Within the scope of this work, we focus on channel assignment algorithm using POCs in the context of both 20 and 40 MHz channels. A 40 MHz channel is a combination of two adjacent 20 MHz channels. The list of bonded channels is demonstrated in Tab. 2. The degree of interference among channels has been widely studied where only few of them reported on bonded channels. This work considers degree of interference among channels for both bonded and non-bonded channels.

The estimation of the degree of interference between two channels is a difficult task. For 20 MHz channel, there has been a popular study for measurement of overlapping degree in [20] which is summarized in Tab. 3. Where Δc represents the difference between two channels and ρΔc does the overlapping degree for the respective Δc.

In this work, we utilized this overlapping degree to derive interference model for proposed channel assignment algorithm. If APi and APj are operating on channels ci and cj respectively, the channel difference Δc=|ci−cj| and degree of interference ρi,j is defined as follows: (1)whenbothciandcjare20MHz:Δc=|ci−cj|,ρi,j=ρj,i=ρΔcwhenandbothciandcjare40MHz:if|ci−cj|≤4:Δc=|ci−cj|,ρi,j=ρj,i=1+ρΔc2else:Δc=|ci−cj−4|,ρi,j=ρj,i=ρΔc2whenciis20MHzandcjis40MHz:Δc={0when(cj−2)≤ci≤(cj+2)|ci−cj−2|otherwiseρi,j=ρΔcandρj,i=ρΔc2

When two APs are interfered each other, the communication is delayed due to collision which results in higher communication time. The definition of communication time will be expounded later in Section 2.2. The transmission distance between devices has a direct impact on interference which gets stronger, the closer the interfered sources. To model such phenomenon, in this work, we consider an inverse linear relationship between degree of interference and distance. It has been observed that for the target link, the interference from the interfering link increases communication time as the rate adaptation mechanism lowers transmission rate resulting throughput drop. The degree of throughput drop is proportionate to inverse of the distance between the devices. Suppose, APi is operating on channel, ci and communication time with its associated host is τi. If APi is interfered by its neighboring APs say APj operating on channel cj and communication time τj then communication time of APi due to interference is calculated as follows: (2)τ^i=τi+τj×(dmax−d)dmax×ρjiwhere d is the distance of interfered AP i.e., APi to APj and dmax is the maximum distance for consideration of interference.

Let, C be the set of all available k channels in underlying WLAN physical layer i.e., C={c1,c2,⋯,ck}. In this work, we consider m APs presented as A={AP1,AP2,⋯,APm} and n hosts presented as H={host1,host2,⋯,hostn}. A host connects to only one AP during communication using the same channel assigned to that AP. The list of hosts associated to APi is presented as Host(i). This work focuses on downlink only, i.e., data is transmitted from APs to hosts. Generally, the traffic in downlink is higher than that of uplink in most of the application [13,21].

The channel assignment algorithm minimizes overall interference Ftot of a network which can be formulated as an optimization problem. (3)Minimize:Ftot=∑i∈A⁡τ^iwhere τ^i is the interfered communication time of APi and A is the set of all APs in the network. If APi is interfered with its neighboring APs, e.g., APj and APk, then L(i) presents the list of APs interfered with APi including itself. Therefore, L(i)={i,j,k}. Then, the interfered communication time, τ^i is derived as follows: (4)τ^i=τi+τj×(dmax−dj,i)dmax×ρji+τk×(dmax−dk,i)dmax×ρki =τi+∑j∈L(i),j≠i τj×(dmax−dj,i)dmax×ρjiwhere τi is the communication time of APi and ρji is the degree of interference on APi from APj as defined in (1). The communication time of APi is defined as the total time when the APi transmits 1-bit to all its associated hosts, Host(i): (5)τi=∑j∈Host(i)1spijwhere spij is the link speed from APi to hostj. Therefore, 1spij presents the link delay for sending 1-bit from the APi to hostj with link speed spij. Here, τi in Eq. (5) can also be interpreted as the estimated latency of APi to send 1-bit to all its associated hostj in Host(i). Consequently, the inverse of this estimated latency becomes the estimated minimum throughput, s^i,j for a host connected to APi. (6)s^i,j=1∑j∈Host(i)⁡1spi,j

To ensure a minimum throughput to every host in the network, we consider a threshold thhost that estimated minimum throughput of any host must satisfy i.e., s^i,j≥thhost. Besides, speed of any link must satisfy a threshold thlink to ensure coverage and a minimum speed in all links. Each AP must be assigned a channel while minimizing overall interference in the network. Finally, the optimization problem of the proposed channel assignment algorithm can be formulated as follows: (7)Minimize:Ftot=∑j∈A∑j∈L(i)⁡τ^i×dmax−dj,idmaxSubject to{:}APjisassignedchannel c:c∈Cs^ij≥thhostspij≥thlink

The speed of a link is affected by many factors like transmission power, distance between devices, modulation and coding schemes (MCS) i.e., modulation type and coding rate, channel interference, and even the receiver design [22–24]. This makes the theoretical computation of link speed is difficult. Therefore, this work considers an alternative approach that conducts extensive measurements to establish a formula for link speed by applying the regression analysis. The formula takes RSSI as single input based on measurements and provides respective link speed using sigmoid function [25]. The formula is extended to support distance as input which utilizes log-distance path loss model [26,27] to derive RSSI. This RSSI is then converted to link speed. Theoretical parameters such as transmission power and MCS are fully characterized in commercial devices, thus, the link speed is solely dependent on adopted devices and physical distance between devices under same network environment. This empirical model for link speed estimation greatly simplifies the optimization complexity.

One commercial AP and two personal computers (PCs) are adopted in this measurement experiment. The configuration of devices is listed in Tab. 4. The experiment was conducted under 32°C and the relative humidity was around 82%. A Wi-Fi channel-scanner [28] was utilized to inspect frequency band activity in 2.4 GHz and avoid unnecessary interference in experiment. TCP traffics were generated using iperf [29] for 180 s. We set TCP window size 477 Kbytes and buffer size 8 Kbytes. Figs. 3 and 4 show link speed and RSSI measurement results for physical distance ranging from 0 to 100 m respectively. We have observed that the RSSI of bonded and non-bonded channels were not synchronized in some cases. This error may be caused by the fluctuation of RSS at measurements.

Both link speed and RSSI drastically drops sharply with the increase of transmission distance. Such stair-like changes in link speed and RSSI are caused by changes of MCS. The relationship between RSSI and link speed in measurement results is analyzed to yield estimated link speed, spRSSI which is presented as: (8)spRSSI=3.5×cw1−e5cw×(RSSI+70)where RSSI is the given RSSI and cw is the operating channel width; cw=20 and cw=40 when operating on 20 and 40 MHz channel respectively. The relationship between estimated link speed and RSSI is shown in Figs. 5 and 6 for 20 MHz channel and 40 MHz channel respectively. RSSI of a host from AP can be easily estimated using indoor path loss model [26,27] from transmission distance. The RSSI of host from AP at distance, d (m) is RSSId (dBm) can be derived as: (9)RSSId=RSSI0−10×α×log10d−∑k⁡nkWkwhere RSSI0 presents RSSI at 0m transmission distance between AP and host, α presents path loss exponent, nk presents the number of k-type obstacles on the transmission path and Wk presents the signal attenuation factor (dB) for the k-type obstacle.

In this section, we model an interference graph from set of APs and interference among them. Let, G=(V,E) be an interference graph where V represents the set of APs and the edge E presents the existence of the interference in any pair of APs in network. An edge, e(i,j)∈E if APi is interfered with APj. In this work, we consider minimum RSSI of −82 dBm for interference. The weight of an edge presents the degree of inference between two interfered APs e.g., the degree of interference between APi and APj is presented by ρij which can be derived using Eq. (1). After interference graph, necessary information of hierarchical optimization in Eq. (7) is derived for further processing as follows:

Step 1: Preprocessing

① Calculate link speed of any pair AP and host in network. Link speed, spij between APi and hostj can be derived from distance or RSSI using Eqs. (8) and (9).

In the second step, an initial solution is generated. A straight forward method is to test all possibilities of channel assignment to every AP in the network. For m number of APs and k number of channels, there will be mk number of possibilities. The complexity becomes exponential for such exhaustive method which is not suitable in practical applications for large network. To avoid such exponential cost, we choose a faster greedy method for generation of initial solution. At every step, select a channel for assigning to an AP that minimizes current communication time considering interference. The detail of the procedure is given below:

Step 2: Initial Solution Generation

① Calculate interfered communication time τ^i for each AP using (4) and sort APs in descending order of τ^i. In case of equality, resolve using τi.

In this step, initial solution generated in previous step is improved iteratively using simulated annealing (SA) algorithm. Greedy method for channel selection makes a locally optimal choice aiming to minimize overall interference. However, this does not yield an optimal solution. Therefore, herein, we utilize probabilistic technique of SA to approximate minimal interference. SA starts with the initial solution, selects an AP and a channel with random choice, and examines the changes in total interference. We keep the new selection if it reduces the total interference. Otherwise, we generate a random value between [0,1] and accept or reject the new selection with some probability based on the generated random value, changes in total interference and a given constant namely temperature TSA for the iterative repeating time RTSA. The detail of the iterative step is given below:

Step 3: Iterative Improvement

① Select initial solution in the previous step as the current solution, Ftotold.

In this step, we aim to balance load on operating channels. After allocation of channels to APs, some APs may become congested and some APs may not. Therefore, balancing of loads on channels is necessary to decrease network traffic and utilize channel capacity more efficiently. For this purpose, it is required to compute network load based on channels. Then, the load of channels can be balanced by changing associations of hosts in intersection of APs to low crowded AP. Thus, network performance achieved in previous step is further improved by applying procedure of averaging load on channels. The detail of the procedure is given below:

Step 4: Balancing Load on Channels

① Select an AP say APi that has the largest interfered communication time, τ^i. A host associated to this AP is considered to change its association to another AP operating on different channel.

The proposed interference model is utilized in channel assignment algorithm where link speed is estimated considering interference of neighbor APs using the proposed interference. Therefore, the accuracy of the interference model is needed to verify before using in channel assignment algorithm. To verify the efficiency, we conduct a measurement campaign to collect actual link speed for different channel differences of an AP and its interfered APs at different physical distances. We considered 1 and 50 m physical distances for this work. Then, we compare the estimated link speed using proposed interference model with the actual link speed. The comparison results are shown in Figs. 8 and 9 non-bonded and bonded channels respectively. The estimated link speed using proposed interference model and measured link speed of the same link are shown in y-axis where channel distance ranging from 0 to 8 is shown in x-axis. The results indicate that the estimated results coincide with the measured results. However, the results deviate noticeably at certain places because of signals from unknown APs in of the building. It should be noted that we conducted the measurement experiment at night in weekend when university is closed aiming to minimize the interference from existing Wi-Fi signals.

First, we consider a random topology as depicted in Fig. 10 where 12 APs and 60 hosts are distributed in a 250 m × 200 m rectangular area. The APs are distributed equally in the grids of this rectangular network field. We assume that each AP has the same coverage of 100 m. The hosts are generated randomly in the field where a host is associated with an AP that provides maximum RSSI.

Next, we consider a regular network topology, as depicted in Fig. 11. The regular topology models 3rd floor of engineering building-3 in Okayama University, Japan. There are 6 rooms with 2 different room sizes: 7 m× 6 m and 3.5 m × 6 m. We allocated 60 hosts and 9 APs in this dense network environment. The APs and hosts in this topology are regularly distributed since students have their own positions in research rooms. RSSI drops when passing through concrete walls. We consider two types of attenuation factor due to obstacles namely concrete wall and door wall in propagation model Eq. (9). We model such phenomenon by reducing the RSSI 15(dBm) and 3(dBm) if a wall exists between two nodes for concrete wall and door wall respectively.

Now, to evaluate proposed channel assignment algorithm, we compare network throughputs obtained by random assignment of channels, greedy channel assignment and proposed algorithm assignment. Here, the random channel assignment experiment is repeated 10 times and average of the network throughputs is considered for evaluation to avoid any kind of biasness. Fig. 12 shows simulation results of network throughput for random and regular topology respectively.

The throughput for combination of bonded and non-bonded channels is expected to be better for flexibility of selection of partial overlapping channels. The numerical results in Fig. 12 demonstrate that the proposed algorithm can successfully assign channels in both topologies to reduce interference compared to random and greedy assignment. The network throughput is comparatively higher in random topology than that of regular topology. This is because, the regular topology is denser than random topology and all the APs are within the coverage of each other. Nonetheless, even in such extreme dense network environment, proposed algorithm works well in assigning channels to improve network performance by reducing overall network interference.