shown in Figure4, the local balance equation at state (i,0) can be expressed as
(18)πi+1,0μ~1,i+1−πi,0λ1,i′=ρ0+α¯3πi,0−α2πi,1−α1πi+1,1i=0μ~1,i+ηeα¯πi,0−α2πi,1−πi−1,0λ1,i−1′−α1πi+1,1i≥1
For the partition function, we rewrite the second section and obtain(19)πi+1,0μ~1,i+1−πi,0λ1,i′=πi,0μ~1,i−πi−1,0λ1,i−1′+ηeα¯πi,0−α2πi,1−α1πi+1,1for all i≥1. Through recursion of (19) and by inserting π1,0μ~1,1−π0,0λ1,0′ given by (18), we can further get
(20)πi+1,0μ~1,i+1−πi,0λ1,i′=ηeα¯∑m=1iπm,0−α2∑m=0iπm,1−α1∑m=1iπm,1+ρ0+α¯3π0,0
By substituting (20) into (17), we can express the average power Pav as
(21)Pav=ρ0+α¯3Qd+ρ2π0,0−α2Qdπ0,1+∑i=1Qdηeα¯Qd−i+ρ2−α0πi,0−∑i=1Qdη¯eαQd−i+α1πi,1which perfectly corresponds to the (14) and (15).Therefore, by eliminating the dependence ofPav on the parameters gi and fi, the Pav have converted into a linear function of the steady-state probabilities πi,j. Since the analytical expressions of both average queuing delay and average power consumption are determined by steady-state probability, we can achieve the excepted scheduling strategy by optimizing πi,j, which is debated in Section 4.


## 3.2. Delay Minimization Problem under Power Constrained

We can find that the average delay is a linear combination of the steady-state probabilities in (13), so it can be minimized by restricting the steady-state probabilities. In addition, the average power consumption is also a linear function of the steady-state probabilities and coefficients are related to the transition probabilities in (21). Furthermore, the relationship between power consumption limitation and delay optimization can be established through the steady-state probabilities. Meanwhile, some transition probabilities can be expressed by the transmission parameters gi and fi, so we can get several constraints.Due togi∈0,1 and fi∈0,1, the transition probabilities λ1,i′=ρ2α1−gi+ρ2α¯ and μ~1,i=ρ1αfi+α0 should satisfy the following inequalities
(22)ρ2α¯≤λ1,i′≤ρ2(23)α0≤μ~1,i≤η¯dαIn (22), we get the minimum of λ1,i′ called ρ2α¯ when gi=1 and the maximum called ρ2 whengi=0. However, we get the minimum of μ~1,i called α0 when fi=0 and the maximum called η¯dα when fi=1 in (23). Consequently, recalling the local balance equation (12), we can obtain the following the steady-state probabilities constraint
(24)γπi−1≤πi≤γπi−1+η¯eπi,0+δη¯eπi−1,0where δ=ηd/η¯d and γ=α¯ηd/αη¯d. Let Pmax denotes the upper bound of allowable power from reliable energy source. Based on the above constraints and the scheduling probability as the optimization variable, the average delay minimum problem under given power restriction is established as follows
(25)minπi,jDav=1kdηd∑i=1Qdiπias.t.∑i=0Qdξiπi,0−∑i=0Qdζiπi,1≤Pmaxγπi−1≤πi≤γπi−1+η¯eπi,0+δη¯eπi−1,0πi,j≥0∑i=0Qd∑j=0Qeπi,j=1Λπ−π=0bcdefIn problem (25), the objective and the first constraint are exactly the average queuing delay and the paid power consumption. Constraint (25.c) represents the mapping from the scheduling probabilities to the steady-state probabilities. Constraints (25.d)- (25.f) indicate the properties of the Markov chain. Problem (25) is a linear programming (LP) problem with the steady-state probabilities being the variables. With the optimal solutionπi,j∗, we can obtain the optimal scheduling probabilities gi∗ and fi∗ as shown in the sequel.


## 4. Solution of Optimization Problem

In this subsection, we will discuss how to seek the optimal solutionπi,j∗ of the 2-D Markov chain to attempt solving the optimization problem (25), and how to further derive all optimal transmission parameters gi∗ and fi∗.


### 4.1. The Optimal Steady-State Probability

The (25) is a formulation of the LP problem with the stead-state probabilities being the variable. However, due to the complexity of 2-D state transitions, it is difficult to derive a closed-form optimal solution of the corresponding LP problem.For the purpose of reducing latency, the scheduling policy should send data packets continually, which makes the queue of data packet not too large. On the other side, for the purpose of cutting down the average power consumption from the reliable energy source, the scheduling policy prefers to wait for the harvested energy when the energy queue is none, which leads to an increase in the length of data queue. As a result, there exists an optimal threshold imposed on the data queue length,i∗∈0,1,⋯,Qd. When the data queue length exceeds the threshold, the source transmits using the reliable energy, otherwise it waits for the harvested energy. Therefore, the transmission could consume more energy from the reliable energy source with the threshold decreasing. It further shows that the optimal threshold is determined by the power constraint Pmax from the reliable energy source.Given all of that, combining with (24), we can obtain the optimal solution πi,j∗ corresponding to the threshold i∗ and Pmax based data transmission scheme [9], as follows
(26)ρ0+α¯3Qd+ρ2π0,0−α2Qdπ0,1=Pmax,γ∑j=0Qeπi−1,j∗−∑j=0Qeπi,j∗=0,1≤i≤i∗−1,γ∑j=0Qeπi−1,j∗+η¯eπi,0∗+δπi−1,0∗−∑j=0Qeπi,j∗=0,i∗+1≤i≤Qd.In order to derive allπi,j∗,0≤i≤Qd,0≤j≤Qe from (26), a multidimensional matrix calculation is constructed by adding two constraints (25.e) and (25.f)
(27)ΓN×N⋅πN×1∗=Pmax,0,⋯,11×NT,where Γ is a N×N coefficient matrix, and N=1+Qd1+Qe. Through (27), we can calculate all optimal steady-state probabilities πi,j∗ by solving N independent linear equations.


### 4.2. The Optimal Transmission Parameters

For ease of discussion, we supposePcon is the power consumption when the RSU always use the reliable power for transmitting. Due to the limit of Pmax, there are two different situations [9], as following.Situation 1:Pcon≤Pmax, which means there are enough reliable power can be used for transmitting when it needs. In this case, the optimal threshold satisfies i∗=0. It presents that the RSU delivers one packet if the data buffer is nonempty and the transmission channel is “on,” without concern for whether the harvested energy queue is empty. Furthermore, the optimal transmission parameters are given by gi∗=1i∗≥0 and fi∗=1i∗>0.Situation 2:Pcon>Pmax, which means the reliable power is limit. Then, it is necessary to balance the use of energy from the reliable energy source to achieve the optimal problem (25). In this case, the utilization of power from the reliable energy source could happen only in the case qn−1=i,0when new data packets arrive and the transmission channel is available but the harvested energy queue is empty. According to the above threshold strategy, if qdn−1+dn>i∗ and sn=⋅,⋅,′on′, the source should transmit data packet with probability “1” even though no harvested energy. So, the optimal transmission parameters gi∗ and fi∗ are set to 1. In contrast, if qdn−1+dn<i∗, the source need to silently wait for the harvested energy whether new data arrives or not. Here, gi∗ andfi∗ are set to 0. But, when qdn−1+dn=i∗, the source transmits using the reliable power with gi∗<1 and fi∗<1 in order to obtain the minimum average delay Dav∗ under the power constraint Pmax.In order to obtaingi∗ and fi∗ at the optimal threshold, we recall the local balance equation with the optimal steady-state probabilities πi,j∗ at state (i, j), and have
(28)η¯dαπi∗∗−ηdα¯πi∗−1∗=λ1,i∗−1′−α¯2πi∗−1,0∗+η¯dα−μ~1,i∗πi∗,0∗By substitutingλ1,i∗−1′=ρ21−αgi∗−1∗ and μ~1,i∗=ρ1αfi∗∗+α0 into (28), we can get
(29)ρ21−αgi∗−1∗πi∗−1,0∗=η¯dαπi∗∗−ηdα¯πi∗−1∗−α11−fi∗∗πi∗,0∗+α¯2πi∗−1,0∗In (29), fi∗∗ denotes the optimal transmission probability when qn−1=i∗,0 and sn=⋅,⋅,′on′. Although the data queue length has reached the threshold and no harvested energy is available at the end of last timeslot, no new data packet arrives at the n-th timeslot. For the convenience, fi∗∗=0 is acceptable and reasonable. So that, we can get all transmission parameters as follow
(30)gi∗=0i<i∗−11−η¯dαπi∗−ηdα¯πi−1∗−α1πi,0∗α2πi−1,0∗i=i∗−11i>i∗−1(31)fi∗=0i≤i∗1i>i∗From (30) and (31), the transmit probabilities are only determined by the comparison between the data queue length and the threshold value. Due to the limit of reliable power consumption, RSU waits for the harvested energy when the data queue length is less than or equal to the threshold, but when the data queue length exceeds the threshold, one packet is sent by the reliable power over downlink wireless channel. Therefore, the optimal scheduling policy can be described as a threshold-based policy.


## 4.1. The Optimal Steady-State Probability

The (25) is a formulation of the LP problem with the stead-state probabilities being the variable. However, due to the complexity of 2-D state transitions, it is difficult to derive a closed-form optimal solution of the corresponding LP problem.For the purpose of reducing latency, the scheduling policy should send data packets continually, which makes the queue of data packet not too large. On the other side, for the purpose of cutting down the average power consumption from the reliable energy source, the scheduling policy prefers to wait for the harvested energy when the energy queue is none, which leads to an increase in the length of data queue. As a result, there exists an optimal threshold imposed on the data queue length,i∗∈0,1,⋯,Qd. When the data queue length exceeds the threshold, the source transmits using the reliable energy, otherwise it waits for the harvested energy. Therefore, the transmission could consume more energy from the reliable energy source with the threshold decreasing. It further shows that the optimal threshold is determined by the power constraint Pmax from the reliable energy source.Given all of that, combining with (24), we can obtain the optimal solution πi,j∗ corresponding to the threshold i∗ and Pmax based data transmission scheme [9], as follows
(26)ρ0+α¯3Qd+ρ2π0,0−α2Qdπ0,1=Pmax,γ∑j=0Qeπi−1,j∗−∑j=0Qeπi,j∗=0,1≤i≤i∗−1,γ∑j=0Qeπi−1,j∗+η¯eπi,0∗+δπi−1,0∗−∑j=0Qeπi,j∗=0,i∗+1≤i≤Qd.In order to derive allπi,j∗,0≤i≤Qd,0≤j≤Qe from (26), a multidimensional matrix calculation is constructed by adding two constraints (25.e) and (25.f)
(27)ΓN×N⋅πN×1∗=Pmax,0,⋯,11×NT,where Γ is a N×N coefficient matrix, and N=1+Qd1+Qe. Through (27), we can calculate all optimal steady-state probabilities πi,j∗ by solving N independent linear equations.


## 4.2. The Optimal Transmission Parameters

For ease of discussion, we supposePcon is the power consumption when the RSU always use the reliable power for transmitting. Due to the limit of Pmax, there are two different situations [9], as following.Situation 1:Pcon≤Pmax, which means there are enough reliable power can be used for transmitting when it needs. In this case, the optimal threshold satisfies i∗=0. It presents that the RSU delivers one packet if the data buffer is nonempty and the transmission channel is “on,” without concern for whether the harvested energy queue is empty. Furthermore, the optimal transmission parameters are given by gi∗=1i∗≥0 and fi∗=1i∗>0.Situation 2:Pcon>Pmax, which means the reliable power is limit. Then, it is necessary to balance the use of energy from the reliable energy source to achieve the optimal problem (25). In this case, the utilization of power from the reliable energy source could happen only in the case qn−1=i,0when new data packets arrive and the transmission channel is available but the harvested energy queue is empty. According to the above threshold strategy, if qdn−1+dn>i∗ and sn=⋅,⋅,′on′, the source should transmit data packet with probability “1” even though no harvested energy. So, the optimal transmission parameters gi∗ and fi∗ are set to 1. In contrast, if qdn−1+dn<i∗, the source need to silently wait for the harvested energy whether new data arrives or not. Here, gi∗ andfi∗ are set to 0. But, when qdn−1+dn=i∗, the source transmits using the reliable power with gi∗<1 and fi∗<1 in order to obtain the minimum average delay Dav∗ under the power constraint Pmax.In order to obtaingi∗ and fi∗ at the optimal threshold, we recall the local balance equation with the optimal steady-state probabilities πi,j∗ at state (i, j), and have
(28)η¯dαπi∗∗−ηdα¯πi∗−1∗=λ1,i∗−1′−α¯2πi∗−1,0∗+η¯dα−μ~1,i∗πi∗,0∗By substitutingλ1,i∗−1′=ρ21−αgi∗−1∗ and μ~1,i∗=ρ1αfi∗∗+α0 into (28), we can get
(29)ρ21−αgi∗−1∗πi∗−1,0∗=η¯dαπi∗∗−ηdα¯πi∗−1∗−α11−fi∗∗πi∗,0∗+α¯2πi∗−1,0∗In (29), fi∗∗ denotes the optimal transmission probability when qn−1=i∗,0 and sn=⋅,⋅,′on′. Although the data queue length has reached the threshold and no harvested energy is available at the end of last timeslot, no new data packet arrives at the n-th timeslot. For the convenience, fi∗∗=0 is acceptable and reasonable. So that, we can get all transmission parameters as follow
(30)gi∗=0i<i∗−11−η¯dαπi∗−ηdα¯πi−1∗−α1πi,0∗α2πi−1,0∗i=i∗−11i>i∗−1(31)fi∗=0i≤i∗1i>i∗From (30) and (31), the transmit probabilities are only determined by the comparison between the data queue length and the threshold value. Due to the limit of reliable power consumption, RSU waits for the harvested energy when the data queue length is less than or equal to the threshold, but when the data queue length exceeds the threshold, one packet is sent by the reliable power over downlink wireless channel. Therefore, the optimal scheduling policy can be described as a threshold-based policy.


## 5. Simulation Results

In simulations, the data packet and energy packet arrival processes in the RSU are modelled by generating two Bernoulli random variables with the parametersηd and ηe at the beginning of each time slot, respectively. The number of newly arrived data and energy packets is separately less than 2 in each time slot. We adopt two-state “on/off” channel model to describe the RSU-to-vehicle downlink communication link state. The packet transmission is scheduled according to the optimal transmission parameters gi∗ and fi∗ in (30) and (31). Each simulation runs over 106 time slots.We simulate and discuss the optimal queuing delay and reliable power consumption trade-off performance in terms of battery capacity, packet arrival rate and channel availability rate. The results are shown in Figures5–7. Figure 5 plots the trade-off under different queue length of harvested energy Qe, where the data packet and energy arrival rates in RSU are same, as ηd=ηe=0.3. In Figure 5, the theoretical results are presented as the lines, while the simulation results are marked by symbol “∘.” In addition, we use the Linprog optimization tool to solve the LP problem (25) in order to further verify the correctness of the theoretical solution, and use the maker “+” to demonstrate the Linprog results. We can see that theoretical results are in good agreement with the simulation and Linprog results from Figure 5. It is observed that the average minimum queuing delay decreases when the average power increases, and when the available power decreases and approaches a critical value, the queuing delay will increase dramatically and grow to infinity. Besides, the decreasing rate grows with the increase of the battery capacity Qe. This means that a larger Qe leads to a much smaller queuing delay, since less harvested energy is wasted due to the limitation of the battery capacity. For the sake of brevity, the Linprog results will be omitted in the following simulation diagrams.Figure 5
The optimal delay-power curves for differentQe. (ηd=ηe=0.3, Qd=45).Figure 6
The optimal delay-power curves for different energy harvesting rateηe (ηd=0.3, Qd=45, Qe=5).Figure 7
The delay-power curve for differentα. (ηd=ηe=0.3).Figure6 focuses on the impact of different energy arrival rates ηe on the trade-off curve, where the data packet arrival rates is set to ηd=0.3. From Figure 6, we show that the comparison between the energy arrival rate and the data package arrival rate has a great influence on the whole delay-power trade-off curve. Given the identical queue length and data packet arrival rate, a greater average power will be induced for lower harvested energy arrival rate, besides that the delay-power curves change steeper. It is because that the RSU must exploit more extra reliable energy source power to transmit backlogged packets waiting when the energy arrival rate is less than the data package arrival rate ηe<ηd. Conversely, the RSU has enough harvested energy to transmit data packets when ηe>ηd. Moreover, there has the less average delay even if Pmax=0.Figure7 shows the optimal queuing delay and reliable power consumption trade-off under different probabilities of RSU-to-vehicle downlink availability rate, α =0.5, 0.6, 0.8 and 1. The average queuing delay decreases with the average power increasing, as expected. For each numerical curve, when the reliable power become large enough which means there always exists energy for transmitting if needed, the average queuing delay of RSU only depends on the data packets arrival and the available rate of the channel and remains same. Meanwhile, we can see that, the smaller the availability rate of the channel is, the greater the average queuing delay is. This is because more timeslots will stay in the “on” state and can be used for transmitting.Next, we verify that the optimal scheduling policy is determined by the data queue thresholdi∗ and the transmission parameters gi∗ and fi∗. Figure 8 shows the optimal threshold i∗ value responding to the assigned average reliable power constraint Pmax when the availability rate of downlink channel α=0.6, 0.7 and 0.8, respectively. The threshold-power curve shows ladder-type decline in whichever given α. With the increasing of the disposable power constraint, the optimal threshold decreases slowly since the RSU could afford to transmit more frequently without harvested energy when the channel is “on” state. Furthermore, as the channel availability rate increases, the optimal threshold i∗ decreases gradually under the same reliable power limit because the RSU has more opportunities to send queuing data to passing vehicle within the coverage of RSU.Figure 8
The threshold structure of optimal scheduling policy.In Figure9, we demonstrate the theoretical results to confirm the threshold-based structure of the optimal scheduling policy expressed by (30) and (31), where ηe=ηd=0.4 and α=0.7. What is more, the transmission parameters bring to light a threshold-based structure imposed on the data queue length under reliable power constraints. From Figure 9(a), we can find that the optimal threshold is queue length “8,” and transmission parameters are set as g7∗=0.597, gi∗=fi∗=1i≥8, which is the idea of threshold-based policy. However, it is queue length “4” and g3∗=0.649, gi∗=fi∗=1i≥4in Figure 9(b). Though it is relatively easier to dispatch data packets when the communication channel is usable in Figure 9(b), it leads to a higher reliable power consumption in RSU at cost. In other words, the RSU makes good use of the power resource from reliable energy source by regulating the optimal threshold point for different power constraints.Figure 9
The threshold-based scheduling policy. (a) Pmax=0.02, (b) Pmax=0.03.
(a)(b)


## 6. Conclusions

In this paper, we investigated the delay-minimal scheduling problem with the supply of hybrid energy sources for RSU-to-vehicle downlink communication in CVIS. The RSU is powered by a limited capacity energy harvesting battery, coupled with a reliable energy constrained by maximum power consumption. By establishing probabilistic scheduling policy and modelling the two-dimensional Markov chain, we formulated the expressions of delay and reliable power consumption, and construct an optimization LP problem with the steady-state probability as variables. For solving the steady-state probability problem caused by the complexity of local equations, the analytical solution of the optimal transmission parameters can be obtained by threshold piecewise optimization. It has been proved that the optimal scheduling policy is threshold-based, that is, the use of reliable power resource for transmission should only occur when the data queue length exceeds optimal threshold while no harvested energy can be resorted if wireless channel is available. The optimal threshold is jointly decided by packet arrivals, downlink availability rate as well as the maximum reliable power constraint. At the end, simulation results confirmed our theoretical analysis. It was shown that there always exists an optimal delay and reliable power limit trade-off in RSU and its decreasing rate depends on the harvested energy arrival rate and the battery capacity. Moreover, the downlink availability rate increases, the optimal threshold decreases gradually under the same power consumption limit.



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*Source: 4147373-2022-04-26.xml*