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In this paper, we propose a new packet routing strategy that incorporates memory information for reducing congestion in communication networks. First, we study the conventional routing strategy which selects the paths for transmitting packets to destinations using the distance information and the dynamical information such as the number of accumulating packets at adjacent nodes. Then, we evaluate the effectiveness of this routing strategy for the scale-free networks. From results of numerical simulations, we conclude that this routing strategy is not effective when the density of the packets increases due to the impermeability of the communication network. To avoid this undesirable problem, we incorporate memory information to the routing strategy. By using memory information effectively, packets are spread into the communication networks, achieving a higher performance than conventional routing strategies for various network topologies, such as scale-free networks, small-world networks, and scale-free networks with community

The transmission rate of data packets in communication networks is drastically increasing due to the widespread use of mobile devices. If the number of packets increases sufficiently, then packet congestion occurs in communication networks. Transmissions of packets to their destinations are then blocked, and in the worst case, the packets may be removed from the network by a buffer constraint or a time-to-live constraint. Therefore, the study and analysis of packet congestion has been given significant attention in recent years [

The shortest path algorithms adopted in real-world systems, such as the Dijkstra [

Understanding the dynamics of packet transmission on networks is also an important task in analyzing the onset of traffic congestion. From this viewpoint, Ohira et al. [

On the other hand, the development of efficient routing methods that drastically reduce the congestion of a network presents another important task for carrying a large volume of data traffic. Recent studies concerning the design of routing strategies have evolved according to two basic ideas. The first is to select paths for transmitting packets based on only local information of the network, such as degree information or the number of packets waiting at adjacent nodes [

From the above viewpoints, our neural-based routing method [

This paper is organized as follows. In Section 2, we first describe how the routing method is constructed based on mutually connected neural networks, and then evaluate the MCNN routing method using the packet congestion rate. The modified version that utilizes transmitting information is then described in Section 3. Furthermore, our proposed neural-based routing method is evaluated in comparison to other conventional neural-based routing methods in Section 4. Finally, we conclude this paper in Section 5.

First, we explain how to construct the original MCNN routing method [

The communication network model has N nodes, where the ith node is connected to

In the MCNN routing method, each node determines one of the adjacent nodes as the transmitting node at each iteration by minimizing the following energy function:

where

In Equation (1), the second term expresses the load distribution of adjacent nodes, and the third term is the distance for a packet at the ith node to the destination. The last term guarantees that only a single neuron fires in each neural network.

The internal state of the mutually connected neural network is updated using the method proposed by Hopfield and Tank [

Moreover, when the ith and lth nodes are adjacent, the internal state of the ilth neuron

Note that Equation (3) is obtained by partially differentiating Equation (1) with respect to

From Equation (2) we obtain that

Thus,

Equation (5) indicates that the energy function E always decreases when

In this method, the firing of the neuron is determined using the threshold

The lth adjacent node of the ith node is selected as the transmitting node if it has the shortest distance to the destination of the packet and a smaller number of packets in its buffer among all of adjacent nodes.

As mentioned above,

Next, we evaluate the effectiveness of the MCNN routing method. Because real communication networks have the scale-free property [

connected to the existing nodes in the network with probability

the ith node

We conducted numerical experiments according to the following procedure. We generated p packets whose sources and the destinations are randomly determined at each iteration. Then, link selection is simultaneously conducted at every node, and the packets that are ready to exit each node are transmitted to adjacent nodes. A packet is removed from the network when it reaches its destination. In addition, the transmission of a packet is canceled if the buffer of the node to which the packet should be transmitted is full.

We set the number of iterations, T, to 3000, and the buffer size of each node, B, to 100. We set

We adopted the packet congestion rate [

In Equation (6), C is the packet congestion rate, T is the number of iterations, and

In Equation (7),

To analyze why the MCNN routing method displays a poor congestion rate if the number of flowing packets increases, we next measure the congestion level of nodes in the computer network. To evaluate the congestion levels of the nodes, we employ a computer network model in which a fixed number of packets are flowing [

In these simulations, we set the buffer size of every node to 100 and limited the number of packet movements between nodes to 20. The congestion level of the ith node at the tth iteration,

In Equation (8),

We have clarified that the conventional neural-based routing method [

One promising approach to overcome these problems is to consider all of the queuing information for every node. However, this requires heavy processing costs because such information constantly changes depending on the transmitting states of the packets. Another effective way might be to utilize limited queuing information for the nodes that lie on the shortest distance path between the packet and its destination. Hereafter, we refer to this limited queuing information as transmitting information. From this viewpoint, we propose a neural-based routing method that uses transmitting information by modifying the original energy function defined by Equation (1) as follows:

Here, we define the subset of nodes

where

In Equation (9),

In Equation (9), the first term expresses the transmitting information of nodes on the shortest distance path through the lth adjacent node, and the second term is the distance of a packet from the lth adjacent node to the destination. The third term guarantees that only one neuron fires in the neural network. If the distance to the destination of a packet from the lth adjacent node is the shortest among all of the adjacent nodes, and the transmitting information of the shortest distance path to the destination starting through the lth adjacent node is the lowest among all of the paths, then the lth adjacent node is selected as the transmitting node.

The cost of calculation for the conventional neural-based routing method [

because it only calculates packet transmitting information along the shortest distance paths, without using

We begin this section by evaluating our proposed routing method compared with the original MCNN routing method. Hereafter, we call the conventional MCNN routing method as the NN-O method in order to di- stinguish it from other neural-based routing methods. For these simulations, we adopted the same experimental assumptions and the same parameter settings as those in Section 2. We set

Next, the performance of the proposed routing method is evaluated against three other conventional neural network based routing methods. The first is the original neural-based routing method (NN-O) [

In the neural-based routing method with reinforcement learning [

In Equation (10),

1)

2) The ith node transmits a packet at the tth iteration using

3)

4) Go to 2.

In Equation (10),

In this paper, we set the values of

In the neural-based routing method including stochastic effects [

In these simulations,

Next, we evaluate the average numbers of arriving and lost packets, using a computer network model in which a fixed number of the packets flow [

where

In this paper, we use the packet congestion rate to evaluate a proposed packet routing method that employs mutually connected neural network and transmitting information. The packet congestion rate is mainly employ-

ed to clarify performance in terms of congestion avoidance as the number of flowing packets gradually increases in a communication network. First, we showe that the original neural-based routing method is unable to alleviate packet congestion, because packets accumulate at hub nodes of large degree in the network, and packet con- gestion is then distributed into the network. To overcome this problem, we propose a neural-based routing method that exploits transmitting information, and evaluate our proposed method against conventional neural- based routing methods, which incorporate either reinforcement learning or stochastic effects. From the results of the numerical simulations, we can conclude that our proposed method is able to reduce the packet congestion rate more effectively than the other neural-based routing methods. We also evaluate our proposed method in terms of the average numbers of arriving and lost packets by using a computer network model in which a fixed number of packets flow. We are able to confirm that our proposed method can maintain a higher arrival rate, or lower lost rate, than the other neural-based routing methods. From these results, we conclude that our proposed neural-based routing method achieves reliable communication by avoiding the congestion of packets through the effective use of transmitting information.

For this study, we assumed that information regarding the number of packets accumulated on a path could instantaneously be obtained by all nodes. In future works, we will consider how to deal with delays in this information in order to implement the proposed neural-based routing method into real-world systems.

The research of T.K. was partially supported by a Grant-in-Aid for Young Scientists (B) from JSPS (No. 16K21327).

Takayuki Kimura,Tohru Takamizawa,Takafumi Matsuura, (2016) Neural-Based Routing Method for Alleviating Congestion in Complex Networks. American Journal of Operations Research,06,343-354. doi: 10.4236/ajor.2016.64032