Ensuring fair access in IEEE 802.11pbased vehicletoinfrastructure networks
 Vettath Pathayapurayil Harigovindan^{1}Email author,
 Anchare V Babu^{1} and
 Lillykutty Jacob^{1}
https://doi.org/10.1186/168714992012168
© Harigovindan et al; licensee Springer. 2012
Received: 12 September 2011
Accepted: 14 May 2012
Published: 14 May 2012
Abstract
IEEE 802.11p is an approved amendment to the IEEE 802.11 standard to facilitate wireless access in vehicular environments (WAVE). In this article, we present an analytical model to evaluate the impact of vehicle mobility on the saturation throughput of IEEE 802.11pbased vehicletoinfrastructure (V2I) networks. The throughput model is then used to investigate an unfairness problem that exists in such networks among vehicles with different mobility characteristics. Assuming a saturated network, if all the vehicles in the network use the same MAC parameters, IEEE 802.11p MAC protocol provides equal transmission opportunity for all of them, provided they have equal residence time in the coverage area of a road side unit (RSU). When vehicles have different mobility characteristics (e.g., extremely high and low speeds), they do not have similar chances of channel access. A vehicle moving with higher velocity has less chance to communicate with its RSU, as compared to a slow moving vehicle, due to its short residence time in the coverage area of RSU. Accordingly, the data transfer of a higher velocity vehicle gets degraded significantly, as compared to that of the vehicle with lower velocity, resulting in unfairness among them. In this article, our aim is to address this unfairness problem that exists among vehicles of different velocities in V2I networks. Analytical expressions are derived for optimal minimum CW (CW_{min}) required to ensure fairness, in the sense of equal chance of communicating with RSU, among competing vehicles of different mean velocities in the network. Analytical results are validated using extensive simulations.
Keywords
IEEE 802.11p fairness residence time saturation throughput vehicletoinfrastructure networks1. Introduction
Vehicular adhoc network (VANET) is an emerging wireless network in which vehicles constitute the mobile nodes in the network. Such networks are aimed at providing support for road safety, traffic management, and comfort applications by enabling vehicletovehicle (V2V) or vehicletoinfrastructure (V2I) communications [1, 2]. The emerging technology for VANETs is the dedicated short range communications (DSRC), for which the Federal Communications Commission in the United States has allocated 75 MHz of spectrum between 5850 and 5925 MHz. The DSRC is based on IEEE 802.11 technology and is proceeding towards standardization under the standard IEEE 802.11p, whereas the entire communication stack is being standardized by the IEEE 1609 working group under the name wireless access in vehicular environments (WAVE). The overall WAVE architecture includes IEEE standards 1609.1 to 1609.4 (for resource management, security architecture, networking service, and multichannel operation, respectively) and IEEE 802.11p (MAC and PHY standard). IEEE 802.11p uses essentially the same PHY defined for 802.11a but operates in a 10 MHz wide channel instead of 20 MHz. The goal of 802.11p standard is to provide V2V and V2I communications over the dedicated 5.9 GHz licensed frequency band and supports data rates of 327 Mbps (3, 4.5, 6, 9, 12, 18, 24, and 27 Mbps) [3, 4].
Future intelligent transportation systems (ITS) will necessitate wireless V2I communications. Besides the delivery of infotainment services, the role of typical V2I systems will include the provisioning of safety related, realtime, local, and situationbased services, such as speed limit information, safe distance warning, lane keeping support, intersection safety, traffic jam, and accident warning, etc. All these services aim to prevent accidents by providing timely information directly to the car and/or to the driver. The main technical challenges for communication in V2I and V2V networks are the very high mobility of the nodes, highly dynamic topology, high variability in node density, and very short duration of communication [1–3]. The IEEE 802.11p uses the enhanced distributed channel access (EDCA) medium access control (MAC) sublayer protocol based on distributed coordination function (DCF) [4]. DCF, which is based on Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA), was originally developed for WLANs [5]. Under DCF, all stations compete for access by using CSMA/CA protocol; they sense the channel before transmitting, and if the medium is found idle for a minimum time equal to DIFS, the packet will be transmitted. Otherwise, the station enters backoff and randomly sets its backoff counter within the range of its Contention Window (CW). The random discrete back off time is selected from the range [0, CW  1]. To transmit packets after DIFS, node first decrements the backoff time counter and transmit only if the backoff counter is 0. The countdown of backoff counter is frozen once the channel becomes busy due to other node transmission, and resumes when the channel is idle for another DIFS. The size of CW depends on the history of transmissions. At the first transmission attempt, it is set to a predefined value CW_{min}, the minimum CW. Upon each unsuccessful transmissions, it is updated to 2^{ s }CW_{min} until it reaches a maximum value CW_{max}. Here s is called backoff stage. Each station maintains a retry counter that indicates the number of retransmission attempts of a data packet. More details of DCF can be found in [5]. The EDCA mechanism assigns four different priority classes for incoming packets at each node which are called Access Categories (AC). Each AC has its own channel access function when compared with 802.11 DCF in which all packets exploit the same access function to acquire the channel. Different access functions for different categories mean assigning different delay times, different minimum contention windows, and different number of backoff stages for each type of service [6].
The DCF protocol was originally developed for low mobility networks such as WLANs and recent studies have shown that it does not operate efficiently for a high mobility communication scenario such as vehicular networks. In static networks, the performance of the IEEE 802.11 DCF depends on network parameters such as the number of communicating nodes, type of data traffic, backoff procedure, packet size, data rates of different nodes, etc. [7–11]. In vehicular communication networks, the performance of DCF protocol is also affected by other factors such as vehicle density and node mobility [12–16]. Node mobility can be characterized by node position, speed, and acceleration, direction of movement, potential communication duration, and potential number of communication neighbors. All these factors are highly dynamic in V2I networks, and difficult to predict especially in an extreme mobility environment.
The problem of unfairness due to vehicles having different velocities has been reported in [17] for the V2I communication scenario, involving fixed RSU. The standard IEEE 802.11p does not take into consideration, the resident time of nodes within the coverage of each RSU. Vehicles having different velocities have different resident times in the coverage area of an RSU. Assuming a saturated network, if all the vehicles in the network use the same MAC parameters, DCF protocol provides equal transmission opportunity for all of them. When vehicles have different mobility characteristics (e.g., extremely high and low speeds), they do not have similar chances of communication with RSU due to the different resident times and, therefore, a fairness problem exists. A fast moving vehicle has less chance to communicate with its RSU and consequently less amount of data transferred as compared to a slow moving vehicle. This problem occurs for each area covered by an RSU. Therefore, the amount of data transferred at each area (useful for next areas) is not equal. The contiguous areas covered by different RSUs and handoffs between them do not solve this problem. Since emergency information are more vital for faster vehicles, the above problem has more degrading effect on the efficiency of VANETs. In this article, our aim is to resolve this unfairness problem by adjusting the transmission probability of each vehicle according to its speed, through changing the minimum contention window size. In this way, the amount of successfully transmitted data of all nodes are made equal regardless of their velocities, while in the coverage area of an RSU. Using Jain's fairness index, we show how fairness in the sense of equal chance of communicating with RSU can be achieved by a judicious choice of minimum CW for the competing vehicles of distinct velocities in the network. The impact of these choices on throughput are also presented. The analytical findings are verified with extensive simulation studies.
The major contributions of this article are as follows:

A modified analytical model for DCF is developed for V2I networks for computing individual vehicle throughput when vehicles are moving at very high velocities. The problem of unfairness that arises due to different vehicle velocities is investigated using this model. Analytical expressions for optimal minimum CW parameters to achieve the desired fairness objectives are derived.

Extensive analytical and simulation results are provided to support the claims.
The rest of this article is organized as follows. Section 2 presents related study. In Section 3, we present an analytical model to compute the saturation throughput of a V2I network. In Section 4, we discuss how fairness in data transfer to RSU for individual node can be ensured. The analytical and simulation results are presented in Section 5. The article is concluded in Section 6.
2. Related work
The performance of DCF has been extensively studied in the literature [6–11]. Furthermore, an extensive body of research has been devoted to the performance evaluation of IEEE 802.11p standard [18–25]. A performance evaluation of IEEE 802.11p WAVE standard, considering collision probability, throughput and delay, is presented in [18] using simulations and analytical means. Studies show that WAVE can prioritize messages; however, in dense and high load scenarios the throughput decreases and the delay increases significantly. Authors of [12, 13] propose analytical model to evaluate performance and reliability of IEEE 802.11abased V2V safetyrelated broadcast services in DSRC system on highway. In [14], simulation results of IEEE 802.11p MAC protocol are presented for the V2I scenario. The authors show that the specified MAC parameters for this protocol can lead to undesired throughput performance under dense and dynamic conditions. Authors of [15] propose a simple but accurate analytical model to evaluate the throughput performance of DCF in the high speed V2I communications. They show that with node velocity increasing, throughput of DCF decreases monotonically due to mismatch between CW and mobility. Using a ppersistent CSMA based model, they analyze the performance when different p parameter values are assigned to nodes with different data rates (determined by the different distances from the RSU). In [19], the same authors used a 3D Markov chain to evaluate the throughput of DCF in the drivethru internet scenario. Their proposals for protocol enhancement are (i) CW_{min} should be adapted to the data rates of the vehicles (according to their distances from RSU) and also to the vehicle velocity; and (ii) the maximum backoff stage should be kept small (m = 1) to mitigate the impact of mobility. In [16], authors propose an analytic model to evaluate the DSRCbased intervehicle communication. The impacts of the channel access parameters associated with different services including arbitration interframe space (AIFS) and contention window (CW) are investigated. In [20], Suthaputchakun and Ganz study the use of IEEE 802.11e for priority based safety messaging for V2V in VANETs. Analytical model for DSRC network that uses the IEEE 802.11 DCF MAC protocol is developed in [21]. In [22, 23], Tan et al. derive analytical models to characterize the average and the distribution of the number of bytes downloaded by a vehicle by the end of its sojourn through an AP's coverage range, in the presence of contention by other vehicles. Authors of [24], propose a new vehicular channel access scheme to compromise the tradeoff between system throughput and throughput fairness in V2I communication scenario.
The problem of unfairness due to vehicles having different velocities has been explained for a V2I scenario in [17] and for a V2V scenario in [25]. Karamad and Ashtiani [17] present an analysis, in which the network that spans the coverage area of RSU is modeled as an M/G/∞ queue. Customers in this queue are the batches of vehicles entering the network, with vehicles in a batch having the same speed. They divide the batches of vehicles according to their speed into P classes. For class i, the service time T_{ i } in the M/G/∞ queuing model is the residence time in the coverage area of RSU. Using this model, they obtain an expression for the saturation throughput. They also approximate the number of packets transmitted by a node during its residence time by a poisson random variable. Using these approximations and results from Bianchi's analysis [7], they derive an approximation for the optimal CW_{min} for fair access. In [25], Alasmary and Zhuang propose two dynamic CW based mechanisms to alleviate the performance degradation caused by vehicle mobility in V2V networks. But the article does not describe the exact procedure for the selection of optimal CW value to achieve the objectives. In this article, we present a simple yet accurate analytical model for DCF in high mobility scenario of V2I networks, and use this model to analyze the problem of unfairness that arises due to different vehicle velocities. We derive expressions for optimal minimum CW for vehicles with different mean velocities to achieve the desired fairness objectives. Extensive studies of the impact of parameters such as vehicle arrival rate, vehicle density, mean vehicle speed, traffic jam density, and number of nodes, on the amount of data transmitted by each vehicle during its sojourn time, are conducted.
3. Analytical model for computing saturation throughput in V2I network
where L denotes the maximum retry limit in DCF protocol.
3.1. The discrete time Markov chain (DTMC) model for class i vehicle
 (i)State transition from zone 0 to zone 1:$P\left(1,0,\phantom{\rule{2.77695pt}{0ex}}k\mathcal{O}\right)=\frac{E\left[{T}_{\text{slot}}\right]}{E\left[{T}_{0,i}\right]{W}_{i,\text{min}}},\phantom{\rule{1em}{0ex}}k\in \left[0,\phantom{\rule{2.77695pt}{0ex}}{W}_{i,\text{min}}1\right]$(4)
where represents zone 0 and E [T_{slot}] is the mean duration of one time slot. $P\phantom{\rule{2.77695pt}{0ex}}\left(1,0,k/\mathcal{O}\right)$ accounts for the transition probability that the node moves from zone 0 to zone 1 and selects the backoff time k from the range [0, W_{i,min} 1]. This is because of the fact that within one time slot, with probability ${\scriptscriptstyle \frac{E[{T}_{\text{slot}}\text{]}}{E[{T}_{0,1}\text{]}}}$, the node moves from zone 0 to zone 1 according to the geometrically distributed sojourn time in each zone. After reaching zone 1, the node selects the initial b_{ i }(t) uniformly from [0, W_{i,min} 1]. As the zone transition and backoff time selection are independent, the overall transmission probability is ${\scriptscriptstyle \frac{E[{T}_{\text{slot}}\text{]}}{E[{T}_{0,1}\text{]}{W}_{i,\mathrm{min}}}}$.
 (ii)
State transitions within zone 1 (the RSU coverage):
Let E [T_{ s }] and E [T_{ c }], respectively, represent average successful and collision time of the class i node in zone 1. The various transition probabilities are as follows:$\begin{array}{ll}\hfill P\left(1,\phantom{\rule{2.77695pt}{0ex}}j,\phantom{\rule{2.77695pt}{0ex}}k11,\phantom{\rule{2.77695pt}{0ex}}j,\phantom{\rule{2.77695pt}{0ex}}k\right)& =\left(1\frac{E\left[{T}_{\text{slot}}\right]}{E\left[{T}_{1,i}\right]}\right),\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}j\in \left[0,\phantom{\rule{2.77695pt}{0ex}}L\right],k\in \phantom{\rule{2.77695pt}{0ex}}\left[1,{2}^{\text{min}\left(j,{L}^{\prime}\right)}{W}_{i,\text{min}}1\right]\phantom{\rule{2em}{0ex}}\end{array}$(5a)$\begin{array}{ll}\hfill P\left(1,\phantom{\rule{2.77695pt}{0ex}}j,\phantom{\rule{2.77695pt}{0ex}}k1,\phantom{\rule{2.77695pt}{0ex}}j1,0\right)& =\left(1\frac{E\left[{T}_{c}\right]}{E\left[{T}_{1,i}\right]}\right)\left(\frac{{p}_{c,i}}{{2}^{j}{W}_{i,\text{min}}}\right),\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}j\in \left[1,\phantom{\rule{2.77695pt}{0ex}}L\right],k\in \left[0,{2}^{\text{min}\left(j,{L}^{\prime}\right)}{W}_{i,\text{min}}1\right]\phantom{\rule{2em}{0ex}}\end{array}$(5b)$\begin{array}{ll}\hfill P\left(1,0,\phantom{\rule{2.77695pt}{0ex}}k1,j,\phantom{\rule{2.77695pt}{0ex}}0\right)& =\left(1\frac{E\left[{T}_{s}\right]}{E\left[{T}_{1,i}\right]}\right)\frac{\left(1{p}_{c,i}\right)}{{W}_{i,\text{min}}},\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}j\in \left[0,\phantom{\rule{2.77695pt}{0ex}}L1\right],k\in \left[0,\phantom{\rule{2.77695pt}{0ex}}{W}_{i,\text{min}}1\right]\phantom{\rule{2em}{0ex}}\end{array}$(5c)$\begin{array}{ll}\hfill P\left(1,0,\phantom{\rule{2.77695pt}{0ex}}k1,\phantom{\rule{2.77695pt}{0ex}}L,\phantom{\rule{2.77695pt}{0ex}}0\right)& =\left(1\left(\frac{{p}_{c,i}E\left[{T}_{c}\right]+\left(1{p}_{c,i}\right)E\left[{T}_{s}\right]}{E\left[{T}_{1,i}\right]}\right)\right)\frac{1}{{W}_{i,\text{min}}},\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}k\in \left[0,{2}^{{L}^{\prime}}{W}_{i,\text{min}}1\right]\phantom{\rule{2em}{0ex}}\end{array}$(5d)Here (5a) accounts for the probability that the node remains in zone 1 after its backoff counter gets decremented by one. The second equation in (5b) accounts for the probability that the node encounters collision and enter the next back off stage, while remaining zone 1. The third equation (5c) accounts for the probability that the node transmits successfully and starts a new backoff and fourth case (5d) accounts for the probability that after the L^{ th } retransmission attempt, the node starts a new back off. Here p_{c,i}E[T_{ c }] + (1  p_{ c,i })E [T_{ s }] represents the mean duration of transmission time (either successful or collided transmission).
 (iii)
State transition from zone 1 to zone 0:
The transition probabilities that the node departs from zone 1 to zone 0 are given by$P\left(\mathcal{O}1,j,\phantom{\rule{2.77695pt}{0ex}}k\right)=\frac{E\left[{T}_{\text{slot}}\right]}{E\left[{T}_{1,i}\right]},\phantom{\rule{1em}{0ex}}j\in \left[0,\phantom{\rule{2.77695pt}{0ex}}L\right],k\in \left[1,{2}^{\text{min}\left(j,{L}^{\prime}\right)}{W}_{i,\text{min}}1\right]$(6)$P\left(\mathcal{O}1,j,\phantom{\rule{2.77695pt}{0ex}}0\right)=\frac{\left(1{p}_{c,i}\right)E\left[{T}_{s}\right]+{p}_{c,i}E\left[{T}_{c}\right]}{E\left[{T}_{1,i}\right]},\phantom{\rule{1em}{0ex}}j\in \left[0,\phantom{\rule{2.77695pt}{0ex}}L\right]$(7)Here (6) represents the probability that the tagged node departs zone 1 and enters zone 0, after decrementing its back off counter, while (7) the probability that the transition from zone 1 to zone 0 occurs after a packet transmission attempt, where (1  p_{c,i}) E [T_{ s }]+p_{c,i}E [T_{ c }] is the mean duration of the transmission time. Since DCF protocol is inactive in zone 0, the backoff counter value will not be inherited when the node enters the coverage area of the next RSU. Accordingly a fresh packet transmission will be initiated. It is assumed that the upper layer protocols will take care of the packets that are dropped during zone transitions. Define the stationary probability distribution of DTMC as follows${\pi}_{i}(z,j,\phantom{\rule{0.25em}{0ex}}k)=\underset{t\to \infty}{\mathrm{lim}}P\{z(t)=z,\phantom{\rule{0.25em}{0ex}}{s}_{i}(t)=j,\phantom{\rule{0.25em}{0ex}}{b}_{i}(t)=k\}$(8)where z ∈ [0, 1], j ∈ [0, L], k ∈ [0, W_{ i,j }  1], i ∈ [1, N]. The following relations can be obtained from the transition probabilities and the global balance equations:$\begin{array}{l}{\pi}_{i}(1,j,0)={\left(1{\scriptscriptstyle \frac{E[{T}_{c}]}{E[{T}_{1,i}]}}\right)}^{j}{p}_{c,i}^{j}{\pi}_{i}(1,0,0);0<j\le L\\ \phantom{\rule{0.50em}{0ex}}\phantom{\rule{0.50em}{0ex}}\phantom{\rule{0.50em}{0ex}}\phantom{\rule{0.50em}{0ex}}={({p}_{c,i}^{\text{'}})}^{j}{\pi}_{i}(1,0,0);0<j\le L\end{array}$(9)${\pi}_{i}\left(1,j,\phantom{\rule{2.77695pt}{0ex}}k\right)=\left(\frac{{W}_{i,j}k}{{W}_{i,j}}\right){\pi}_{i}\left(1,j,\phantom{\rule{2.77695pt}{0ex}}0\right);\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}0\le j\le L,1\le k\le {W}_{i,j}1$(10)$\sum _{j=0}^{L}\sum _{k=0}^{{W}_{i,j}1}{\pi}_{i}\left(1,\phantom{\rule{2.77695pt}{0ex}}j,\phantom{\rule{2.77695pt}{0ex}}k\right)\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\frac{{d}_{1}}{{d}_{1}+{d}_{0}}$(11)where ${p}_{c,i}^{{}^{\prime}}=\left(1\frac{E\left[{T}_{c}\right]}{E\left[{T}_{1,i}\right]}\right){p}_{c,i}$ and (11) represents the limiting probability that a node resides within zone 1. Combining (4) and (9)(11), the following relation can be obtained for steady state probability π_{ i }(1, 0, 0):${\pi}_{i}\left(1,0,0\right)=\left(\frac{{d}_{1}}{{d}_{1}+{d}_{0}}\right)\frac{2\left(1{p}_{c,i}^{\text{'}}\right)\left(12{p}_{c,i}^{\text{'}}\right)}{\left(\begin{array}{c}\hfill \left(12{p}_{c,i}^{\text{'}}\right)\left(1{\left({p}_{c,i}^{\text{'}}\right)}^{L+1}\right)+{W}_{i,\text{min}}\left(1{\left(2{p}_{c,i}^{\text{'}}\right)}^{{L}^{\prime}+1}\right)\left(1{p}_{c,i}^{\text{'}}\right)\hfill \\ \hfill +{W}_{i,\text{min}}{2}^{{L}^{\prime}}{\left({p}_{c,i}^{\text{'}}\right)}^{{L}^{\prime}+1}\left(12{p}_{c,i}^{\text{'}}\right)\left(1{\left({p}_{c,i}^{\text{'}}\right)}^{L{L}^{\prime}},\right)\hfill \end{array}\right)}$(12)A frame transmission will occur when the back off counter is equal to zero, regardless of the back off stage, while the vehicle is in zone 1. Here τ_{ i } is the conditional probability that the class i vehicle transmits a frame in a time slot, given that the vehicle is in zone 1:$\begin{array}{ll}\hfill {\tau}_{i}& =\frac{{\sum}_{j=0}^{L}{\pi}_{i}\left(1,j,0\right)}{\left(\frac{{d}_{1}}{{d}_{1}+{d}_{0}}\right)}=\frac{{\sum}_{j=0}^{L}{\left({p}_{c,i}^{\text{'}}\right)}^{j}{\pi}_{i}\left(1,0,0\right)}{\left(\frac{{d}_{1}}{{d}_{1}+{d}_{0}}\right)}\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1{\left({p}_{c,i}^{\text{'}}\right)}^{L+1}}{\left(1{p}_{c,i}^{\text{'}}\right)}\right)\phantom{\rule{2.77695pt}{0ex}}\left(\frac{{\pi}_{i}\left(1,0,0\right)}{\left(\frac{{d}_{1}}{{d}_{1}+{d}_{0}}\right)}\right)\phantom{\rule{2em}{0ex}}\end{array}$(13)Combining (12) and (13), we get the following expression for τ_{ i } :${\tau}_{i}=\frac{2\left(1{\left({p}_{c,i}^{\text{'}}\right)}^{L+1}\right)\left(12{p}_{c,i}^{\text{'}}\right)}{\left(\begin{array}{c}\hfill \left(12{p}_{c,i}^{\text{'}}\right)\left(1{\left({p}_{c,i}^{\text{'}}\right)}^{L+1}\right)+{W}_{i,\text{min}}\left(1{\left(2{p}_{c,i}^{\text{'}}\right)}^{{L}^{\prime}+1}\right)\left(1{p}_{c,i}^{\text{'}}\right)\hfill \\ \hfill +{W}_{i,\text{min}}{2}^{{L}^{\prime}}{\left({p}_{c,i}^{\text{'}}\right)}^{{L}^{\prime}+1}\left(12{p}_{c,i}^{\text{'}}\right)\left(1{{\left({p}_{c,i}^{\text{'}}\right)}^{LL}}^{\prime}\right)\hfill \end{array}\right)}$(14)The conditional collision probability for the class i node, p_{ c,i }, can be expressed as,${p}_{c,i}=1{(1{\tau}_{i})}^{{n}_{i}1}{\displaystyle \prod _{j=1,j\ne i}^{N}{(1{\tau}_{j})}^{{n}_{j}}}$(15)Let p_{ tr } be the probability that at least one node transmits in a given slot time and is given by,${p}_{tr}=1{\prod}_{j=1}^{N}{\left(1{\tau}_{j}\right)}^{{n}_{j}}$(16)The probability p_{ s,i } that a class i node transmits and it is successful is given by,${p}_{s,i}=\frac{{n}_{i}{\tau}_{i}{\left(1{\tau}_{i}\right)}^{{n}_{i}1}{\prod}_{j=1,j\ne i}^{N}{\left(1{\tau}_{j}\right)}^{{n}_{j}}}{{p}_{tr}}$(17)The average successful payload information transmitted for class i nodes that are within the coverage area of RSU is computed as follows$\begin{array}{ll}\hfill {Z}_{i}& =\frac{\left(\begin{array}{c}\text{Average}\phantom{\rule{2.77695pt}{0ex}}\text{payload}\phantom{\rule{2.77695pt}{0ex}}\text{information}\phantom{\rule{2.77695pt}{0ex}}\text{for}\\ \text{class}\phantom{\rule{2.77695pt}{0ex}}i\phantom{\rule{2.77695pt}{0ex}}\text{transmitted}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}\text{a}\phantom{\rule{2.77695pt}{0ex}}\text{slot}\phantom{\rule{2.77695pt}{0ex}}\text{time}\end{array}\right)}{\text{Averagelengthofaslottime}}\times \text{Mean}\phantom{\rule{2.77695pt}{0ex}}\text{residence}\phantom{\rule{2.77695pt}{0ex}}\text{time}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{class}\phantom{\rule{2.77695pt}{0ex}}i\phantom{\rule{2em}{0ex}}\\ {Z}_{i}=\frac{{p}_{tr}{p}_{s,i}E\phantom{\rule{2.77695pt}{0ex}}\left[M\right]}{\left(1{p}_{tr}\right)\sigma +{p}_{tr}{p}_{s}E\phantom{\rule{2.77695pt}{0ex}}\left[{T}_{s}\right]+{p}_{tr}\left(1{p}_{s}\right)E\phantom{\rule{2.77695pt}{0ex}}\left[{T}_{c}\right]}\times E\left[{T}_{1,i}\right]\phantom{\rule{2em}{0ex}}\end{array}$(18)where E [M] is the average payload length (assumed to be equal for all nodes), p_{ s } is the probability that a transmission that occur in a time slot is successful, σ is the duration of a empty time slot, E[T_{1,i}] is the mean sojourn time for class i within the coverage of RSU, E [T_{ s }] and E [T_{ c }], respectively, represent the mean duration of successful and collision slots. Assuming basic access, these are computed as follows [7]:$\begin{array}{ll}\hfill E\left[{T}_{s}\right]& ={T}_{H}+{T}_{E\left[M\right]}+SIFS+\delta +{T}_{ACK}+DIFS+\delta \phantom{\rule{2em}{0ex}}\\ \hfill E\left[{T}_{c}\right]& ={T}_{H}+{T}_{E\left[M\right]}+DIFS+\delta \phantom{\rule{2em}{0ex}}\end{array}$(19)Here T_{ H }, T_{E[M]}represent the transmission times of header and pay load, δ is the propagation delay, T_{ ACK } is the transmission time of the ACK packet and SIFS and DIFS are defined according to IEEE 802.11p standard. To compute the bits transferred for class i node using (18), τ_{ i } and p_{ c,i } are first determined using (14) and (15). It may be noted that (14) and (15) form a set of nonlinear equations which can be solved by using numerical techniques [7]. The bits transferred can be determined by using (16)(18), if the no. of nodes corresponding to class i, i ∈ (1, N) are given.
In V2I networks, the no. of vehicles on the highway depends on parameters such as vehicle arrival rate, vehicle density, and vehicle speed. The total arrival rate λ_{ i } of class i vehicles to the RSU can be determined as${\lambda}_{i}={k}_{i}{\mu}_{{v}_{i}}$(20)where k_{ i } is the vehicle density (veh/meter) in lane i along the highway segment and ${\mu}_{{v}_{i}}$ is the mean vehicle speed (m/sec). According to Greenshield's model [23], the node density k_{ i } linearly changes with the mean velocity ${\mu}_{{v}_{i}}$ as${k}_{i}={k}_{\text{jam}}\left(1\frac{{\mu}_{{v}_{i}}}{{v}_{\text{free}}}\right)$(21)where k_{jam} is the vehicle jam density at which traffic flow comes to a halt, v_{free} is the free moving velocity, i.e., the maximum speed with which vehicle can move, when the vehicle is driving alone on the road (usually taken as the speed limit of the road). The mean number of class i nodes, N_{ i } in the highway segment, is then determined using Little's theorem as follows [15]:$\begin{array}{ll}\hfill {N}_{i}& =\frac{{\lambda}_{i}\left({d}_{1}+{d}_{0}\right)}{{\mu}_{{v}_{i}}}\phantom{\rule{2em}{0ex}}\\ ={k}_{\text{jam}}\left(1\frac{{\mu}_{{v}_{i}}}{{v}_{\text{free}}}\right)\phantom{\rule{2.77695pt}{0ex}}\left({d}_{1}+{d}_{0}\right)\phantom{\rule{2em}{0ex}}\end{array}$(22)The number of class i nodes within the coverage area of RSU is given by$\begin{array}{ll}\hfill {n}_{i}& ={N}_{i}\frac{{d}_{1}}{{d}_{1}+{d}_{0}}\phantom{\rule{2em}{0ex}}\\ ={k}_{\text{jam}}\left(1\frac{{\mu}_{{v}_{i}}}{{v}_{\text{free}}}\right){d}_{1}\phantom{\rule{2em}{0ex}}\end{array}$(23)
4. Ensuring fairness in V2I networks
where U is the total number of nodes in the network, and y_{ i }'s are the individual node share. It may be noted that F ≤ 1 and equality holds i f f y_{ i } = y ∀i.
4.1. Selection of minimum CW for fair service in V2I networks
In the following section, we derive expression for the minimum CW required, for vehicles belonging to different classes of mean velocities, to meet the desired fairness objective. Initially we consider a network with two velocity classes. The analysis is then extended for a V2I network with three velocity classes.
4.1.1. Two classes of mean velocities
Under the default parameter settings (where all MAC parameters are equal irrespective of node velocities), the ratio of bits transferred per node for slow and fast nodes is obtained from (33) as z_{ S } / z_{ F } ≅ E[T_{1,S}]/E[T_{1,F}]. When optimal minimum CW is chosen according to (34) or (35), the ratio of bits transferred per node for slow and fast station becomes equal to unity, thus resulting in bitbased fairness.
4.1.2. Three classes of mean velocities
In this section, we extend our analysis to a V2I network in which there are three classes of mean velocities: slow (S), medium (M) and fast (F). Let n_{ S }, n_{ M }, n_{ F }, respectively, denote the number of vehicles corresponding to the three categories. ${\mu}_{{v}_{S}}$, ${\mu}_{{v}_{M}}$, and ${\mu}_{{v}_{F}}$, respectively, be their mean velocities; and E[T_{1,S}], E[T_{1,M}], and E[T_{1,F}], respectively, be their mean residence time. Clearly, E[T_{1,S}] > E[T_{1,M}] > E[T_{1,F}]. Further, let τ_{ S }, τ_{ M }, and τ_{ F }be the conditional frame transmission probabilities and let p_{c,S}, p_{c,M}, and p_{c,F}be the frame collision probabilities of slow, medium and fast vehicles, respectively.
Note that ${W}_{F,\text{min}}^{*}$ required to achieve bitbased fairness in a network with three classes of mean velocities is same as that of two classes case. Also, ${W}_{M,\text{min}}^{*}$ required to achieve bit based fairness in network with three classes of mean velocities is same as that required in a network two velocity classes, where the mean velocities are ${\mu}_{{v}_{M}}$ and ${\mu}_{{v}_{S}}$. Thus the optimal value of minimum CW required to achieve bitbased fairness in a network with two velocity classes, hold for network with three mean velocity classes as well. For a V2I network with N number of mean velocity classes, the results of (37) can be extended for all the higher velocity classes, provided we consider the slowest vehicle to be the reference node.
5. Analytical and simulation results
System parameters
Parameter  Value 

Packet payload  8184 bits @ 6 Mb/s 
MAC header  256 bits @ 6 Mb/s 
PHY header  192 bits @ 3 Mb/s 
ACK  112 bits + PHY header @ 3 Mb/s 
Channel bit rate  6 Mb/s 
Propagation delay  2 µ s 
Slot time  13 µ s 
SIFS  32 µ s 
DIFS  58 µ s 
Network size: two classes of mean velocities
${\mathit{\mu}}_{{\mathit{v}}_{\mathit{S}}},{\mathit{v}}_{\mathit{F}}\mathbf{\left(}\mathbf{km/hr}\mathbf{\right)}$  ^{ k }_{jam} = 80 veh/km/lane  ^{ k }_{jam} = 160 veh/km/lane  

n _{ S }  n _{ F }  n _{ S }  n _{ F }  
60, 120  12  5  25  10 
80, 120  10  5  20  10 
Network size: three classes of mean velocities
${\mathit{\mu}}_{{\mathit{v}}_{\mathit{S}}},{\mathit{\mu}}_{{\mathit{v}}_{\mathit{M}}},{\mathit{v}}_{\mathit{F}}\phantom{\rule{2.77695pt}{0ex}}\left(\mathit{km/hr}\right)$  ^{ k }_{jam} = 80 Veh/km/lane  ^{ k }_{jam} = 160 Veh/km/lane  

n _{ S }  n _{ M }  n _{ F }  n _{ S }  n _{ M }  n _{ F }  
40, 80, 120  15  10  5  30  20  10 
30, 90, 150  16  8  1  32  17  2 
5.1. Network with two classes of mean velocities
Data transferred (per node and total): default CW_{min} and optimal CW_{min} $\mathbf{\left(}\mu {\mathit{v}}_{\mathit{S}}\mathbf{=}\mathbf{60}\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}{\mu}_{{\mathit{v}}_{\mathit{F}}}\mathbf{=}\mathbf{120}\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{\mathit{v}}_{\mathit{S}}}\mathbf{=}\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{\mathit{v}}_{\mathit{F}}}\mathbf{=}\mathbf{5}\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}{\mathit{k}}_{\mathbf{jam}}\mathbf{=}\mathbf{80}\phantom{\rule{2.77695pt}{0ex}}\mathbf{and}\phantom{\rule{2.77695pt}{0ex}}\mathbf{160}\phantom{\rule{2.77695pt}{0ex}}\mathbf{veh/km/lane}\mathbf{\right)}$
Number of vehicles  CW_{min} settings  Slow vehicle (Mb)  Fast vehicle (Mb)  Total (Mb)  

Analytical  Simulation  Analytical  Simulation  Analytical  Simulation  
Default  W_{S,min}= 16 W_{F,min}= 16  3.1035  3.0754  1.5517  1.5487  45.008  44.6495  
n_{ S } = 12  W_{S,min}= 32 W_{F,min}= 32  3.3499  3.3373  1.6749  1.6671  48.5738  48.3886  
n_{ F } = 5  Bitbased fairness  ${W}_{S,\text{min}}^{*}=30$ W_{F,min}= 16  2.5594  2.4681  2.5239  2.5765  42.7313  42.5433 
${W}_{S,\text{min}}^{*}=62$ W_{F,min}= 32  2.6636  2.6433  2.7026  2.7428  45.4772  45.5108  
Default  W_{S,min}= 16 W_{F,min}= 16  1.3442  1.3545  0.6710  0.6806  40.3263  40.6703  
n_{ S } = 12  W_{S,min}= 32 W_{F,min}= 32  1.4941  1.4863  0.7470  0.7317  44.8250  44.4757  
n_{ F } = 10  Bitbased fairness  ${W}_{S,\text{min}}^{*}=30$ W_{F,min}= 16  1.1130  1.0940  1.1267  1.1487  39.0941  38.8381 
W_{S,min}= 16 ${W}_{F,\text{min}}^{*}=9$  1.3189  1.2732  1.3014  1.2602  45.9882  44.4360  
${W}_{S,\text{min}}^{*}=62$ W_{F,min}= 32  1.2259  1.2042  1.2286  1.2408  42.9354  42.5139 
Data transferred (per node and total): default CW_{min} and optimal CW_{min} $\mathbf{\left(}{\mu}_{{\mathit{v}}_{\mathit{S}}}\mathbf{=}\mathbf{80}\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}\mu {\mathit{v}}_{\mathit{F}}\mathbf{=}\mathbf{120}\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{\mathit{v}}_{\mathit{S}}}\mathbf{=}{\sigma}_{{\mathit{v}}_{\mathit{F}}}\mathbf{=}5\phantom{\rule{2.77695pt}{0ex}}\mathbf{km/hr},\phantom{\rule{2.77695pt}{0ex}}{\mathit{k}}_{\mathbf{jam}}\mathbf{=}\mathbf{80}\phantom{\rule{2.77695pt}{0ex}}\mathbf{and}\phantom{\rule{2.77695pt}{0ex}}\mathbf{160}\phantom{\rule{2.77695pt}{0ex}}\mathbf{veh/km/lane}\mathbf{\right)}$
Number of vehicles  CW_{min}Settings  Slow vehicle (Mb)  Fast vehicle (Mb)  Total (Mb)  

Analytical  Simulation  Analytical  Simulation  Analytical  Simulation  
Default  W_{S,min}= 16 W_{F,min}= 16  2.6806  2.6829  1.7870  1.7749  35.7415  35.7043  
n_{ S } = 10  W_{S,min}= 32 W_{F,min}= 32  2.8965  2.8893  1.9376  1.8891  38.7538  38.3355  
n_{ F } = 5  Bitbased fairness  ${W}_{S,\text{min}}^{*}=23$ W_{F,min}= 16  2.3618  2.3313  2.3679  2.3837  35.4588  35.2317 
${W}_{S,\text{min}}^{*}=47$ W_{F,min}= 32  2.5426  2.5071  2.5662  2.5551  38.2578  37.8477  
Default  W_{S,min}= 16 W_{F,min}= 16  1.2076  1.2223  0.8050  0.8032  32.2028  32.4785  
n_{ S } = 20  W_{S,min}= 32 W_{F,min}= 32  1.3351  1.3359  0.8900  0.8803  35.6032  35.2223  
n_{ F } = 5  Bitbased fairness  ${W}_{S,\text{min}}^{*}=23$ W_{F,min}= 16  1.0797  1.0771  1.0630  1.0663  32.2245  32.2068 
${W}_{S,\text{min}}^{*}=47$ W_{F,min}= 32  1.1787  1.1609  1.1800  1.1920  35.3755  35.1402 
5.1.1. Evaluation of optimal CW_{min}for Slow and fast vehicles
It can be observed that, with optimal CW_{min} values, the data transferred for the nodes are almost equal irrespective of their velocities; thus ensuring bitbased fairness. Further, the optimal CW_{min} values do not depend on the number of slow or fast stations, thus bitbased fairness is maintained always, irrespective of the network size. However, we observe a slight reduction in the total amount of data transferred (last column in Tables 4 and 5) for the bitbased fairness case compared to the default case. One possible reason for this reduction is that we used default value for W_{F,min}and optimal value for W_{S,min}which is larger than the corresponding default. On the other hand, when we used default value for W_{S,min}and optimal value for W_{F,min}which is smaller than the default, we got improved value. In our future study, we will be choosing optimal values for the windows of all classes, so as to maximize the total data transferred while also providing the bitbased fairness.
5.2. Network with three classes of mean velocities
Variation of fairness index with CW_{min} of slow and medium Vehicle for W_{F,min}= 16
Window size  Fairness index  

W _{ M ,min }  W _{ S ,min }  n _{ F } = 5, n _{ M } = 10, n _{ S } = 15  n _{ F } = 10, n _{ M } = 20, n _{ S } = 30  
Analysis  Simulation  Analysis  Simulation  
4  4  0.7960  0.7633  0.7949  0.7549 
8  8  0.8223  0.7846  0.8217  0.7745 
16  16  0.8681  0.8314  0.8677  0.8205 
24  24  0.9017  0.8703  0.9013  0.8671 
24  46  0.9998  0.9618  0.9998  0.9502 
32  32  0.9213  0.8732  0.9211  0.8627 
64  64  0.8822  0.8318  0.8862  0.8162 
128  128  0.6504  0.5986  0.6504  0.5915 
Data transferred (per node and total): default CW_{min} and optimal $C{W}_{\mathrm{min}}\phantom{\rule{0.25em}{0ex}}({\mu}_{{\mathit{v}}_{\mathit{S}}}=40\phantom{\rule{0.25em}{0ex}}\mathbf{km/hr},\phantom{\rule{0.25em}{0ex}}{\mu}_{{v}_{M}}=80\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},\phantom{\rule{0.25em}{0ex}}{\mu}_{{\mathit{v}}_{\mathit{F}}}=120\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},\phantom{\rule{0.25em}{0ex}}{\mathit{k}}_{\mathrm{jam}}=80\phantom{\rule{0.25em}{0ex}}\mathrm{veh/km/lane}$ and ${\mu}_{{\mathit{v}}_{\mathit{S}}}=80\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},\phantom{\rule{0.25em}{0ex}}{\mu}_{{\mathit{v}}_{\mathit{M}}}=105\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},\phantom{\rule{0.25em}{0ex}}{\mu}_{{v}_{F}}=140\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},\phantom{\rule{0.25em}{0ex}}{\mathit{k}}_{\mathrm{jam}}=80\phantom{\rule{0.25em}{0ex}}\mathrm{veh/km/lane},\phantom{\rule{0.25em}{0ex}}{\sigma}_{{\mathit{v}}_{\mathit{S}}}={\sigma}_{{\mathit{v}}_{\mathit{M}}}={\sigma}_{{\mathit{v}}_{\mathit{F}}}=5\phantom{\rule{0.25em}{0ex}}\mathrm{km/hr},)$
Number of vehicles  CW_{min}Settings  Slow vehicle (Mb)  Medium vehicle (Mb)  Fast vehicle (Mb)  Total (Mb)  

Analytical  Simulation  Analytical  Simulation  Analytical  Simulation  Analytical  Simulation  
n_{ S } = 15  Default  W_{S,min}= 16 W_{M,min}= 16 W_{F,min}= 16  2.4152  2.3398  1.2070  1.1592  0.8050  0.7728  52.3294  50.5451 
n_{ M } = 10  W_{S,min}= 32 W_{M,min}= 32 W_{F,min}= 32  2.6702  2.5213  1.3351  1.2751  0.8900  0.8330  57.8550  54.7367  
n_{ F } = 5  Bitbased fairness  ${W}_{S,\text{min}}^{*}=46$ ${W}_{M,\text{min}}^{*}=24$ W_{F,min}= 16  1.5682  1.4642  1.5565  1.4903  1.6187  1.5509  47.1824  44.6424 
${W}_{S,\text{min}}^{*}=92$ ${W}_{M,\text{min}}^{*}=47$ W_{F,min}= 32  1.7066  1.5730  1.7151  1.6187  1.7243  1.6521  51.3728  48.8124  
n_{ S } = 10  Default  W_{S,min}= 16 W_{M,min}= 16 W_{F,min}= 16  2.1775  2.0989  1.6590  1.5998  1.2444  1.1899  34.2181  32.9676 
n_{ M } = 6  W_{S,min}= 32 W_{M,min}= 32 W_{F,min}= 32  2.3719  2.3211  1.8071  1.7888  1.3553  1.3212  37.2734  36.5862  
n_{ F } = 2  Biotbased fairness  ${W}_{S,\text{min}}^{*}=28$ ${W}_{M,\text{min}}^{*}=22$ W_{F,min}= 16  1.8168  1.7918  1.8001  1.7788  1.9010  1.7912  32.9506  32.1732 
${W}_{S,\text{min}}^{*}=56$ ${W}_{M,\text{min}}^{*}=44$ W_{F,min}= 32  1.9813  1.9711  1.9474  1.9299  1.9166  1.9098  35.3306  35.0123 
5.3. Impact of standard deviation of vehicle speed
6. Conclusion
In this article, we have investigated the issue of fairness in IEEE 802.11pbased vehicle to Infrastructure networks. We presented a simple yet accurate analytical model to compute the data transferred for contending vehicles in V2I network, taking into account their residence time within the coverage area of RSU. Classifying the vehicles according to their mean velocities, the model can be used to find the data transferred for a class i vehicle with mean velocity ${\mu}_{{v}_{i}},i\in \left[1,N\right]$. We, then addressed an unfairness problem that occur in V2I networks because of distinct vehicle velocities. The communication with RSU for a vehicle with higher velocity is affected significantly owing to their reduced residence time within the coverage area of RSU. It was proved that ratio of data transferred for vehicles with two different mean velocities is equal to the ratio of their mean residence times, assuming that the vehicles use the same MAC parameters and frame size. This implies that average amount of data transferred for each node is inversely proportional to its velocity, meaning that a fast moving vehicle, which naturally needs more recent information, has less chance to communicate with the RSU. We proposed to adapt CW_{min} according to node velocity, and determined optimal CW_{min} values required to achieve fairness (in the sense that all nodes with different speeds have same chance of communicating during their residence time in the coverage area of an RSU). It was proved that these optimal CW_{min} values are independent of number of vehicles in the network. Analytical and simulation results were presented for the data transferred for vehicles belonging to different velocity classes. The impact of variability of vehicle speed were also analyzed. Adjusting TXOP rather than minimum CW, to achieve the desired fairness objective, is for the future study.
Endnote
^{a}We adopt the short notation: P(z_{1}, s_{1}, b_{1}z_{0}, s_{0}, b_{0}) = P (z_{ i }(t+1) = z_{1}, s_{ i }(t+1) = s_{1}, b_{ i }(t+1) = b_{1}z_{ i }(t) = z_{0}, s_{ i }(t) = s_{0}, b_{ i }(t) = b_{0})
Declarations
Authors’ Affiliations
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