Abstract But with the growth of different factors

 

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.

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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 (Hurst, Black, & Sinaika, 1965)
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. So to get 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. Thus 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:

 

 

 

 

Table 1: Summary statistics for daily dropped call and
call initiation signal

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

 

         

               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) 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)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 (W.Wang, 2006).
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) 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(Shoeb & Clifford, 2006):

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 three methods, VGA, HFD and GHE which are being tabulated below in Table 2.

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

 

Hurst exponent (H)

Methods

Daily dropped calls

Daily busy hour
Initiated calls

R/S

0.2707

0.2405

GHE

0.2461

0.1565

 

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 claim the anti-persistent behaviour of both of them i.e. their
future values have the tendency to revert to their long-term mean with the
daily busy hour initiated calls profile has more tendency to return to its mean
compared to the daily dropped calls profile. Since there are the tendencies for
both the profiles to return to their respective mean, it can be said that there
must be some driving forces which bring back the series towards their means
when the profiles deviate from the mean (the most stable position of any
fluctuation). This implies that some negative feedback system must be working
which continuously try to stabilise the profiles. Moreover 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 evident from this SRD phenomenon of them.

The SPWVD 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

Figure
3 CWT for daily call drop

Figure 3
undoubtedly indicates that the daily dropped calls frequency is varying with
time.So, daily dropped calls data set is non-stationary in nature.

Figure
2 shows that this signal is nearly stationary as the frequency contents do not
change with time. So it can be concluded that busy hour initiated calls data
are stationary. In a non-stationary signal the frequency contents are the
functions of time i.e. they are not independent of time change. Frequency any
event signifies the number of events happen per unit time. So, it can be
inferred that the number call drops per unit time is not independent of time
but varies with time. In case of busy hour call initiation profile there are
nearly eight types of frequency contents as is evident
from figure 3 but all of them remains constant with respect to time. This can
be interpreted as the rates of busy hour call initiation is not varying with
time and hence proper modelling and forecasting of the busy hour call
initiation can be made easily.

5. Conclusion

One
of the statistical measures used in to classify the time series is Hurst
exponent. Using the value of H, the attributes within the time series can be
predicted: H=0: The time series is a white noise whose autocorrelation function
decreases rapidly with lag, a value of H in the range 0 – 0.5 indicates a
time series with long-term switching between high and low values in adjacent
pairs, meaning that a single high value will probably be followed by a low
value and that the value after that will tend to be high, with this tendency to
switch between high and low values lasting a long time into the future. 
The persistency of the signal is negative (or anti) where the probability of
opposite trend between any two successive data intervals is very high. This
means that future values have a tendency to return to a long-term mean and
hence it is slower than classical/standard Brownian motion.  If this
tendency is more in the time series, the value of H will be found to be closer
to 0. A value of H=0.5 can indicate a completely uncorrelated series, but
in fact it is the value applicable to series for which the autocorrelations at
small time lags can be positive or negative but where the absolute values of
the autocorrelations decay exponentially quickly to zero. Whereas H=1 denotes time
series ideally smooth and the autocorrelation function does not vary with lag
but settle to constant level signal has arbitrary fluctuation.  If the
value of H  is in this range 0.5–1, indicates a time series with long-term
positive autocorrelation, meaning both that a high value in the series will
probably be followed by another high value and that the values a long time into
the future will also tend to be high.  The persistency of the signal is
positive where the probability of related trend between any two successive data
intervals is very high. The stronger the trend, the H value moves towards 1.

                                                                

 

 

 

 

 

 

 

 

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