Introduction

Recent advancement toward new generation cellular communication networks provides ubiquitous subscriber with high-quality multimedia services. Emerging problems from increasing demand of individual access to land-mobile cellular networks have so far been intensively studied for quality-of-services (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 non-linear 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 multi-directional flow problem in the given network in which the user may move in any direction. For the multi-directional 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 multi-directional 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 one-directional 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 bumper-to-bumper 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 x-direction 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. Non-Linear Initial Value Problem in Traffic Flow

For non-linear 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 time-dependent-statistical-distribution of the mobile user density in a fixed length of the network path produce a non-linear 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 x-axis represents the different traffic point of the total network path and the y-axis 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 y-axis. 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 Runge-Kutta 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 flow-density 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 multi-directional problem in the mobile network traffic performance considering the velocity flux of the mobile users at each direction of network layout.