Design and Analysis of Weighted Frequency-Domain Contention in Wireless LANs

The time to frequency protocol is the first frequency-domain contention protocol in wireless LANs, and has attracted a great deal of attention, since it can improve channel efficiency significantly. However, this protocol mainly provides a uniform service by letting each user conduct channel contention over the same frequency-domain range. In this paper, we first propose a novel weighted frequency-domain contention (WFC) scheme for service differentiation. In WFC, different frequency-domain ranges are assigned to different users to achieve weighted channel access opportunity, and a signature-assisted method is employed to completely exclude collisions. We then develop a theoretical framework to study the performance of WFC. With this framework, we can evaluate system throughput, optimize system parameter settings, and achieve proportional fairness. Extensive simulations verify that our theoretical model is very accurate.

SECTION I. Introduction

With IEEE 802.11 wireless LANs (WLANs) being widely deployed worldwide, its channel efficiency has received growing concern. The 802.11 networks perform the time-domain contention. In 802.11, each user must wait for a random time, before packet transmission. When multiple users are simultaneously backing off, the channel must remain idle, naturally leading to under-utilization. It has been shown in [1] that more than 30% reduction in throughput due to the time wasted in back-off operators.

Recently, Sen et al. [2] first proposed a time to frequency (T2F) protocol to improve the channel efficiency of wireless LANs. In T2F, users employ OFDM subcarriers to perform channel contention in the frequency domain, instead of the time domain. T2F arbitrates channel contention in two slots. In slot 1, each user signals on one subcarrier randomly chosen from a pool of subcarriers, and listens to this subcarrier pool at the same time by employing a second antenna. Then each user checks all subcarriers to determine the winners, who signal on the smallest subcarrier. In slot 2, the winners perform the 2nd-round frequency-domain contention for further arbitration. Finally, the winner in slot 2 transmits a packet in the next slot. By limiting the channel-contention time to two slots, T2F shortens the contention time greatly and hence improve the channel efficiency, compared to the 802.11.

T2F has attracted a great deal of attention [3]–​[8], [10], [11] since it has been proposed. Among them, [3]–​[5] are the most relevant to this study. Feng et al. [3] proposed a novel MAC design called REPICK. REPICK partitions all OFDM subcarriers into two groups: identification subcarriers (each of which is assigned to a unique user) and contention subcarriers (which are used by all users for channel contention). By allowing users to simultaneously transmit ACK over identification subcarriers and contend for channel over contention subcarriers like T2F, REPICK can exclude the DIFS time and significantly reduce the random backoff time, and the ACK transmission time in conventional 802.11 networks. Sen et al. [4] proposed Back2F protocol. Back2F extended T2F by creating backoff operations in the frequency domain. Like T2F, each user in Back2F first randomly picks a subcarrier from a specified range to contend for the channel, and then start the transmission, where the subcarrier chosen by a user is regarded as the current backoff value of the user. However, unlike T2F, after the transmission is finished, all other users will continue contending for the channel using the updated backoff value (rather than re-choosing a new subcarrier randomly), where the updated backoff value of a user is the result that its chosen subcarrier number subtracts the minimum subcarrier number in the last contention.

However, in the above related work, each user uniformly selects a subcarrier from the same subcarrier pool. As a result, each user has the same channel access opportunity. However, in reality, different applications have different quality of service (QoS) requirements. For example, a voice packet should have a more stringent delay requirement than a data packet and therefore should be assigned a higher transmission opportunity. Clearly, T2F, B2F and REPICK cannot fulfill the QoS requirements of real-time applications.

Recently, prioritized frequency-domain contention scheme [5] has been proposed. The authors in [5] proposed a scheme called WiFi-BA. WiFi-BA introduces a binary-mapping scheme to make collision detection in frequency domain, and a bitwise-arbitration (BA) mechanism to contend for channel. What WiFi-BA provides is absolute priority, where high-priority (HP) users will occupy all available bandwidth for immediate transmission, thereby starving low-priority (LP) users. In addition, WiFi-BA cannot completely exclude collisions (for example, a collision will occur when more than one user selects the same random value in binary mapping scheme) and the low-priority users often need several slots to contend for channel.

Instead of providing absolute priority, in this paper, we propose a novel weighted frequency-domain contention (WFC) scheme to provide relative priority (where HP and LP users coexist and share the total available bandwidth with different proportions). Our contributions are summarized as follows.

• In the design, WFC makes signification modifications in the two-slot contention process.

  • In slot 1, WFC allows HP and LP users, respectively, to choose subcarriers from [1, $S$ ] and [$F+1$ , $L$ ], where $F\leq S\leq L$ , and they delimit the range of the selectable subcarriers. Through suitably setting different values of $S$ and $F$ , WFC can provide general and fine-grained priority differentiations. For example, when $S=L$ and $F=0$ , WFC reduces to T2F.

  • In slot 2, the winners in slot 1 enter the second round contention. In this paper, we adopt a signature-assisted method to detect all these winners (which is feasible because the number of winners is not too large as shown in Fig. 8). This enables us to exclude collisions totally. In contrast, most existing schemes (such as WiFi-BA and T2F) still adopted the similar frequency-contention scheme as that in slot 1, and therefore there might exist collisions in slot 2.

• In the analysis, we propose a theoretical framework to study the throughput and the proportional fairness of WFC. With this framework, we can evaluate the system performance, and optimize the settings of the design parameters S and F, and achieve proportional fairness. In contrast, the most previous works (such as T2F, B2F, and WiFi-BA) just evaluate their design via simulation.

• In the simulation, we run extensive experiments to study how the system performance varies with the user number and various design parameters. These experiments verify that our theoretical model is very accurate.

FIGURE 8.

$\mathrm {E(\xi )}$ vs. m.

The rest of this paper is organized as follows. Section II outlines the T2F protocol. Section III presents the proposed WFC protocol. Section IV theoretically analyzes the performance of WFC. Section V presents simulation results to validate the accuracy of the proposed model. Section VI concludes this paper. In addition, Table 1 lists all used notations.

TABLE 1 Notations Used in This Paper

SECTION II. T2F

In this section, we present the basic ideas and the drawbacks of T2F [2].

A. Basic Ideas

T2F is a scheme that provides fair channel access via frequency-domain contention for WLANs.

In T2F, each user has two antennas: one for regular data transmission and another for listening to the channel. T2F uses the OFDM-based physical layer techniques. In OFDM, the whole channel is divided into $L$ subcarriers (e.g. $L=52$ in 802.11a/g).

Consider a star-topology WLAN illustrated in Fig. 1, where each user, $i =1, 2, 3, 4$ can hear each other and contend for channel for data transmission. With the help of Fig. 2, we now explain the data transmission process in T2F. In T2F, each user first senses channel idle for a DIFS (Distributed Inter Frame Space) time, then performs 2-round channel contention (i.e., R1 and R2 in Fig. 2) in two consecutive slots, and finally executes the data transmission. We next explain how R1 and R2 work as follows.

FIGURE 1.

A star-topology WLAN for T2F.

FIGURE 2.

Two-round contention in T2F.

In R1, each user signals on one subcarrier (via the transmission antenna) randomly chosen from a pool of subcarriers, and at the same time listens to this subcarrier pool via the listening antenna. T2F users treat the subcarriers as integer numbers. Then each user can determine the winners, who signal on the smallest subcarrier. In the example of Fig. 2, in R1, U1 and U4 select No.5 subcarrier, U3 and U2 select No.8 and No.11 subcarriers, respectively. Then U1 and U4 win because their selected subcarrier is the minimum.

In R2, the users choosing the smallest subcarrier perform the frequency-domain contention in the same way as R1 contention, and then transmit data in the ascending order of the chosen subcarriers. In the example of Fig. 2, in R2, U1 and U4 select No. 4 and No. 8 subcarriers, respectively. After that, U1 and U4 transmit data sequentially.

B. Drawbacks

In T2F, each user uniformly selects a subcarrier from the same subcarrier pool. Therefore, each user has the same channel access opportunity. However, in reality, different applications have different QoS requirements. For example, a voice packet should have more stringent delay requirement than a data packet and therefore should be assigned a higher transmission opportunity. Clearly, T2F cannot fulfill the QoS requirements of real-time applications. In addition, T2F cannot exclude the collisions fully. For example, if more than one user enters R2 and chooses the same subcarrier, this will cause collisions in data transmission.

SECTION III. The Proposed WFC

In this section, we present the weighted frequency-domain contention (WFC). We first make an overview of the WFC protocol in Subsection A, then, present its contention process in Subsections B and C.

A. Overview of WFC

WFC is an amendment of T2F. In T2F, it provides the same service for every contention user. In contrast, WFC would provide weighted services for different users based on their different demands. In this paper, we consider a one-hop star-topology WLAN where each user can hear each other as shown in Fig. 3, and assume that the system has two priority classes: high-priority (HP) class and low-priority (LP) class. In WFC, HP users have higher channel access probabilities than LP users.

FIGURE 3.

A star-topology WLAN for WFC.

Like T2F, in WFC, each user has two antennas (one for transmission and the other for listening signals), and performs the frequency-domain contention. As illustrated in Fig. 4, when a user wants to transmit data, it first senses the channel for a DIFS time, and then enters a two-round contention process consisting of R1 and R2, and chooses the winners. At last, the winners transmit data.

FIGURE 4.

Overview of WFC MAC design.

However, unlike T2F, we assume that each user is assigned to a unique signature. Each signature is associated with a sequence number, and is known to all users. WFC has the following key differences in the R1 and R2 contention stages.

1) Difference in R1

In WFC, different users signal on different subcarrier pools (i.e., different subcarrier ranges). As a result, different users will have different channel access opportunities. For example, assume that U1 and U2 are HP users, whereas U3 and U4 are LP users. Fig. 4 illustrates that the HP users (i.e., U1 and U2) can select a random subcarrier in subcarrier pool [1], [10], while the LP users (i.e., U3 and U4) choose in [4, $L$ ]. Since the subcarrier pools of HP and LP users are different, they will have different probabilities to select the smaller subcarriers. In this example, U1 and U2 respectively choose subcarriers 5 and 8, while U4 and U3 respectively choose subcarriers 5 and 11. Because U1 and U4 choose the same subcarrier 5, they will enter R2 for further contention resolution.

2) Difference in R2

Our purpose is to schedule multiple data transmissions upon one contention. Therefore, we aim at detecting all winners in R1 (which is feasible because the number of winners is not too large as shown in Fig. 8). In R2, while sending its signature on the whole channel, each user also executes correlation to detect all transmitted signatures, and then determines the transmission order by the sequence numbers of these signatures. For example, in Fig. 4, we assume that U1 first transmits, and then U4 transmits, according to the associated sequence number.

In the following subsections, we will detail the R1 and R2 contention stages.

B. Weighted Frequency-Domain Contention in R1

We assume that the whole channel is divided into $L$ OFDM subcarriers.

As shown in Fig. 5, in R1, we let the HP and LP users, respectively, uniformly choose subcarrier from subcarrier pool [1, $S$ ] and [$F+1$ , $L$ ], where $0 \leq F \leq S \leq L$ . Then, the smaller the S and the larger the $F$ , the higher the HP channel access probability is. By carefully tuning $S$ and $F$ , we can assign different channel access probability to HP and LP users, according to their QoS requirements. Note that when $F= 0$ and $S= L$ , R1 in WFC is the same to that in T2F.

FIGURE 5.

HP and LP subcarrier pools in R1.

WFC assigns different channel access probabilities to different priority classes, but it does not exclude the collision - multiple users simultaneously choose the same “minimum” subcarrier (namely, there are multiple winners). For example, in Fig. 4, both U1 and U4 choose subcarrier 5, which is the minimum subcarrier among all chosen subcarriers. Once collisions happen, multiple winners must enter R2 for further contention resolution.

C. Signature-Assisted Contention Resolution in R2

WFC introduces a signature set (e.g. $\text {G} = \{\mathrm {s}_{1},\mathrm {s}_{2},\ldots ,\mathrm {s}_{N}\}$ ), where $N$ is the total number of users, each unique signature $\mathrm {s}_{i}$ is mapped into a two-tuple, $\langle$ user_ID, transmission_order$\rangle$ . For example, the top-subfigure in Fig. 6 illustrates that each signature $\mathrm {s}_{i}$ , $i=1,\ldots ,4$ , is mapped to the user $i$ (i.e., $U_{i})$ and the transmission order $i$ . Before DATA transmission, the AP will broadcast the signature set and the mapping relationship. R2 consists of three procedures: signature sending, signature detection and DATA transmission.

FIGURE 6.

Contention resolution in R2 and DATA transmission.

1) Signature Sending

Each R1 winner transmits its assigned signature on the transmission antenna, and at the same time, it keeps sensing on its listening antenna.

2) Signature Detection

Each winner uses $N$ correlators to execute signature detection ${corr}_{i}\left ({ L_{s} }\right )$ shown in (1), where ${corr}_{i}\left ({ L_{s} }\right )$ , $i=1,\ldots ,N$ , denotes the correlation value between the received signal and the user $i$ ’s signature $\mathrm {s}_{i}$ . When ${corr}_{i}\left ({ L_{s} }\right )$ is higher than a predefined threshold, this means user$i$ participates in R2.

$$\begin{equation} corr_{i} (L_{S} )=\sum \limits _{k=1}^{L_{S}} {s_{i}^{\ast } (k)y(k+\Delta )} /\sum \limits _{k=1}^{L_{S}} {\left |{ {s_{i} (k)} }\right |^{2}} \end{equation}$$ where $L_{s}$ denotes the length of the signature, $y(k+\Delta )$ , $1\leq k\leq L_{s}$ , denotes the $k$ -th symbol of the received signal, from a shifted position $\Delta$ , $s_{i}(k)$ denotes the $k$ -th symbol of user$i$ ’s signature, and $s_{i}^{\ast }(k)$ denotes the complex conjugate of $s_{i}(k)$ .

3) Data Transmission

After the signature detection, each winner knows how many users will transmit in R2, which is first transmitted and which is then transmitted by the transmission order (mapped from the corresponding signature).Taking Fig. 6 as an example, winners in R1, say U1 and U4, enter R2 and contend for the channel. According to the transmission order, U1 first transmits and U4 then transmits.

4) Signatures Detection With Similar SNRs

In a typical WLAN, each user is near to each other. Therefore, when all winners in R1 (whose population is not too large as shown in Fig. 8) transmit their respective signatures in R2, each of them will receive these signatures with similar SNRs. This makes the accurate signature detection feasible. In WLAN environments, similar methods of simultaneously detecting all collided nodes via signatures have already been used in related work [8], [9].

SECTION IV. Performance Analysis

In this section, we develop a theoretical model to analyze the WFC throughput and study how to achieve the proportional fairness in saturation operation (where each user always has packets to transmit).

In this model, we consider a one-hop WLAN consisting of one AP, $m$ HP users, and $n$ LP users, where $m+n=N$ . We focus on the uplink traffic (i.e., traffic is only transmitted from users to the AP) and assume that the channel is perfect.

A. Throughput

In the following, we first express the per-user throughput and the total system throughput, and then calculate relevant parameters.

Per-user throughput: In WFC, the DATA transmission time can be divided into a series of transmission periods. As shown in the bottom-subfigure in Fig. 6, each transmission period consists of a DIFS interval, $T_{DIFS}$ , an R1 contention interval, $T_{R1}$ , an R2 contention interval, $T_{R2}$ , and a DATA transmission interval, $T_{DATA}\times E(\xi )$ , where $E(\xi )$ denotes the mean of the number $\xi$ of the winners in R1, and $T_{DATA}$ is one packet transmission time (including ACK time).

Define the throughput of one type-$I$ user, $\Gamma _{I}$ , to be the number of bits that the type-$I$ user successfully transmits during a whole transmission period, where $I=H$ denotes type-HP user and $I=L$ denotes type-LP user. $\Gamma _{I}$ can be expressed as

$$\begin{equation} \Gamma _{I} =P_{I} \times \frac {L_{DATA}}{T_{DATA} \times E(\xi )+T_{R1} +T_{R2} +T_{DIFS}} \end{equation}$$ where $P_{I}$ denotes the successful transmission probability of type-$I$ user, and $L_{DATA}$ denotes the packet length.

Total system throughput: The total system throughput,$\Gamma$ , is the sum of each user’s throughput $\Gamma _{I}$ . Therefore, $\Gamma$ is given by

$$\begin{equation} \Gamma =m\Gamma _{H} +n\Gamma _{L} \end{equation}$$ To calculate $\Gamma _{I}$ and $\Gamma$ , the remaining tasks are to calculate $P_{I}$ and E($\xi )$ .

1) Calculation of $P_{I}$

To calculate $P_{I}$ , we first consider a simple case that there is only one user in each class (i.e.,$m = 1$ and $n = 1)$ , and then we extend it to the general case (i.e., $m > 1$ and $n > 1)$ .

The case of $m = 1$ and $n = 1:$ We assume that user 1 is an HP user and user 2 is an LP user. According to the design in R1, user 1 selects a random subcarrier from the subcarrier pool [1, $S$ ], while user 2 selects from [$F+1$ , $L$ ], as shown in Fig. 7. In this figure, only user 1 can select one subcarrier from [1, $F$ ]; only user 2 can select one subcarrier form [$S+1$ , $L$ ]; nevertheless, both users 1 and 2 can select subcarriers from [$F+1$ , $S$ ]. In WFC, a user will win the contention if its selected subcarrier number is the smallest. Therefore, from Fig. 7, user 1 will win the contention absolutely if its selected subcarrier number belongs to [1, $F$ ]. User 2 will always lose the contention if its selected subcarrier number belongs to [$S+1$ , $L$ ]. Meanwhile, both user 1 and 2 have certain probabilities to win the contention if their selected subcarrier numbers belong to [$F+1$ , $S$ ].

FIGURE 7.

Illustration on calculation of $\mathrm {P}_{\mathrm {I}}$ when m = 1 and n = 1.

In short, the successful transmission probabilities, $P_{H}$ and $P_{L}$ when $m=n=1$ , are respectively by (4) and (5)

$$\begin{align} P_{H}=&P(user~1~win~absolutely\,in~[1,~F]) \notag \\&+\,P(user~1~win~in~[F+1,~S]) \notag \\=&\sum \limits _{i=1}^{F} {f_{H} +f_{H} \sum \limits _{i=F+1}^{S} {(L+1-i)f_{L}}} \\ P_{L}=&P(user~2~win~in~[F+1,~S]) \notag \\=&f_{L} \sum \limits _{i=F+1}^{S} {(S+1-i)f_{H}} \end{align}$$ where $f_{H}=1/S$ and $f_{L}=1/(L-F)$ , respectively, denote the probabilities that users 1 and 2 choose an arbitrary subcarrier from their subcarrier pools. In (4), $\sum _{i=1}^{F} f_{H}$ denotes the probability that user 1 wins the contention absolutely in the range of [1, $F$ ], since only user 1 can select the subcarrier from this range, while user 2 cannot; $f_{H}\sum _{i=F+1}^{S} {(L+1-i)f_{L}}$ denotes the probability that user 1 wins when it selects the subcarrier $i$ from [$F+1$ , $S$ ], since user 1 wins the contention in this range only when user 2 chooses the subcarrier from [$i$ , $L$ ] with a probability of $(L+1-i)f_{L}$ . Likewise, in (5), $f_{L}\sum _{i=F+1}^{S} {(S+1-i)f_{H}}$ denotes the certain probability that user 2 wins when it selects the subcarrier number $i$ from [$F+1$ , $S$ ], since that only user 1 chooses the subcarrier from [$i$ , $S$ ] can ensure that user 2 wins the contention.

The case of $m > 1$ and $n > 1$ : We now focus on [1, $F$ ] to explain the extension ideas, as shown in Fig. 7. From this figure, we know that a tagged HP user will win the contention absolutely if its selected subcarrier number $i$ belongs to [1, $F$ ], and is the smallest compared with the selected subcarrier numbers of the rest $m-1$ HP users. That is, the rest $m-1$ HP users would choose the subcarrier from [$i$ , $S$ ] with a probability of $\left ({ S+1-i }\right )f_{H}$ . Since $i$ varies from 1 to $F$ , the probability that the tagged HP user wins the channel is $\sum _{i=1}^{F} {f_{H}(\left ({ S+1-i }\right )f_{H})}^{m-1}$ . Similarly, we can get the probabilities that any HP user and LP user win the contention when they select the subcarriers from [$F+1$ , $S$ ], respectively. So the probability of each HP user ($P_{H})$ and LP user ($P_{L})$ win the contention can be expressed as

$$\begin{align}&\hspace {-1.2pc}P_{H} (m,n,F,S,L)\notag \\=&P(a~tagged~HP~user~wins~in~[1,~F]) \notag \\&+\,P(a~tagged~HP~user~wins~in~[F+1,~S]) \notag \\=&f_{H} \left({\sum \limits _{i=1}^{F} {(S+1-i)f_{H}} }\right)^{m-1} \notag \\&+\sum \limits _{i=F+1}^{S} {(((S+1-i)f_{H} )^{m-1}\times ((L+1-i)f_{L} )^{n})}\qquad \\&\hspace {-1.2pc}P_{L} (m,n,F,S,L)\notag \\=&P(a~tagged~LP~user~wins~in~[F+1,~S]) \notag \\=&f_{L} \sum \limits _{i=F+1}^{S} {(((S+1-i)f_{H} )^{m}\times ((L+1-i)f_{L} )^{n-1})} \end{align}$$

Note that, when $m=n=1$ , (6) and (7) reduce to (4) and (5) respectively.

2) Calculation of E($\xi )$

In this section, we calculate $E(\xi )$ . To do so, we express the total throughput from two perspectives.

On one hand, the total system throughput is the sum of each user’s throughput. Let $\Gamma _{H}$ and $\Gamma _{L}$ respectively be the throughput of each HP and LP user, since the users of each class have the same subcarrier pools, and then have the same transmission probabilities and the same throughput. From (2) and (3), we can also express the system throughput as follow

$$\begin{align} \Gamma=&m\times \Gamma _{H} +n\times \Gamma _{L} \notag \\=&(m\times P_{H} +n\times P_{L} )\notag \\&\times \frac {L_{DATA}}{T_{DATA} \times E(\xi )+T_{R1} +T_{R2} +T_{DIFS}} \end{align}$$

On the other hand, the total system throughput can be expressed as the ratio of the average data length to the average transmission period. Since $E(\xi )$ is the average number of the winners in R1, in a transmission period, the total length of data transmission is $L_{DATA}\times E(\xi )$ , then we can also express the system throughput in terms of $E(\xi )$ as

$$\begin{equation} \Gamma =\frac {L_{DATA} \times E(\xi )}{T_{DATA} \times E(\xi )+T_{R1} +T_{R2} +T_{DIFS}} \end{equation}$$

Combining (8) and (9), we have

$$\begin{align}&\hspace {-1.2pc}E(\xi )\notag \\=&m\times P_{H} +n\times P_{L} \notag \\=&m\times f_{H} \Big (\sum \limits _{i=1}^{F} {\prod \limits _{j=2}^{m} { \left (S+1-i \right )f_{H}}} \notag \\&+\sum \limits _{i=F+1}^{S} \left (\prod \limits _{j=2}^{m} {(S+1-i)f_{H} \times \prod \limits _{k=1}^{n} {(L+1-i)f_{L}}} \right ) \Big ) \notag \\&+\,n\times f_{L} \sum \limits _{i=F+1}^{S} \left (\prod \limits _{j=1}^{m} {(S+1-i)f_{H} \times \prod \limits _{k=2}^{n} {(L+1-i)f_{L}}} \right )\notag \\ {}\end{align}$$

B. Proportional Fairness

In our model, HP and LP users can, respectively, choose subcarriers from [1, $S$ ] and [$F+1$ , $L$ ]. In this section, we consider how to set S and F so as to achieve a predefined proportional throughput fairness ratio, $\mathrm {\gamma }$ .

From (6) and (7), we have

$$\begin{align} \gamma=&\frac {\Gamma _{H} (m,n,F,S,L)}{\Gamma _{L} (m,n,F,S,L)} \notag \\=&\frac {\sum \limits _{i=1}^{F} \!{(S\!+\!1\!-\!i)^{m-1}} \!+\!f_{L}^{n}\sum \limits _{i=F\!+\!1}^{S} {((S\!+\!1\!-\!i)^{m-1}\!\times \! (L\!+\!1\!-\!i)^{n})}}{f_{L}^{n}\!\sum \limits _{i=F+1}^{S}\! {((S+1-i)^{m}\times (L+1-i)^{n-1})}}\notag \\ {}\end{align}$$ Then, given $m$ , $n$ , $L$ , and $\gamma$ , we can find the desired $S$ and $L$ by numbering solving (11).

SECTION V. Simulation Verification

In this section, we verify the accuracy of the proposed WFC model in a single-hop star-topology. For verification, we develop a simulator based on C++ language. The simulator is developed by using eclipse CDT and based on the simulation frameworks in [12]. In the framework, we replace the contention process in CSMA/CA by frequency-domain contention. The default parameter settings are shown in Table 2. Each simulation run lasts 200 seconds. In all figures, the labels “ana” and “sim”, respectively, denote the theoretical and simulation results.

TABLE 2 Parameter Settings in Simulation

A. E($\xi$ ) Verification

In this section, we verify the accuracy of E($\xi )$ , which plays an important role in modeling throughput. Fig. 8 plots the theoretical and simulation results of $E(\xi )$ as m varies from 1 to 50, where the theoretical result of $E(\xi )$ is plotted by (10), $m=n$ , $S=40$ , $F=10$ , and $L=52$ . From this figure, we can see that 1) the sim curve matches the ana curve very well, indicating the theoretical result of $E(\xi )$ is very accurate; and 2) as $m$ increases from 1 to 50, the value of $E(\xi )$ only changes from 1 to 1.8, implying that in most cases, only 1 or 2 users will enter the 2nd-round contention.

B. Throughput Verification

In this section, we verify the system throughput and per-user throughput, where L is set to 52.

Fig. 9 plots the system throughput, where the theoretical results is plotted by (3), $m = n$ , $S = 40$ , and $F = 10$ . From the figure, we can see that 1) the sim curve closely match the corresponding ana curve; and 2) the system throughput is a quasi-constant regardless of how m varies, for example, as m increases from 1 to 50, the system throughput varies around 0.8. The reason why WFC can achieve high efficiency of 0.8 (i.e., the overhead is about 0.2) is because WFC just uses two slots for two round frequency-domain contentions and therefore limits the contention overheads. In contrast, CSMA/CA will consume more time for the time-domain contention; the results in [1] show that the overhead of CSMA/CA is more than 30%.

FIGURE 9.

System throughput vs. m.

Fig. 10 plots per-user throughput when F, S, n and m respectively change, where the theoretical result is plotted by (2), as shown in Fig. 10(a)–​10(d).

FIGURE 10.

Per-user throughput in each class when (a) F varies, (b) S varies, (c) n varies, (d) m varies.

Fig. 10 (a) plots the per-user throughput when $F$ changes from 0 to 20. In this figure, we set $S = 40$ and $m = n = 10$ . From this figure, we can see that 1) the sim curve closely match the corresponding ana curve; 2) the per-user throughput changes dramatically with $F$ . For example, as $F$ increases from 0 to 10, the throughput of each HP and LP user respectively increases and decreases from 0.046 to 0.08, and from 0.035 to a low level. This is because with increasing $F$ , the probability that each HP user wins the channel will get bigger in [1, $F$ ], hence increasing the throughput; in contrast, the probability that each LP user wins the channel will get smaller in [$F+1$ , $S-1$ ], hence lowering the throughput.

Fig. 10 (b) plots the per-user throughput when $S$ varies from 30 to 50. In this figure, we set $F = 10$ and $m = n = \textit {10}$ . From this figure, we can see that 1) the sim curve closely match the corresponding ana curve; 2) the per-user throughput keeps almost unchanged with $S$ . For example, as $S$ increases from 30 to 50, the throughput of each HP user decreases slowly from 0.08 to 0.075, while the throughput of each LP user increases very slowly from 0.0005 to 0.0046. This is because increasing $S$ can lead to increase the probability that each LP user wins the channel, but decrease the probability that each HP user chooses a subcarrier from [1, $F$ ].

Fig. 10 (c) plots the per-user throughput when n varies from 2 to 20. In this figure, we set $m = 10$ , $F = 10$ , and $S = 40$ . From this figure, we can see that 1) the sim curve closely match the corresponding ana curve; 2) the per-user throughput is almost unchanged regardless of how $n$ changes. For example, as $n$ increases from 2 to 20, the throughput of each HP and LP user always keep unchanged as like in Fig. 10(b).

Fig. 10 (d) plots the per-user throughput when $m$ varies from 2 to 20. In this figure, we set $n = 10$ , $F = 10$ , and $S = 40$ . From this figure, we can see that 1) the sim curve closely match the corresponding ana curve; 2) the per-user throughput is decreasing with the increasing of $m$ . For example, as $m$ increases from 2 to 20, the throughput of each HP user decreases from 0.21 to 0.04, and the throughput of each LP user also decreases from 0.04 to a lower value.

C. Proportional Fairness

In this section, we verify the accuracy of our proportional-fairness model. We first present the theoretical and simulation results of the proportional ratio $\gamma$ in terms of $F$ and $S$ , respectively, where $\gamma$ is the ratio of the throughput of HP user to the throughput of LP user, as shown in Fig. 11. And then, we verify that WFC can achieve the target $\gamma$ .

FIGURE 11.

${\gamma }$ when (a) F varies and (b) S varies.

Fig. 11 (a) plots the value of $\gamma$ when $F$ varies from 0 to 20, where $S = 40$ and $m = n = 10$ . From this figure, we can see that as $F$ increases from 0 to 20, the value of $\gamma$ increases from a very low value to 1400. This is because with increasing $F$ , as explained for Fig. 10(a), the throughput of each HP user and LP user will respectively increase and decrease, therefore leading to a drastic increase of the ratio $\gamma$ .

Similarly, Fig. 11 (b) plots the value of $\gamma$ when $S$ varies from 30 to 50, where $F = 10$ and $m = n = 10$ . From this figure, we can see that when $S$ increases from 30 to 50, $\gamma$ decreases from 150 to 16. This is because as $S$ increases, as explained for Fig. 10 (b), the throughput of each HP user and LP user can respectively decrease and increase, therefore leading to a drastic decrease of the ratio $\gamma$ .

Table 3 compares the target value of $\mathrm {\gamma }$ with its simulation result, where we set $m = n = 10$ . In this table, given the target value of $\gamma$ , we can obtain a pair of $F$ and $S$ by (11), and then use them for obtaining the simulation value of $\gamma$ . From this table, we can see that the error between the target value and the simulation value is much small. This manifests that WFC can well achieve the proportional fairness.

TABLE 3 Comparison Between the Target and Simulation Values of $\gamma$

SECTION VI. Conclusion

In this paper, we propose WFC, a novel weighted frequency-domain contention scheme. WFC provides priority differentiation by limiting OFDM subcarrier access ranges in frequency-domain contention, and resolves collision by signature differentiation. We then analyze the throughput and proportional fairness of WFC. Finally, we verify the accuracy of the proposal theoretical model via extensive simulations.