Throughput and delay analysis for IEEE 802.11 multi-hop networks considering data rate

In an IEEE 802.11 Distributed Coordination Function–based wireless network with multiple hops, a node operates on its own with several predefined data rates (i.e. following modulation and coding schemes). Moreover, the IEEE 802.11 Distributed Coordination Function node’s communication is characterized by transmission and carrier-sensing distances. The transmission one is, in general, reverse proportional to the data rate. Meanwhile, the carrier distance keeps constant regardless of the modulation and coding scheme. Therefore, when a node has a high transmission rate, within its carrier-sensing range, the number of nodes may increase. The previous works have not yet extensively investigated the impact of data rates on such a scenario. This article addresses that issue aiming to quantify the network performance of the multi-hop IEEE 802.11 networks. As a solution, we propose the mathematical expressions, which consider data rates, for end-to-end throughputs, as well as delays in the network with string topology. We confirm the expressions’ correctness by presenting the quantitative agreements between the analytical and simulation results.


Introduction
Recently, there has been renewed interest in adopting wireless multi-hop networks, which will be useful in various applications ranging from the traditional wireless sensor networks [1][2][3] to the Internet of Things 4-6 and vehicular ad hoc networks. 7-9 Such a system can relax the complexity of underlay infrastructure, hence improving the easiness and flexibility in network construction. More importantly, the wireless multi-hop networks enable the autonomous operations of network nodes, each of which is predetermined, for example, in a standard. This article focuses on the multi-hop networks in which each node's operation relies on the IEEE 802.11 Distributed Coordination Function (DCF). The IEEE 802.11 DCF defines the behaviors of medium access control (MAC) and physical (PHY) layers of a wireless node. Moreover, the physical data rates depend on predefined modulation and coding schemes.
The operation of an IEEE 802.11 DCF node is following Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA); hence, the transmission and carrier-sensing ranges are essential. In general, the onehop transmission distance is inversely proportional to the data rate in the IEEE 802.11 standards. As a result, in a complete transmission within an identical end-toend distance (ED), the number of nodes increases when the data rate increases, and vice versa. That results in the inherent trade-off between transmission speed and transmission distance when selecting the data rate. However, the node's carrier-sensing range is identical under the same transmission power. Therefore, the higher the data rate is, the more nodes exist in the carrier-sensing range, wherein the impact of frame collisions is not negligible. Motivated by that, this work aims to establish mathematical models considering the data rates accounting the issue.
In theory, several articles theoretically analyze the network performance of multi-hop wireless, mostly the string topology, using the airtime concept. [10][11][12][13][14][15][16][17][18][19][20] First introduced in Chung and Liew 10 for saturation throughput in IEEE 802.11 networks, the concept combines a contention graph to derive the saturated throughput model for a given source-destination pair in Gao et al. 11 In Sugimoto et al., 13 the authors have extended the model to account for the throughput of two-way flows (of a VoIP traffic). Other IEEE 802.11 features such as Request To Send/Clear To Send (RTS/CTS), hidden node collision, long-frame communications, and back-off-stage synchronization have been investigated in previous studies, [14][15][16] respectively. In Qinjuan et al., 17 the integration of airtime Markov chain analysis has well handled the IEEE 802.11 multihop network's unsaturated condition. The proposed model in Shimoyamada et al. 18 can formulate multihop networks' performance using the IEEE 802.11e Enhanced Distributed Channel Access (EDCA). In Sanada et al., 19 the authors successfully derive the models that can manipulate varying hop number, frame length, and offered load. Besides the throughput, the airtime concept is also applicable for the delay analysis in multi-hop networks as in Sanada et al. 20 The previous works, however, often assume the carriersensing range is as twice as the transmission one. Hence, they have considered the varying number of nodes within a carrier-sensing range. As a result, the previous models cannot formulate the aforementioned trade-off caused by the data rates.
This article introduces the new theoretical analysis, which derives expressions for end-to-end throughputs and delays for IEEE 802.11 multi-hop networks. The expressions, which are formed from the airtime concept, efficiently assess the data rate parameter. The proposed model newly introduces the parameter h, which is the ratio of the carrier-sensing distance to the transmission one, to support any data rate. Another novelty is quantifying the duplication time of frame transmissions caused by the concurrent transmission, which often occurs at a high data rate. We have validated the proposed models by comparing the analytical results to simulation ones. The comparative results show that both results are well conformed.
The remainder of this article is organized as follows. Section ''Background and related works'' describes the related works about conventional models for multi-hop networks. Our analytical approach, which considers data rates, is presented in section ''Proposed analytical models.'' In section ''Evaluation results,'' we validate the proposed model by comparing its results to the simulation ones and discuss the network performance. Finally, we draw the conclusion for this work in section ''Conclusion.'' Background and related works IEEE 802.11 data rates There are multiple data rates (i.e. transmission speeds) in the physical layer of the IEEE 802.11 standards. Each defined data rate follows a modulation and coding scheme (MCS) index that includes a modulation type, a coding rate, several spatial streams, and an operation band. When the data rate increases, the signal of associated MCS becomes more sensitive to noise. Hence, the higher the data rate, the shorter the transmission distance in general. In a multi-hop IEEE 802.11 network, considering one-hop transmission, there is a similar trade-off between the transmission speed and the transmission distance. However, in such a scenario, the carrier-sensing range of a node is independent of the transmission speed. When a node operates at a high data rate, the number of nodes within its carrier-sensing range is more significant than that in a lower data rate operation. Therefore, to comprehend the trade-off within the carrier-sensing range of a node, it is necessary to consider the impact of hidden node and concurrent frame transmissions among the nodes.

Airtime concept
The airtime concept is a useful theory to understand the behaviors of the IEEE 802.11 wireless networks. During a duration, the theory can be used to characterize each IEEE 802.11 node's state corresponding to the IEEE 802.11 operation. More specifically, the IEEE 802.11 node is always in a transmission state, a carriersensing state, or an idle state. The first one indicates that the nodes' frames are being transmitted. The second one represents the condition in which the node detects frame transmissions of other nearby nodes. The last state means there is no transmission in the network. All the three states could be formulated in consideration of the neighboring nodes' states. Hence, the airtime analysis is possible to express each network node's operation and the mutual effects between the nodes, and that is effective in the context of the multi-hop IEEE 802.11 network. The fundamental of airtime concept can be represented via the transmission airtime as follows. At Node i, the transmission airtime is defined as where s i is the sum of several periods taken by frame transmissions from Node i to Node (i + 1) in Time duration. More specifically, the periods include the durations of Distributed Inter-Frame Space (DIFS), a DATA frame transmission (DATA), Short Inter-Frame Space (SIFS), and an ACKnowledgement frame (ACK) transmission of Node i. The carrier-sensing and the idle airtime models can be accordingly derived following the network scenarios and other conditions.

Airtime-based analysis
The theoretical analysis using airtime concept has shown the applicability in IEEE 802.11 wireless networks under various conditions. 10-20 The airtime concept was initially proposed in Chung and Liew 10 to construct a model for the maximum saturation throughput. However, the model assumed that simultaneous frame transmissions are negligible. Moreover, it considers only collisions with hidden nodes for the sake of simplicity. The authors in Gao et al. 11 proposed the analytical model that combines airtime and contention graph to derive the throughput of a given source and destination pair. In Inaba et al. 12 and Sugimoto et al., 13 the authors additionally considered two-way flows and RTS/CTS handshake, respectively. Both works have got the maximum throughput in multi-hop wireless networks. The proposed model in Zhao et al. 14 can analyze the intra-flow contention problem by expressing hidden node operation using airtime. In Sekiya et al., 15 the applicable range of the airtime model has been extended from short-frame communication to longframe communication. In Sanada et al., 16 the mechanism of back-off-stage synchronization, which is efficient in multi-hop networks, was revealed. In Qinjuan et al., 17 19 The same authors have proposed the analytical models for the end-to-end delay in multi-hop IEEE 801.11 networks in Sanada et al. 20 However, all these previous works assume a small number of network nodes in the carrier-sensing range, and that is because the carriersensing distance is assumingly the same or twice as the transmission distance. Therefore, all of them have not considered the condition where many nodes coexist in the same range. One of the recent works, which uses airtime concept, addresses the dense nodes but within a transmission range (i.e. single-hop), focusing on the capture effect. 21 In Liu et al., 22 the airtime-based analysis has been also applied in dense wireless networks aiming to quantify the inter-network interference. The earlier results of this work have been presented at IEEE VTC Spring 2019. 23 In this article, we newly include the delay analysis, besides the throughput one. Moreover, the evaluation results are more extensive.

Proposed analytical models
This analysis expresses the difference in transmission distances that depends on the data rate using the ratio of the carrier-sensing distance to the transmission distance. We aim to derive the end-to-end throughput and delay as functions that take the ratio into account when modeling each network node's operation. Moreover, the airtime concept is used to quantify relationships among network nodes. Figure 1 shows the investigated network, which is an H-hop string-topology IEEE 802.11 one. In such a network, the wireless nodes are located at an identical distance, while the total distance from the source to the destination node is denoted as ED. The following analysis is based on several assumptions: 1. Node 0 is the only data source, which generates traffic to Node H (i.e. the destination). The traffic including fixed sized data frames follows the Poisson distribution. 2. The intermediate node (e.g. Node i) only relays the frame from Node (i À 1) to Node (i + 1). 3. All links have ideal channel conditions. In other words, the transmission failures occur only due to the MAC layer. 4. Node i can sense frame transmissions from Node (iÀh) to Node (i + h), and that means Node (i+h + 1) is the hidden node of Node i.

Transmission airtime
The transmission airtime, which is defined as the timeshare of frame transmissions, includes both the successful and failure transmission durations. 10 Using the general expression of the transmission airtime in equation (1), the throughput of Node i can be expressed as where g i is the collision probability of Node i; P is the payload size of data frame; and T = DIFS + DATA + SIFS + ACK.

Carrier-sensing airtime
The carrier-sensing airtime is known as the timeshare of the frame transmissions of the nodes in the carriersensing range. Considering the duplication time of frame transmissions, the carrier-sensing airtime of Node i is where D j, i is the duplication time of Node j to be considered when calculating the carrier-sensing airtime of Node i. Duplications of transmission time are caused by concurrent frame transmissions or hidden node frame transmissions. In the previous works, the case of h = 2 is often assumed. When h is small, the main cause of the duplication is hidden node frame transmissions, and the duplication time caused by concurrent frame transmissions has been ignored in the conventional models. Our model, however, newly considers the duplication time due to concurrent transmissions to make it adaptable for the situation where many network nodes exist in the carrier-sensing range. Therefore, D j, i is expressed as where g i, j is the probability that Node i and Node j concurrently start transmitting a frame. The concrete expression of g i, j is later described in section ''Transmission probability.'' m(i, j) = c(i) \ c(j), where c(i) is a set of nodes in the carrier-sensing range of Node i. In equation (4), the first term of the right-hand side means the duplication time due to concurrent frame transmissions, which is expressed using g i, j .

Channel-idle airtime
The channel-idle airtime is the timeshare that no neighbor node transmits a frame, and that means the node associated with the channel is not in either the transmission state or the carrier-sensing state. Therefore, the idle airtime is

Collision probability
In the investigated topology, the frame collisions occur due to either the concurrent frame transmissions or the hidden node frame ones. Those collision events are, however, independent. Therefore, at Node i, the probability of frame collision is where g C i is the probability of concurrent-transmission collision, and g H i is the one of the hidden node collisions. The concurrent transmissions occur at Node i when at least one node within its carrier-sensing range transmits a frame at the same moment as Node i begins a fame transmission. Hence, the concurrent-transmission collision probability of Node i is From the assumption 4, we know that the hidden node of Node i is Node (i + h + 1). The hidden node collisions of Node i occur when Node i starts transmitting a frame while Node (i + h + 1) is transmitting or when Node (i + h + 1) starts transmitting a frame while Node i is transmitting. Because these two types of collision happen independently, Node i's hidden node collision probability is expressed as where g (1) H i and g (2) H i are the hidden node collision probabilities of the former type and the latter one, respectively.
The collision probabilities can be formed from a Markov chain model as in Figure 2. 19 In the model, d is the number of slots required for a DATA frame transmission; W s is the contention window at back-off stage s. W s is calculated as where CW min and CW max are the minimum and maximum values of contention window, respectively. L is the maximum back-off stage and L 0 = log 2 (CW max + 1=CW min + 1). S is the smallest number of back-off stages that satisfies d ł W s . This Markov chain model includes the back-off timer (BT) decrement state and transmission state. In the two dimensional parameters ½s, t, where s means the back-off stage, and t means the BT in BT-decrement state and the number of elapsed slots in the transmission state. Using the model, the relationship among network nodes in the local time duration can be expressed. 19 From the Markov chain model in Figure 2, g (1) H i and g (2) H i are expressed as where n(i + h + 1, i) = c(i + h + 1) \ c(i). u i means the timeshare that Node i belongs to the state represented by the Markov chain model. Hence, u i is expressed as where a = DATA=T and q i is the frame-existence probability of Node i, which is described in detail in the next section. a i and b i are the sum of stationary distributions of Node i that belong to the transmission state and the state with 0 ł t ł d on the Markov chain model, respectively. Using the stationary distribution of Node i in the Markov chain model, b½s, t i , we have

Frame transmission probability
At Node i, the probability of frame transmission is the probability that the node starts frame transmission when being in the idle state. More specific, the node attempts to transmit a frame when its BT is zero. From the Markov chain model in Figure 2, the frame transmission probability of Node i in the saturated condition is In the case of expressing the non-saturated condition, we need to define the frame-existence probability, q i . q i is the probability that Node i has at least one frame when Node i's state is idle. When a node has frames in the idle state, it decreases BT. Therefore, the airtime for BT decrements is where q 0 i is the frame-existence probability in the nonsaturated condition. The expected spending time for BT decrement with one successful frame transmission is expressed as V i s, where s is the system slot time. Besides, V i , which is the expected slot number of BT decrement for one successful frame transmission, is expressed as Therefore, the airtime for the BT decrement is where l i is the frame reception rate of Node i, which can be expressed by the throughput of Node (i À 1) following the assumption 2 as follows In equation (19), E À1 is the offered load of the stringtopology network. By equating right-hand side of equations (16) and (18), q 0 i is obtained as We know that the frame-existence probability in the saturated condition is one. Therefore, the frameexistence probability in the whole time can be expressed as Using the probability in equation (21), the frame transmission probability is which is also valid for both the non-saturated and saturated states. Considering the number of frame transmissions per unit time and the time for one frame transmission success, the transmission airtime of Node i is expressed by In addition, the probability that Node i and Node j simultaneously start transmitting a frame is expressed as where End-to-end throughput In the string-topology network with H hop, the end-toend throughput can be derived as End-to-end delay The definition of end-to-end delay in this work is similar to the one in Sanada et al. 20 The delay is the duration from the moment that the DATA frame is generated at the source node to the instant when the destination node receives the DATA frame. In other words, the end-to-end delay is the sum of multiple delays of single-hop transmission from the source node to the destination node. However, the single-hop transmission delay includes two types of delays, which are the MAC access delay and the queueing one. The frame-existence probability, q i , is previously defined in the channel-idle state. In the carrier-sensing state, the probability is assumingly as same as the total time. Therefore, the frame-existence probability in the total time is expressed as From equation (27), we have Figure 3 shows the M=M=1 buffer queueing model of Node i in the multi-hop network. In the model, the service time is the duration from the instant that a frame reaches the top of node buffer to the instant that a frame transmission is successful. The service time consists of the durations of BT decrements, BT freezing, and frame transmissions. Therefore, the MAC access delay of Node i is The queueing delay is the time duration from the instant that Node i receives a frame to the instant that the frame reaches to the top of Node i's buffer. Using the service time in equation (29), the service rate of the model in Figure 3 can be as follows Since we know the frame reception rate from equation (19) and the service rate from equation (30), the buffer utilization rate of Node i can be derived in equation (31) From the model in Figure 3, we can derive the steady-state probability that the Node i has l frames as Because b i:0 is the steady-state probability in which Node i does not have any frames, it is expressed as Besides, since the sum of the steady-state probability that Node i has l frames is one, we obtain By combining equations (32)-(34), we have Using the steady-state probability that the Node i has l frames from the above, the queueing delay of Node i is From equations (33) and (36), the total transmission delay at Node i is calculated as From equation (37), we can obtain the end-to-end delay in the H-hop string network From the above analysis, when the offered load is given, we can obtain the end-to-end throughput and delay by fixing 4H unknown parameters including

Evaluation results
This section presents the validation of the proposed analytical models. We compare the analytical results to simulation ones, as well as evaluate the network performance. We use the simulator, which has been developed in our laboratory. The simulator efficiently describes the operations of IEEE 802.11 wireless nodes. Moreover, the validity of our simulator is carefully discussed in Ikuma et al. 24 In our evaluation, the system parameters have been associated with the IEEE 802.11a standard. Table 1 gives the values of typical investigated parameters. We investigate the throughput and delay performance with the variations of h and the distance ED.
Throughput performance evaluation Figure 4 shows comparison between the analytical and simulation results of the end-to-end throughput with different values of h (i.e. 2 and 5). In the figure, the throughput is investigated following the variation of offered load. With each value of h, we evaluate four different networks. From Figure 4(a), we can see that in the case of h = 2, the analytical predictions (in lines) well match with the simulation results (in plots). These results show that the proposed model covers the previous one in Sanada et al., 19 which considers only the simple carrier-sensing range with h = 2. The comparison in Figure 4(b) is for the case of h = 5 with four networks (i.e. eight-hop, 12-hop, 17-hop, and 23-hop). The results in the figure again indicate that the analytical results agree with the simulation ones quantitatively, and that also confirm the applicability of proposed model in the condition where many nodes are within the carrier-sensing range. In other words, we have successfully applied the analysis with a wide range of h (other than only 2). From both figures, it can be seen that the proposed model can express the throughput values not only in the non-saturation condition but also the saturation one, and that shows the effectiveness of the proposed models under any conditions of offered loads. Figure 5 shows the comparison of end-to-end throughput when the ED is fixed while the value of h is varied from 1 to 5. We investigate two different scenarios with 350 m and 1200 m ED. Figure 5(a) and (b) shows the throughput results as a function of offered load different h for the case of ED = 350 m and ED = 1200 m, respectively. From the both figures, we can observe that in case of h = 2, the largest end-toend throughput (for the five-hop and 16-hop network) has been achieved in the evaluation. The reason is with h = 2, we got the best balance of transmission speed and transmission distance. Moreover, in all cases, we always see the agreements between the simulation and analytical results.

Delay performance evaluation
Similar to the previous evaluation, this section investigates the network performance when the offered load varies, but focusing on the end-to-end delay. Figure 6 shows the comparison of end-to-end delay values, which are collected from the simulation and analytical results. Wherein, Figure 6 Figure 6, we can observe that the delay increases significantly as the hop number increases. The networks with a smaller number of hops tend to reach the high delay values sooner at the same condition of offered load. This is because network nodes close to the source node are a bottleneck where a large delay occurs compared with other network nodes. However, we can always see the good agreement means that the proposed model can express the end-to-end delay correctly in different conditions of h. Figure 7 shows the end-to-end delay with different EDs, at each of which five fixed values of h are investigated. The results with ED = 350 m and ED = 1200 m are reported in Figure 7(a) and (b), respectively. In the figures, the analytical predictions again well match with the simulation results. Especially, we can observe the matched saturation points (i.e. between the saturated and unsaturated conditions of the networks). In the unsaturated condition, it is seen when the h increases, the end-to-end delay also increases under the identical end-to-end distance when h is large, and that is because the number of hops dramatically increases with an increase in h.

Conclusion
In this work, we have proposed the new theoretical models for the IEEE 802.11 string-topology multi-hop networks' performance, including the end-to-end throughputs and delays. The models newly take into account the data rate by introducing the parameter h, which is the ratio of the carrier-sensing distance to the transmission one. By doing so, the applicability of models has been extended to a wide range of data rate values. Another novelty is that the modes consider the duplication time of frame transmissions, which are caused by the simultaneous transmission, especially in high-rate conditions. The analytical results in various topologies have been confirmedly matched with the simulation results to affirm the validity of the proposed models.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the JSPS KAKENHI (grant no. 19K20251). In addition, K.N. is supported by the Leading Initiative for Excellent Young Researchers (LEADER) program from MEXT, Japan.