Ant Colony Optimization for Social Utility Maximization in a Multiuser Communication System

Abstract

In a multiuser communication system such as cognitive radio or digital subscriber lines, the transmission rate of each user is affected by the channel background noise and the crosstalk interference from other users. This paper presents an efficient ant colony optimization algorithm to allocate each user’s limited power on different channels for maximizing social utility (i.e., the sum of all individual utilities). The proposed algorithm adopts an initial solution that allocates more power on the channel with a lower background noise level. Besides, the cooling concept of simulated annealing is integrated into the proposed method to improve the convergence rate during the local search of the ant colony optimization algorithm. A number of experiments are conducted to validate the effectiveness of the proposed algorithm.

1. Introduction

In a modern communication system such as cognitive radio or digital subscriber lines (DSL), multiple users share the same frequency band and how to mitigate interference is a major design and management concern. The frequency division multiple access (FDMA) mechanism is a standard approach to eliminate interference by dividing the spectrum into multiple tones and preassigning them to the users on a nonoverlapping basis. However, this approach may lead to high system overhead and low bandwidth utilization. Therefore, how to allocate spectrum dynamically to enhance overall system utility in a multiuser communication system has become a topic of intensive research in the signal processing and digital communication community.

From the optimization perspective, the study of the dynamic spectrum management problem can be divided into three kinds of categories [1, 2]: noncooperative Nash equilibrium, cooperative social utility maximization, and competitive economy equilibrium. Several approaches were proposed to compute a Nash equilibrium solution such as iterative waterfilling algorithm (IWFA) [3, 4] and linear complementarity problem (LCP) [5]. Some algorithms [6–9] were developed to compute a globally optimal power allocation for the cooperative game. Walras [10] proposed the competitive equilibrium concept which dynamically balances by the supply and demand to model the market operation. Ye [2] proposed a competitive economy equilibrium solution that may achieve both social economic efficiency and individual optimality in dynamic spectrum management. Lin et al. [11] further observed that adjusting budget can meet each user’s physical power demand or balance all individual utilities in a competitive spectrum market. Xie et al. [13] proposed a decentralized tâtonnement process for adjusting the prices to achieve a competitive equilibrium. They showed that competitive equilibrium is the solution of a linear complementarity problem and can be computed efficiently.

It is difficult to optimize the overall system utility if only each user’s individual utility is improved. Yu et al. [14] proposed a method which refined the optimal spectrum management with less complexity and generalized the multiuser power management for orthogonal frequency division multiplex systems maximized the overall utility. Due to the nonconvexity of this problem, the CPU time required to solve this problem by the conventional mathematical programming techniques increases greatly as the number of users and channels increases. Luo and Zhang [1] presented a systematic study of the dynamic spectrum management problem, covering two key theoretical aspects: complexity and duality. They have determined the complexity status of the spectrum management problem under various practical settings as well as different choices of system utility functions and have identified subclasses which are polynomial time solvable. They proved that the sum-rate maximization problem is NP-hard even when there are only two users in the system. Ma et al. [15] proposed an efficient scheduling algorithm for concurrent transmissions with power control to maximize the spectrum utilization and thus the network throughput in broadband wireless networks. The original nonconvex problem is converted into a series of convex problems using a two-step approximation. Then a power and channel allocation with fairness algorithm is proposed to solve these problems. Simulation results showed that both network throughput and fairness are improved by their method compared to the previously developed scheduling algorithms.

Although the heuristic algorithms cannot guarantee global optimality, they have the advantages of robustness and easy implementation for complex problems. In this paper, we propose an efficient ant colony optimization algorithm to allocate each user’s limited power on different channels for maximizing social utility (i.e., the sum of all individual utilities). In order to improve the solution efficiency, the proposed algorithm adopts an initial solution that allocates more power on the channel with a lower background noise level. Besides, the cooling concept of simulated annealing is integrated into the proposed method to improve the convergence rate during the local search of the ant colony optimization algorithm. A number of experiments are conducted to validate the effectiveness and contribution of the proposed algorithm. Comparing with the simulated annealing approach, computational results indicate that the proposed algorithm spends less CPU time but finds a solution with a higher social utility. Comparing with mathematical programming solver LINGO, the differences in social utility obtained between the proposed algorithm and LINGO are within 0.5% for all cases, and the proposed algorithm obviously decreases CPU time for solving large-scale problems.

The rest of this paper is organized as follows. Section 2 gives an overview of the problem formulation of the social utility maximization problem as well as ant colony optimization techniques. Section 3 describes the proposed ant colony optimization algorithm. Section 4 presents the extensive experimental results and analysis of the proposed algorithm. Finally, concluding remarks are made in Section 5.

2. Background Information

2.1. Social Utility Maximization Problem

Consider a communication system consisting of 𝑚 users and 𝑛 channels indicated in Figure 1. Multiple users may use the same channel at the same time and cause interference to each other. Suppose the power allocated by user 𝑖 at channel 𝑗 is 𝑥_𝑖𝑗>0, the total power allocated by user 𝑖 is bounded by the power budget 𝑃_𝑖, ∑^𝑛_𝑗=1𝑥_𝑖𝑗≤𝑃_𝑖. To evaluate each user’s utility, a commonly recognized utility for user 𝑖 in communication is the Shannon utility [2, 16]: 𝑢_𝑖𝐱_𝑖,𝐱_𝑖=^𝑛_𝑗=1𝑥log1+^𝑖𝑗𝜎_𝑖𝑗+∑_𝑘≠𝑖𝑎^𝑖_𝑘𝑗𝑥_𝑘𝑗, (1) where variable 𝐱_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛) is the power allocation of user 𝑖 on each channel, variables in 𝐱_𝑖=(𝐱₁,…,𝐱_𝑖−1,𝐱_𝑖+1,…,𝐱_𝑚) are the power units allocated by all other users on each channel, parameter 𝜎_𝑖𝑗 denotes the normalized background noise power for user 𝑖 at channel 𝑗, and parameter 𝑎^𝑖_𝑘𝑗 is the normalized crosstalk ratio from user 𝑘 to user 𝑖 at channel 𝑗. Due to multiuser interference, user 𝑖’s utility evaluated by Shannon utility depends on not only its own power allocation on each channel but also those of others. This study aims to maximize the overall system throughput, which is the sum of individual rates in each frequency carrier. Since each user has a fixed transmit power budget 𝑃_𝑖, the social utility maximization problem (sum-rate maximization problem) can be formulated as follows:

Figure 1 

The power allocation problem in a multiuser communication system.

Model 1: maximize^𝑚𝑛_𝑖=1𝑗=1𝑥log1+^𝑖𝑗𝜎_𝑖𝑗+∑_𝑘≠𝑖𝑎^𝑖_𝑘𝑗𝑥_𝑘𝑗subjectto^𝑛_𝑗=1𝑥_𝑖𝑗≤𝑃_𝑖𝑥,𝑖=1,…,𝑚,_𝑖𝑗≥0,𝑖=1,…,𝑚,𝑗=1,…,𝑛, (2) where 𝑥_𝑖𝑗 are variables and the other are parameters.

Consider a communication system with two channels (𝑓₁ and 𝑓₂) and two users, each of them has a physical power constraint 𝑃₁=𝑃₂=1. Let the Shannon utility function for user 1 be 𝑥log1+¹¹1+𝑥₂₁𝑥+log1+¹²4+𝑥₂₂ (3) and one for user 2 be 𝑥log1+²¹2+𝑥₁₁𝑥+log1+²²4+𝑥₁₂, (4) and let the aggregate social utility be the sum of the two individual user utilities. The social utility maximization problem for the above system can be formulated as follows.

Model 2: 𝑥maximizelog1+¹¹1+𝑥₂₁𝑥+log1+¹²4+𝑥₂₂𝑥+log1+²¹2+𝑥₁₁𝑥+log1+²²4+𝑥₁₂subjectto𝑥₁₁+𝑥₁₂≤1,𝑥₂₁+𝑥₂₂𝑥≤1,₁₁,𝑥₁₂,𝑥₂₁,𝑥₂₂≥0. (5)

The Nash equilibrium of Model 2 is (𝑥₁₁,𝑥₁₂)=(1,0) and (𝑥₂₁,𝑥₂₂)=(1,0), where the social utility has value 0.3010. However, the solution for maximizing social utility is (𝑥₁₁,𝑥₁₂)=(1,0) and (𝑥₂₁,𝑥₂₂)=(0,1), where the social utility has value 0.3979. This paper aims to find the power allocation for each user to achieve the maximal social utility, that is, the highest system throughput. The above sum-rate maximization problem (Model 1) has been proven to be NP-hard even when only two users are available in the system. In this paper, we propose an efficient ant colony optimization algorithm to allocate each user’s limited power on different channels for maximizing social utility.

2.2. Ant Colony Optimization Techniques

Although a wide variety of deterministic optimization techniques are proposed to solve optimization problems, many of these methods have inherent disadvantages because they require continuous evaluation of derivatives and become prohibitively ill conditioned for large-scale problems [17]. Several heuristic methods inspired from physics, biology, and social science have been developed in recent years such as genetic algorithm, simulated annealing, particle swarm optimization, tabu search, neural nets, and ant colony systems. Ant colony optimization (ACO) which simulates the action of ants was first introduced by Dorigo [18]. ACO aims at mimicking the way almost blind ants establish the shortest route from their nests to the food source and back. It was observed that a moving ant deposits some pheromone on the paths they traverse. New ants joining the search, attracted by the pheromone, further reinforce the path. With time, a series of feasible routes are traversed by teams of ants. As the pheromone evaporates, solutions with no substantial travelers will have difficulty in maintaining their pheromone trails and infeasible routes are gradually eliminated [19]. The complete surveys of applications, models, and algorithmic advances for ACO through 2006 are provided by Blum [20], Cordon et al. [21], Dorigo et al. [22], Dorigo and Stützle [23], and Lim et al. [19].

The ant colony algorithm was originally used for solving combinatorial optimization problems including the traveling salesman problem. Then it has been applied to other optimization problems such as quadratic assignment, vehicle routing, sequential orderings, job shop scheduling, power system optimization, telecommunication network routing, and logic circuit design [17, 19]. Although ACO has been proven to be one of the best metaheuristics in some combinatorial optimization problems, the application to the real-parameter optimizations appears more challenging, since the pheromone laying method is not straightforward. The algorithms for continuous function optimization were first developed by Bilchev and Parmee [24]. Attention in these works was devoted only on local search processes. Wodrich and Bilchev [25] extended and modified the earlier algorithm to make it an effective global optimization procedure for continuous problem spaces. Later, several approaches [26–31] have been proposed for continuous optimization in recent years. The complete surveys and comparisons of using ACO for continuous cost function optimization can be found in de França et al. [26] and Korošec and Šilc [28].

3. The Proposed Algorithm

This study proposes an efficient ant colony optimization algorithm to allocate each user’s limited power on different channels for maximizing social utility. To improve the solution efficiency, the proposed algorithm uses an initial solution that allocates more power on the channel with a lower background noise. Moreover, the cooling concept of simulated annealing is incorporated into the proposed method to improve the convergence rate during the local search of the ant colony optimization algorithm. The flowchart of the proposed simulated annealing-based ant colony optimization (SAACO) algorithm is shown in Figure 2.

Figure 2 

Flowchart of the proposed SAACO algorithm.

Let 𝑣 be the number of ants in the colony; the location of ant 𝑘, 𝑘=1,…,𝑣 at time 𝑡 represents a solution 𝑋^𝑡_𝑘={𝐱^𝑡_𝑘,1,…,𝐱^𝑡_𝑘,𝑚} of the social utility maximization problem, where 𝐱^𝑡_𝑘,𝑖=(𝑥^𝑡_𝑖1,𝑥^𝑡_𝑖2,…,𝑥^𝑡_𝑖𝑛) and 𝑥^𝑡_𝑖𝑗 is the power units allocated by user 𝑖 at channel 𝑗 at time 𝑡. More detail of the proposed SAACO algorithm in this study is summarized as follows.

(1) Initialization. 𝑣 ants are placed in solution space. According to the Shannon utility function, allocating power on the channel with a lower background noise level contributes more to the user utility. The proposed algorithm considers an initial solution based on the background noise level. One half of ants adopt an initial solution considering background noise level, another half generate the initial solution randomly.

(2) Pheromone Updating Rule. Since this study aims to find a solution with maximal social utility (i.e., ∑^𝑚_𝑖=1∑^𝑛_𝑗=1log(1+𝑥_𝑖𝑗/(𝜎_𝑖𝑗+∑_𝑘≠𝑖𝑎^𝑖_𝑘𝑗𝑥_𝑘𝑗))), the pheromone laid on each location is set in proportion to the value of social utility. That is, the pheromone level of visited location by ant 𝑘 at time 𝑡, ph^𝑡_𝑘, is updated as follows: ph^𝑡_𝑘=1−𝑝ph_𝑘^𝑡−1+𝑔𝑈𝑋^𝑡_𝑘, (6)  where 𝑝 is the pheromone evaporation rate, 𝑔 is a proportional ratio, and 𝑈(𝑋^𝑡_𝑘) is the social utility of solution 𝑋^𝑡_𝑘. The ant deposits more pheromone on a location with a higher social utility value. The pheromone level is updated only when a better solution is found.

(3) State Transition Rule. After objective function (social utility) evaluation, the best solution (i.e., the highest pheromone) location uncovered by the ants is used to guide the search in subsequent generations. Assume that the ant with the highest social utility deposits the most pheromone at time 𝑡, BestPh^𝑡=max_𝑘{ph^𝑡_𝑘}, on its location, then each ant 𝑘 on the location with its pheromone level Ph^𝑡_𝑘 uses the following state transition rule to determine the next move: 𝑋_𝑘^𝑡+1=𝑋^𝑡_𝑘+min_step∗𝑑,if𝑤≤𝐺₀,𝑋exploitation_𝑡𝑘+max_step∗𝑑,if𝑤>𝐺₀,exploration, (7) where 𝑤=(BestPh^𝑡−Ph^𝑡_𝑘)/BestPh^𝑡, 𝑑 is a random variable between 0 and 0.5, and min_step, max_step, and 𝐺₀ are prespecified constants. To facilitate both exploitation and exploration in the proposed method, an ant searches locally when 𝑤 is found to be lower than or equal to the user-specific threshold 𝐺₀ and searches globally when 𝑤 is greater than 𝐺₀.

(4) Solution Adjustment. After an ant uses the transition rule to move next, the solution on next location is verified so that it meets the problem constraint (∑^𝑛_𝑗=1𝑥_𝑖𝑗≤𝑃_𝑖). If the problem constraint is violated, the proposed method performs solution adjustment to make the next solution a feasible solution.(i)If 𝑥_𝑖𝑗 is not in the domain (𝑥_𝑖𝑗,𝑥_𝑖𝑗), where 𝑥_𝑖𝑗 and 𝑥_𝑖𝑗 are lower bound and upper bound of 𝑥_𝑖𝑗, then adjust 𝑥_𝑖𝑗 in its range by the following rule: 𝑥_𝑖𝑗=𝑥_𝑖𝑗,if𝑥_𝑖𝑗>𝑥_𝑖𝑗,𝑥_𝑖𝑗,if𝑥_𝑖𝑗<𝑥_𝑖𝑗,𝑥_𝑖𝑗,otherwise. (8)(ii)If ∑^𝑛_𝑗=1𝑥_𝑖𝑗>𝑃_𝑖, then randomly select one of the following three conditions for adjusting 𝑥_𝑖𝑗.(a) 𝑥_𝑖𝑗 are adjusted proportionally such that the total power allocation does not exceed its limitation 𝑃_𝑖, that is, 𝑥_𝑖𝑗=(𝑥_𝑖𝑗/∑^𝑛_𝑗=1𝑥_𝑖𝑗)𝑃_𝑖.(b) Remain the power allocation for the channel with maximal power allocation, and allocate 𝑃_𝑖−max_𝑗{𝑥_𝑖𝑗} on a randomly selected channel.(c) Allocate all power on the channel with the lowest background noise level.(iii)If ∑^𝑛_𝑗=1𝑥_𝑖𝑗<𝑃_𝑖, then allocate unused power 𝑃_𝑖−∑^𝑛_𝑗=1𝑥_𝑖𝑗 to the channel that gets the lowest power allocated.

(5) Local Search Range Reduction. If there is no improvement in solution quality after a user-defined number of iterations in the searching process, the searching process is stable and the proposed method incorporates the concept of simulated annealing to reduce the local search range from min_step to min_step∗𝑅_step, where 𝑅_step is between 0.8 and 0.99.

(6) Stopping Criteria. After min_step is below the user-specific threshold 𝑆𝑞, terminate the searching process.

4. Experimental Results and Discussions

In this section, the effectiveness of the proposed SAACO method is validated using experimental data. All experiments are implemented using C++ language under the circumstance of Windows XP on an Intel CPU P4 1.7 G computer with 1 GB memory. In all simulated cases, each user’s power constraint 𝑃_𝑖=1, the channel background noise level 𝜎_𝑖𝑗 are chosen randomly from the interval [1,9], and the normalized crosstalk ratio 𝑎^𝑖_𝑘𝑗 are chosen randomly from the interval [0.1,0.9].

4.1. Parameters Settings

Since different parameter settings significantly affect the performance of the proposed SAACO algorithm, a number of experiments are run to determine suitable parameter settings for the algorithm listed in Table 1.

Table 1 

Parameter setting for the proposed SAACO algorithm.

4.2. Comparison between SAACO and SA

We compare the proposed SAACO approach with the simulated annealing (SA) approach proposed by Lin et al. [12] in the case of 𝑚 users (𝑚=10,20,30,40) and 𝑛 channels (𝑛=10,20,30,40,50). Figures 3 and 4 compare social utility and CPU time, respectively, by SAACO and SA. We find that the proposed SAACO algorithm spends less CPU time but obtains better solutions with higher social utilities than the SA approach Lin et al. [12].

Figure 3 

Comparison of social utility by SAACO and SA Lin et al. [12].

Figure 4 

Comparison of CPU time by SAACO and SA Lin et al. [12].

4.3. Comparison between SAACO and LINGO

In this section, some experiments are conducted to compare the proposed SAACO approach with the optimization software LINGO [32]. First we consider small-scale problems of 𝑚 users (𝑚=10,20) and 𝑛 channels (𝑛=10,20,30,40,50). Ten problems are randomly generated for each group. Table 2 compares average performance of the proposed SAACO approach and LINGO. To compare SAACO and LINGO more clearly, we normalize the data and plot it in Figures 5 and 6. In Figure 5, we observe that the difference between social utility by LINGO and by SAACO is between +0.5% and −0.2%. Besides, the proposed SAACO approach uses less CPU time than LINGO in obtaining an optimal solution when 20 users and over 20 channels exist in the communication system.

Table 2 

Comparisons of CPU time and social utility between the proposed SAACO approach and LINGO.

Figure 5 

Comparison of social utility by SAACO and LINGO.

Figure 6 

Comparison of CPU time by SAACO and LINGO.

To demonstrate effectiveness of the proposed method in large-scale problems, we compare the proposed SAACO approach with LINGO in the cases of 𝑚 users (𝑚=10,20) and 𝑛 channels (𝑛=100,128,256). Three problems are randomly generated for each group. Table 3 compares average performance of SAACO and LINGO measured by the resulting social utility and the required CPU time. To compare SAACO and LINGO more clearly, we normalize the data and plot it in Figures 7 and 8. Figure 8 indicates that the proposed SAACO algorithm performs better than LINGO in CPU time as the number of users and channels increases. Compared with LINGO, the proposed SAACO approach uses roughly 50% of CPU time to obtain the solution in the case of 20 users and 256 channels. Although the required CPU time by the proposed SAACO approach is much less than LINGO, the proposed SAACO approach can get a better social utility or an approximate social utility not less than 1 percent from the solution reported by LINGO as indicated in Figure 7.

Table 3 

Comparisons of CPU time and social utility between the proposed SAACO approach and LINGO in large scale problem.

Figure 7 

Comparison of social utility by SAACO and LINGO in large-scale problems.

Figure 8 

Comparison of CPU time by SAACO and LINGO in large-scale problems.

5. Conclusions

With the rapid development of wireless networks and devices, the demand for spectrum resource is increasingly growing. Therefore, how to allocate spectrum dynamically to enhance total system throughput in a multiuser communication system has become an important issue in recent years. Due to the nonconvexity of this problem, the CPU time required to solve this problem by the conventional mathematical programming techniques increases greatly as the number of channel and user increases. This paper proposes an efficient ant colony algorithm for allocating spectrum to maximize social utility. Computational results show that the proposed algorithm significantly outperforms the simulated annealing approach in solution efficiency and solution quality in most cases. Comparing with the mathematical programming solver LINGO in solving large-scale problems, the proposed SAACO algorithm spends much less CPU time obtaining an approximate social utility close to the social utility reported from LINGO.

Acknowledgment

The research is supported by Taiwan NSC Grants NSC 101-2410-H-158-002-MY2 and NSC 102-2410-H-027-012-MY3.