Analytical models and performance evaluation of vehicular-to-infrastructure networks with optimal retransmission number

Vehicle-to-infrastructure and vehicle-to-vehicle communications has been introduced to provide high rate Internet connectivity to vehicles to meet the ubiquitous coverage and increasing high-data rate internet and multimedia demands by utilizing the 802.11 access points (APs) used along the roadside. In order to evaluate the performance of vehicular networks over WLAN, in this paper, we investigate the transmisison and network performance of vehicles that pass through AP by condidering contention nature of vehicles over 802.11 WLANs. Firstly, we derived an analytical traffic model to obtain the number of vehicles under transmision range of an AP. Then, incorporating vehicle traffic model with Markov chain model and for arrival packets, MG1K queuing system, we developed a model evaluating the performance of DCF mechanism with an optimal retransmission number. Based on traffic model, we also derived the probability of mean arrival rate to AP. A distinctive aspect of our work is that it incorporates both vehicular traffic model and backoff procedure with M/G/1/K queuing model to investigate the impact of various traffic load conditions and system parameters on the vehicular network system. Based on our model, we show that the delay and througput performance of the system reduces with the increasing vehicle velocity due to optimal retransmision number m, which is adaptively adjusted in the network with vehicle mobility.


Introduction
Demand for on the move access to internet, for mobile users in the vehicles has stimulated the research study on vehicular networks (Xu, et. Al, 2019). Vehicular communications can mainly classified into vehicular-to-network (V2N), vehicular to vehicular (V2V), vehiclar-to-infrastructure (V2I) and vehicleto-pedestrian (V2P) communications. Several research has been made to study [Wu and Zheng, 2014;Chen et. al, 2018) V2I and V2V communications and in particular, vehicular ad-hoc networks (VANETs) have recently started to promose as a promising solution for vehicular communications.
5. 85-5.925 GHz band frequency is allocated to short range communication band as a emerging radio standart for promoting the communication in safe and efficient highway or high-speed freeways. Basically, wireless access devices used in vehicles can be utilized permanently scan for signals from avaliable access points (AP) and report the real-time traffic conditions signal and data captured along the way (e.g. data dissemination latency, roadway congestion levels) and also to enable drivers and on board-passengers with permanently enhanced safety and entertaintment.
Internet access requests and real-time traffic to roadside unit (RSU), are conveyed by users traveling by car usually in the range of multiple WiFi access points (APs). At roadsides and intersections WiFi networks can be deployed along the highways with their low-cost high-capacity low coverage nature. In WLANs, contention-based medium access control (MAC) has been widely adopted such as IEEE 802.11 Distributed Coordination Function (DCF). The IEEE 802.11 DCF networks, defines a maximum retransmission number for each arrival packet, i.e, the retry limit (Bianchi, 2000). At ecah retransmission, the packet is transmitted and then after the value of retransmission number reaches the maximum limit, vehicles may retransmit a packet and then an optimal retransmission limit will be on effect of DCF mechanism and the time pasted in retransmission will be shortened, thereby delay and throuhgput performance of packets transmitted in vehicular networks would be improved.
Extensive works have been studied on the performance of vehicular networks (Tzanakaki, et. Al, 2019;Pinnaka, 2015;Azpilicueta et. Al, 2016;Soto et.al, 2019). However, tehese works don't consider the network performance with the optimal retransmission limit through constructing on corporate analytical Markov model and queuing network. Most of these studies focus on numerical evaluation of network throughput and delay performance for an initial given network sets, i.e, retransmission number is usually assumed to be fixed number in which larger number is enlargeing the delay of successfully transmitted packets.
In this paper, we provide an analytical model to evaluate the network thgroughput, access delay and impact of optimal retransmission number, M on performance of vehicular-toinfrastructure networks over WLAN. By incorporating vehicle traffic model with Markov chain model, and for arrival packets, M/G/1/K queue system, we devoleped to model evaluating the performance of DCF mechanism with an optimal retransmission number. Based-on traffic model, we derive the number of vehicles in the network, i.e, network size, the mean arrival rate λ to AP. Then, relationship between network size and transmission and collision probability of a vehicle is derived. Based on developed model, for arrival packets patterns, with an iterative algorithm we derive the optimal retransmission number associated with Markov model's derived parameters.
The rest of paper is organized as follows: In section II, we discuss the related work. In section III, we describe the our proposed analytical traffic model, Markov chain model and queuing analysis in detail and in Section IV we provide numerical studies to evaluate the performance of our analytical models. We conclude the paper in Section V.
Studying the performance of the vehicle-toinfrastructure communications for the 802.11 MAC has drawn considerable attention of many researchers in the literature. Based only on slowly varying large-scale fading channel information, (Guo et. al, 2019), has investigated an optimal power allocation for vehicular-to-X(everything) communications to maximize the sum throughput of V2N links while quaranteing V2X links' reliability and latency. In highway environment, a number of performance models of the IEEE 802.11p, enhanced distributed channel access (EDCA) has been proposed [Zheng and Wu, 2016;Yao et. al, 2013;Wang et. al, 2014, Zheng et. al, 2015.
Taking into account all major factors that could affect the access performance of the IEEE 802.11p ADCA mechanism including the saturation condition, standart parameters, backoff counter freezing and internal collision, a 2-D Markov and 1-D discretetime Markov chain has been developed to model backoff procedure of an access category (AC) queue and establish a relationship between transmission probability and collision probability of the AC queue. Moreover, considering with hidden terminal problem and message strict priorities, a twodirectional vehicular ad-hoc network (VANET) model has also been proposed (Wang et. al, 2014;Zheng et. al, 2015). By direct (or one-hop) broadcast approach on the control channel (CCH), analysis of the 802.11p safety-critical broadcast in VANET enviroment has been studied (Yao et. al, 2013). To show impact of performance anomaly at the intersections in WiFi-based vehicular networks (Kim et. al, 2011) has developed a signal-to-noise (SNR) based admission control scheme that excludes vehicles with bad channel qualities and an analytical model. (Hadaller, 2006), also has studied performance anomaly problem to improve overall system throughput by proposing a simple and intuitive opportunustic medium access that grants wireless access only to vehicles with good SNR.
By taking into account the randomness of the vehicle taffic and statistical variation of the distrupted communication channel, (Abdrabou adn Zhuang, 2009) has studied an analytical framework in order to obtain the maximum distance between adjacent RSU's that stocastically limits the worst case packet delivery delay to a certain maximum value. For drive-thru internet, that heaviliy affected by the contention overhead of the uplink, upload performance (from vehicles to the AP) has been studied (Soto et. al, 2019;Guo et. al, 2019) by considering the contention nature of the uplink and realistic vehicle traffic model.
Using three different measurement settings, network characteristics has been investigated and reference parameters for equipment with UDP-based test tools has been obtained and possibilities and limitations for the use of scattered WLAN by devices has been discussed (Ott, 2004).
To explore QoS performance in term of throughput and delay, a comprehensive analytical model that takes into account both QoS features of EDCA and the vehicle mobility (velocity and moving directions), has been developed (Zhuang et.al, 2012). To optimally adjust parameters of EDCA towards the controllable QoS provision to vehicles. For impact of mobility on the resultant throughput, and accurate analytical model that incorporates the high-node mobility with the modeling of DCF was proposed (Luan et. al, 2012). For vehicle-toroadside communications, mobility pattern is very different. Different nodes do not have similiar channel access the channel, which is called fairness problem. As a solution to this problem, the authors (Karamad and Ashtiani, 2008) a modified 802.11 DCF channel access scheme was proposed by changing the probability of transmission through changing the minimum contention window size.
These proposed models, however, assumes that maximum retransmission number is constant or infinite, which would detoriate the delay and throughput performance of overall system due to non-optimal or consequetive and unlimited retransmissions, which can not capture the impact of retransmission number in the network performance. Impact of retransmissions in the vehicular transmission performance, a number of models for IEEE 802.11 DCF networks have been proposed (Wu et. al, 2018;Woo et. al, 2013;Le et. al, 2015). However, they don't construct the proposed analytical models extensively to perform with MAC queuing model taking the packet arrival rates, average packet size, average network size considerations caused by the queue in the vehicular enviroment.
In order to study effect of the time wasted in retransmissions with a finite retry-limit, an analytical framework (Sun and Dai, 2016) has been proposed on the performance optimization of CSMA networks and with a finite retry limit, it is shown that the network performance could be highly sensitive to the value of retransmission number M. The authors (Hassan et. al, 2010) obtained explicit expressions for the mean of the total packet delivevey ratio in an unsaturated network formed by moving on highway to investigate the performance of a modified DCF that uses a fixed number of sequential retransmissions to improve the reliability of a packet delivery.

Analytical Model
In this section, we will model the real-time transmission processes over 802.11 Wireless LAN, quantitatively analyze the performance of the system, and derive the parameters that can guide the engineering of the system for the optimal performance. In deriving the optimal throughput, we consider that the network works in the saturation condition and each vehicle always has packets to transmit, i.e., the probability of an empty queue is zero. Under a high traffic load, a packet queue may be full and additional arrival packets will be blocked from entering the queue thus the WLAN system.

Traffic Model
In particular, one consider the following types of descriptions.
-Microscopic description: All vehicles are individually idendified.
In the literature, traffic flow is analyzed by experimental data (Daganzo, 1997;Button, 1993) which refers to flow and averaged velocity of vehicles as function of the vehicular density.
Relationship between flow-density results in fundemental diagrams. The mean speeddensity curve resuşts in velocity diagram.
The traffic flow rate λtag, which refers to the arrival rate of vehicles to the road segment, to AP can be express as (1) where k denotes the vehicle density that corresponds to the number of vehicles per unit distance in each lane along the road segments.
Based on Greenshield's model (Greenshields, 1935), speed-flow-density diagrams,can be constructed. Figure 1 shows the relationsheeps between traffic flow(λ), density (k), and speed (v). A number of important points exist on this diagrams. Traffic flow is zero, when there are no vehicle on the road segment and when the vehicle density take its highest value and vehicle speed limit, Vf. If speed increases density starts to decrease, when speed is zero, density takes its highest value because all vehicles on the road stop and traffic flow λ, is also stop.
Traffic flows in vehiclar networks can be generally divided into two major types: Uninterupted and interupted (Transportation Research Board, 1996).
First type uninterupted is defined as all the flows individually regulated by vehicle-tovehicle interactions and interactions between vehicles and the roadway. The second type, interupted flow is regulated bu an external means, such as a traffic signal.
In this paper, we chose to consider only uninterupted flows. In particular, we also chose to consider microscopic model representation, in which all vehicles seperately described and position and velocity of each vehicle determine the state of the system.
The vehicle density k linearly cchanges with the average velocity v as (2) where vehicle jam density at which traffic flow rate is kjam and free-flow speed is Vf, which corresponds to highest speed when the vehicle is driving on the road segment (usually taken as the road's vehicle speed limit).
The mean sojourn time of each vehicle in the coverage of AP is given by (3) where L is the lenght of the AP's coverage range.
The network capacity of the system (C), which is maximum number of vehicles that can be accomodated by the AP's coverage range is given by (4) The mean vehicle population in the road segment N, is kv tag Based on M/D/C/C queuing model, PN, the steady-state probability that there are N vehicles simultaneously under the coverage of the AP is given by (Tan et. al, 2011) As mentinoied before, we consider the uninterupted flows, M/D/C/C queuing model is for uninterupted vehicular flow, (not interupted stop-and-go traffic flows), i.e, there aren't any external means).

Markov Chain model
In DCF mechanism, each packet contends for the channel transmission randomly. According to traffic model, N number of vehicles contend for the transmission to the channel. The channel status is monitored during the idle period and if a channel is sensed idle period and if a channel is sensed idle with a duration of distributed interframe space (DIFS) time, when a packet arrives at queue in a vehicle, a packet can be transmitted. If the channel is sensed busy, a vehicle node will backoff and will continoue to sense the channel again idle and not immediately compete for the channel access again and the backoff counter will be start up with the initial value set to random period within the backoff window selected from [0, W] where W=Wmin. The backoff window size, CWmin initially set, will be doubled with each additional collision. Then, if channel sensed idle in a slot time, the backoff counter will be decremented by 1. When backoff counter becomes zero, the packet will be transmitted. We denote the number of backoff stages as j, and the size of contention window at the jth backoff stage is CWj=2j*CWmin, 0<j<m, m being the maximum number of retransmissions limit that will be reached, which is allowed with our derivation. If the m limit is reached, the value of CW will be reset to CWmin. If the channel is sensed busy and collision happened, if there is at least one of other vehicles also initiates the transmision at the same time, then probability of a packet being collided is denoted as pc.
Whenever there is a collision, the Markov chain moves from collision stage j-1, and waits for a random backoff time to avoid collisions and the counter will reduce by one after each time slot if the medium is sensed idle. .

NT tag
To find the one-step transition probabilities of the Markov chain bj,i form the transition diagram, we will first establish the balance equations. We have (7) (8) For each i in [0,CWj-i], the steady-state probability bj,i is given by The probability of transmission is successful given in a slot time defined as Ps. If at least one of vehicle transmits during a slot time, when the channel is busy and probability of transmission is denoted as Ptran.
The collision probability pc, the success probability Ps, and the transmission probability Ptran are given respectively as follows (12) (13) The parameter  is the probability that a station would transmit a packet in a given time slot. At the steady-state, the collision probability pc depends on , while  also depends on the backoff duration thus pc,  can be calculated fom the Markov chain as the total probability that a counter reaches 0 from any of the m transmission attempts and is represented as follows: (15) Equations 14 and 15 are solved iteratively to determine the unknown parameters  and pc until a converging condition is met. After the m-th backoff stage, a packet is discarded in the Markov chain model.

M/G/1/K queuing analysis
We apply the M/G/1/K queuing model, where K represents the maximum capacity of the queue at a station. Packets arriving after K packets are already in the queue are dropped. Call arrivals of real time applications are assumed to follow the Poisson process and the arrival rate to a station n is given by λ (packets/s) and the arrival rate matrix for all N stations is λ =diag (λ1, λ2, ,... λN) The steady-state probability of the queue with k packets is π(k) where k=0,1,2,....K. Each packet is transmitted using the full channel capacity C.
Let E[L] denote the average packet size (in bytes) in the MAC layer protocol. Average MAC service time is defined as the time from a packet reaches the head of M/G/1/K queue at the station to the time it successfuly departs from the queue, and is calculated as the time to transmit the average payload size within the retransmission limit m. The average service time E[Ts] is calculated as: The mean offered traffic load or traffic intensity from the each vehicles N is (17) Average data payload size L is given by (18) The average duration it takes to transmit a packet in one transmission stage, E[Tslot], is given as (19) where δ is the duration of the empty slot time, Tsuc and Tc represent the duration for succesful transmission and duration of collision, respectively. The probability that the channel is empty for slot time is given by Pı=(1-Ptran).
The duration due to succesful transmission is calculated as: The time cost in the collision Tc is calculated as: where Theader and Td represent respectively the transmissiom time of the packet header and propogation delay.
In M/G/1/K queuing model, data packets are produced according to Poisson arrival process. Packets arriving at finite-size queu buffer accomodating K packets follow a ,0 0 Poisson distribution. In one slot time, packet arrival probability is determined by The steady-state probability π(k) can be calculated as (Cao, et.al 2009;Cooper, 1981) Packet blocking probability from the WLAN server is given by (25) By adaptively adjusted optimal retransmission limit M, after reaching the retransmisison limit, the system has maximum network throughput in satırated conditions. The system network can be obtained as (26) According to Little's Law, packets arrival to the system, is calculated by the average sojourn (delay) time in the queue. E[Tdelay]can be expressed as the summation of the average number of idle backoffslots multiplied by yhe average idle slot duration as a result of backoff at state (j,i), the average busy duration as the result of both succesful transmission and failed transmission due to collisions, and also retransmission duration as follows: (27)

Numerical Results
In this section, we eveluate the network performance in an 802.11 WLAN based on our analytical models and demonstrate how to optimize the network performance of IEEE 802.11 DCF network in vehicular network enviroment. We present the numerical results under various traffic load conditions. The default parameters of WLAN are set as following the current standarts of 802.11 DCF protocol: ACK=50µS, SIFS=10µS , DIFS=50µS, and initial backoff window is set as CW=32. Wireless network channel rate is set as 2 Mbps. Packet size is fixed as 1000 byte. We set the traffic jam density kjam as 120 veh/km and free way speed Vf as 160 km/h. The time slot is set as 50µS. Figure 2 shows the system throughput of M/G/K/1 queu under different arrival rates of vehicles in the network, respectively. Offered load rates are q0=0.01, q1=0.02, q2=0.05 for optimal and q3=0.01 for fixed scheme respectively. At each different arrival rate changes, the mean offered load also changes with fixed packet size. System throughput increase with the total system capacity increasing when capacity approach to C=2000Kb/s linearly.

Figure 2 Total capacity, (Kbits/s) versus systm throughput
This is because, as the system capacity increases, more vehicles would contend for packet transmission in the network, resulting in usage of the system capacity by each vehicle in the whole range of system capacity given. In fixed m scheme, average throughput is lower than that of optimal scheme as fixed m scheme can not adapt the retransmission number to maintain the targeted throughput and requirements. Figure 3 shows the backoff stage number when minimum contention window size increases with vehicle velocity for v=20km/h, v=80km/h, and v=140km/h.
As can be seen in Figure 3, backoff stage number increases when the initial window size increases due to increased delay resulting in more often retransmission attempt and larger adapted retransmission number. Window size is set according to 802.11 DCF maximum retransmission number m=7, whereas in optimal scheme, it is adapted to required targeted delay and throughput settings. Due to increased mobility of vehicles, backoff time, tehrefore, average delay increase, backoff stage number will be larger than that of non-optimal scheme.  For the default size of initial window size, CW=32 transmission probability is 0.01. As initial window size varies from 0 to 80, transmission probability of fixed scheme decreases linearly with the initial window size, while transmission probability of optimal scheme remains in the range of 0.001-0.005, there is optimal m value for the minimum contention window size for which transmission probability for the mean number of vehicles in the system, is optimized. In non-optimal scheme, variations in the window size cannot be adapted to maximize the transmission probability. Figure 5 and Figure 6 also depict the impact of initial window size for different traffic rate λ=2 (vehicle/sec) and λ =4 veh/sec. For large λ, the initial window size has larger impact and transmisison probability decreases much faster with the increase of initial window size. At the default size of initial window size, CWmin=32, transmission probabilityfor optimal scheme is 0.0075 and 0.0040 for λ=2 veh/sec and λ=4 veh/sec respectively.
Transmission probability of fixed scheme is much more smaller than that of optimal scheme. Difference for two traffic rates, much increases as traffic rate becomes larger.  Figure 7 shows collision probability performance when vehicles velocity increases. As we can see, when velocity increased, vehicles with high mobility and fast transmission have smaller backoff time, this result in more backoff stages and collision more frequently. As mean backoff stage increases, the collision probability increases, which leads to the reduced mean system throughput. The number of contending vehicles that can be accomodated by the network is a random variable. Based on M/D/C/C queuing model, probability that there are N vehicles on the roadway segment and under the AP's coverage range can be obtained according to Poisson process with mean vehicle arrival rate λ, e.g, the traffic flow rate. Figure 8 shows the effect of vehicle density is zero (λ =0) and thete are vehicles (cars) on the road, flow is zero and a point that shows the lowest value of network size occurs. As vehicle density becomes high, network size also increase. At the other points, that is, highest value of network size is shown, vehicle density is so high that all vehicles stop. Between these two points, network size increase linearly and take its highest value at kjam=60 vehicle/km.  Figure 9 shows the average transmission delay with increasing the number of vehicles, N. As we can observe, when the number of vehicles increases, the average transmision delay increases in different adapted retransmission limit number, average delay is optimally remains in the range of targeted delay requirements. When the number of vehicles increases with the traffic rate=1,5 veh/seconds, for the N=60, average delay is 0.xxx1 s. As the number of vehicles varies from 0 to 120, average delay of fixed scheme stays linearly in the targeted delay upperbounds (0.5 seconds) with the number of vehicles, while average delay of optimal m scheme remains in the 0.40-0.55 seconds is lower than the fixed m scheme. m is optimized to result in reduced transmission delay and its impact to average delay is obvious. Figure 9 The number of vehicles, N versus average delay Figure 10 also shows the average tranmission delay with the increasing the number of vehicles, N for different value of the traffic rate λ=1.5 vehicle/seconds. For smaller traffic rate, the number of vehicles has a smaller effect and average delay remains in the range of 0.39-0.41 seconds . For the N=60, average delay is 0.38 s and 0.58 s for optimal and fixed m scheme respectively. For higer number of vehicles, adjusting the optimal retransmission limit number m, severely effects the average transmission delay and average transmission delay remains at bounded delay limit below (<0.5s). In contrast, the average delay of fixed m scheme remains at above the delay limit.

Conclusion
In this paper, we provide an analytical traffic model to investigate the effect of retransmission number M on the performance of vehicular networks.

Figure 10
The number of vehicles, N versus average delay Our derived analytical model novelly evaluates backoff procedure, transmission probability, the number of vehicles that AP can accomodate, average delay (sojourn times) and throughput that a vehicle can transmit a packet by taking into account the impact of maximum retransmission number on the service time of a packet. We used M/G/1/K queuing model and relationship between queuing model and Markov chain model derived to show the network performance highly sensitive to maximum retransmission number. Based on queuing model, we evaluate the delay and througput performance optimally under various traffic rate and other system parameter changes. Analytical results show the optimal scheme can improve the network delay and throughput performance much better than fixed m scheme with a fixed retransmission number.