Abstract number and daily call drop number,




 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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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

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

 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:


Call Initiation

Call Drop Signal




























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








plotting of the initiated calls and dropped calls



3. Hurst Exponent

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

 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.

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


,  the R/S
statistic is defined as:




 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:


= 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




 is the
time series.pis the
order of the moment of distribution and

is the lag which ranges between


. 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

the value of original Hurst Exponent


, i.e.


3. Test for
Stationarity of Non-Stationarity:

(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.

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                                                                                                                                                           


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:

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:


) =


 ) x(t) dt



) 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 &

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)


Daily dropped calls

Daily busy hour Initiated calls







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









Figure 3: CWT for daily call drop


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

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.