## 1 Introduction
All the above research works used the queueing theory approach or the Markov approach to model the performance of the 802.11 MAC layer. In addition, in [15; 16; 17; 18], the use of machine learning methods (supervised learning [15], reinforcement learning [16], deep reinforcement learning [17], joint learning [18], etc. ), which have been hotter in recent years, is proposed to study the performance of the MAC layer of 802.11 networks in terms of the throughput and channel utilization. However, a great deal of research has been conducted on modeling the service delay of 802.11 networks. These studies generally conducted modeling in the form of non-closed expressions or individual statistical moments. For example, the authors in [19] deduced the queuing delay and channel access delay of the 802.11 p DCF, but which were the non-closed model. In [20], the author used the limiting probability, \(p\), of successful transmission head-of-line (HOL) grouping to represent the saturated or unsaturated state of the network to study the performance of homogeneous cached IEEE 802.11 DCF networks, such as the delay, but the delay model was in the form of a single statistical moment. In [21], the authors derived the access delay of the IEEE 802.11 DCF in the saturated state, as well as in [22], where the authors derived the MAC delay and queuing delay using z-transform, which all investigated the mean, variance, and probability distributions of the MAC delay, but none of the resulting models could estimate the total delay spent in the transmitting packets. Furthermore, for the delay analysis of the point coordination function (PCF), see [23; 24]. In this paper, we comprehensively consider four primary factors affecting the performance of IEEE 802.11 networks, including the ICS procedure, the backoff counter freezing, up-to-date standard parameters, and computational complexity, and constructed a novel statistical model for the service delay in 802.11 networks; it is a 2-D Markov model including the ICS procedure and the backoff procedure. The model can identify the service delay's behavior and fully characterize the network performance. By analyzing the ICS delay and the backoff delay, we obtained the probability generating function (PGF) of the average idle time. The analytical expressions of the other service delays, such as the successful transmission time and collided transmission time, were derived to obtain the PGF of the total service delay. It was found that our delay evaluation results were significantly better than the traditional evaluation results. The average service delay of the Nav model in all the scenarios was larger than that of the analysis model due to the lack of the ICS procedure in the Nav model. Since a DIFS duration is generally much shorter than a random backoff duration, our model saves the bandwidth and improves transmission efficiency. The rest of the paper is organized as follows. In Section 2, we briefly analyze the transmission procedure of the IEEE 802.11 DCF mechanism and put forward the corresponding modeling ideas. In Section 3, a 2-D Markov model is established to simulate the operating mechanism of the 802.11 DCF, and the transmission probability is obtained. In Section 4, we calculate the average idle time experienced by a node between two consecutive successful transmissions. In Section 5, we establish the total service delay model in the IEEE 802.11 network. In Section 6, the validity and accuracy of the proposed model are verified by numerical simulation. Finally, the paper is concluded in Section 7.