Abstract

In this paper, the fractality and stationarity

of a usual wireless network has been investigated by exposing the scaling

pattern and nature of frequency fluctuation of the two crucial parameters, the daily peak hour call arrival number and daily call drop number, allied

with a wireless network. The time series of these parameters between 3rd March,2005

to 31st October, 2015, of a sub-urban local mobile switching centre have

been considered for revealing the nature of scaling (fractality) and stationary

behaviour using statistical methodologies. Having the knowledge about the

fractality, Hurst Exponent for the time series have been considered using the

methodologies like General Hurst Estimation (GHE) and R/S. It has been observed

that both the

time series show Short Range Dependent (SRD) anti-persistent behaviour. Continuous

Wavelet Transform (CWT) method has been used to find out the stationarity/non-stationarity

of the data-series where both the time series exhibit the nonstationarity. These observations conclude that the both the time series are not a

random phenomenon but complex. However both the series found to have non-linearity

and stability.

1. INTRODUCTION

With the rapid growth in wireless technology

different applications are vividly applied in smartphone. Now a day’s

smartphones are widely used as the simple and most common devices for

communication. The multi-featured attributes of smartphone devices are widely

acceptable across the world for various ways of communications like data

services and voice. With the repeated use of these services the demand for

wireless networks increases rigorously. It becomes a tricky task for the service

providers to maintain the Quality of Service (QoS) and cost effectiveness by

upgrading the technical and infrastructural features of the wireless network

system. So various issues consisting of system design, congestion control, and

admission control should be addressed more efficiently to provide multi-class

services through desired wireless networks.

To upgrade the service quality and to achieve the optimum performance

there is a need to understand the nature of the fluctuation and underneath

pattern (particularly the scaling, self-similarity property and stationarity)

of the wireless network traffic data. But with the growth of different factors

like call drop rate and call arrival rate, the performance of network traffic

in mobile is highly affected. So it has become a necessity to understand the

nature of fluctuations of these two parameters. In this paper an initiative has

been taken to uncover the nature of the scaling behaviour and time dependency

of the frequency (stationarity or non-stationarity) of occurrence of the two

parameters, daily busy hour call arrivals and dropped calls, of a local mobile

switching centre during 3rd March, 2005 to 31st October,

2015 as shown in Figure 1 which can be treated as the signatory representative of

any wireless network traffic.The maximum number of call attempts in the peak

hour of a day is defined as busy hour call initiation. The resource of a

network can be limited to or can be upgraded as per requirements depending on

the maximum call arrival and the call drop caused due to congestion. A

concurrent study of busy hour call initiation and daily dropped call time

series may give a feasible nature of the incoming traffic pattern, the call

congestion, grade of service and blocking probability.

In this work Hurst exponent has been calculated for

revealing the scaling behaviour of the time

series, daily busy hour arrival call and call drop. Two different methods like

Rescaled Range analysis (R/S) method and General Hurst Estimation (GHE) method have

been used to calculate the Hurst Exponents to understand the nature of the

signals with respect to different scales to identify the signals as fractional

Brownian motion i.e. whether they are stationary or non-stationary. There are

many limitations of calculating Hurst exponent using other methods. For getting

a non-controversial conclusion about the scaling property of the time series,

it will be useful to apply more than one method to estimate the Hurst Exponent.

Hence two methods (mentioned above) have been chosen to calculate the Hurst

Exponent for confirming the authenticity of the conclusions taken out of the

results.

Stationary or non-stationary behaviour of the data series

could be completed by analysing the fluctuating nature of the busy hour call

initiation rate and call drop rate. A non-stationary signal has changing

frequency whereas stationary signal has constant frequency. The signals are

checked with respect to time. The analysis for non-stationary behaviour is necessary

due to: 1) asymptotic analysis which will not be applicable for the regression

model with non-stationary variables. Usually “t-ratios” does not follow a

t-distribution, and hence valid tests about the regression parameters cannot be

undertaken. 2) The properties of the signal are highly affected by the

stationary or non-stationary behaviour. Different methods can be used to check

the stationary/ non-stationary behaviour of the signals. Continuous Wavelet

Transform (CWT) based method has been implanted in this paper to determine the

nature of frequency dependency of the wireless network traffic. The advantages of using CWT are: a)

simultaneous localization in time and frequency domain and is computationally

fast. ii) Wavelets have the great advantage of being able to separate the fine

details in a signal. Very small wavelets can be used to isolate very fine

details in a signal, while very large wavelets can identify coarse details. It

decomposes a signal into component wavelets.

2. Experimental

dataset:

First and foremost the real time data are recorded in the Server

positioned in the Mobile Switching Centre (MSC) of the ISP. The recorded data

sets collected from the ISP sited in our city for the period 3rd March, 2005 to

31st October, 2015used for exclusively research purpose. The data

can not be exported commercially, it comprises of call initiation, call holding

time, call drops and its causes, time and delay of hand-off etc. From these

dataset the call initiation and the call drop statistics for each hour of a day

have been considered such as the peak hour call initiation and the call drop

statistics have been taken for analysis. The summary statistics and plotting of

original data set of signal are described in table1 and figure1 respectively:

Scores

Call Initiation

Signal

Call Drop Signal

Mean

168.3746

0.2596

Median

171

0

Mode

171

0

Standard

Deviation

14.7867

1.3505

Variance

218.6452

1.8238

Maximum

197

23

Minimum

14

0

Skewness

-3.9571

10.0416

Kurtosis

29.2970

133.6682

Table 1: Summary statistics for daily

dropped call and call initiation signal

Figure1:

plotting of the initiated calls and dropped calls

3. Hurst Exponent

Estimation

One of

the statistical measures used in to classify the time series is Hurst exponent.

Random series is recognised by H=0.5 while H>0.5 indicated reinforcing

series in trends. When two consecutive data intervals are very high then the

consistance of the signal is negative. The value of H=0 denotes that the time

series is a white noise whose autocorrelation function (ACF) decreases rapidly

with delay.. For this, the upcoming values have

a tendency to return to a long-term mean. Hence it becomes slower than

standard Brownian motion. With an increase in the tendency in the time series,

the value of H will tend to 0. The signal contains short-range

dependent (SRD) memory that exhibits fractal behaviour. The ACF decreases

exponentially with lag and is relatively slower than that of the white noise,

and H=0.5 denotes that the time series will show Standard Brownian motion

through Markov chain feature. The ACF decay is slow compared to the

anti-persistent time series. Arbitrary fluctuations are seen in the signal. Irregularity in

behaviour will appear with the difference in the various data points of the

time series. When the value of H lies within the range of 0.5-1.0

then it shows that with an increase in the successive data intervals the

persistency of the signal shows positive behaviour.

The Hurst value will tend towards 1. The signal shows long-range

dependence (LRD) and non-periodical cycle. LRD unlike the SRD series exhibits

similar statistical properties at different scales (lower or higher). The ACF

decays hyperbolically and is slower compared to standard Brownian motion. The

consistency of the signal is smooth.When the value of H is equal to 1.0 then the

time series appears to be perfectly smooth and the ACF comes to a constant

level.

Different estimators for the estimation of the

Hurst Exponent of any signal or data are available. In this paper, two Hurst

estimation methods have been used. The very recent method, Rescaled Range (R/S)

analysis has been used along with traditional Generalized Hurst Exponent (GHE)

estimation method. The Rescaled Range method is used for statistical

measurement of a time series. Its aim is to provide an estimation of how the

variability of a series changes with the length of the time-period. GHE

provides the best finite sample behaviour among all the methods in respect of

the bias and lowest variance. GHE is suitable for any data series/signal

irrespective of the size of its distribution tail.

3.1.

R/S Analysis:

R/S analysis (Rescaled Range

analysis) was initially coined by Edwin Hurst in the year 1951. This method can

be implemented in a program by providing a direct estimation of the Hurst

Exponent. The Hurst Exponent is a precious indicator of the state of randomness

of a time-series.

Given a time-series

with n elements

X

, X

,…,X

, the R/S

statistic is defined as:

=

Where

,

is the

arithmetic mean and

is the standard deviation from the mean.

With this R/S value, Hurst found

a generalization of a result in the following formula:

E

= C

as n

Where H is the Hurst exponent.From

there, it is clear that an estimation of the Hurst exponent can be obtained

from an R/s analysis.

3.2. Generalized Hurst Exponent (GHE) method:

This method was coined by (Hurst, Black, & Sinaika,

1965) (Morales, Di

Matteo, Gramatica, & Tomaso, 2012) (Mandelbrot, 1997) (Mukherjee, Ray,

Samanta, Moffazal, & Sanyal, 2017) defines a function

as

Where

is the

time series.pis the

order of the moment of distribution and

is the lag which ranges between

and

. Generalised Hurst Exponent (GHE), is related to

through a power law:

Depending upon whether it is

independent of p

or not, a time series can be judged as uni-fractal or multi-fractal (Matteo, 2007) respectively. The GHE h

yields

the value of original Hurst Exponent

for

, i.e.

.

3. Test for

Stationarity of Non-Stationarity:

3.1.Kwiatkowski–Phillips–Schmidt–Shin

(KPSS) tests:

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests (Kwiatkowski, Phillips, Schmidt, & Shin, 1992) (W.Wang, 2006) are used for testing a null hypothesis to check whether the observable time series is stationary or termed stationary or is non-stationary.This test is used as a complement to the

standard tests in analyzing time series properties.

The

KPSS test is based on linear regression. The time

series is broken down into three parts: a deterministic trend (?t), a random walk (rt), and a stationary

error (?t), with the regressio equation:

xt = rt + ?t + ?1

If

the data is stationary, it will have a fixed element for an intercept or the

series will be stationary around a fixed level. The test uses OLS to find the equation,

which differs slightly depending on whether you want to test for level

stationarity or trend stationarity. A simplified version, without the time

trend component, is used to test level stationarity.

3.2. Continuous Wavelet Transform (CWT) test:

Realword

data or signals are frequently exhibit slowly changing trend or oscillations

punctuated with transient. Though Fourier Transform (FT) is a powerful tool for

data analysis, however it does not represent abrupt changes efficiently. FT

represents data as sum of sine waves which are not localized in time or space.

These sine waves oscillate forever, therefore to accurately analyse signals

that have abrupt changes, need to use new class of functions that are well

localized with time and frequency. These bring the topic of wavelets.

The primary objective of the Continuous Wavelet Transform

(CWT) (Shoeb & Clifford, 2006)

is to get the signal’s energy distribution in the time and frequency domain

simultaneously.The continuous wavelet transform is a generalization of the

Short-Time FourierTransform (STFT) that allows for the analysis of

non-stationary signals at multiple scales.Key features of CWT are time

frequency analysis and filtering of time localized frequency components. The

mathmetical equation for CWT is given below:

C

(a,

) =

(

) x(t) dt

Where

C(a,

) is the function of the parameter a,

.

The a

parameter is the dilation of wavelet (scale) and

defines a

translation of the wavelet and indicates the time localization,?(t)

is the wavelet. The coefficient

is an energy

normalized factor (the energy of the wavelet must be the same for different a

value of the scale).

4. Results &

Discussion

The values of Hurst exponents for the two time series a) daily dropped calls and b) daily busy hour

call initiated has been calculated using the two methods, GHE and R/S which are

being tabulated below in Table 2.

Hurst exponent (H)

Methods

Daily dropped calls

Daily busy hour Initiated calls

R/S

0.2707

0.2405

GHE

0.2461

0.1565

Table 2: Hurst parameter values for daily dropped calls and daily busy hour call initiation

The Hurst exponents for both the series are less than

0.5. The Hurst exponent for daily busy hour initiated calls is lower than that

of the daily dropped calls. These results state the anti-persistent behaviour of both of them i.e.

their future values have the trend to regress to their long-term mean with the

daily busy hour initiated calls profile has more tendency to come back to its

mean compared to the daily dropped calls system. As there are the tendencies

for both the profiles to back again to their respective mean, it can be said

that there must be some motivating forces which bring back the series towards

their means when the profiles deviate from the mean .This signifies that some

negative feedback system must be functioning which constantly try to stabilise

the systems. However these low values of H signify that both the signals have short-range dependent (SRD)

memory. The self similar nature in short scale for both the times series is apparent

from this SRD phenomenon of them.

The CWT based

time-frequency spectrum for the two time series are shown in Figure 2 and

Figure 3 respectively.

Figure 2: CWT for daily call initiation

statistics

.

Figure 3: CWT for daily call drop

statistics

Figure 2 and 3 certainly denotes that the both time

serties are varying with time. So, they exihibit the non-stationarity. In a

non-stationary signal the frequency contents are the functions of time i.e.

they are not independent of time change. So, it can be inferred that the both

time series are not independent of time but varies with time. Call inintiation and call drop are random in

nature, its not depend upon the users choice.

5. Conclusion

The value of Hurst Exponent of any system is

greater than 0.5 or less than 0.5 but not equal to 1 are normally being

supposed to prove nonlinear dynamics. It can be concluded to consider that both

time series are nonlinear in their dynamics as the Hurst vale less than 0.5. Additionally

the anti-persistent behaviour of both call drop and busy hour call initiation

give the outline of the existence of some negative feedback system which needs

to be exposed more in the successive works. The low value of H denotes more

steadiness of the daily busy hour call initiation than that of the daily call

drop. Again it is found that there is non-Stationarity in both the time series.

So, it can be concluded that the call drop rate and peak hour call initiation

are not a random trend rather it is much more complex and non-linear, stable

process.

As the call

drop rate and Call initiation rate are used as the vital figures of worth to

evaluate the quality of service (QoS) in mobile wireless networks, the

consequent works will be to explore the nature of the non-linear dynamics and

originate the model depending on the present work findings.