INTRODUCTION
In recent years with the rapid development of some delaysensitive realtime
network applications, the Internet needs to provide more reliable and stable
network services. Thus, delay characteristics of the network have been paid
more and more attention to. Network delay is influenced by multiple factors,
such as network topology, forwarding node and background traffic. These factors
change randomly over time which can reflect the current performance of the network
link (Paxson, 1999; Allman and Paxson,
1999). The use of network resources and performance trends can be obtained
by forecasting the dynamic changes of the network delay which can provide a
reference for balancing the network load and optimize network performance (Johari
and Tan, 2001).
Wong (1978), based on the traditional queuing analysis
theory, holds the view that there is no correlation between time intervals of
data transmission but the predicted results have a great difference with the
network measured data. The conclusion from these articles (Jiao
et al., 2006; Li and Millis, 2001; Yang
and Li, 2003) is that the accurate model of the time series based on network
delay can be established by AR or ARIMA model of low level which provides predictive
values with high accuracy. However, the model cannot meet the requirement of
dynamic prediction of network delay. Parlos (2002) and
Srikar (2004) have respectively carried out prediction
and analysis of the variation of network delay by using Multilayer Perceptron
(MLP) neural network with different algorithms. However, there are some questions
for the traditional neural network, for example, falling into the local extremum
easily and the limited capacity of extrapolation.
Compared with multilayer perceptron, RBF neural network overcomes many disadvantages of the traditional neural network which has its own unique characteristics in terms of network structure, learning algorithm and status updating rules. For this reason, RBF neural network is widely used in many aspects, such as nonlinear function approximation, time series analysis, pattern recognition, information processing, data classification, image processing and system modeling. The main contribution of this study is the design and realization of a model based on RBF neural network to predict the network delay. The model is trained by actual statistical data and then its effectiveness is analyzed and researched through contrasting the actual network delay value with the predicted delay value.
Delay characteristics: The transmission delay of the network constantly
changes with operating state of the entire network system. There are some factors
that have a great impact on network delay, for example, the amount of data in
the network and processor performance of the nodes. In order to find out the
characteristics of network delay we should measure network delay continuously
for a long time. After carrying out statistical analysis of a large number of
experimental data, we summarize that (Zhang, 2003):
• 
The delay values will be very different at different times
and in different places 
• 
Network delay is a randomly varying value. The delay, which is discrete,
exits at any time 
• 
In any network, the transmission delay between two nodes is within a certain
range 
These features will provide assistance to the following simulation and analysis
of the results.
Delay prediction model based on RBF neural network: Among popular neural network models, the theory used to build RBF neural network model not only has a solid mathematical foundation but also has many advantages. The most basic form of RBF neural network consists of three layers Fig. 1. The task of its input layer nodes is simply to pass the input data to the hidden layer. In RBF neural network, the basis function of the hidden layer is generally the distance function and its activation function is the Gaussian function in most instances.
APCIII (Hwang and Bang, 1994) algorithm is extended
from APCI algorithm. This method can calculate the center of the radial basis
function through a rapid clustering on the collected samples. This algorithm
is widely used in pattern classification, because of its high computing speed
and small amount of calculation. APCIII algorithm can determine the number
of basis functions and the center vector and the width of the radial basis function
is a fixed value.
The sole parameter needed to be predetermined is the clustering radius denoted as R in APCIII algorithm. Generally, R is computed via Eq. 1.
where, S is the number of samples and α is the adjustment factor. The
value of α can affect the result of R. The amount of calculation is too
much, especially when there are a large number of samples, by using the mentioned
equation.
If we just take a subset of samples to compute R approximately, the accuracy
of the neural network will be affected. In order to improve the accuracy, we
can take the arithmetic mean of R computed from a plurality of different subsets.
The specific implementations of the algorithm are as follows.
At first, define some variables. The input of the algorithm is X = {X_{1},
X_{2}, …, X_{S}} and the output is the center of each class.
We denote the center of class j as b_{j}, the number of samples in class
j as n_{j}, the number of clusters as m, the distance between the sample
X_{i} and the center of class j as D_{ij}.
Step 1: 
Initialize the parameters. The initial value of m is set to
be 1 and b_{1} is set to be X_{1}; the value of n_{1}
is 1. Also, there has to be an appropriate value for α 
Step 2: 
Calculate the value of R via Eq. 1 
Step 3: 
Compare D_{ij}, the distance between X_{i} which is a
new input sample and the existent cluster center, with R. When D_{ij}
= R, X_{i }should be classified into class j and the data center
of class j should be updated via Eq. 2. Also, the number
of samples in class j increases, that is, n_{j }= n_{j}+1.
When D_{ij}>R, we need create a new class and increase the number
of clusters, that is, m = m+1. Then, b_{m}, the center of class
m, is set to be X_{i} and n_{m} is set to be 1: 
Step 4: 
Execute step (2) and step (3) repeatedly until all samples
are trained 
After the end of the algorithm, the hidden layer structure of RBF neural network
will be determined. Then, we can carry out a supervised learning for RBF neural
network by using the method of least squares to get the optimal weight of the
hidden layer.
So far, the delay prediction model based on RBF neural network has been established.
Simulation experiments of delay prediction based on RBF neural network: Network delay is the time interval that a data packet transmits from a node to another which is oneway delay. However, it is very difficult to establish an accurate computational model for network delay, because network load and dynamic routing change over time. Nowadays, Round Trip Time (RTT) is used in actual experimental research process, instead of oneway delay. RTT is the time interval that a packet is transmitted from one host to another and back to the original host.
We selected a host from campus networks of North China Electric Power University
and Central South University respectively for measuring network delay. We continuously
measured the delay during the daytime and recorded these measurements for several
days. The measurements of each day were regarded as a sample space. During measurement,
network delay was measured every 25 sec. Four data packets were sent each time
and the size of each packet was 128 bytes. The average of each measurement was
recorded as a delay datum. We carried out the simulation experiment by using
two sample spaces selected optionally from the sample set. The delay graphs
of these two sample spaces are presented in Fig. 2. It is
revealed that the trends and ranges of the delay are roughly the same. Network
delay is small during the daytime; however, it becomes big to some extend in
the evening, due to the increase in the amount of network usage. The change
of network delay shows certain regularity.
In the simulation, the delay prediction model was established by using the
method elaborated in the previous section. At first, we need select a center
vector from the training sample set and calculate the variance for the basis
function of the hidden layer. Then, we could compute the optimal weight of the
hidden layer by using the method of least squares. Under the premise of not
affecting prediction accuracy and training speed, the delay prediction model
was trained by the delay data measured in the network. In order to predict the
time delay of a certain moment accurately, we selected seven network delays
in previous times and another three variables which were the variations of network
delay in previous times as the input of the delay prediction model.
We conducted our simulation experiment by using these selected samples in MATLAB. The delay data of one day were used to train the neural network as initial sample. After the training, we validated the effect of the neural network model by using three data sets which were selected from another sample in different time periods. There were 300 consecutive data in each data set. The results of prediction are presented in Fig. 3.
The curves of predicted delay and actual delay are very similar to each other. When the actual delay fluctuates smoothly or dramatically, the variation of the predicted delay becomes small or big accordingly.

Fig. 2: 
Delay of two days 

Fig. 3(af): 
Results of prediction, (a) Delay in the morning, (b) Delay
prediction in the morning, (c) Delay in the afternoon, (d) Delay prediction
in the afternoon, (e) Delay in the evening and (f) Delay prediction in the
evening 

Fig. 4(ac): 
Prediction errors, (a) Error in the morning, (b) Error in
the afternoon and (c) Error in the evening 
Therefore, RBF neural network can well predict the variation of network delay,
especially in the condition that the change of delay is random and nonlinear.
Prediction errors are presented in Fig. 4 to show the performance
of the delay prediction model based on RBF neural network. Most of the prediction
errors are less than 2 m sec and nearly half of the prediction error rates are
beyond 5%. Moreover, when delay variation is violent, the prediction error becomes
bigger.
CONCLUSION
In this study, we establish a time delay prediction model based on RBF neural network by using APCIII algorithm and carry out the simulation experiment in MATLAB. The results show that this time delay prediction model can predict the trend of network time delay accurately and the value of network delay with relatively high accuracy. However, dramatic changes in delay will reduce prediction accuracy to some extent. In order to obtain a better effect, we need seek a better algorithm for the delay prediction model based on RBF neural network to further improve the accuracy and reduce the impact of dramatic changes on prediction accuracy.
ACKNOWLEDGEMENT
The study is supported by the Fundamental Research Funds for the Central Universities.