From our SAS analysis in Figure 3, three indicators (Juser: J of user; Jap: J of AP; Tpt: relative throughput) show that J of AP can represent throughput (value > 0.8, so it is highly linear related).

Through our statistics calculation, PCAP focus on J of AP only, which means it only focus on throughput. This is a deficiency of Target Function design, which is not well reflecting our topic.

The problem is defined as a NP-hard problem since we apply a practical scenario that includes 20 APs, each AP has 10 levels of power, so the state space of the problem will be 10²⁰, making it neither solvable nor verifiable in polynomial time, so it is NP-hard.

From the computation theory, we know that we cannot get an accurate solution. Comparing with other NP- hard problems such as TSP (Traveling Salesman Problem), we get some heuristic methods. Since existing models either using complex definition such as utility or involving many parameters such as channel gain, those models are not clear enough to apply informed search techniques, so first we need to build a clear, simplified model SCV, and then apply the Ant Colony algorithm to solve the model.

Since our topic is: “J of user (fairness of users) & throughput”, which means to make balance between these two parameters. Obviously the two parameters have different units, then we have to convert the “throughput” to “J of AP” (already explained, it can represent throughput, with high linear relation).

Then our SCV gives a new designed target function:

which reflects the balance of two parameters (J of user & throughput), and we will rewrite to get its final form f.

The user will select the strongest Received Signal Strength Indicator (RSSI) as default. In the model [27] , where “a” is a constant factor, “P” is received power by user, “X” is distance between user and selected AP, while “” has different value in different scenarios, generally between 1.6 and 6.5 [28] . The formula only determines the association matrix of user-AP. In practice, the general power range of the AP is 10 dBm - 30 dBm, i.e. 1 mw ~1 w, here we adopt for indoor case. From the formula, the value of “a” does not affect the association results, to simplify the mathematical form, we take, so our model adopts a simplified form:

We use r_ij to denote SINR (Signal to Interference plus Noise Ratio). Assuming the user [i] connects to AP[j], the power of AP[j] is P_j. Wherein “g” are channel gains, A_i is a set of all APs within the same channel of AP[j]. N_0j is an additive white Gaussian noise generated by AP[j].

It is worth noting that N_0j can be adjusted to an exact value [29] [30] . So we can set a constant,

From Table 1 in [13] , v_i denotes user’s transmission rate and r_ij is SINR. We see the monotonically increasing relationship between the two variables. Here we might assume that two variables meet the linear relationship as an approximation, , is a constant of proportionality. Then connect this to (2) and (3) we have:

So λ is a constant:

Let N[j] denote the total number of users which connect with AP[j]. P_j is AP[j]’s power. Because the users are time based sharing the chance of AP[j], so we use to denote the effective speed of v_i:

According to the simulation result in [13] , we know that usually 10 Levels of AP power will be enough to achieve a good result. Therefore, in our model, p_max and p_min have relationship as following: p_max/p_min=10, p_max will be the basis of calculation, since we need to increase the p_j, so the 10 APs’ power value is in Table 2.

Note here the unit of power is “mw”, not “dBm”. Since (), here l_j denotes the level of AP’s power, so the Formula (6) can be rewritten as follows:

Let M be the total number of users and N be the total number of APs. From statistics we know that the Expectation of V̅_i for all users is denoted as E(V̅_i), and Variance of V̅_i for all users is denoted as S²(V̅_i). We have the following (;):

Let b[i] denote the effective transmission speed from user[i] to AP[j], we have b[i] = V̅_i. Moreover, let U[j] denote the transmission speed from the AP[j] to backbone. The Expectation of b[i] is denoted as:, and Variance of b[i] is denoted as:, the Expectation of U[j] is denoted as:, and Variance of U[j] is denoted as:, so continue we have formulas as following:

Here cvusers denotes the coefficient of variation of transmission speed of all users, and cvAPs denotes the coefficient of variation of transmission speed of all APs, we have:

Note, here we adopt the definition of J in [13] , where we have, above is the relationship between J and the square of coefficient of variation.

According to our topic, we need a function that can describe the tradeoff between fairness of users and throughput of network. In [13] , the algorithm is divided into two steps: increase average value and decrease variance value of AP utility to increase throughput of network. They are equal to decreasing or. So increasing J of users is equal to decreasing.

Let F denote a target function as follows:, is a weight proportion factor, it is very important reflecting our requirement that how to make the balance between fairness and throughput, it is a quantifiable indicator

Here we do some mathematical derivation to illustrate how we get a reasonable value of. Considering the static grouping problem: m numbers are average divided by n groups, therefore each group has m/n numbers. Given that the expectation of total numbers is, and their variance is. Cvnumbers denotes the coefficient of variation of m numbers. Cvgroups denotes the coefficient of variation of n groups. Defined as before, we have:

From the above, it means is much smaller than, comparing this example to our function “F”, in function “F” we should amplify the small part since two parts have relationship. So we decide to give value to, let.

where in

Note that M and N are constants as defined before. M is total number of users, N is total number of APs. When “F” goes to minimum, it is equal to “” goes to minimum. So (18) will be our simplified target function to achieve the purpose of the tradeoff between Fairness (users) and throughput (network).

From Formula (12) we know that:

throughput_real is real throughput, and throughput_relative is relative throughput. Since the λp_min is constant, we use throughput_relative to represent throughput_real.

In this part we are going to place a total number of N = 20 APs on a 4 by 5 grid, with each AP on a grid point. The coverage area of each AP can stretch across the whole area. The distance between two adjacent APs is set to 100 meters. The maximum transmission power of each AP is set to 20 dBm (100 mw), so the minimum transmission power of each AP is set to 100/10 = 10 mw = 10 dBm.

We arrange M = 200 users randomly distributed in the whole area. According to [31] , a separation of four channels can be used without reducing the performance, so the possibilities could be opened to channels 1, 5, 9 and 13. In this paper we decide to use these channels in order to get a bigger.

Let AP_j → Ci denote AP_j using channel i, we use 1, 5, 9, 13 these channels to configure the network as in Table 3.

(a) To initialize 200 users’ and 20 APs’ coordinates. NC_max is the maximum number of iterations, m is the number of ants, Alpha is a parameter to characterize the importance of pheromone, Beta is a parameter to characterize the importance of heuristic information, and Rho is a pheromone evaporation coefficient. Also we need to initialize pheromone matrix Tau: Tau = ones (10, 10, 19). We calculate the average distance for each AP to all users then to construct a row vector called xaverage (1 × 20): xaverage = sum (Distance)/200.

(b) Heuristic Matrix Eta Design: is the average distance from AP to all the users, then in order to obtain an attractive balance, based on our attractive model in Formula (1), the ratio of two power levels is equal to the

ratio of the two, so that both APs may have the similar chance to attract users. Heuristic matrix is Eta (10

× 10 × 19), from the 1st AP to 20th AP, which has total of 19 pages, and each page is a matrix (10 level × 10 level) describing the heuristic information from all power levels of current AP to all power levels of the next AP, and we set the value of these heuristic transition points as “1”. We introduce a new mechanism: we give such heuristic matrix a quantitative parameter defined as “similar”, namely the degree of similarity between non-heuristic points and the heuristic points, and we set the value of non-heuristic points in the heuristic matrix as “similar×1”. Then we finish the design of heuristic matrix Eta.

(c) We set totally 10 ants at, and each ant is in a grid. Here the grid denotes one power level, so there are 10 grids (10 power levels). Initially, all the ants are in (city1), then 10 ants will move to (city2) together but choose the different grid (power level) according to the probability function: P = (Tau.^Alpha). * (Eta.^Beta), then repeat the procedure until moving to (city20) to complete their travel. Then we start to calculate the cost function of 10 paths (a path is defined as the route from to, and the value of each AP is a power level) and select the minimum cost, then record such path. After this, finally we will update the pheromone which includes decreasing all of the paths’ pheromone according to parameter Rho and increasing all of the paths’ pheromone traveled by ants.

(d) Repeat step(c) until reaching the NC_max (a constant), and output all cycles’ results.

(a) The solution of ant colony algorithm has many parameters, so we need to run multiple times on the same data set and same parameters and select the average result, also to search for the more suitable parameters’ values. Here we use to represent. We know that the maximum of is:, but it will never be achieved, at least because of users’ distribution.

(b) All parameters can affect the calculation result. Generally, it is suitable to set Alpha = 1; Beta = 2; Rho = 0.1; and similar = 0.9 to get better results.

(c) Strictly speaking the convergence of ant colony algorithm is defined as that the pheromone in the iteration process without change, or at least the obtained path without change. This is usually in the theoretical analysis. In practical applications due to a number of random factors, it is very difficult to achieve this status. So as long as the actual optimal solutions have vibration around the average straight line after each iteration, this is considered as convergence, and such average line is called convergence line. It can be seen from Figure 1 and Figure 2, after traveling 200 times, all of the parameters have been around the horizontal line with slight vibration, and these mean convergence.

We use the samples from experimental data to study the correlation coefficients among these indicators. Wherein Juser denotes J of user, Jap denotes J of AP, Tpt denotes, cost denotes the f in (18).

Figure 3 shows that at Alpha = 0.05 significance level, all the p-values are less than 0.05, we reject the H₀ and accept H₁ that these variables are linearly related, wherein the Tpt-(Jap, Juser) have highly significant linear correlations, while correlations of Jap-Juser is weak. We compared the degree of concentration of those data points in Figure 4 and Figure 5. It is clear that data points are more concentrated in Figure 5. This means the linear correlation of Tpt-Jap is much higher than the linear correlation of Tpt-Juser, which also proves the effectiveness of SCV model (coefficient of Tpt-Jap > 0.8, Tpt-Juser = 0.17, so it is more effective to use J of AP whereas not J of user to represent the throughput).

We select average case in Figure 1, at the 100th time, the J of User is almost equal to 0.75, and corresponding J of AP is almost equal to 0.94, the is almost equal to 145, since its maximum value is 200 as mentioned before, then the throughput of the network is equal to 145/200 = 72.5% of the network bandwidth. And the corresponding cost of f is almost equal to 0.061.

Here we want to compare our solution with PCAP in Table 4 from [13] , since we use different definitions to denote throughput of AP and throughput of network, we have to use the indirect method to illustrate some issues. According to [13] , we can transfer and calculate their J of AP and their throughput percentage of network bandwidth:

So their

And we have:

Then their throughput percentage of network bandwidth is:

In Figure 1, our J of AP is superior to theirs in (22). From the throughput point of view, our throughput percentage of network bandwidth is 72.5% > 61.3% in (24), so our method is better than PCAP. But from the fairness of users (J of user) point of view, PCAP is better than ours since 0.75 < 0.9 in (22).

According to (17), we convert (22) into our function F, we have:

So the overall performance depends on the requirement of administrators, which indicator they most concern. We define the value of “F” as the overall performance criteria of algorithm, note smaller “F” is better, and then from (25) and (26) we know that our SCV model is much better than PCAP. The above comparison analysis result is in Table 5.

Theoretically, our design of target function “F” in (17) is more simple and rational than PCAP algorithm, since we joint consider the J of user and throughput (represented by J of AP), we regard them as two variables to reflect our topic which is a balance problem. While the target of PCAP is the throughput, the authors used two sub-algorithms to achieve J of AP only, and then got their by-product: J of user.

Technically, our SCV math model is a door that leads this problem to AI algorithms. The clear target function “F” is easy to be applied to other AI algorithms, while PCAP cannot.