An accurate and complete performance modeling of the IEEE 802.11p MAC sublayer for VANET

Computer Communications 149 (2020) 107–120

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Computer Communications journal homepage: www.elsevier.com/locate/comcom

An accurate and complete performance modeling of the IEEE 802.11p MAC sublayer for VANET✩ Shengbin Cao, Victor C.S. Lee ∗,1 Department of Computer Science, City University of Hong Kong, Hong Kong

ARTICLE

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Keywords: IEEE 802.11p Performance modeling Vehicular ad-hoc networks (VANETs)

ABSTRACT This paper proposes an analytical model for the throughput, channel access delay, and queueing delay of the IEEE 802.11p enhanced distributed channel access (EDCA) mechanism in the medium-access control (MAC) sublayer. In order to make the proposed analytical model explicitly solvable and to avoid computation complexity caused by most of existing 3-D or 4-D Markov-chain-based analytical models, a combination of 2-D and 1-D Markov chain model is used. The 2-D Markov chain is established to model the backoff procedure of each access category (AC) queue, deriving a relationship between the transmission probability and collision probability for each AC queue. Afterwards, a 1-D discrete-time Markov chain is established to model the contention period caused by the distinction of four different arbitration inter-frame space (AIFS) and contention window (CW) values, deriving another relationship between the transmission probability and collision probability of each AC queue. In order to consider both saturated and non-saturated cases, an infinitelong 1-D Markov chain model is used. In the proposed analytical model, all the features in the EDCA such as different CW and AIFS for each AC, various saturation status, internal collision, backoff counter freezing, up-to-date standard parameters, and initial-carrier-sensing procedure are taken into consideration. Based on the two Markov models, we further derive performance models which describe the relationships between the network parameters and channel access performance metrics in terms of normalized throughput, channel access delay, and queueing delay, respectively. The proposed analytical model applies to both basic access mode and request-to-send/clear-to-send (RTS/CTS) access mode, and is suitable for four access categories of traffic in the IEEE 802.11p. The proposed model can be employed to analyze the performance of real-world vehicular network. Simulation results are shown to verify the accuracy and effectiveness of the proposed analytical model.

1. Introduction Recently, the intelligent transportation system (ITS) is drawing dramatic attention due to its ability to improve safety and effectiveness of the future transportation system [1–3]. ITS attempts to apply information and communication technologies (ICT) to establish the vehicular ad-hoc network (VANET) to enable access to the Internet, which will contribute to effectiveness of infotainment and safety service provisions. The key components of VANET are roadside unit (RSU) [4], which is located on the road and provides vehicles with the access to broad-band Internet, and onboard unit (OBU) [5], which is mounted on a vehicle and acts as moving node. Communication among OBUs is categorized into Vehicle-to-vehicle (V2V) communication. Vehicle-toinfrastructure (V2I) is a form of communication between an OBU and an RSU.

In order to support both ad-hoc and infrastructure-based communications in the environment that has rapidly changing network topology and special characteristic of propagation environment, the IEEE 802.11p [6] has been proposed for the medium-access control (MAC) scheme for VANET. In the IEEE 802.11p, frequent authorization and handshake are restricted in order to deal with the fast movement and dramatically changing trajectory of VANET. For the sake of the importance of safety and considering the unique characteristics of VANET, low latency and high reliability need to be guaranteed for safety-related applications, whereas high throughput and resource utilization are the major concerns for infotainment applications [7]. Hence, in order to provide different quality-of-service (QoS), IEEE 802.11p employs enhanced distributed channel access (EDCA) mechanism based on the IEEE 802.11e [8] standard. The 802.11p EDCA mechanism specifies four different access categories (AC) with different parameters to contend for the channel access.

✩ The work described in this paper was substantially supported by a grant from City University of Hong Kong (Project No. 7004888). ∗ Corresponding author. E-mail addresses: [email protected] (S. Cao), [email protected] (V.C.S. Lee). 1 Member, IEEE.

https://doi.org/10.1016/j.comcom.2019.08.026 Received 22 April 2019; Received in revised form 30 July 2019; Accepted 30 August 2019 Available online 27 September 2019 0140-3664/© 2019 Elsevier B.V. All rights reserved.

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120

to avoid computation complexity and make the proposed analytical model explicitly solvable, two Markov chain models are combined to derive the transmission and collision probabilities for each AC queue, based on which we derive accurate models for performance metrics in terms of normalized throughput, channel access delay, and queueing delay. Since the prime objective of VANET is to provide safety services, which need strict delay requirement, not only channel access delay, but queueing delay is the major concern, which, however, is omitted by other papers. Hence, in our paper, we propose an accurate model for queueing delay of the IEEE 802.11p. The remainder of this paper is organized as follows: In Section 2, a comprehensive review of the related works and the motivation of the work are provided. In Section 3, a brief introduction of MAC scheme in the IEEE 802.11p is introduced. In Section 4, the analytical model of the IEEE 802.11p EDCA mechanism is provided. In Section 5, accuracy validation of the proposed model is shown by comparing the analytical and simulation results. The conclusion is provided in Section 6.

These parameters include the minimum contention window (CWmin), maximum contention window (CWmax), arbitration inter-frame space (AIFS), and retransmission limit. In order to reveal the shortcomings and investigate the performance on the medium-access scheme of the IEEE 802.11p, it is worthy to develop an analytical model for the IEEE 802.11p. In order to make the analytical model accurate, when constructing the model, all major factors have to be taken into consideration, which are described as follows: • Initial-carrier-sensing procedure: As specified by the IEEE 802.11p standard, if a packet is generated when the channel is idle, the packet will start initial-carrier-sensing procedure, i.e., if the channel keeps idle during an AIFS period, the packet will be sent directly at the end of the AIFS duration without the backoff procedure. Otherwise, if the channel is sensed busy before the AIFS duration ends, the packet will start a new backoff procedure. The analytical model should incorporate the initial-carrier-sensing procedure. • Saturation status: An AC queue can be either saturated or nonsaturated, i.e., the condition in which there are always packets to send in the AC queue and the case where packets occasionally arrive at the AC queue. In order to shed light on the performance in terms of different packet arrival rate, the analytical model should apply to both saturated and non-saturated conditions. • Backoff counter freezing: In the IEEE 802.11p standard, it is specified that the backoff counter needs to be suspended upon channel is sensed busy. As soon as the channel becomes idle, the backoff counter will be resumed. The analytical model should incorporate this factor. • Up-to-date standard parameters: In order to apply the model to analyze the performance of the IEEE 802.11p in a large-scale vehicular network system, the parameters, namely, contention window (CW) size, AIFS value, retransmission limit, slot time duration, short inter-frame space (SIFS), distributed inter-frame space (DIFS) need to be most up-to-date. This factor should also be taken into account as well. • Internal collision: The internal collision occurs when more than one AC queue in the same station attempts to transmit packets. As specified by the IEEE 802.11p, in this situation, the transmission opportunity is granted to the AC queue with the highest priority. The model should contain this factor. • Computational complexity: The model should avoid computation complexity in order to explicitly solvable. This factor should also be taken into consideration when establishing analytical model.

2. Related work and motivation Several recent publications [9–32] have studied the analytical model for EDCA. Based on the model of [10], Malone et al. [9] proposed a mathematical model for the DCF mechanism, which is suitable for both saturation and non-saturation conditions. In [14], a Markov-chainbased analytical model was proposed to discover the performance of the IEEE 802.11e. In [15], by combining several approaches, a unified approximation model is derived to study the throughput and delay performances in the saturated condition. In [16], a model considering distinct collision probabilities during different contention periods was proposed to study the throughput performance of the IEEE 802.11p. However, the common limitation of these models is that the backoff counter freezing mechanism was not taken into consideration. In the work of [12], Kaabi et al. proposed an analytical model, which is based on Markov chain, for the EDCA mechanism. In [18], a discrete-time Markov chain model was employed to describe the performance of the EDCA mechanism. In [19], an analytical model was proposed for the IEEE 802.11e, considering contention period after a busy channel. In [20], a discrete-time Markov-chain-based analytical model was proposed to discover the performance of service channel reservation scheme in the IEEE 802.11p. However, the common limitation of them is that only two ACs were taken into consideration rather than 4, as specified in the standard. In [13], an analytical model was proposed to discover the performance of the IEEE 802.11e with consideration of transmission opportunity specified in the IEEE 802.11e. In [21], an analytical model was proposed, taking into account the distinctions of AIFSs, CWs, and retransmission limits, based on 3-D Markov chain. In [22], a 2-D Markov-chain-based model was proposed in order to maximize the throughput. However, the common limitation of [13,15,16,21], and [22] is that they only considered saturated condition. In [17], a discrete-time Markov chain model was used to model the IEEE 802.11e EDCA mechanism, considering AIFS and CW differentiation. In [23], the throughput of the IEEE 802.11e was discovered using a combination of two Markov chains. In [24], a frame-transmissioncycle approach was employed to derive the throughput performance of the IEEE 802.11e on the saturated condition. In [25], a scalable analytical model was proposed to capture the performance of the EDCA mechanism under non-saturated condition. In [26], an analytical model based on 3-D Markov chain was proposed to analyze the EDCA mechanism in terms of delay, throughput, and packet-drop rate. However, the proposed models in [17,19], and [23–26] did not consider the internal collision factor. In [27], Xiang et al. proposed an analytical model employing signal transfer function to analyze the performance of the EDCA mechanism on both saturated and non-saturated conditions. In [28], a discrete-time Markov chain model was proposed in order to capture the performance

Although there are a number of works focusing on the analytical modeling of the IEEE 802.11p [9–13], to the best of our knowledge, none of them has taken all the aforementioned factors into consideration. For instance, in [9] a mathematical model is proposed to analyze the performance of the IEEE 802.11 distributed coordination function (DCF). However, the backoff counter freezing factor has not been taken into account in the model. In [11], a Markov chain model for the IEEE 802.11p is proposed, which is suitable for both saturated and nonsaturated conditions. However, the initial-carrier-sensing procedure is not considered in the model. In the model of [12], only two ACs are considered. The proposed model of [13] only applies to saturation condition. Therefore, motivated by these shortcomings, we propose an accurate and computationally tractable analytical model, which takes into consideration all the major factors. The contributions of our paper are: our work aims to propose an analytical model for the IEEE 802.11p MAC sublayer, taking into account all the major factors that affect the performance, including distinct CWs and AIFSs for different ACs, initial-carrier-sensing procedure, flexible saturation status, backoff counter freezing, most up-to-date standard parameters, internal collision resolution mechanism. Strong approximations are avoided to assure the accuracy of our model. In order 108

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Computer Communications 149 (2020) 107–120 Table 1 EDCA parameters. AC AC_BK AC_BE AC_VI AC_VO

𝐶𝑊𝑚𝑖𝑛 𝑎𝐶𝑊𝑚𝑖𝑛 𝑎𝐶𝑊𝑚𝑖𝑛 (𝑎𝐶𝑊𝑚𝑖𝑛 + 1)∕2 − 1 (𝑎𝐶𝑊𝑚𝑖𝑛 + 1)∕4 − 1

𝐶𝑊𝑚𝑎𝑥 𝑎𝐶𝑊𝑚𝑎𝑥 𝑎𝐶𝑊𝑚𝑎𝑥 𝑎𝐶𝑊𝑚𝑖𝑛 (𝑎𝐶𝑊𝑚𝑖𝑛 + 1)∕2 − 1

AIFSN 9 6 3 2

will be directly transmitted if the channel keeps idle for 𝑇𝐴𝐼𝐹 𝑆[𝜎] . Otherwise, if the channel is sensed busy during the initial-carriersensing procedure, the procedure is stopped and an ordinary backoff procedure will begin. Note that the packet immediately after the headof-line (HOL) needs to go through an ordinary backoff procedure after the packet in the HOL is dequeued. If the channel becomes idle and keeps the idle state for a 𝑇𝐴𝐼𝐹 𝑆 , an initial backoff counter will be uniformly selected in the range of [0, 𝐶𝑊𝑚𝑖𝑛 ]. During the backoff procedure, if the channel keeps idle for a time slot, the backoff counter will be decremented by 1. Otherwise, the counter will be frozen at the current value and resumed if the channel becomes idle and keeps idle for a 𝑇𝐴𝐼𝐹 𝑆 . When the backoff counter becomes zero, the packet will be transmitted. If a packet collision occurs, i.e., there is no acknowledgment (ACK) frame received, the packet will be retransmitted. At each retransmission, the value of the current CW will be doubled and a new backoff counter is uniformly selected from the new range. If the value of CW reaches 𝐶𝑊𝑚𝑎𝑥 , it keeps this value until the retransmission limit is reached. After the retransmission limit is reached, the packet is dropped and the value of CW will be set to zero. If an internal collision occurs, i.e., more than one AC queue in the same station attempts to transmit, the highest-priority AC queue will be granted the chance to transmit packet. At the same time, the other colliding AC queues with lower priorities behave as if there were an external collision.

Fig. 1. EDCA mechanism.

of the IEEE 802.11e under heterogeneous environment. In [29], a 3-D Markov chain was used for establishing an analytical model to derive probability distribution of channel access delay and maximum throughput of the IEEE 802.11e EDCA mechanism under saturated condition. In [30], an analytical model was proposed to derive throughput and channel access delay based on 3-D discrete-time Markov chain. In [31], a 3-D Markov-chain-based model was proposed to capture the effect of distinct AIFSs and CWs on the performance of the IEEE 802.11e EDCA. However, the common limitation of the proposed models in [17, 27–31] is the unnecessary computational complexity caused by 3-D Markov-chain-based models. In [11], an analytical model combining two discrete-time Markov chains was proposed to calculate throughput and delay for the IEEE 802.11p, which is suitable for both saturated and non-saturated conditions. In [32], an analytical model was proposed to derive throughput of the IEEE 802.11p, taking into account distinct transmission opportunities, using two Markov chain models. However, none of them considered initial-carrier-sensing procedure.

4. Analytical model In this section, in order to derive a performance model, we use a combination of a 2-D Markov chain representing backoff procedure of each AC queue and a 1-D infinite-long discrete-time Markov chain that represents the contention period. We first use 2-D Markov chain to obtain the relationship between the transmission probability and collision probability of each AC queue. Due to different channel access priority, during the backoff procedure, time slots are grouped into different contention zones, which we use 1-D Markov chain to model in order to obtain another relationship between the transmission probability and collision probability. Then, the combination of the two Markov models allows us to derive the transmission and collision probabilities of each AC queue for an arbitrary time slot. Finally, an accurate model is established to obtain the performance of each AC queue in terms of throughput, average channel access delay, and queueing delay. We consider both saturated and non-saturated conditions and take into consideration the most up-to-date standard parameters, different AFISs and CWs for different ACs, backoff counter freezing, internal collision, and initial-carrier-sensing, when establishing the analytical model. Furthermore, both RTS/CTS and basic modes are taken into consideration in our model. Following the most related works [10,11], and [16], we consider idle channel condition, i.e., the only factor that causes the transmission error is packet collision, and sing-cell environment, i.e., all nodes are in the transmission range of each other. We assume that packets arrive at an 𝐴𝐶𝜎 queue according to the Poisson process with rate 𝜆𝜎 . In addition, we apply an M/G/1/∞ queueing model for each station to obtain queueing delay. The major notations used in the analysis models are summarized in Table 2.

3. MAC sublayer in the IEEE 802.11p In this section, we give a brief introduction of the MAC sublayer in the IEEE 802.11p standard. The EDCA mechanism is a contentionbased channel access mechanism, which aims to support prioritized QoS services. It specifies four ACs, each of which corresponds to an AC queue. Each AC queue operates as if it were an independent station to contend for the channel access based on enhanced distributed channel access function (EDCAF). As shown in Fig. 1, each station contains four AC queues and each AC queue acts independently using their own AC-related parameters, including CWmin, CWmax, AIFS number, and retransmission limit. Different ACs are assigned different 𝐶𝑊𝑚𝑖𝑛 and 𝐶𝑊𝑚𝑎𝑥 values, i.e., shorter CW for higher priority AC. The parameters of the most up-to-date IEEE 802.11p standard are shown in Table 1. The value of 𝐶𝑊𝑚𝑖𝑛 and 𝐶𝑊𝑚𝑎𝑥 are set to 15 and 1023, respectively [6]. The AC_VO, i.e., access category for voice-type message, is assigned the highest priority while, AC_BK, i.e., access category for background-type message, is provided with the lowest priority. In order to provide prioritized QoS service, distinct AIFS values are used for different ACs, with lower value for higher priority. The value of AIFS for the 𝐴𝐶𝜎 (𝜎 = 1, 2, 3, 4), i.e., 𝑇𝐴𝐼𝐹 𝑆[𝜎] can be expressed as 𝑇𝐴𝐼𝐹 𝑆[𝜎] = 𝐴𝐼𝐹 𝑆𝑁[𝜎] ⋅ 𝑇𝑠𝑙𝑜𝑡 + 𝑇𝑆𝐼𝐹 𝑆 , where 𝐴𝐼𝐹 𝑆𝑁[𝜎], 𝑇𝑠𝑙𝑜𝑡 , and 𝑇𝑆𝐼𝐹 𝑆 stand for AIFS number, duration of a time slot, and duration of an SIFS, respectively. If a packet arrives at the 𝐴𝐶𝜎 queue when the channel state is idle, an initial-carrier-sensing procedure is triggered, in which the packet 109

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Computer Communications 149 (2020) 107–120

Table 2 Notations used in the model. Notation

Definition

𝐴𝐶𝜎

Access category 𝜎, 0 ≤ 𝜎 ≤ 3

𝜎 𝐶𝑊𝑚𝑖𝑛

Minimum contention window for the 𝐴𝐶𝜎

𝜎 𝐶𝑊𝑚𝑎𝑥

Maximum contention window for the 𝐴𝐶𝜎

retransmission limit, the packet will be discarded and the value of 𝑚 will be reset to 0. In addition, once the transmission is successful, the value of 𝑚 will be reset to 0. The initial value of 𝑘 is uniformly drawn from [0, 𝑊𝑖𝜎 − 1], if the state is in backoff stage 𝑖. The value of 𝑘 will be decremented by 1 at the end of a time slot, if the channel is sensed idle during the slot. The HOL packet will be transmitted when the value of backoff counter reaches zero. When there is no packet inside the tagged 𝐴𝐶𝜎 queue, the queue is in the idle state. It can be denoted as following:

𝑚𝜎

Maximum backoff stage for the 𝐴𝐶𝜎

𝑚𝜎 + 𝑓𝜎

Retransmission limit for the 𝐴𝐶𝜎

𝑝𝜎𝑏𝑢𝑠𝑦

𝑋𝜎 (𝑡) = (𝑖𝑑𝑙𝑒).

𝑝𝜎𝑐𝑜𝑙

Probability that the channel is sensed busy by the 𝐴𝐶𝜎 during a time slot Probability that there is a packet in the 𝐴𝐶𝜎 queue after a transmission at the end of a backoff procedure Probability that there is a packet in the 𝐴𝐶𝜎 queue at the end of an initial-carrier-sensing procedure Collision probability for the 𝐴𝐶𝜎

𝑝𝜎𝑠𝑢𝑐

Successful probability for the 𝐴𝐶𝜎

𝑋𝜎 (𝑡) = (𝑖𝑐, 𝑑),

𝑝𝜎𝑖 𝑐𝑜𝑙

Collision probability for the 𝐴𝐶𝜎 in zone 𝑖

𝜏𝜎

Transmission probability for the 𝐴𝐶𝜎

𝑝𝑖𝑧

Transition probability in zone 𝑖

𝑝𝜎𝑞 𝑝̂𝜎𝑞

If a packet is generated during idle state, the packet will enter the initial-carrier-sensing procedure. If the channel is sensed idle for the 𝑇𝐴𝐼𝐹 𝑆[𝜎] , the HOL packet will be transmitted at the end of the period. This case is denoted by 𝑑 = 0, 1, 2, … , 𝐷𝜎 ,

where 𝐷𝜎 ≜ 𝐴𝐼𝐹 𝑆𝜎 ∕𝑇𝑠𝑙𝑜𝑡 . Here, we use 𝑇𝑠𝑙𝑜𝑡 to denote the length of a time slot. 𝑋𝜎 (𝑡) = (𝑖𝑐, 𝑑) means that the channel is sensed idle for (𝐷𝜎 − 𝑑)𝑇𝑠𝑙𝑜𝑡 period during the initial-carrier-sensing procedure. For the sake of simplification, we define following:

𝑡𝑖

Stationary probability for state 𝑖 in 1-D Markov chain

𝛷𝑖

Stationary probability of staying in zone 𝑖

1 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

𝑝̂𝜎𝑠𝑢𝑐

Expected transmission time when a packet is transmitted at the end of a backoff procedure Expected time when a packet is transmitted at the end of an initial-carrier-sensing procedure Probability that a packet is generated from upper layer during idle slot in the 𝐴𝐶𝜎 queue Probability that a packet is generated from upper layer during busy slot in the 𝐴𝐶𝜎 queue Probability that the 𝐴𝐶𝜎 observes a successful transmission

𝑝̂𝜎𝑖 𝑠𝑢𝑐

Probability that the 𝐴𝐶𝜎 observes a successful transmission in zone 𝑖

𝑃 (𝑚 + 1, 𝑘|𝑚, 0) =

𝑝̃𝑗𝑖 𝑠𝑢𝑐[𝜎]

Probability that the 𝐴𝐶𝜎 observes a successful transmission of the 𝐴𝐶𝑗 in zone 𝑖

for 𝑚 = 𝑚𝜎 + 𝑓𝜎 ,

𝜆𝜎𝑖

Packet arrival rate for the 𝐴𝐶𝜎

𝑃 (𝑖𝑑𝑙𝑒|𝑚𝜎 + 𝑓𝜎 , 0) = 1 − 𝑝𝜎𝑞 ,

𝐵𝑚𝜎

Average waiting time in the 𝑚th backoff stage

𝜎 𝑇̂𝑎𝑣𝑒

Average duration of a time slot observed by the 𝐴𝐶𝜎

2 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

𝛼𝜎 𝛽𝜎

𝑃 (𝑘|𝑙) ≜ 𝑃 (𝑋𝜎 (𝑡 + 1) = 𝑘|(𝑋𝜎 (𝑡) = 𝑙). From the Markov chain given in Fig. 2, the one-step transition probabilities can be given as following: for 0 ≤ 𝑚 < 𝑚𝜎 + 𝑓𝜎 , 𝑃 (𝑖𝑑𝑙𝑒|𝑚, 0) = (1 − 𝑝𝜎𝑐𝑜𝑙 )(1 − 𝑝𝜎𝑞 ), 𝑃 (0, 𝑘|𝑚, 0) =

(1 − 𝑝𝜎𝑐𝑜𝑙 )𝑝𝜎𝑞 ∕𝑊0𝜎 𝜎 𝑝𝜎𝑐𝑜𝑙 ∕𝑊𝑚+1

𝑃 (0, 𝑘|𝑚𝜎 + 𝑓𝜎 , 0) =

𝑝𝜎𝑞 ∕𝑊0𝜎

𝜎 𝑇𝑎𝑣𝑒

Average duration of a time slot

for the case of 𝑋𝜎 (𝑡) = (𝑖𝑑𝑙𝑒),

𝑇𝑖𝑐𝜎

Average initial-carrier-sensing delay

𝜎 𝑇𝑎𝑐𝑐𝑒𝑠𝑠

Average access delay

𝑃 (𝑖𝑐, 𝐷𝜎 |𝑖𝑑𝑙𝑒) = 𝛼 𝜎 ,

𝑄𝜎

Queueing delay for the 𝐴𝐶𝜎

𝜎 𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒

Service time for the 𝐴𝐶𝜎

𝑋𝑏𝜎

Delay for the backoff procedure

𝑋𝑖𝑐𝜎

Delay for the initial-carrier-sensing procedure

𝑚𝑖𝑐

The number of time slots elapsed from the entrance of the initial-carrier-sensing until the instant the procedure terminates The probability of successful transmission of the 𝐴𝐶𝜎 in a time slot

𝑣𝜎𝑠𝑢𝑐 𝛩

𝜎

𝑃 (0, 𝑘|𝑖𝑑𝑙𝑒) = 𝛽

𝜎

∕𝑊0𝜎 𝜎

(1) 𝑊0𝜎

− 1,

(2)

𝜎 𝑊𝑚+1

− 1.

(3)

0≤𝑘≤ 0≤𝑘≤

(4) 0≤𝑘≤

𝑊0𝜎

0≤𝑘≤

𝑊0𝜎

(5)

− 1.

(6) (7)

− 1,

𝑃 (𝑖𝑑𝑙𝑒|𝑖𝑑𝑙𝑒) = 1 − 𝛼 − 𝛽 𝜎 .

(8)

for the case of 1 ≤ 𝑑 ≤ 𝐷𝜎 during initial-carrier-sensing procedure, 𝑃 (𝑖𝑐, 𝑑 − 1|𝑖𝑐, 𝑑) = 1 − 𝑝𝜎𝑏𝑢𝑠𝑦 , 𝑃 (0, 𝑘|𝑖𝑐, 𝑑) = 𝑝𝜎𝑏𝑢𝑠𝑦 ∕𝑊0𝜎

Throughput of the 𝐴𝐶𝜎

(9) 0 ≤ 𝑘 ≤ 𝑊0𝜎 − 1.

(10)

for the case of 𝑑 = 0 during initial-carrier-sensing procedure, 𝑃 (𝑖𝑑𝑙𝑒|𝑖𝑐, 0) = (1 − 𝑝𝜎𝑐𝑜𝑙 )(1 − 𝑝̂𝜎𝑞 ), 𝑃 (0, 𝑘|𝑖𝑐, 0) =

4.1. Analysis of the backoff procedure for each AC queue

[𝑝𝜎𝑐𝑜𝑙

+ (1 − 𝑝𝜎𝑐𝑜𝑙 )𝑝̂𝜎𝑞 ]∕𝑊0𝜎

(11) 0≤𝑘≤

𝑊0𝜎

− 1.

(12)

In order to represent backoff counter freezing, for 0 ≤ 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 ,

We first use 2-D Markov chain model to describe dynamic behavior of the EDCA backoff procedure for an individual 𝐴𝐶𝜎 queue, which is shown in Fig. 2. It shall be noted that Y and Z in Fig. 2 denote not states, but the sums of probabilities for the sake of computational simplicity. We use 𝑋𝜎 (𝑡) to denote the state of the tagged 𝐴𝐶𝜎 queue at 𝑡th (𝑡 = 1, 2, 3, … ) embedded point. There are three types of states in our model: (i) backoff state, (ii) idle state, and (iii) initial-carrier-sensing state. The backoff state means the case that tagged 𝐴𝐶𝜎 queue performs backoff procedure during a time slot. It is denoted by following:

𝑃 (𝑚, 𝑘|𝑚, 𝑘) = 𝑝𝜎𝑏𝑢𝑠𝑦 𝑃 (𝑚, 𝑘 − 1|𝑚, 𝑘) =

1 − 𝑝𝜎𝑏𝑢𝑠𝑦

0 ≤ 𝑘 ≤ 𝑊𝑚𝜎 − 1, 1≤𝑘≤

𝑊𝑚𝜎

− 1.

(13) (14)

where each equation implies as follows: • Eq. (1) accounts for the fact that before the retransmission limit is reached, if the HOL packet is successfully transmitted without collision and the 𝐴𝐶𝜎 queue is empty, the 𝐴𝐶𝜎 enters the idle state. • Eq. (2) accounts for the fact that before the retransmission limit is reached, if the HOL packet is successfully transmitted and the 𝐴𝐶𝜎 queue is not empty, a backoff procedure is initiated. • Eq. (3) stands for the case that before the retransmission limit is reached, if it fails to transmit the HOL packet, the packet is retransmitted, leading to increment of backoff stage.

𝑋𝜎 (𝑡) = (𝑚, 𝑘), where 𝑚 and 𝑘 represent backoff stage and backoff counter of HOL packet, respectively. The value of 𝑚 is initiated as 0 and incremented by 1 after each transmission failure until it reaches the retransmission limit. If the packet transmission fails when the value of 𝑚 reaches 110

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120

Fig. 2. 2-D Markov chain.

• Eq. (4) corresponds to the fact that when the retransmission limit is reached and the tagged 𝐴𝐶𝜎 queue is empty, the 𝐴𝐶𝜎 enters the idle state regardless of the transmission result. • Eq. (5) accounts for the fact that when the retransmission limit is reached and the tagged 𝐴𝐶𝜎 queue is not empty, the 𝐴𝐶𝜎 enters the backoff procedure regardless of the transmission result. • Eq. (6) accounts for the fact that if a packet is generated when the channel is idle, the 𝐴𝐶𝜎 enters the initial-carrier-sensing procedure. • Eq. (7) accounts for the fact that if a packet is generated when the channel is busy, the 𝐴𝐶𝜎 enters the backoff procedure. • Eq. (8) accounts for the fact that if no packet is generated during a time slot, the 𝐴𝐶𝜎 stays in the idle state. • Eq. (9) accounts for the fact that during initial-carrier-sensing procedure, if channel is idle during a time slot, then the counter decreases by one at the end of the slot. • Eq. (10) accounts for the fact that during initial-carrier-sensing procedure, if channel is sensed busy, the backoff procedure is initiated. • Eq. (11) corresponds to the case that when the HOL packet is successfully transmitted at the end of initial-carrier-sensing procedure, if no packet is generated during the transmission period, the 𝐴𝐶𝜎 enters idle state. • Eq. (12) corresponds to the case that the 𝐴𝐶𝜎 enters backoff procedure if the transmitted HOL packet encounters a collision at the end of initial-carrier-sensing procedure or if a packet is generated during the successful transmission period. • Eq. (13) corresponds to the case that during backoff procedure, if the channel is sensed busy, the backoff counter freezes. • Eq. (14) corresponds to the case that during backoff procedure, if the channel is sensed idle, the backoff counter decreases by one.

In addition, the contention window size of the 𝐴𝐶𝜎 at backoff stage 𝑚 can be expressed as following: ⎧𝐶𝑊 𝜎 𝑚𝑖𝑛 ⎪ 𝜎 +1 𝐶𝑊𝑚𝜎 = ⎨2𝐶𝑊𝑚−1 ⎪𝐶𝑊 𝜎 𝑚𝑎𝑥 ⎩

𝑚 = 0, 1 ≤ 𝑚 ≤ 𝑚𝜎 − 1, 𝑚𝜎 ≤ 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 ,

(15)

from which we can express backoff counter of the 𝐴𝐶𝜎 , which is: ⎧𝐶𝑊 𝜎 + 1 𝑚𝑖𝑛 ⎪ 𝑊𝑚𝜎 = ⎨2𝑚 𝑊0𝜎 ⎪𝐶𝑊 𝜎 + 1 𝑚𝑎𝑥 ⎩

𝑚 = 0, 1 ≤ 𝑚 ≤ 𝑚𝜎 − 1, 𝑚𝜎 ≤ 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 .

(16)

4.2. Analysis of the contention period Next, we establish a mathematical model to describe contention period by using a 1-D Markov chain. During the backoff procedure, if channel is sensed busy, the backoff counter freezes for the packet transmission period. After the packet transmission is finished, the 𝐴𝐶𝜎 keeps sensing the channel during 𝑇𝐴𝐼𝐹 𝑆[𝜎] period. If the channel is sensed idle for 𝑇𝐴𝐼𝐹 𝑆[𝜎] period, the 𝐴𝐶𝜎 continues backoff procedure. According to the IEEE 802.11p [6], the values of AIFS for different ACs are distinct, leading to different contention zones, as shown in Fig. 3. The values of AIFS for 𝐴𝐶0 , 𝐴𝐶1 , 𝐴𝐶2 , and 𝐴𝐶3 are defined as 2, 3, 6 and 9 times 𝑇𝑠𝑙𝑜𝑡 , respectively. From Fig. 3, it can be observed that the distinction of AIFS values results in different contention zones. We use 𝑙𝑖 to denote the number of time slots in contention zone 𝑖 (𝑖 = 1, 2, 3, 4). Hence, from the figure, it can be observed that 𝑙1 = 1, 𝑙2 = 3, 𝑙3 = 3, and 𝑙4 = ∞. Due to distinct AIFS values, different ACs contend for channel access in different contention zones: only 𝐴𝐶0 contends in contention zone 1; in contention zone 2, 𝐴𝐶0 and 𝐴𝐶1 contend; in zone 3, 𝐴𝐶0 , 𝐴𝐶1 , and 𝐴𝐶2 join contention group; 𝐴𝐶0 , 𝐴𝐶1 , 𝐴𝐶2 , and 𝐴𝐶3 contend 111

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120

− 𝑝𝜎𝑐𝑜𝑙 ⋅ 𝑝̂𝜎𝑞 − 1)] + 𝛽 𝜎 } ⋅ 𝜎 + 𝜋𝑖𝑑𝑙𝑒

⋅𝛼

𝜎

1 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +𝑓𝜎 +1 1 − 𝑝𝜎𝑐𝑜𝑙

(1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 ,

(19)

𝜎 𝜋𝑖𝑑𝑙𝑒

where denotes the stationary probability of idle state, Eq. (20) is shown in Box I. Here, 𝑝𝜎𝑞 determines whether the station is saturated or non-saturated, i.e., the condition that 𝑝𝜎𝑞 < 1 (resp. 𝑝𝜎𝑞 = 1 ) means non-saturated (resp. saturated) condition. The derivation of (19) and (20) is shown in Appendix A, in detail. Now, we need another relationship between 𝜏 𝜎 and 𝑝𝜎𝑐𝑜𝑙 . Since any packet transmission results in a busy channel sate and there are different ACs contending for the channel access in the different contention zones, according to the 1-D Markov chain model, the transition probabilities in Fig. 4 can be obtained as following: ∏ 𝑝𝑖𝑧 = (1 − 𝜏 𝜎 )𝑁𝜎 1 ≤ 𝑖 ≤ 4, 0 ≤ 𝜎 ≤ 3, (21) 𝜎<𝑖

where 𝑁𝜎 denotes the number of the 𝐴𝐶𝜎 queues. We use 𝑡𝑣 to denote the stationary probability distribution of state 𝑣 in Markov chain of Fig. 4, i.e.,

Fig. 3. Contention zone.

𝑡𝑣 ≜ lim 𝑃 (State 𝑣) 𝑡→∞

According to (17), we can derive the following:

𝑡𝑣+1

∞ ∑

(22)

𝑙1 + 𝑙2 + 𝑙3 + 1 ≤ 𝑣.

(23)

(24)

𝑡𝑣 = 1.

𝑣=1

Thus, according to (24), we can express 𝑡1 in terms of transition probabilities ( 𝑝2 − (𝑝2𝑧 )𝑙2 +1 1 − (𝑝1𝑧 )𝑙1 +1 + (𝑝1𝑧 )𝑙1 ⋅ 𝑧 𝑡1 = 1 − 𝑝1𝑧 1 − 𝑝2𝑧

𝑙1 + 1 ≤ 𝑣 ≤ 𝑙1 + 𝑙2 , (17)

+ 𝑙2 + 𝑙3 , 𝑙1 + 𝑙2 + 𝑙3 + 1 ≤ 𝑣.

+ (𝑝1𝑧 )𝑙1 (𝑝2𝑧 )𝑙2 ⋅

4.3. Transmission and collision probability of each AC queue

𝑝3𝑧 − (𝑝3𝑧 )𝑙3 +1 1 − 𝑝3𝑧

+ (𝑝1𝑧 )𝑙1 (𝑝2𝑧 )𝑙2 (𝑝3𝑧 )𝑙3 ⋅

In this section, the transmission and collision probability are derived based on the established Markov chain models. We denote the stationary probability distribution of the Markov chain as following: For 0 ≤ 𝑑 ≤ 𝐷𝜎 , 0 ≤ 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 , and 0 ≤ 𝑘 ≤ 𝑊𝑚𝜎 − 1,

𝑝4𝑧 1 − 𝑝4𝑧

)−1 .

(25)

Since the stationary probabilities of a zone is the sum of all stationary probabilities of states in that zone, we can obtain ⎧ ⎪ 𝛷1 ⎪ ⎪ ⎪ ⎪ 𝛷2 ⎪ ⎨ ⎪ ⎪ 𝛷3 ⎪ ⎪ ⎪ ⎪ 𝛷4 ⎩

(18)

By using 2-D Markov model that represents the backoff procedure, (1)–(18), we can obtain the relationship between the transmission probability 𝜏 𝜎 and collision probability 𝑝𝜎𝑐𝑜𝑙 for the 𝐴𝐶𝜎 as following: 𝜎 𝜏 𝜎 = 𝜋𝑖𝑑𝑙𝑒 ⋅

𝑙1 + 𝑙2 + 1 ≤ 𝑣 ≤ 𝑙1 + 𝑙2 + 𝑙3

According to the balance equation, we have

1 ≤ 𝑣 ≤ 𝑙1 ,

⎧𝜋 𝜎 ≜ lim 𝑃 (𝑋 (𝑡) = 𝑖𝑑𝑙𝑒), 𝜎 ⎪ 𝑖𝑑𝑙𝑒 𝑡→∞ ⎪ 𝜎 lim 𝑃 (𝑋𝜎 (𝑡) = (𝑖𝑐, 𝑑)), ⎨𝜋(𝑖𝑐,𝑑) ≜ 𝑡→∞ ⎪ 𝜎 lim 𝑃 (𝑋𝜎 (𝑡) = (𝑚, 𝑘)), ⎪𝜋(𝑚,𝑘) ≜ 𝑡→∞ ⎩

𝑙1 + 1 ≤ 𝑣 ≤ 𝑙1 + 𝑙2 ,

⎧𝑡1 ⋅ (𝑝1 )𝑖−1 , if 1 ≤ 𝑖 ≤ 𝑙1 + 1, 𝑧 ⎪ 𝑙 2 1 1 ⎪𝑡1 ⋅ (𝑝𝑧 ) ⋅ (𝑝𝑧 )𝑖−𝑙1 −1 , if 𝑙1 + 2 ≤ 𝑖 ≤ 𝑙1 + 𝑙2 + 1, ⎪ ⎪𝑡1 ⋅ (𝑝1𝑧 )𝑙1 ⋅ (𝑝2𝑧 )𝑙2 ⋅ (𝑝3𝑧 )𝑖−𝑙1 −𝑙2 −1 , 𝑡𝑣 = ⎨ ⎪ if 𝑙1 + 𝑙2 + 2 ≤ 𝑖 ≤ 𝑙1 + 𝑙2 + 𝑙3 + 1 ⎪ 1 𝑙1 2 𝑙2 3 𝑙3 4 𝑖−𝑙1 −𝑙2 −𝑙3 −1 , ⎪𝑡1 ⋅ (𝑝𝑧 ) ⋅ (𝑝𝑧 ) ⋅ (𝑝𝑧 ) ⋅ (𝑝𝑧 ) ⎪ if 𝑙 + 𝑙 + 𝑙 + 2 ≤ 𝑖. 1 2 3 ⎩

in zone 4. To this end, the contention period can be represented as 1D Markov chain shown in Fig. 4. We use 𝑣 to denote the state that 𝑣th (𝑣 = 1, 2, 3, … ) time slot is visited in Fig. 4. Since any packet transmission results in a busy channel, leading to re-initialization of contention period, transition from one state to subsequent state means that the channel has been sensed idle until the subsequent time slot. Otherwise, if the channel is sensed busy, the state will enter state 1. The state transition probabilities are different in different contention zones. Hence, from Fig. 4, we can derive

𝑙1 + 𝑙2 + 1 ≤ 𝑣 ≤ 𝑙1

1 ≤ 𝑣 ≤ 𝑙1 ,

By (22), we can express all stationary probability distribution of 1-D Markov chain in terms of 𝑡1 ,

Fig. 4. 1-D Markov chain.

⎧𝑃 (𝑣 + 1|𝑣) = 𝑝1𝑧 , 𝑃 (1|𝑣) = 1 − 𝑝1𝑧 ⎪ ⎪𝑃 (𝑣 + 1|𝑣) = 𝑝2𝑧 , 𝑃 (1|𝑣) = 1 − 𝑝2𝑧 ⎪ 3 3 ⎨𝑃 (𝑣 + 1|𝑣) = 𝑝𝑧 , 𝑃 (1|𝑣) = 1 − 𝑝𝑧 ⎪ ⎪ ⎪𝑃 (𝑣 + 1|𝑣) = 𝑝4 , 𝑃 (1|𝑣) = 1 − 𝑝4 ⎩ 𝑧 𝑧

⎧𝑡𝑣 ⋅ 𝑝1𝑧 ⎪ ⎪𝑡𝑣 ⋅ 𝑝2𝑧 =⎨ 3 ⎪𝑡 𝑣 ⋅ 𝑝 𝑧 ⎪ 4 ⎩𝑡 𝑣 ⋅ 𝑝 𝑧

1 ⋅ {𝛼 𝜎 [1 + (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 ⋅ (𝑝𝜎𝑐𝑜𝑙 + 𝑝̂𝜎𝑞 1 − 𝑝𝜎𝑞 112

=

=

𝑙1 ∑

𝑡𝑣 , 𝑣=1 𝑙1 +𝑙2 ∑

𝑡𝑣 ,

𝑣=𝑙1 +1

(26)

𝑙1 +𝑙2 +𝑙3

=

=

∑

𝑡𝑣 ,

𝑣=𝑙1 +𝑙2 +1 ∞ ∑ 𝑣=𝑙1 +𝑙2 +𝑙3 +1

𝑡𝑣 ,

S. Cao and V.C.S. Lee

{ 𝜋𝑖𝑑𝑙𝑒 = +

1 + 𝛼𝜎

Computer Communications 149 (2020) 107–120

1 − (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 +1 𝑝𝜎𝑏𝑢𝑠𝑦

} { 1 − (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +1 { [ ] 1 𝑐𝑜𝑙 𝛼 𝜎 1 + (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 (𝑝𝜎𝑐𝑜𝑙 + 𝑝̂𝜎𝑞 − 𝑝𝜎𝑐𝑜𝑙 𝑝̂𝜎𝑞 − 1) + 𝛽 𝜎 𝜎 1 − 𝑝𝑞 1 − 𝑝𝜎𝑐𝑜𝑙 ] }} 𝑝𝜎 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +1 (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +1 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +𝑓𝜎 +1 𝜎 𝜎 𝑚𝜎 −1 − 𝑐𝑜𝑙 + 𝑊 + (𝑊 2 − 1) − 1 . 0 0 1 − 𝑝𝜎𝑐𝑜𝑙 1 − 𝑝𝜎𝑐𝑜𝑙

+

[ 2𝑝𝜎 − (2𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +1 1 1 𝑊0𝜎 𝑐𝑜𝑙 𝜎 2 1 − 𝑝𝑏𝑢𝑠𝑦 1 − 2𝑝𝜎𝑐𝑜𝑙

(20)

Box I. 1 by To this end, we can express 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

where 𝛷𝑖 denotes the stationary probability of zone 𝑖. Now, we attempt to derive the collision probability experienced by the 𝐴𝐶𝜎 in zone 𝑖, which we use 𝑝𝜎𝑖 to denote. Note that the 𝐴𝐶𝜎 queue experiences a 𝑐𝑜𝑙 collision when there is at least one simultaneously transmitting 𝐴𝐶𝑟 , with 𝑟 < 𝜎 (resp. 0 ≤ 𝑟 ≤ 3), if 𝐴𝐶𝑟 queue is in the same station (resp. different station). For 0 ≤ 𝜎 ≤ 3, 0 ≤ 𝑟 ≤ 3, and 1 ≤ 𝑖 ≤ 4, ⎧1 − ∏(1 − 𝜏 𝑟 )𝑁𝑟 ⋅ ∏ (1 − 𝜏 𝑟 )𝑁𝑟 −1 𝜎 < 𝑖 ⎪ 𝑟<𝜎 𝜎≤𝑟<𝑖 𝑝𝜎𝑖 (27) 𝑐𝑜𝑙 = ⎨ ⎪0 𝜎 ≥ 𝑖. ⎩

1 𝜎 𝜎 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎] = 𝑝̃𝜎𝑠𝑢𝑐 ⋅ 𝑇𝑠𝑢𝑐 + 𝑝̃𝜎𝑐𝑜𝑙 ⋅ 𝑇𝑐𝑜𝑙 .

Similarly, due to the Poisson process, we can express 𝑝̂𝜎𝑞 by 2 −𝜆𝜎 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

𝑝̂𝜎𝑞 = 1 − 𝑒

2 𝜎 𝜎 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎] = 𝜋𝑖𝑐,0 (1 − 𝑝𝜎𝑐𝑜𝑙 ) ⋅ 𝑇𝑠𝑢𝑐 ,

,

Then, using (26) and (36), we can calculate 𝑝𝜎𝑏𝑢𝑠𝑦 by For 0 ≤ 𝜎 ≤ 3 and 1 ≤ 𝑖 ≤ 4, ∑ 𝜎𝑖 𝜎<𝑖 𝛷𝑖 𝑝𝑏𝑢𝑠𝑦 𝑝𝜎𝑏𝑢𝑠𝑦 = ∑ . 𝜎<𝑖 𝛷𝑖

(29)

1 where 𝜆𝜎 and 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎] denote the packet arrive rate of the 𝐴𝐶𝜎 and expected transmission time when a packet is transmitted at the end of a backoff procedure, respectively. It can be observed that (29) depends 1 on the unknown parameter 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎] . There are two events regarding the state transition from state (𝑚, 0) to state (𝑖𝑑𝑙𝑒), namely, successful transmission event and packet discard event, whose probabilities we 𝜎 and 𝑇 𝜎 to denote use 𝑝̃𝜎𝑠𝑢𝑐 and 𝑝̃𝜎𝑐𝑜𝑙 to denote, respectively. We use 𝑇𝑠𝑢𝑐 𝑐𝑜𝑙 the durations of successful and collision transmissions, respectively. In 𝜎 𝜎 the basic mode, 𝑇𝑠𝑢𝑐 and 𝑇𝑐𝑜𝑙 can be expressed by { 𝜎 =𝑇 +𝑇 +𝛿+𝑇 𝑇𝑠𝑢𝑐 𝐻 𝐹 𝑆𝐼𝐹 𝑆 + 𝑇𝐴𝐶𝐾 + 𝛿 + 𝑇𝐴𝐼𝐹 𝑆[𝜎] , (30) 𝜎 =𝑇 +𝑇 𝑇𝑐𝑜𝑙 + 𝛿 + 𝑇𝐴𝐼𝐹 𝑆[𝜎] , 𝐻 𝐹𝑚𝑎𝑥

(37)

Next, we attempt to derive the expressions of 𝛼 𝜎 and 𝛽 𝜎 , which denote the probability that a packet is generated from upper layer during idle and busy time slot, respectively. The probability that a packet is generated from upper layer during idle time slot can be derived by 𝛼 𝜎 = (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 ) ⋅ (1 − 𝑒−𝜆

𝜎𝑇 𝑠𝑙𝑜𝑡

(38)

),

Similarly, the probability that a packet is generated during busy time slot can be derived as

where 𝑇𝐻 , 𝑇𝐹 , and 𝑇𝐴𝐶𝐾 denote the time required for frame header, frame body, and ACK frame transmission, respectively; 𝛿 denotes propagation time; 𝑇𝐹𝑚𝑎𝑥 denotes transmission time of the largest frame body involved in a collision; 𝑇𝑆𝐼𝐹 𝑆 and 𝑇𝐴𝐼𝐹 𝑆[𝜎] denote SIFS and AIFS duration for the 𝐴𝐶𝜎 , respectively. On the other hand, in the RTS/CTS mode 𝜎 =𝑇 ⎧𝑇𝑠𝑢𝑐 𝑅𝑇 𝑆 + 𝛿 + 𝑇𝑆𝐼𝐹 𝑆 + 𝑇𝐶𝑇 𝑆 + 𝛿 ⎪ + 𝑇𝑆𝐼𝐹 𝑆 + 𝑇𝐻 + 𝑇𝐹 + 𝛿 + 𝑇𝑆𝐼𝐹 𝑆 ⎪ ⎨ + 𝑇𝐴𝐶𝐾 + 𝛿 + 𝑇𝐴𝐼𝐹 𝑆[𝜎] , ⎪ ⎪ 𝜎 ⎩𝑇𝑐𝑜𝑙 = 𝑇𝑅𝑇 𝑆 + 𝛿 + 𝑇𝐴𝐼𝐹 𝑆[𝜎] ,

(35)

Next, we attempt to calculate 𝑝𝜎𝑏𝑢𝑠𝑦 . An 𝐴𝐶𝜎 queue senses the channel as busy when at least one other AC queue transmits simultaneously. Due to contention zone, the probability that 𝐴𝐶𝜎 senses the channel as busy in zone 𝑖, i.e., 𝑝𝜎𝑖 , can be obtained as following: 𝑏𝑢𝑠𝑦 For 0 ≤ 𝜎 ≤ 3, 0 ≤ 𝑟 ≤ 3, and 1 ≤ 𝑖 ≤ 4, ∏ ⎧1 − (1 − 𝜏 𝑟 )𝑁𝑟 ⋅ (1 − 𝜏 𝜎 )𝑁𝜎 −1 𝜎 < 𝑖 ⎪ 𝑟<𝑖,𝑟≠𝜎 (36) 𝑝𝜎𝑖 = ⎨ 𝑏𝑢𝑠𝑦 ⎪0 𝜎 ≥ 𝑖. ⎩

Given a set of parameters, the relationship between 𝜏 𝜎 and 𝑝𝜎𝑐𝑜𝑙 can be obtained by (19)–(28). In order to calculate 𝜏 𝜎 and 𝑝𝜎𝑐𝑜𝑙 , we need to express 𝑝𝜎𝑞 , 𝑝̂𝜎𝑞 , 𝑝𝜎𝑏𝑢𝑠𝑦 , 𝛼 𝜎 , and 𝛽 𝜎 , in terms of 𝜏 𝜎 and/or 𝑝𝜎𝑐𝑜𝑙 . Then, we can obtain the equation system that only contains 𝜏 𝜎 and 𝑝𝜎𝑐𝑜𝑙 with 𝑊0𝜎 that can be calculated by (16), 𝑚𝜎 and 𝑓𝜎 that are given parameters. We first attempt to express 𝑝𝜎𝑞 . Since we expect packet is generated from upper layer based on Poisson process, 𝑝𝜎𝑞 can be obtained by 1 −𝜆𝜎 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

(34)

,

2 denotes the expected time experienced when a packet where 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎] is transmitted at the end of an initial-carrier-sensing procedure. Since 2 can be calculated by there is no packet discard event occurring, 𝑇𝑡𝑟𝑎𝑛𝑠[𝜎]

To this end, we can calculate the collision probability of the 𝐴𝐶𝜎 as following: For 0 ≤ 𝜎 ≤ 3 and 1 ≤ 𝑖 ≤ 4, ∑ 𝜎𝑖 𝜎<𝑖 𝛷𝑖 𝑝𝑐𝑜𝑙 𝑝𝜎𝑐𝑜𝑙 = ∑ . (28) 𝜎<𝑖 𝛷𝑖

𝑝𝜎𝑞 = 1 − 𝑒

(33)

𝜎 −𝜆𝜎 𝑇𝑏𝑢𝑠𝑦

𝛽 𝜎 = 𝑝𝜎𝑏𝑢𝑠𝑦 ⋅ (1 − 𝑒

(39)

).

𝜎 where 𝑇𝑏𝑢𝑠𝑦 denotes the expected duration of a busy time slot. In order 𝜎 . The channel busy probability 𝑝𝜎 to derive 𝛽 𝜎 , we need to derive 𝑇𝑏𝑢𝑠𝑦 𝑏𝑢𝑠𝑦 is the sum of successful and collision transmission probabilities that is observed by the 𝐴𝐶𝜎 . Hence, we first attempt to derive successful transmission probability observed by the 𝐴𝐶𝜎 . We use 𝑝̂𝜎𝑖 𝑠𝑢𝑐 to denote the successful transmission probability observed by the 𝐴𝐶𝜎 in zone 𝑖, which can be calculated as following: For 0 ≤ 𝜎 ≤ 3, 0 ≤ 𝑟 ≤ 3, and 1 ≤ 𝑖 ≤ 4,

(31)

∑ ⎧ 1 𝑁𝑗 ⋅ 𝑝̃𝑗𝑖 ⎪∑ 𝑠𝑢𝑐[𝜎] 𝑁 = ⎨ 0≤𝑤<𝑖 𝑤 𝑗<𝑖 ⎪0 ⎩

where 𝑇𝑅𝑇 𝑆 and 𝑇𝐶𝑇 𝑆 represent transmission time of RTS and CTS packets, respectively. We can express 𝑝̃𝜎𝑠𝑢𝑐 and 𝑝̃𝜎𝑐𝑜𝑙 by

𝑝̂𝜎𝑖 𝑠𝑢𝑐

⎧ ∑ 𝜎 ⎪𝑝̃𝜎𝑠𝑢𝑐 = 𝜋(𝑚,0) ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 ), ⎨ 𝑚=0 ⎪𝑝̃𝜎 = 𝜋 𝜎 ⎩ 𝑐𝑜𝑙 (𝑚𝜎 +𝑓𝜎 ) ⋅ 𝑝𝑐𝑜𝑙 .

where 𝑝̃𝑗𝑖 denotes the probability that the 𝐴𝐶𝜎 observes successful 𝑠𝑢𝑐[𝜎] transmission of 𝐴𝐶𝑗 in zone 𝑖. We can derive 𝑝̃𝑗𝑖 as following: 𝑠𝑢𝑐[𝜎]

𝑚𝜎 +𝑓𝜎

(32)

113

𝜎<𝑖 (40) 𝜎 ≥ 𝑖,

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120 𝜎 denotes the average duration of a time slot observed by the where 𝑇̂𝑎𝑣𝑒 𝜎 ̂ 𝐴𝐶𝜎 . 𝑇𝑎𝑣𝑒 can be calculated by

For 0 ≤ 𝜎 ≤ 3, 0 ≤ 𝑟 ≤ 3, 0 ≤ 𝑗 ≤ 3, and 1 ≤ 𝑖 ≤ 4, ∏ ⎧𝑁𝑗 𝜏 𝑗 (1 − 𝜏 𝑗 )𝑁𝑗 −1 ⋅ (1 − 𝜏 𝑟 )𝑁𝑟 ⎪ 𝑟<𝑗 ⎪ ∏ (1 − 𝜏 𝑟 )𝑁𝑟 −1 𝑗 < 𝑖, 𝑗 ≠ 𝜎, ⎪⋅ ⎪ 𝑟>𝑗 ∏ ⎪ = ⎨𝑁𝑗 𝜏 𝑗 (1 − 𝜏)𝑁𝑗 −2 ⋅ 𝑝̃𝑗𝑖 (1 − 𝜏 𝑟 )𝑁𝑟 𝑠𝑢𝑐[𝜎] ⎪ 𝑟<𝑗 ⎪ ∏ (1 − 𝜏 𝑟 )𝑁𝑟 −1 𝑗 < 𝑖, 𝑗 = 𝜎, ⎪⋅ ⎪ 𝑟>𝑗 ⎪ 𝑗 ≥ 𝑖. ⎩0

𝜎 𝜎 𝑇̂𝑎𝑣𝑒 = (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 ) ⋅ 𝑇𝑠𝑙𝑜𝑡 + 𝑝𝜎𝑏𝑢𝑠𝑦 ⋅ 𝑇𝑏𝑢𝑠𝑦 .

Note that, the value of backoff stage is incremented by one, when a collision occurs. If the backoff stage reaches the retransmission limit, and a collision occurs, the packet will be discarded. Therefore, 𝐷1𝜎 can be calculated by

(41)

𝜎 ) ⋅ (1 − 𝑝𝜎 ) 𝐷1𝜎 = (𝑇𝐴𝐼𝐹 𝑆[𝜎] + 𝑇𝑠𝑢𝑐 𝑐𝑜𝑙 𝑚𝜎 +𝑓𝜎

By using (26), (40), and (41), we can derive 𝑝̂𝜎𝑠𝑢𝑐 , which represents the probability that the 𝐴𝐶𝜎 observes a successful transmission as following: For 0 ≤ 𝜎 ≤ 3 and 1 ≤ 𝑖 ≤ 4, ∑ 𝜎𝑖 𝜎<𝑖 𝛷𝑖 𝑝̂𝑠𝑢𝑐 𝜎 . (42) 𝑝̂𝑠𝑢𝑐 = ∑ 𝜎<𝑖 𝛷𝑖

+

𝑝̂𝜎𝑐𝑜𝑙 = 𝑝𝜎𝑏𝑢𝑠𝑦 − 𝑝̂𝜎𝑠𝑢𝑐 .

which includes three cases: (i) the case on which the packet is successfully transmitted at the end of an initial-carrier-sensing procedure, (ii) the case on which the packet is successfully transmitted after 𝑚 + 1 collisions, and (iii) the case on which the packet is discarded after 𝑚𝜎 + 𝑓𝜎 + 2 collisions. Since 𝐷2𝜎 represents the case where it fails to go through the initialcarrier-sensing procedure, we need to derive the expression of expected 𝜎, delay for incomplete initial-carrier-sensing procedure, denoted by 𝐷𝑖𝑐 which can be calculated by

(44)

𝜎 𝐷𝑖𝑐 =

4.4. Average channel access delay The channel access delay of an 𝐴𝐶𝜎 , denoted by is the duration from the time a packet in the 𝐴𝐶𝜎 queue starts to contend for the channel access to the time the packet is transmitted without 𝜎 collision or discarded. Based on Markov chain shown in Fig. 2, 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 falls into four cases, denoted by 𝐷𝑞𝜎 (𝑞 = 1, 2, 3, 4), with probability 𝛤𝑞𝜎 , respectively

= = =

𝛥

𝑚𝜎 +𝑓𝜎 𝜎 𝐷3𝜎 = E[𝑇𝑅𝐸𝑀[1] ]+

𝛥 𝜎 𝛽𝜎 𝜋𝑑𝑖𝑙𝑒

𝛥

𝜎 E[𝑇𝑅𝐸𝑀[1] ]=

𝜎 𝜆𝜎 − 1 + 𝑇𝑠𝑢𝑐

(1 − 𝑒−𝜆

𝜎𝑇𝜎 𝑠𝑢𝑐

)𝜆𝜎

(52)

.

𝑚𝜎 +𝑓𝜎 𝜎 𝐷4𝜎 = E[𝑇𝑅𝐸𝑀[2] ]+

∑

𝜎 (E[𝐵𝑚𝜎 ] + 𝑇𝑠𝑢𝑐 ) 𝑚=0 ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 ) + (E[𝐵𝑚𝜎 +𝑓 ] 𝜎 𝜎 𝜎 ) ⋅ (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +1 , + 𝑇𝑐𝑜𝑙 𝑐𝑜𝑙 𝜎 where E[𝑇𝑅𝐸𝑀[2] ] can be calculated, following

,

(53)

similar derivation pro-

cess of (52), by 𝜎 −𝜆𝜎 𝑇𝑏𝑢𝑠𝑦

From now, we attempt to derive the expression of 𝐷𝑞𝜎 . Since the backoff counter is uniformly drawn from [0, 𝑊𝑚𝜎 −1], if the backoff stage is 𝑚, the average waiting time in the backoff stage 𝑚, denoted by E[𝐵𝑚𝜎 ], can be expressed by 𝜎 ⋅ 𝑇̂𝑎𝑣𝑒 ,

𝜎𝑇𝜎 𝑠𝑢𝑐

(45)

𝜎 𝜎 𝛥 = 𝜋𝑖𝑑𝑙𝑒 (𝛼 𝜎 + 𝛽 𝜎 ) + (𝑝̃𝜎𝑠𝑢𝑐 + 𝑝̃𝜎𝑐𝑜𝑙 )𝑝𝜎𝑞 + 𝜋𝑖𝑐,0 (1 − 𝑝𝜎𝑐𝑜𝑙 )𝑝̂𝜎𝑞 .

2

𝑒−𝜆

Similarly, we can calculate 𝐷4𝜎 by

,

𝑊𝑚𝜎 − 1

(51)

of an ongoing transmission from the time on which the packet is generated. Then, as shown in 𝜎 Appendix B, we can express E[𝑇𝑅𝐸𝑀[1] ] as following:

where

E[𝐵𝑚𝜎 ] =

∑

𝜎 (E[𝐵𝑚𝜎 ] + 𝑇𝑠𝑢𝑐 ) 𝑚=0 ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 ) + (E[𝐵𝑚𝜎 +𝑓 ] 𝜎 𝜎 𝜎 ) ⋅ (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +1 , + 𝑇𝑐𝑜𝑙 𝑐𝑜𝑙 𝜎 where E[𝑇𝑅𝐸𝑀[1] ] is the remaining time

𝜎 𝛼 𝜎 [1 − (1 − 𝑝𝜎 )𝐷𝜎 ] 𝜋𝑖𝑑𝑙𝑒 𝑏𝑢𝑠𝑦

(𝑝̃𝜎𝑠𝑢𝑐

(50)

the packet is generated during its own transmission, the 𝐴𝐶𝜎 enters backoff procedure, skipping the initial-carrier-sensing procedure. Hence, 𝐷3𝜎 is calculated by

,

, 𝛥 𝜎 𝜎 𝜎 + 𝑝̃𝑐𝑜𝑙 )𝑝𝑞 + 𝜋𝑖𝑐,0 (1 − 𝑝𝜎𝑐𝑜𝑙 )𝑝̂𝜎𝑞

∑

𝜎 (E[𝐵𝑚𝜎 ] + 𝑇𝑠𝑢𝑐 ) ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚 𝑚=0 𝜎 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 ) + (E[𝐵𝑚𝜎 +𝑓 ] + 𝑇𝑐𝑜𝑙 ) 𝜎 𝜎 ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +𝑓𝜎 +1 . With respect to the case of 𝐷3𝜎 , since

Therefore, according to the Markov chain of Fig. 2, 𝛤𝑞𝜎 can be calculated as following: 𝜎 𝛼 𝜎 (1 − 𝑝𝜎 )𝐷𝜎 𝜋𝑖𝑑𝑙𝑒 𝑏𝑢𝑠𝑦

(49)

𝑚𝜎 +𝑓𝜎 𝜎 𝐷2𝜎 = 𝐷𝑖𝑐 +

• 𝐷1𝜎 : The case in which, a packet is generated from the upper layer in the 𝐴𝐶𝜎 queue during an idle time slot and completes the initial-carrier-sensing procedure. • 𝐷2𝜎 : The case in which, a packet is generated from the upper layer in the 𝐴𝐶𝜎 queue during an idle time slot and fails to complete the initial-carrier-sensing procedure. • 𝐷3𝜎 : The case in which, a packet is generated from the upper layer in the 𝐴𝐶𝜎 during its own transmission duration. • 𝐷4𝜎 : The case in which, a packet is generated from the upper layer in the 𝐴𝐶𝜎 during the transmission time of other ACs.

=

𝐷𝜎 ∑ 1 𝜎 (𝑝𝜎 )𝑖−1 𝑝𝜎𝑏𝑢𝑠𝑦 [(𝑖 − 1)𝑇𝑠𝑙𝑜𝑡 + 𝑇𝑏𝑢𝑠𝑦 ], 𝜎 𝐷 1 − (𝑝𝑖𝑑𝑙𝑒 ) 𝜎 𝑖=1 𝑖𝑑𝑙𝑒

where 𝑝𝜎𝑖𝑑𝑙𝑒 stands for the idle channel state observed by the 𝐴𝐶𝜎 . By using (49) we can derive 𝐷2𝜎 by

𝜎 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 ,

⎧ ⎪𝛤 𝜎 ⎪ 1 ⎪ ⎪ 𝜎 ⎪𝛤2 ⎨ ⎪ 𝜎 ⎪𝛤3 ⎪ ⎪ 𝜎 ⎪𝛤4 ⎩

(48)

𝜎 ) ⋅ (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +2 , ] + 𝑇𝑐𝑜𝑙 𝑐𝑜𝑙

𝜎 +𝑓𝜎

by

𝜎 𝜎 𝜎 𝑇𝑏𝑢𝑠𝑦 = 𝑝̂𝜎𝑐𝑜𝑙 ⋅ 𝑇𝑐𝑜𝑙 + 𝑝̂𝜎𝑠𝑢𝑐 ⋅ 𝑇𝑠𝑢𝑐 .

𝜎 (E[𝐵𝑚𝜎 ] + 𝑇𝑠𝑢𝑐 ) ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚+1 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 )

+ (E[𝐵𝑚𝜎

(43)

To this end, we can derive expression of

∑

𝑚=0

Hence, the probability that a collision transmission is observed by the 𝐴𝐶𝜎 , i.e., 𝑝̂𝜎𝑐𝑜𝑙 can be derived straightforwardly

𝜎 𝑇𝑏𝑢𝑠𝑦

(47)

𝜎 E[𝑇𝑅𝐸𝑀[2] ]=

𝑒

(1 − 𝑒

𝜎 𝜆𝜎 − 1 + 𝑇𝑏𝑢𝑠𝑦 𝜎 −𝜆𝜎 𝑇𝑏𝑢𝑠𝑦

.

(54)

)𝜆𝜎

𝜎 By using (45)–(54), we can derive 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 by 𝜎 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 = 𝛤1𝜎 𝐷1𝜎 + 𝛤2𝜎 𝐷2𝜎 + 𝛤3𝜎 𝐷3𝜎 + 𝛤4𝜎 𝐷4𝜎 .

(46) 114

(55)

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120

Since the selected number of time slots follows uniform distribution, we have 𝜎 ⎧E[𝑉 𝜎 ] = 𝑊𝑚 − 1 , ⎪ 𝑚 2 (68) ⎨ 𝜎 2 ⎪Var[𝑉 𝜎 ] = (𝑊𝑚 ) − 1 . ⎩ 𝑚 12 By using (47), we have

4.5. Queueing delay In this section, we attempt to derive the calculation of queueing delay for an 𝐴𝐶𝜎 , denoted by 𝑄𝜎 . The queueing delay is the duration from the instant a packet arrives at the queue to the instant the packet starts to receive the service, i.e., contents for the channel access. According to well-known property of M/G/1∞ queueing theory [33], we can derive 𝑄𝜎 as 𝑄𝜎 =

𝜎 𝜎 𝜆𝜎 (Var[𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ] + E[𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ]2 )

𝜎 ] = 𝑇̂ 𝜎 , ⎧E[𝑋𝑠𝑙𝑜𝑡 𝑎𝑣𝑒 ⎪ 𝜎 𝜎 𝜎 2 2 𝜎 ⎨Var[𝑋𝑠𝑙𝑜𝑡 ] = (𝑇𝑠𝑙𝑜𝑡 ) (1 − 𝑝𝑏𝑢𝑠𝑦 ) + (𝑇𝑏𝑢𝑠𝑦 ) 𝑝𝑏𝑢𝑠𝑦 ⎪ 𝜎 2 − (E[𝑋𝑠𝑙𝑜𝑡 ]) . ⎩

(56)

𝜎 2(1 − 𝜆𝜎 E[𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ])

𝜎 where 𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 stands for the service time for the 𝐴𝐶𝜎 . Since the service time is the sum of channel access delay and transmission time, we have

{

𝜎 𝜎 𝜎 E[𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ] = E[𝑇𝑎𝑐𝑐𝑒𝑠𝑠 ] + E[𝑇𝑠𝑢𝑐𝑐𝑒𝑠𝑠 ], 𝜎 𝜎 𝜎 Var[𝑇𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ] = Var[𝑇𝑎𝑐𝑐𝑒𝑠𝑠 ] + Var[𝑇𝑠𝑢𝑐𝑐𝑒𝑠𝑠 ].

Due to the fixed-length of the packets for all vehicles, we have { 𝜎 𝜎 E[𝑇𝑠𝑢𝑐𝑐𝑒𝑠𝑠 ] = 𝑇𝑠𝑢𝑐𝑐𝑒𝑠𝑠 , 𝜎 Var[𝑇𝑠𝑢𝑐𝑐𝑒𝑠𝑠 ] = 0.

𝜎 According to Appendix B, we can derive Var[𝑇𝑅𝐸𝑀[1] ] by 𝜎 Var[𝑇𝑅𝐸𝑀[1] ]=

(57)

(58)

𝜎 Var[𝑇𝑅𝐸𝑀[2] ]

𝛩𝜎 =

𝑖𝑐

1 (𝑃 𝜎 )𝑥−1 𝑝𝜎𝑏𝑢𝑠𝑦 . 1 − (𝑝𝜎𝑖𝑑𝑙𝑒 )𝐷𝜎 𝑖𝑑𝑙𝑒

(62)

Var[𝑚𝜎𝑖𝑐 ] =

From (50), the expected value of

(64) is

𝑚𝜎 +𝑓𝜎

E[𝑋𝑏𝜎 ] =

∑

𝜎 (E[𝐵𝑚𝜎 ] + 𝑇𝑠𝑢𝑐 ) ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 )

𝑚=0

+ (E[𝐵𝑚𝜎

𝜎 +𝑓𝜎

𝜎 ) ⋅ (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +1 , ] + 𝑇𝑐𝑜𝑙 𝑐𝑜𝑙

𝑝𝑖𝑑𝑙𝑒 =

𝜎 𝑚𝜎 +𝑓𝜎 +1 − E[𝑋𝑏𝜎 ])2 ] + (𝑃𝑐𝑜𝑙 ) 𝜎 + (E[𝐵𝑚 +𝑓 ] − E[𝑋𝑏𝜎 ])2 ]. 𝜎 𝜎

⋅ [Var[𝐵𝑚𝜎 +𝑓 ] 𝜎 𝜎

𝑝𝑠𝑢𝑐 =

(66)

𝑇𝑎𝑣𝑒

4 ∑

(1 − 𝑒

𝜎 −𝜆𝜎 𝑇𝑏𝑢𝑠𝑦

.

(71)

)(𝜆𝜎 )2

(72)

,

𝛷𝑖 𝑣𝜎𝑖 ,

4 ∑

4 ∑ 𝑖=1

(73)

𝛷𝑖 𝑝𝑖𝑧 .

(75)

𝛷𝑖

3 ∑

𝑣𝑗𝑖 .

(76)

𝑗=0

Hence, the collision probability, denoted by 𝑝𝑐𝑜𝑙 , can be derived by

Using well-known property for the variance of random sum [33], we have 𝜎 𝜎 Var[𝐵𝑚𝜎 ] = E[𝑉𝑚𝜎 ]Var[𝑋𝑠𝑙𝑜𝑡 ] + (E[𝑋𝑠𝑙𝑜𝑡 ])2 Var[𝑉𝑚𝜎 ],

𝜎 𝜆𝜎 )2 + 2 − 2𝑇 𝜎 𝜆𝜎 + (𝑇𝑏𝑢𝑠𝑦 𝑏𝑢𝑠𝑦

By using (26), (74), the probability of successful transmission, i.e., 𝑝𝑠𝑢𝑐 , can be calculated by

𝜎 𝑚 𝜎 (𝑃𝑐𝑜𝑙 ) (1 − 𝑃𝑐𝑜𝑙 ) ⋅ [Var[𝐵𝑚𝜎 ] + (E[𝐵𝑚𝜎 ]

𝑚=0

=

𝑖=1

𝑚𝜎 +𝑓𝜎

Var[𝑋𝑏𝜎 ] =

𝜎 −𝜆𝜎 𝑇𝑏𝑢𝑠𝑦

By using (21), (26), the probability of idle time slot, i.e., 𝑝𝑖𝑑𝑙𝑒 , can be calculated by

(65)

where 𝐵𝑚𝜎 stands for backoff delay in 𝑚th backoff stage. The variance of 𝑋𝑏𝜎 can be derived by ∑

(70)

.

where 𝑣𝜎𝑖 stands for the probability that the 𝐴𝐶𝜎 successfully transmits a packet in zone 𝑖. We can express 𝑣𝜎𝑖 as following: For 0 ≤ 𝜎 ≤ 3, 0 ≤ 𝑟 ≤ 3, and 1 ≤ 𝑖 ≤ 4, ⎧𝑁 𝜏 𝜎 ∏(1 − 𝜏 𝑟 )𝑁𝑟 −1 ∏(1 − 𝜏 𝑟 )𝑁𝑟 𝜎 < 𝑖 ⎪ 𝜎 𝜎 𝜎<𝑟 𝑟≤𝜎 𝑣𝑖 = ⎨ (74) ⎪0 𝜎 ≥ 𝑖. ⎩

By using (61)–(63), we can derive the variance of 𝑋𝑖𝑐𝜎 as

𝑋𝑏𝜎

)(𝜆𝜎 )2

𝑖=1

(63)

Var[𝑋𝑖𝑐𝜎 ] = (𝑇𝑠𝑙𝑜𝑡 )2 ⋅ Var[𝑚𝜎𝑖𝑐 ].

−2𝑒

𝑣𝜎𝑠𝑢𝑐 𝐸[𝐹 ]

𝑣𝜎𝑠𝑢𝑐 =

By using (62), the variance of 𝑚𝜎𝑖𝑐 can be derived by 𝜎 [1 − (𝑃 𝜎 )𝐷𝜎 ] − 1 𝑃𝑖𝑑𝑙𝑒 𝑖𝑑𝑙𝑒 . 𝜎 )2 (1 − (𝑃 𝜎 )𝐷𝜎 )2 (1 − 𝑃𝑖𝑑𝑙𝑒 𝑖𝑑𝑙𝑒

𝜎𝑇𝜎 𝑠𝑢𝑐

where 𝑣𝜎𝑠𝑢𝑐 , E[F], and 𝑇𝑎𝑣𝑒 denote the probability that a successful transmission of the 𝐴𝐶𝜎 in a time slot, the time needed to transmit a frame body, and the average duration of a time slot. From Markov chain of Fig. 4, 𝑣𝜎𝑠𝑢𝑐 can be obtained by

According to the well-known property of geometric process [33], we can obtain the calculation of probability mass function (PMF) of 𝑚𝜎𝑖𝑐 by 𝑃𝑚𝜎 (𝑥) =

(1 − 𝑒−𝜆

In this section, we attempt to derive the calculation of throughput for the 𝐴𝐶𝜎 , denoted by 𝛩𝜎 . Since the normalized throughput is the fraction of the time from an expected duration of a time slot used for a successful transmission of bits [10], it can be calculated by

(61)

− 1) ⋅ 𝑇𝑠𝑙𝑜𝑡 .

𝜎 𝜆𝜎 )2 + 2 − 2𝑇 𝜎 𝜆𝜎 + (𝑇𝑠𝑢𝑐 𝑠𝑢𝑐

4.6. Normalized throughput

denote the time slot on which the initial-carrier-sensing proceLet dure terminates, i.e., the time channel is sensed busy. We can derive 𝑋𝑖𝑐𝜎 by =

𝜎𝑇𝜎 𝑠𝑢𝑐

Thus, by using (49), (56)–(71), we can derive the calculation of queueing delay.

𝑚𝜎𝑖𝑐

(𝑚𝜎𝑖𝑐

−2𝑒−𝜆

𝜎 Similarly, we can obtain Var[𝑇𝑅𝐸𝑀[2] ] by

𝜎 We can derive Var[𝑇𝑎𝑐𝑐𝑒𝑠𝑠 ] by (59), shown in Box II, where 𝑋𝑏𝜎 and 𝜎 𝑋𝑖𝑐 stand for the delay caused by backoff procedure and uncompleted initial-carrier-sensing procedure, respectively. Considering fixed-length duration of 𝑇𝐴𝐼𝐹 𝑆[𝜎] , we obtain { E[𝑇𝐴𝐼𝐹 𝑆[𝜎] ] = 𝑇𝐴𝐼𝐹 𝑆[𝜎] , (60) Var[𝑇𝐴𝐼𝐹 𝑆[𝜎] ] = 0.

𝑋𝑖𝑐𝜎

(69)

𝑝𝑐𝑜𝑙 = 1 − 𝑝𝑖𝑑𝑙𝑒 − 𝑝𝑠𝑢𝑐 .

(77)

Therefore, 𝑇𝑎𝑣𝑒 can be derived as

(67)

𝜎 𝜎 𝑇𝑎𝑣𝑒 = 𝑇𝑠𝑙𝑜𝑡 ⋅ 𝑝𝑖𝑑𝑙𝑒 + 𝑇𝑠𝑢𝑐 ⋅ 𝑝𝑠𝑢𝑐 + 𝑇𝑐𝑜𝑙 ⋅ 𝑝𝑐𝑜𝑙 .

𝜎 and 𝑉 𝜎 denote the length of one time slot and the number where 𝑋𝑠𝑙𝑜𝑡 𝑚 of time slots selected in 𝑚th backoff stage for the 𝐴𝐶𝜎 , respectively.

Hence, we can derive the throughput by using (72)–(78). 115

(78)

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Computer Communications 149 (2020) 107–120

[ ] 𝜎 𝜎 )2 Var[𝑇𝑎𝑐𝑐𝑒𝑠𝑠 ] = 𝛤1𝜎 ⋅ Var[𝑇𝐴𝐼𝐹 𝑆[𝜎] ] + Var[𝑋𝑏𝜎 ] + (E[𝑇𝐴𝐼𝐹 𝑆[𝜎] ] + E[𝑋𝑏𝜎 ] − 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 [ ] 𝜎 + 𝛤2𝜎 ⋅ Var[𝑋𝑖𝑐𝜎 ] + Var[𝑋𝑏𝜎 ] + (E[𝑋𝑖𝑐𝜎 ] + E[𝑋𝑏𝜎 ] − 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 )2 [ ] 𝜎 𝜎 𝜎 + 𝛤3𝜎 ⋅ Var[𝑇𝑅𝐸𝑀[1] ] + Var[𝑋𝑏𝜎 ] + (E[𝑇𝑅𝐸𝑀[1] ] + E[𝑋𝑏𝜎 ] − 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 )2 [ ] 𝜎 𝜎 𝜎 + 𝛤4𝜎 ⋅ Var[𝑇𝑅𝐸𝑀[2] ] + Var[𝑋𝑏𝜎 ] + (E[𝑇𝑅𝐸𝑀[2] ] + E[𝑋𝑏𝜎 ] − 𝑇𝑎𝑐𝑐𝑒𝑠𝑠 )2 .

(59)

Box II. Table 3 Parameter setting. 0 𝐶𝑊𝑚𝑖𝑛

3

1 𝐶𝑊𝑚𝑖𝑛

7

2 𝐶𝑊𝑚𝑖𝑛

15

3 𝐶𝑊𝑚𝑖𝑛

15

0 𝐶𝑊𝑚𝑎𝑥

7

1 𝐶𝑊𝑚𝑎𝑥

15

2 𝐶𝑊𝑚𝑎𝑥

1023

3 𝐶𝑊𝑚𝑎𝑥

1023

𝑓0 𝑓1 𝑓2 𝑓3 𝐴𝐼𝐹 𝑆0 𝐴𝐼𝐹 𝑆0 𝐴𝐼𝐹 𝑆0 𝐴𝐼𝐹 𝑆0 Packet payload Data rate Slot time SIFS DIFS Propagation time

6 5 0 1 2 3 6 9 512 bytes 11 Mbps 13 μs 32 μs 58 μs 2 μs

Fig. 5. Normalized throughput in terms of the number of vehicles.

5. Simulation results In this section, the simulation results are presented to validate the effectiveness of the proposed analytical model. The well-known simulation tool NS-2 with implementation of the IEEE 802.11e EDCA mechanism established by the TKN group from Technical University of Berlin [34] is used. In addition, we revised some parts of the original source codes accordingly to fulfill the characteristic of the most upto-date IEEE 802.11e standard [6]. The simulation scenario under consideration is a segment of a one-way highway, with a RSU located in the central position. The length of the highway segment is set to the communication range, i.e., 1000 m, so that all the vehicles on the highway including the RSU are in the communication range of each other. Each vehicle moves with constant speed drawn from a Gaussian distribution with mean value 70 mi/h [16]. In order to keep the number of vehicles moving on the highway constant, when a vehicle arrives at the end of highway, it re-enters the other end of highway segment with the same direction. The data packets are generated in each AC queue according to a Poisson process with the mean value of 𝜆. The simulation time is set to 1000 s in each simulation. In order to achieve a confidence interval of 0.95 with half-widths of less than 5% about the mean, we ran 100 simulations with different random seeds and averaged the results. In addition, in order to show the accuracy improvement achieved by our model, we compare our model with the state-of-theart model [11], which is referred to as the Jun’s model herein. The rest of main simulation parameters used in the simulation experiment are listed in Table 3. Figs. 5–7 show the normalized throughput, channel access delay, and queueing delay, in terms of the number of vehicles, respectively, with packet arrival rate set as 0.45 Mbps and packet size set to 512 B. Note that, the normalized throughput is the fraction of the time used for

Fig. 6. Channel access delay in terms of the number of vehicles.

a successful transmission of bits from an expected duration of a time slot; the channel access delay is the duration from the time a packet starts to contend for channel access to the time the packet is transmitted without collision or discarded; the queueing delay is the duration from the instant a packet arrives at the queue to the instant the packet starts to receive the service. From the figures, it can be observed that the analytical results are very close to the simulation results, leading to the conclusion that the analytical model is effective and accurate. From Fig. 5, it can be observed that the 𝐴𝐶0 enjoys highest throughput. It also can be seen from the figure that the throughput of each AC queue keeps increasing until a turning point, after which it starts to decrease. This is caused by the fact that the increase in the number of vehicles results in increasing packet collision among the stations. In addition, the throughput of Jun’s model is less than that of our model due to 116

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Computer Communications 149 (2020) 107–120

Fig. 7. Queueing delay in terms of the number of vehicles.

Fig. 9. Channel access delay in terms of packet arrival interval.

Fig. 8. Normalized throughput in terms of packet arrival interval.

Fig. 10. Queueing delay in terms of packet arrival interval.

the reason that the Jun’s model does not include the initial-carriersensing procedure. With the initial-carrier-sensing procedure, a packet can be sent directly after an AIFS duration, which is much less than the duration of backoff procedure. Hence, the amount of bandwidth wasted by the backoff procedure can be saved, leading to increased throughput. As the number of vehicles increases, the gap between our model and Jun’s model decreases. This is because the increase of channel contention results in higher channel busy probability and packet collision probability, which prevent direct packet transmission after an AIFS. Fig. 6 shows the channel access delay in terms of the number of vehicles. The channel access delay remains at a low level for the two higher priority AC queues, whereas the channel access delay increases fast for the other two AC queues with lower priorities. The reason of this phenomenon is the shorter CW and AIFS values of higher priority ACs. Since the channel access delay of the highest priority is almost negligible, the delay requirement for ITS safety-related services can be completely guaranteed. Non-time-critical services, such as weather forecast service, can be allocated to the other AC queues. In addition, it can be observed that channel access delay of Jun’s model is larger than our model due the lack of initial-carrier-sensing in Jun’s model. With initial-carrier-sensing procedure, a packet can be transmitted directly without going through a backoff procedure. Thus the channel access delay can be dramatically decreased. Fig. 7 shows the queueing delay in terms of the number of vehicles. From the figure, it can be observed that the queueing delay for the

highest priority remains at a low level, compared to the other three priorities, which increase fast as the number of vehicles increases. The reason this occurs is that higher priority ACs are assigned shorter CW and AIFS values compared to lower priority ACs. From the figure, it can be observed that the ITS safety-related service can be completely fulfilled. In addition, it can be noticed that queueing delay of Jun’s model is larger than our model. This is because with initial-carrier-sensing procedure, the channel access delay can be significantly decreased, leading to decreased queueing delay. Figs. 8–10 show the normalized throughput, channel access delay, and queueing delay, in terms of packet arrival interval, i.e., reciprocal of the packet arrival rate, respectively, with the number of vehicles of 5 and packet size set to 512 B. From Fig. 8, it can be discovered that when the packet arrival interval is very short, i.e., near saturated condition, the channel resource is almost monopolized by the AC queue with the highest priority. This occurs since the saturation condition enlarges the advantage of the AC queue with the highest priority in the channel access contention. In addition, we can see that the throughput of Jun’s model is less than our model due to the reason that the Jun’s model does not include the initial-carrier-sensing procedure. With the initial-carrier-sensing procedure, a packet can be sent directly after an AIFS duration, hence saving the amount of bandwidth wasted by the backoff procedure to increase the throughput. Also, we can see that as packet arrival interval increases, the gap between our model and Jun’s model increases. This is because the increase of packet arrival interval 117

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Computer Communications 149 (2020) 107–120

results in lower contention intensity, which leads to lower channel busy probability and packet collision probability. Hence a packet has more chance to be sent directly only after an AIFS duration rather than going through a whole backoff procedure. Fig. 9 shows channel access delay in terms of packet arrival interval. From the figure, it can be discovered that the channel access delays of the AC queues with higher priority, i.e., 𝐴𝐶0 and 𝐴𝐶1 , keep at a low level. The reason this occurs is the shorter CW and AIFS values of higher priority ACs. However, the other AC queues have large delay when the packet arrival interval is short, i.e., near saturated condition. Also, we can see that channel access delay of Jun’s model is larger than our model due to the lack of initial-carrier-sensing procedure. Fig. 10 shows queueing delay in terms of packet arrival interval. From the figure, it can be observed that the queueing delay of the AC queue with the highest priority keeps at a low level so that the delay requirement of the safety-related services can be guaranteed. However, the other AC queues with lower priorities have large queueing delay when the channel condition is dense. In addition, we can observe that the queueing delay of Jun’s model is larger than our model due to the lack of initial-carrier-sensing procedure.

Also, according to Markov chain, it is obtained that For 0 < 𝑚 ≤ 𝑚𝜎 ,

6. Conclusion

𝜎 𝜋(𝑚,𝑘) = 𝜋(𝑚,0) ⋅

𝑝𝜎𝑐𝑜𝑙 ⎧ 𝜎 𝜎 𝜎 𝜎 ⎪𝜋(𝑚,1) = 𝜋(𝑚−1,0) ⋅ 𝑊 𝜎 + 𝜋(𝑚,2) ⋅ (1 − 𝑝𝑏𝑢𝑠𝑦 ) 𝑚 ⎪ ⎪ + 𝜋(𝑚,1) ⋅ 𝑝𝜎𝑏𝑢𝑠𝑦 ⎪ 𝑝𝜎𝑐𝑜𝑙 ⎪ 𝜎 𝜎 𝜎 + 𝜋(𝑚,3) ⋅ (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 ) ⎪𝜋(𝑚,2) = 𝜋(𝑚−1,0) ⋅ 𝜎 𝑊 ⎨ 𝑚 ⎪ + 𝜋(𝑚,2) ⋅ 𝑝𝜎𝑏𝑢𝑠𝑦 ⎪ ⎪⋮ ⎪ ⎪ 𝑝𝜎 𝜎 𝜎 𝜎 ⎪𝜋(𝑚,𝑊 = 𝜋(𝑚−1,0) ⋅ 𝑐𝑜𝑙𝜎 + 𝜋(𝑚,𝑊 ⋅ 𝑝𝜎𝑏𝑢𝑠𝑦 . 𝜎 𝜎 𝑚 −1) 𝑚 −1) ⎩ 𝑊𝑚

(81)

Thus, we have For 0 < 𝑚 ≤ 𝑚𝜎 and 0 < 𝑘 ≤ 𝑊𝑚𝜎 − 1, 𝑊𝑚𝜎 − 𝑘

𝜎 𝜋(𝑚,𝑘) = 𝜋(𝑚,0) ⋅

𝑊𝑚𝜎

1 . 1 − 𝑝𝜎𝑏𝑢𝑠𝑦

⋅

(82)

Similarly, we have For 𝑚𝜎 < 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 and 0 < 𝑘 ≤ 𝑊𝑚𝜎 − 1, 𝑊𝑚𝜎 − 𝑘 𝜎

𝑊𝑚𝜎

⋅

𝜎

In this paper, an analytical model of the IEEE 802.11p EDCA mechanism has been proposed and evaluated by comprehensive simulations. The model is established by combining two Markov chain models, i.e., a 2-D Markov chain model to represent the backoff procedure of each AC queue and a 1-D Markov chain model to describe contention period caused by different AIFSs and CWs. Since both saturated and nonsaturated conditions were taken into consideration, infinite-long 1-D Markov chain model was employed in our model. Based on the two Markov chain models, the transmission probability and collision probability for each AC queue were achieved. Then, by using the achieved probabilities, the analytical models for performance metrics, namely, normalized throughout, channel access delay, and queueing delay are derived. The proposed analytical model takes into account all the major factors that affect the performance, including distinct CWs and AIFSs for different ACs, initial-carrier-sensing procedure, flexible saturation status, backoff counter freezing, most up-to-date standard parameters, internal collision resolution mechanism, and computational tractability. The simulation results show the effectiveness and accuracy of the proposed analytical model.

1 . 1 − 𝑝𝜎𝑏𝑢𝑠𝑦

(83)

According to the Markov chain it can be observed that ⎧ 𝑝𝜎 ⎪𝜋 𝜎 = 𝜋 𝜎 ⋅ 𝑐𝑜𝑙 + 𝜋 𝜎 ⋅ (1 − 𝑝𝜎 ) 𝑏𝑢𝑠𝑦 (1,1) (1,0) (0,0) ⎪ 𝑊1𝜎 ⎪ 𝜎 𝑝𝑐𝑜𝑙 ⎪ 𝜎 𝜎 𝜎 𝜎 ⎪𝜋(2,0) = 𝜋(1,0) ⋅ 𝑊 𝜎 + 𝜋(2,1) ⋅ (1 − 𝑝𝑏𝑢𝑠𝑦 ) 2 ⎪ ⎨⋮ ⎪ 𝑝𝜎𝑐𝑜𝑙 ⎪ 𝜎 𝜎 ⋅ ⎪𝜋(𝑚 +𝑓 ,0) = 𝜋(𝑚 𝜎 𝜎 𝜎 +𝑓𝜎 −1,0) 𝑊𝑚𝜎 +𝑓 ⎪ 𝜎 𝜎 ⎪ 𝜎 + 𝜋(𝑚 ⋅ (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 ). ⎪ +𝑓 ,1) 𝜎 𝜎 ⎩

(84)

Hence, we have For 0 < 𝑚 ≤ 𝑚𝜎 + 𝑓𝜎 , 𝜎 𝜎 𝜋(𝑚,0) = 𝜋(0,0) ⋅ (𝑝𝜎𝑐𝑜𝑙 )𝑚 .

(85)

According to the Markov chain, we have For 0 ≤ 𝑘 ≤ 𝐷𝜎 , 𝜎 𝜎 𝛼 𝜎 ) ⋅ (𝑝𝜎 )𝐷𝜎 −𝑘 . 𝜋(𝑖𝑐,𝑘) = (𝜋𝑖𝑑𝑙𝑒 𝑖𝑑𝑙𝑒

Declaration of competing interest

Next, we attempt to express 𝑌 , 𝑍, and Markov chain, it can be obtained that

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

𝑌 =

𝑊0𝜎 ⋅ (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )

.

From the

𝜎 𝜎 𝜋(𝑖𝑐,𝑖) ⋅ 𝑝𝜎𝑏𝑢𝑠𝑦 + 𝜋(𝑖𝑐,0) ⋅ [𝑝𝜎𝑐𝑜𝑙 + (1 − 𝑝𝜎𝑐𝑜𝑙 )𝑝̂𝜎𝑞 ]

(87)

⋅ 𝛽 𝜎 + 𝑍 ⋅ 𝑝𝜎𝑞 .

𝑚𝜎 +𝑓𝜎 −1 ∑ 𝜎 𝜋(𝑖,0) 𝑖=0

𝑍=

𝜎 ⋅ (1 − 𝑝𝜎𝑐𝑜𝑙 ) + 𝜋(0,𝑚

𝜎 +𝑓𝜎 )

.

(88)

By substituting (85) into (88), we can find out that 𝜎 𝑍 = 𝜋(0,0) .

(89)

According to balance equation of the Markov chain with respect to the 𝜎 , we have state 𝜋(0,0)

(79)

𝜎 𝜋(0,0) =𝑌 ⋅

1 𝜎 + 𝜋(0,2) ⋅ (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 ). 𝑊0𝜎

(90)

𝜎 Hence, from (79), (86)–(90), we can obtain 𝜋(0,0) as following: 𝜎 𝜋(0,0) =

− 𝑘)

in terms of

𝜎 . 𝜋𝑖𝑑𝑙𝑒

Also, 𝑍 can be expressed by

Thus, we have For 0 < 𝑘 ≤ 𝑊0𝜎 − 1, 𝜎 𝜋(0,𝑘)

𝐷𝜎 ∑

𝑖=1 𝜎 + 𝜋𝑖𝑑𝑙𝑒

From the Markov chain shown in Fig. 2, we can obtain For 0 < 𝑘 ≤ 𝑊0𝜎 − 1,

⋅ (𝑊0𝜎

=

𝑌

Appendix A. Detailed derivation process of (19) and (20)

⎧ 𝜎 1 𝜎 𝜎 𝜎 ⎪𝜋(0,1) = 𝑌 ⋅ 𝑊 𝜎 + 𝜋(0,2) ⋅ (1 − 𝑝𝑏𝑢𝑠𝑦 ) + 𝜋(0,1) ⋅ 𝑝𝑏𝑢𝑠𝑦 0 ⎪ ⎪ 𝜎 1 𝜎 𝜎 𝜎 ⎪𝜋(0,2) = 𝑌 ⋅ 𝑊 𝜎 + 𝜋(0,3) ⋅ (1 − 𝑝𝑏𝑢𝑠𝑦 ) + 𝜋(0,2) ⋅ 𝑝𝑏𝑢𝑠𝑦 0 ⎨ ⎪⋮ ⎪ 1 ⎪ 𝜎 𝜎 ⎪𝜋(0,𝑊0𝜎 −1) = 𝑌 ⋅ 𝑊 𝜎 + 𝜋(0,𝑊0𝜎 −1) ⋅ 𝑝𝑏𝑢𝑠𝑦 . ⎩ 0

(86) 𝜎 𝜋0,0

𝜎 𝜋𝑖𝑑𝑙𝑒

1 − 𝑝𝜎𝑞 𝜎 𝜎 +𝑝̂𝑞 − 𝑝𝑐𝑜𝑙 ⋅ 𝑝̂𝜎𝑞

(80) 118

{

𝛼 𝜎 ⋅ [1 + (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 ⋅ (𝑝𝜎𝑐𝑜𝑙 } − 1)] + 𝛽 𝜎 .

⋅

(91)

S. Cao and V.C.S. Lee

Computer Communications 149 (2020) 107–120

1 − (1 − 𝑝𝜎𝑏𝑢𝑠𝑦 )𝐷𝜎 +1

𝜎 𝜎 1 = 𝜋𝑖𝑑𝑙𝑒 + 𝜋𝑖𝑑𝑙𝑒 ⋅ 𝛼𝜎 ⋅

⋅

𝑝𝜎𝑏𝑢𝑠𝑦

[ 2𝑝𝜎 − (2𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +1 1 ⋅ 𝑊0𝜎 ⋅ 𝑐𝑜𝑙 𝜎 1 − 𝑝𝑏𝑢𝑠𝑦 1 − 2𝑝𝜎𝑐𝑜𝑙

{

1 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +𝑓𝜎 +1

1 1 1 + ⋅ ⋅ (𝑊0𝜎 − 1) + 1 − 𝑝𝜎𝑐𝑜𝑙 2 1 − 𝑝𝜎𝑏𝑢𝑠𝑦 2 ] 𝜎 )𝑚𝜎 +1 − (𝑝𝜎 )𝑚𝜎 +𝑓𝜎 +1 } 𝑝𝜎𝑐𝑜𝑙 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +1 (𝑝 1 1 𝑐𝑜𝑙 − + ⋅ ⋅ (𝑊0𝜎 ⋅ 2𝑚𝜎 − 1) ⋅ 𝑐𝑜𝑙 . 1 − 𝑝𝜎𝑐𝑜𝑙 2 1 − 𝑝𝜎𝑏𝑢𝑠𝑦 1 − 𝑝𝜎𝑐𝑜𝑙

𝜎 + 𝜋(0,0) ⋅

(93)

Box III.

References

According to normalizing equation of the Markov chain, we have 𝐷𝜎

∑

𝜎 1 = 𝜋𝑑𝑖𝑙𝑒 +

𝜎 𝜋(𝑖𝑐,𝑖) +

𝑖=0

𝑚𝜎 +𝑓𝜎 𝑊𝑚𝜎 −1 ∑ ∑ 𝑚=0

𝜎 𝜋(𝑚,𝑘) .

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(92)

𝑘=0

By substituting, (80), (92), (83), (85), and (86) into (92), we have (93) shown in Box III. By substituting (91) into (93), we can obtain the 𝜎 expression of 𝜋𝑖𝑑𝑙𝑒 as (20). Since the transmission probability 𝜏 𝜎 of the 𝐴𝐶𝜎 is 𝑚𝜎 +𝑓𝜎

𝜏𝜎

∑

=

𝜎 𝜋(𝑖,0) + 𝜋(𝑖𝑐,0)

𝑖=0 𝜎 = 𝜋(0,0) ⋅

1 − (𝑝𝜎𝑐𝑜𝑙 )𝑚𝜎 +𝑓𝜎 +1 1 − 𝑝𝜎𝑐𝑜𝑙

(94) 𝜎 + 𝜋𝑖𝑑𝑙𝑒 ⋅ 𝛼 𝜎 ⋅ (𝑝𝜎𝑖𝑑𝑙𝑒 )𝐷𝜎 .

By substituting (91) into (94), we can express the transmission probability of the 𝐴𝐶𝜎 shown in (19). Appendix B. Derivation process of mean and variance of 𝑻 𝝈

𝑹𝑬𝑴

The remaining time of an ongoing transmission counted from the time on which a packet is generated can be expressed as 𝜎 𝜎 𝜎 𝑇𝑅𝐸𝑀 = 𝑇𝑠𝑢𝑐 − 𝑇𝑔𝑒𝑛 ,

(95)

𝜎 is the time on which a packet is generated in the 𝐴𝐶 . Thus, where 𝑇𝑔𝑒𝑛 𝜎 𝜎 cumulative distribution function (CDF) of 𝑇𝑅𝐸𝑀 can be calculated by

𝐹𝑇 𝜎

𝑅𝐸𝑀

𝜎 (𝑥) = 𝑃 (𝑇𝑅𝐸𝑀 ≤ 𝑥) 𝜎 − 𝑇 𝜎 ≤ 𝑥) = 𝑃 (𝑇𝑠𝑢𝑐 𝑔𝑒𝑛

(96)

𝜎 − 𝑥 ≤ 𝑇𝜎 ) = 𝑃 (𝑇𝑠𝑢𝑐 𝑔𝑒𝑛 𝜎 𝜎 (𝑇 = 1 − 𝐹𝑇𝑔𝑒𝑛 𝑠𝑢𝑐 − 𝑥).

By differentiating (96) with respect to 𝑥, we can obtain the probability 𝜎 density function (PDF) of 𝑇𝑅𝐸𝑀 as 𝑓𝑇 𝜎

𝑅𝐸𝑀

𝜎 𝜎 (𝑇 (𝑥) = 𝑓𝑇𝑔𝑒𝑛 𝑠𝑢𝑐 − 𝑥).

(97)

Since it is expected that the packet is generated according to Poisson process and due to memorylessness property of the exponential distri𝜎 bution, the PDF of 𝑇𝑅𝐸𝑀 on the condition that the packet is generated during transmission time can be calculated by 𝜎

𝜎

𝜆𝜎 ⋅ 𝑒−𝜆 (𝑇𝑠𝑢𝑐 −𝑥) . 𝜎 𝜎 1 − 𝑒−𝜆 𝑇𝑠𝑢𝑐 𝜎 Therefore, we can calculate E[𝑇𝑅𝐸𝑀 ] by 𝜎 (𝑥) |𝑇 𝜎 ≤𝑇𝑠𝑢𝑐 𝑅𝐸𝑀 𝑔𝑒𝑛

𝜎 𝑇𝑠𝑢𝑐

𝜎 E[𝑇𝑅𝐸𝑀 ]=

∫0 𝜎 −𝜆𝜎 𝑇𝑠𝑢𝑐

=

(98)

=

𝑓𝑇 𝜎

𝑒

𝑥 ⋅ 𝑓𝑇 𝜎

𝑅𝐸𝑀

𝜎 ≤𝑇 𝜎 (𝑥)𝑑𝑥 |𝑇𝑔𝑒𝑛 𝑠𝑢𝑐

𝜎 𝜆𝜎 − 1 + 𝑇𝑠𝑢𝑐 𝜎 −𝜆𝜎 𝑇𝑠𝑢𝑐

(1 − 𝑒

)𝜆𝜎

(99) .

𝜎 Similarly, we can derive Var[𝑇𝑅𝐸𝑀 ] as following: 𝜎 Var[𝑇𝑅𝐸𝑀 ]=

=

𝜎 𝑇𝑠𝑢𝑐

∫0

−2𝑒−𝜆

𝑥2 ⋅ 𝑓 𝑇 𝜎

𝑅𝐸𝑀

𝜎𝑇𝜎 𝑠𝑢𝑐

𝜎 ≤𝑇 𝜎 (𝑥)𝑑𝑥 |𝑇𝑔𝑒𝑛 𝑠𝑢𝑐

𝜎 𝜆𝜎 )2 + 2 − 2𝑇 𝜎 𝜆𝜎 + (𝑇𝑠𝑢𝑐 𝑠𝑢𝑐 𝜎𝑇𝜎 𝑠𝑢𝑐

(1 − 𝑒−𝜆

)(𝜆𝜎 )2

(100) .

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Computer Communications 149 (2020) 107–120

[26] P.E. Engelstad, O.N. sterb, Non-saturation and saturation analysis of IEEE 802.11e EDCA with starvation prediction, in: Proc. ACM MSWiM, New York, NY, USA, 2005, pp. 123-133. [27] B. Xiang, M. Yu-Ming, X. Jun, Performance investigation of IEEE 802.11e EDCA under non-saturated condition based on the M/G/1/K model, in: Proc. IEEE ICIEA, Harbin, China, 2007, pp. 298-304. [28] S. Pan, J. Wu, Throughput analysis of IEEE 802.11e EDCA under heterogeneous traffic, Comput. Commun. 32 (5) (2009) 935–942. [29] Z. Tao, S. Panwar, Throughput and delay analysis for the IEEE 802.11e enhanced distributed channel access, IEEE Trans. Commun. 54 (8) (2006) 596–603. [30] Z. Kong, D.H.K. Tsang, B. Bensaou, D. Gao, Performance analysis of IEEE 802.11e contention-based channel access, IEEE J. Sel. Areas Commun. 22 (10) (2004) 2095–2106. [31] J.W. Tantra, H.F. Chuan, A.B. Mnaouer, Throughput and delay analysis of the IEEE 802.11e EDCA saturation, in: Proc. IEEE ICC, Seoul, South Korea, 2005, pp. 3450-3454. [32] M.A. Togou, L. Khoukhi, A.S. Hafid, Throughput analysis of the IEEE 802.11p EDCA considering transmission opportunity for non-safety applications, in: Proc. IEEE ICC, Kuala Lumpur, Malaysia, 2016, pp. 1-6. [33] W. Feller, An Introduction to Probability Theory and its Applications, second ed., Wiley, New York, 1971. [34] S. Wiethölter, M. Emmelmann, C. Hoene, A. Wolisz, TKN EDCA Model for ns-2, Telecommun. Netw. Group, Technische Univ. Berlin, Berlin, Germany, Tech. Rep. TKN-06-003, 2006.

Shengbin Cao received the Ph.D. degree in computer science from the City University of Hong Kong, Hong Kong, in 2018. His research interest is vehicular ad hoc networks, including medium-access control (MAC) protocol design and performance analysis.

Victor C. S. Lee (M’92) received the Ph.D. degree in computer science from the City University of Hong Kong, Hong Kong, in 1997. He is currently an Assistant Professor with the Department of Computer Science, City University of Hong Kong. His current research interests include data dissemination in vehicular networks, real-time databases, and performance evaluation. Dr. Lee was the Chairman of the IEEE Hong Kong Section Computer Chapter from 2006 to 2007. He is a member of ACM and the IEEE Computer Society.

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