PDE
based Analysis for Propagation of Disturbance by Users Mobility in Mobile
Network System
Shathya
Pranav Sujithra Rajesh Kannan^{*a}
^{*a}Fahaheel AL Watanieh Indian Private School ( AL Nouri Teaching Est.Co.)
*Corresponding Author:
Email Address:
Article available
online at:
https://esciencesspectrum.com/AbstractView.aspx?PID=2022221
ARTICLE INFO


ABSTRACT

Original
Research Article
Received:
1 November 2022
Accepted:
10 November 2022
DOI
10.55878/SES2022221
KEYWORDS
Mobile
user flow rate,
Flux
density,
Flow
density wave,
Mobile
jamming.


Analyzing
the nature of mobile user mobility in a given network system is one of the
basic metric for the traffic performance analysis in landmobile cellular
communications. This paper focuses on the partial differential equation (PDE)
based analysis on the mobile user mobility; and proposes a new concept of
user flow density wave in the mobile network system. The theoretical
formulation for measuring the characteristics of network traffic performance
deals with the propagation of disturbance produced by the velocity and
density characteristics of the mobile users in a given network system; while
most of the recent works uses the idea mobile users’ velocity characteristics
and the network traffic layout. The proposed flow density wave concept is
also used to characterize the mobile jamming.

Introduction
Recent
advancement toward new generation cellular communication networks provides
ubiquitous subscriber with highquality multimedia services. Emerging problems
from increasing demand of individual access to landmobile cellular networks
have so far been intensively studied for qualityofservices (QoS) guarantee as
well as for efficient mobility and resource management [1,2,3,7]. It has been
considered [4,5] that the nature of traffic performance in cellular networks is
affected by user mobility.
The
effects of user mobility on traffic performance are studied in various
literature [3,6,8,9], and the need for consideration of user mobility effect in
traffic analysis system is addressed. This is mainly because reasonable
mobility models capable of providing sufficient accuracy for traffic
performance analysis and traffic flow estimation are required in cellular
communication systems due to both increasing bandwidth requirement and limited
available resources.
In
this paper a new concept of differential equation based formulation [13,14] is
proposed to analyze the variable user mobility require considering many factors
that affect the real traffic performance in the given mobile network system.
The performance analysis of the variable user mobility depends on the moving
directions of the users as well as the velocity with randomness which is
partially affected by network layout and traffic flow condition [9,10].
In
this paper, mobile user flow model is analyzed based on the differential form
of flux and density modeling which is unique and different from some
statistical methods proposed in some literature [1,2,5,6,10,11,12]. The concept
of flow density wave is proposed in the domain of user mobility analysis in a
given mobile network system. The approach focuses on the behavior of the
network performance using the measure of disturbance produced by mobile users
in the given network due to the user density and different flow rate. The
partial differential equation (PDE) for the conservation of mobile user in the
given network system is formed, and the equation is solved both in small
disturbance of network traffic and using nonlinear initial value. Since the
differential form of mobile user flow problem lies on the strong platform of
analytical formulation, this theory may be able to provide a solution with
sufficient accuracy. The solution will also provide new insight for better characterization
of the various aspects of mobile user mobility. The curve of flow density
characteristics obtained from the partial differential equation may be used as
behavior measuring of mobile jamming where the users get low or no signal due
to high density flow characteristics and fixed availability of signal power by
the network provider. For a given mobile network, the time and the position of
the mobile jamming may be analytically measure by the statistical
characteristics of mobile user density and the mobile user flow rate.
The
paper is organized in five sections. Section II deals with the basic theory.
The analysis is carried out in section III. The simulated result reflecting the
nature of mobile jamming is given in section IV. Finally, the conclusion is
drawn in section V.
II. Preliminaries
In
this work, the analysis is carried out considering one directional flow of the
mobile user, for simplicity, in a fixed direction of the mobile network system.
The same approach can be easily extended to multidirectional flow problem in
the given network in which the user may move in any direction. For the
multidirectional flow, the component of each direction may be considered as
one directional flow model using the constraints of conservation of flow rate
and density of the mobile users. Let us define the local mobile user density,
, as the number of mobile user per unit length of the network
path. Here we do not want to analyze the motion of the mobile user at every
point, but a small but finite averaging area surrounding the point of
consideration is taken and we define the mean density in this length to be the
effective density at this point. Here the mobile user density is proportional
to the number of mobile users in that small but finite length of the network
path and is the function of x and t, i.e.
, which defines for every position x and t. The local flow
rate of the mobile user,
, is also function of the local mobile user density ie
.
Consider
a finite length of network path,
. The rate of change of number of mobile user in this
interval of network path is equal to the flux of mobile users in at
minus the flux of the
mobile users out at
, or
(1)
where
is the mobile user
flow rate which may be characterized as the mobile user velocity in the fixed
direction of the given network path.
In
terms of
and v,
can be represented as
(2)
Equation
(1) is the integral expression for the conservation of the mobile user in the
interval
of the network path,
and is independent of the choice of
and
; and hence must hold for any value of
and
. For continuous densities, taking the limit
, the differential form for conservation of the mobile users
is obtained as
(3)
This
one directional version of the differential equation can be transformed into
multidirectional form as
(4)
where
and
are the vectors whose
components are densities and flux at different direction, respectively. Here we
are concentrated on the behavior model of the onedirectional flow. On the
functional form of
, it may be assumed that,
a) When
the mobile user density at a fixed point is maximum valued, the flow rate,
which is the function of the density, becomes zero due to the fact that the
movement of the users ceases for high density population of mobile users. In
other word there is an upper limit,
, on the possible densities of the mobile users,
corresponding to bumpertobumper traffic in the network traffic path, so that
when the user density is at maximum the flow rate at that point becomes zero,
i.e.,
.
b) As
the mobile user density increases, it may be assumed that the flow rate of the
user decreases due to high population density of the mobile users at the
neighborhood region of the point of consideration; i.e. as the user density
increases the user flow rate
decreases
monotonically for
due to the fixed
capacity of the network provider. If the density increases there will be a high
possibility of congestion which may produce network jamming, i.e. some of the
user will not be able to get good signal from the network service provider.
. The
nature of mobile network traffic characteristic depends on the mobility of the
users as well as the number of users present at the network path. Hence, the
equation on the conservation of mobile users (refer equation 3) depends on the
mobile users flow rate as well as the users flux at the point of consideration.
The flux of users may be used to measure the disturbance produced by the users.
But the density is also an independent parameter for consideration. Hence the
relative measure of user flux with respect to the density is used here as a
measure of disturbance in the network path produced by the mobile users.
Here equation (3) can be transformed as
(5)
Where
(6)
Here
is a relative measure
of disturbance produced and may be defined as flow density wave speed which
reflects the disturbance due to the change in mobile user flow rate with
respect to the mobile user density at a given network path. This curve gives different
characteristics curve at different network traffic point for different time
periods.
A. Small Disturbance in Traffic Flow
In
reality statistical nature of traffic shows that most of the time the mobile
user density at a fixed interval remain unchanged but shows some small
fluctuation; which may be treated as small disturbance traffic flow due to
which the density shows a small fluctuation. In small disturbance traffic flow
model, the mobile user density may be assumed with fluctuates with very small
magnitude over a constant density value.
For
the solution of small disturbance of the uniform state
, where
is a constant. Using
(7)
(where
is the small
disturbance in the mobile user density)
and for the equation (4) we obtain
(8)
For
any function f,
gives the general
solution of the differential equation(7), as
(9)
is the flow density
wave that propagates in the positive xdirection without change of form at the
flow density wave speed appropriate to the uniform state,
. If
is the maximum value
of the flux function at
, then
for
and
for
for a given network path. This means that flow density waves
propagate in the opposite direction to the traffic flow when
. This explains that what is propagating is not the mobile
users, but the disturbances in the medium made by the users. There is no
backward travel of mobile user when
.
Let
the curves
when
is constant. Since the solution of the PDE varies at the rate
of
without change of
form, which can be represented in a linear form as
(10)
Equation
8 can be defined as characteristic curves or characteristics of the mobile
users in the given network system.
B. NonLinear Initial Value Problem in Traffic Flow
For
nonlinear initial value problem, the change in mobile user density is not only
simply fluctuation but it follows some statistical distribution; as well as it
may be considered as the mobile user velocity, which is a function of the
mobile user density, follows a specific statistical distribution depending on
the density
. Here it is required
to solve the partial differential equation
(11)
subject
to the initial condition
(12)
i.e.,
at time t = 0, the mobile user
density is a function of the network path, x.
To
find the set of characteristics curve
, on which
is constant, it may be assumed that
(13)
Hence
. (14)
Comparing
this equation with the equation (11), we have
(15)
However,
is constant on each
characteristic. Hence on each characteristic
is constant,
is constant and each
characteristic is a straight line given by
(16)
The
solution is given implicitly by Equation
16, and
(17)
Thus
the density characteristic depends on the initial consideration of the partial
differential equation in which the density is a function of time.
Now,
if
is variable, then the
phenomenon of jamming has to be
considered in the theoretical analysis; since
is a function of flow rate and as well as the density of the
mobile users. The timedependentstatisticaldistribution of the mobile user
density in a fixed length of the network path produce a nonlinear nature of
the characteristics.
III. Analysis
Consider
a model where
(18),
This
is the simplest possible form for the velocity function, consistent with
earlier assumptions about its behavior. In this case
(19)
and
. (20)
Considering
the exponential initial conditions for the mobile user density, we have
.(21)
for
positive constants
,
and
.
Consider
, here
as
and
as
, with the change between these two states occurring over a
distance of L. Since the flow density
wave speed,
, is a decreasing function of
, and the initial condition have
a decreasing function
of x,
is an increasing
function of x, with
(22)
There
is a unique characteristic through every point in the domain of solution.
Qualitatively, the spreading out of the characteristics leads to a spreading
out of the initial density profile of the mobile users. Each point on the
initial profile is shifted to the right by a distance
. Here, the equation does not mean that the mobile users are
actually moving with speed
, rather a disturbance propagates at this speed.
Consider
another model in which the mobile user flow rate is exponentially distributed
with the user density in the given network such as,
(23)
Then
the flow density wave becomes,
(24)
These
two models are based on the statistical nature of traffic and show much more
theoretical concepts on the solution of the mobile user conservation based
partial differential equation.
IV. Mobile jamming
The
partial differential analysis on the network traffic characteristics through
the disturbance due to flow rate and density of the mobile users reflects the
nature of mobile jamming in the network. In this work we are using the flow
density characteristics of the mobile users in a fixed network path. In a fixed
network path the flow rate and the density of the users varies with time and
hence different characteristics are obtained through the analysis of flow
density wave. At different traffic point of the total network path gives
different characteristics varying with time. If the disturbance is highly
concentrated then it can be called as jamming
in mobile network. In figure 1, the xaxis represents the different traffic
point of the total network path and the yaxis represents the time. Here total
length of the network path is taken 200 unit and is then subdivided into 100
fixed bounded length. The total time in the simulation is for 1000 unit plotted
in yaxis. The mobile user density distribution at different traffic point is
taken as random Gaussian distribution and then the partial differential
equation is numerically solved through RungeKutta method [13]. The bold curves
shows the mobile jamming at different time span and different traffic point of
the network.
Figure1:
Characteristics curve depending on the flowdensity characterization. The bold
line shows the congestion in the mobile network traffic.
V. Conclusion
This
paper proposes a novel differential equation (PDE) based approach defines the
concept of flow density wave in the domain of mobile user mobility model. It
has been shown that the disturbance in the network traffic performance due to
mobile user mobility is measured using the characteristics of the flow density
wave. The velocity model of a given network system depends on the mobile users
and the network traffic layout. Further, if the velocity model is estimated
statistically, the network traffic performance can be measured through the
statistical characteristics of the flow density propagation. Here, in this
paper, the propagation of disturbance due to mobile users in the given mobile
network is measured through partial differential equation as one direction
problem, rather than measuring the actual characteristics of the mobile users
and the network traffic. The propagation of disturbance gives the scope of
measuring the mobile jamming characterization traffic performance with
sufficient accuracy. This analysis can also be used as a multidirectional
problem in the mobile network traffic performance considering the velocity flux
of the mobile users at each direction of network layout.