Optimization Algorithm of Control Channel Selection for Wireless Networks

Abstract

Control channel is used to transmit protocol or signal information between wireless network nodes and is a key component of wireless network. Compared with data information, protocol or signal information is usually much less, so the spectrum bandwidth requirement of control channel is also much less than that of data channel. In order to optimize the usage of the limited spectrum resources, this paper focuses on the issue of control channel selection. We propose a greedy algorithm which minimizes the total spectrum bandwidth of the set of control channels. Theoretical analysis proves that the proposed algorithm can achieve the optimal set of control whose sum of the spectrum bandwidth is the minimum. Simulation results also show that the proposed algorithm consumes less spectrum resources than other algorithms in the same wireless network environment.

1. Introduction

Recently, with the rapid development of the wireless communication technology, wireless network has been the most important infrastructure in the daily life. In order to organize each node in wireless networks to work together, control channel is used to transmit protocol or signal information between wireless network nodes and is a key component of wireless network. Control channel is usually divided into two categories: in-band control channel and out-of-band control channel. The in-band control channel means that protocol or signal information is transmitted in the same channel with data information. On the contrary, out-of-band control channel is different from the channel used to transmit data information. Channel assignment [1–4] is a very important issue for reducing interference and improving throughput in wireless networks, and the generation and selection of control channel are the base for solving this issue. In this paper, we study the issue of optimal selection of out-of-band control channel.

In wireless networks, each node needs to select one channel from its available channels as the control channel to transmit protocol or signal information with its neighbor nodes. To transmit protocol or signal information, each control channel needs to occupy spectrum bandwidth. The problem is how to select the set of control channels for the whole wireless network which minimizes the total spectrum bandwidth of the control channels. If all nodes in wireless networks have the same available channels, the channel with the minimum spectrum bandwidth will be selected as the control channel. However, in real wireless networks, the available channel set of each node may be different. For example, in cognitive wireless networks, due to the geographical location difference and activity of primary user, the available channel set of each node is quite different, and the spectrum bandwidth of each available channel is also different. In this paper, we set the spectrum bandwidth as channel weight and propose a greedy algorithm to solve the problem of optimal selection of control channel under the constraint of different nodes with different available channel sets. The proposed algorithm is composed of two parts. In the first part of the algorithm, we delete the wireless network node whose accessible channel set completely includes that of some other nodes. Then, in the second part of the algorithm, we iteratively choose the channel which has the minimum spectrum bandwidth in an accessible channel set containing the maximum spectrum bandwidth in the current iteration.

The problem of the optimal control channel selection and related research methods which are also akin to self-organization and evolutionary game theory have been paid attention widely [5–20]. Zhong has proposed an optimal algorithm for this problem [21]. But it assumes that all of the channels have the same spectrum bandwidth. However, in reality, different channels maybe have different spectrum bandwidths. To address this issue, we design a greedy algorithm to minimize the total spectrum bandwidth of the control channels. Theoretical analysis shows that our proposed algorithm can achieve the optimal control channels, where sum of the spectrum bandwidth is the minimum with the time complexity of 𝑂(𝑛²).

2. Preliminaries

(1) To solve the selection of the control channel, we make the following assumptions. (a) Each of the wireless network nodes has a set of the available channels. In this paper, the spectrum bandwidth occupied by the channel is monotonous. We assume that, as long as the index of the channel is bigger, the spectrum bandwidth it occupies is bigger. (b) Wireless network nodes can transmit protocol or signal information with one another through their common control channels. Note that our objective is to find the optimal set of control channels with the sum of spectrum bandwidth as small as possible.  Let 𝐷 be the set of the wireless network nodes; that is, 𝐷=1,2,…,𝑛, and let 𝑇 be the set of available channels; that is, 𝑇=𝐿,𝐿+1,…,𝑈 (here 𝐿<𝑈). For each node 𝑖∈𝐷, the channels that can be accessed are 𝑙_𝑖 through 𝑢_𝑖; that is, 𝑙_𝑖,𝑙_𝑖+1,…,𝑢_𝑖. A function for the spectrum bandwidth of each channel 𝜔∶𝑇→𝑄 has the monotonously increasing property. (2) The objective is to find a subset 𝐶 of control channels 𝑇 with the minimum sum of spectrum bandwidths 𝐵; that is, forall𝑖∈𝐷, ∃𝑐∈𝐶, such that 𝑙_𝑖≤𝑐≤𝑢_𝑖 holds and 𝐵=∑_𝑐∈𝐶𝜔(𝑐) is minimum.

3. Algorithm Overview

We describe the algorithm in this section. For an arbitrary instance, the wireless network nodes 𝐷 and the channels set 𝑇 with the associated function 𝜔 are all given. The pseudocode of the algorithm is presented as Algorithm 1.

Optimization Algorithm of Control Channels Selection Input: 𝐷,𝑇,𝜔∶𝑇→𝑄+; Output: The optimal control channels set 𝐶 with the minimum sum of bandwidth 𝐵; (1)   For ∀𝑖,𝑗∈𝐷, 𝑖≠𝑗; (2)    If 𝑙𝑖≤𝑙𝑗≤𝑢𝑗≤𝑢𝑖 then (3)       𝐷←𝐷−{𝑖}; (4)    End If; (5)   End For; (6)   𝐶←∅ and the initial sum of the spectrum bandwidth 𝐵∶=0; (7)   While 𝐷≠∅ do (8)   𝑖←argmax𝑖∈𝐷{𝑢𝑖}; (9)   𝐶←𝐶∪{𝑙𝑖}, 𝐷←𝐷−{𝑖} and 𝐵∶=𝐵+𝜔(𝑙𝑖); (10)   For ∀𝑗∈𝐷 and 𝑗≠𝑖; (11)    If 𝑢𝑗≥𝑙𝑖 then (12)      𝐷←𝐷−{𝑗}; (13)    End If; (14)   End For; (15) End While;

Algorithm 1  

4. Analysis of the Algorithm

For showing that the output of the algorithm is optimal, we first give the following lemma.

Lemma 1. When the algorithm terminates, 𝐶 is a subset of 𝑇 such that, for any 𝑘∈𝐷, there exists 𝑐∈𝐶, s.t., 𝑙_𝑘≤𝑐≤𝑢_𝑘.

Proof. Let the value of 𝐷 be 𝐷₀ after Steps (1)--(5). We first prove that, for any 𝑘∈𝐷₀, there exists 𝑐∈𝐶 such that 𝑙_𝑘≤𝑐≤𝑢_𝑘. Note that, for any 𝑘∈𝐷₀, there is an iteration which removes 𝑘 from 𝐷₀. Then there are two possible cases in this iteration from Steps (8)--(14). (1)If 𝑖=𝑘, 𝑘 will be removed from 𝐷₀ and 𝑙_𝑘 is added to 𝐶. Hence 𝑙_𝑖=𝑙_𝑘∈𝐶. (2)If 𝑖≠𝑘, then we have 𝑢_𝑘≥𝑙_𝑖 since 𝑢_𝑘 is removed from 𝐷₀ in this iteration according to the algorithm. Furthermore, we should have 𝑙_𝑘≤𝑙_𝑖, otherwise we would have 𝑙_𝑖≤𝑙_𝑘≤𝑢_𝑘≤𝑢_𝑖, since 𝑖←argmax_𝑖∈𝐷{𝑢_𝑖}, where 𝐷 is the left nodes in the current iteration. However, in such a case 𝑖 should have been removed in Steps (1)--(5), which is a contradiction. To sum up, we have 𝑙_𝑘≤𝑙_𝑖≤𝑢_𝑘 and 𝑙_𝑖∈𝐶.
To conclude the above two possible cases, we get that, for any 𝑘∈𝐷₀, we have 𝑐∈𝐶 such that 𝑙_𝑘≤𝑐≤𝑢_𝑘. For any 𝑘∈𝐷−𝐷₀, there must exist 𝑖∈𝐷₀ such that 𝑙_𝑘≤𝑙_𝑖 and 𝑢_𝑖≤𝑢_𝑘 from Steps (1)--(5). According to the above conclusion, we get that there must exist 𝑐∈𝐶 such that 𝑙_𝑘≤𝑙_𝑖≤𝑐≤𝑢_𝑖≤𝑢_𝑘. This completes the proof of the lemma.

Then the main result of the paper is presented as follows.

Theorem 2. The subset 𝐶 of 𝑇 is the optimal control channels with the minimum spectrum width and the time complexity of the algorithm is 𝑂(𝑛²).

Proof. We first claim that 𝐶 is a subset of 𝑇 with the smallest cardinality such that, for all 𝑖∈𝐷, there exists 𝑐∈𝐶, such that 𝑙_𝑖≤𝑐≤𝑢_𝑖.
From the proof of Lemma 1, 𝐷₀⊆𝐷 and the algorithm outputs a subset 𝐶 such that, for all 𝑘∈𝐷₀, there exists 𝑐∈𝐶 such that 𝑙_𝑘≤𝑐≤𝑢_𝑘. Considering any subset 𝐶_𝑥 of 𝑇 such that, for all 𝑖∈𝐷₀, there exists 𝑐∈𝐶_𝑥 such that 𝑙_𝑖≤𝑐≤𝑢_𝑖. Now sort the elements of 𝐶_𝑥 in decreasing order: 𝑒₁,𝑒₂,…,𝑒_|𝐶𝑥|.
By induction, we show that 𝑒_𝑖≥𝑓_𝑖, forall𝑖=1,2,…,|𝐶_𝑥|, where 𝑓_𝑖 is the element added into 𝐶 in the 𝑖th iteration of the “While” cycle in the algorithm.
From the algorithm we define 𝑓₁=𝑙_𝑖1, where 𝑖₁=argmax_𝑖∈𝐷0𝑢_𝑖. Provided that 𝑒₁<𝑓₁, then, for all 𝑖>1, 𝑒_𝑖<𝑓₁ according to the algorithm. Hence, there is no 𝑐∈𝐶_𝑥 such that 𝑙_𝑖1≤𝑐≤𝑢_𝑖1, which is a contradiction. Then 𝑒₁≥𝑓₁. Suppose that 𝑒_𝑘≥𝑓_𝑘, where 𝑘≥1. Let the value of 𝐷₀ be 𝐷_𝑘 at the end of the 𝑘th iteration. Therefore, 𝑓_𝑘+1=𝑙_𝑖𝑘+1, where 𝑖_𝑘+1=argmax_{𝑖∈𝐷𝑘}𝑢_𝑖. Suppose that 𝑒_𝑘+1<𝑓_𝑘+1; then we have 𝑒_𝑘+1<𝑙_𝑖𝑘+1. On the other hand, for all 𝑗≤𝑘, then 𝑒_𝑗≥𝑒_𝑘≥𝑓_𝑘=𝑙_𝑖𝑘. According to the algorithm, we have 𝑙_𝑖𝑘>𝑢_𝑖𝑘+1 (otherwise 𝑖_𝑘+1 would have been removed from 𝐷₀ in the 𝑘th iteration). Then, for all 𝑗≤𝑘,𝑒_𝑗>𝑢_𝑖𝑘+1. In summary, we have that there is no 𝑒_𝑗∈𝐶_𝑥 such that 𝑙_𝑖𝑘+1≤𝑒_𝑗≤𝑢_𝑖𝑘+1, which is a contradiction. Thus, So for any 𝑖=1,2,3,…,|𝐶_𝑥|, we have 𝑒_𝑖≥𝑓_𝑖. Therefore, assuming that |𝐶_𝑥|<|𝐶|, due to 𝑒₁>𝑓₁,𝑒₂>𝑓₂,…,𝑒_|𝐶𝑥|>𝑓_|𝐶𝑥|, we have 𝑓_|𝐶𝑥|=𝑙_{𝑖𝑥||𝐶}>𝑢_{𝑖𝑥|𝐶|+1}. Consequently, there is no 𝑐∈𝐶_𝑥 such that 𝑙_{𝑖𝑥|𝐶|+1}≤𝑐≤𝑢_{𝑖𝑥|𝐶|+1}, which is a contradiction. Thus |𝐶_𝑥|≥|𝐶|. The claim holds. And since the spectrum bandwidth occupied by the channel is monotonous, we have 𝜔(𝑒₁)≥𝜔(𝑓₁), 𝜔(𝑒₂)≥𝜔(𝑓₂),…,𝜔(𝑒_|𝐶|)≥𝜔(𝑓_|𝐶|), and then ∑^𝐶𝑥_𝑖=1𝜔(𝑒_𝑖)≥∑^𝐶_𝑖=1𝜔(𝑒_𝑖)≥∑^𝐶𝑥_𝑖=1𝜔(𝑓_𝑖). So the algorithm outputs the optimal control channels 𝐶 with the minimum spectrum width when it terminates.
At each iteration of Steps (1)--(5) in the algorithm, for each wireless network node, there are (𝑛−1) times comparison. So there are a total of 𝑛(𝑛−1) times.
Obviously, the number of iterations in Step (7) is not more than 𝑛. And it takes 𝑂(𝑛) time to find argmax_𝑖∈𝐷{𝑢_𝑖} in Step (8) and 𝑂(𝑛) time for to find comparison in Steps (10)--(13).
To conclude, the time complexity of the algorithm is 𝑂(𝑛²). This completes the proof.

5. Evaluation

In this section, we use the numerical simulation method to evaluate the proposed algorithm compared with the proposed algorithm by Zhong [21].

The simulation topology is as shown in Figure 1. The set of available channels of each node at time 𝑡 is also shown in Figure 1, and the set will change over time. The left column of each table shows the index of each channel, and the right column shows the spectrum bandwidth of each channel. For easy of explanation, we set the total number of available channels to be 6, and index all channels by the ascending order of spectrum bandwidth of each channel, which means that the bigger the index of the channel, bigger the spectrum bandwidth of the channel.

Figure 1 

Topology of simulation networks.

In the simulation, for each run, we use the random number generator to generate the random set of available channels for each node. The simulation executes 60 times of runs in total, sums all simulation results (number of control channels, total spectrum bandwidth of control channels) of each run, and calculates the average values.

Figure 2 shows that number of control channels between our proposed algorithm and Zhong’s algorithm is same. The reason behind this result is that these two algorithms use similar method to find the minimum number of control channels. However, Figure 3 shows that there is a big difference of total spectrum bandwidth of control channels between these two algorithms. The results show that our proposed algorithm not only tries to find optimal number of control channels but also reduces the total spectrum.

Figure 2 

Comparison of number of control channels.

Figure 3 

Comparison of total spectrum bandwidth of control channels.

6. Concluding Remarks

In this paper, we design an optimal algorithm to find the set of control channels with the minimum spectrum bandwidth for wireless networks. We also present simulation results to show that the proposed algorithm consumes less spectrum resources than Zhong’s algorithm in the same wireless network environment. Furthermore, in some other applications, such as spectrum allocation which usually sorts all channels by descending order, the function for spectrum bandwidth of each channel is monotonously decreasing. It is easy to modify the proposed algorithm and obtain similar results for the case.

Acknowledgments

The authors thank the referees for their helpful comments and suggestions. This work is supported by the National Science Foundation of China under Grant no. 61301159, Open Fund of Lab of Military Network Technology, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (13KJB1100188).