# Network monitoring

### Abstract

The process of networking monitoring is the constant observation and tracking of data traffic. Network monitoring can alert a network administrator to problems caused by overloaded systems, crashed servers, lost network connections, virus or malware infections, and power outages, among other Network monitoring is commonly done by sending a “ping,” or test, to each computer or system on the network. If the system does not respond or takes too long to respond, the network monitoring system does its job by notifying the network administrator of the problem. Pings are intended to be replied to instantaneously; anything else hints at a larger problem than just the absence of a ping. Thus Network Analysers are introduced to keep a constant check on the performance of the network.

### References

[1]Computer Networks,Andrew Tanenbaum ,Prentice hall ,Inc.1981

[2] http://www.cisco.com

### Appendix A

numx = 101; %number of grid points in x

numt = 2000; %number of time steps to be iterated over

dx = 1/(numx - 1);

dt = 0.00005;

x = 0:dx:1; %vector of x values, to be used for plotting

T = zeros(numx,numt); %initialize everything to zero

%specify initial conditions

t(1) = 0; %t=0

T(1,1) = 0; %T=0 at x=0

T(1,numx) = 0; %T=0 at x=1

mu = 0.5;

sigma = 0.05;

for i=2:numx-1

T(i,1) = exp(-(x(i)-mu)^2/(2*sigma^2)) / sqrt(2*pi*sigma^2);

end

%iterate difference equation - note that T(1,j) and T(numx,j) always remain 0

for j=1:numt

t(j+1) = t(j) + dt;

for i=2:numx-1

T(i,j+1) = T(i,j) + (dt/dx^2)*(T(i+1,j) - 2*T(i,j) + T(i-1,j));

end

end

figure(1);

hold on;

plot(x,T(:,1));

plot(x,T(:,11));

plot(x,T(:,101));

plot(x,T(:,1001));

plot(x,T(:,2001));

xlabel('x');

ylabel('T(x,t)');

%calculate the flux at x=0 and x=1

for j=1:numt+1

flux0(j) = -(T(2,j) - T(1,j))/dx;

flux1(j) = -(T(numx,j)-T(numx-1,j))/dx;

end

figure(2);

hold on;

plot(t,flux0,'b');

plot(t,flux1,'r');

xlabel('t');

ylabel('flux');

%calculate approximation to the integral of T from x=0 to x=1

for j=1:numt+1

s(j) = sum(T(1:numx-1,j))*dx;

end

%calculate the amount of T that leaves through the boundaries due to flux

% s0 is the amount of T that leaves through x=0

% s1 is the amount of T that leaves through x=1

s0(1) = 0;

s1(1) = 0;

for j=1:numt

s0(j+1) = s0(j) - flux0(j)*dt;

s1(j+1) = s1(j) + flux1(j)*dt;

end

figure(3);

hold on;

plot(t,s,'g');

plot(t,s0+s1,'r');

plot(t,s+s0+s1,'b');

title('title')

xlabel('t');

ylabel('T_{total}(so-red s1-green)');

### 4.1 Conclusion

The Heat and Diffusion Equations are of fundamental importance in diverse scientific fields.Use of Matlab provides matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages.For an efficient and erroe free output representation we make use of Matlab functions in designing diffusion equation.

### 4.2 Future Scope

The following modifications can be done in the given project:

1.Heat both the ends of the object simultaneously at the same rate of temperature and plot the temperature profile for the simultaneous increment of heat from both the ends.

2.Solve equation for a 2 Dimentional object and verify its temperature profile.

3.Calculate the increment of flux at each values of x and t.

### 1.1 Computer Networking

Computer networking is all about connecting a group of computers with each other for the purpose of communication. Computer networks can be classified on the basis of a wide variety of characteristics.

### 1.2 Importance of Network Monitoring

Every business of any size today operates by using the network. Even though computers and software comes with all the perks, network monitoring plays a vital role in the efficient maintenance and productivity of the business.

Efficient networking monitoring is about maintaining the overall integrity of the network which involves proper functioning of the involved components, hardware and software used.

From practical point of view network monitoring also involves restricting the access of employees to the sites which invites viruses, spywares and all other elements that slow down and damage the functioning of the network.

With proper network monitoring, problems can be spotted early on and a step taken to correct the issue before it has a chance to cause major havoc.

Knowing how the network is being used will help to eliminate the chances of any of these sorts of events, and allow you to get more productivity from your network and your employees.

### Discretization of the Diffusion Equation by Finite Differences

The basic idea of the finite differences method of solving PDEs is to replace spatial and time derivatives by suitable approximations, then to numerically solve the resulting difference equations. Specifically, instead of solving for with and continuous, we solve for where ,

The point labelled corresponds to .

Derivatives of T are approximated in terms of the values of at grid points. For example, we know that

This derivative evaluated at the grid point can be approximated in many different ways, the simplest being the following:

* Forward Difference:

* Backward Difference:

### Central Difference:

The second derivative at the grid point may be approximated by using

Instead of using approximations for in terms of the values of at as for the forward difference, or at the points as for the backward difference, let's imagine instead that we evaluate it at the (fictitious) points , defined in the obvious way. Then, using central difference approximations for the spatial derivatives evaluated at these points,

Then ,

(3)

We can approximate derivatives with respect to time in the same way. For example, the forward difference approximation for at the grid point is

(4)

It should be noted that these finite difference approximations are only valid to some order in or .The error in the approximations is called the truncation error. It is possible to get approximations which are valid to higher order by using more grid points in the approximations. This is all quite important, but for our purposes the approximations given above will be sufficient.

Using the approximations(3) and (4) in (2), and rearranging, we get the following difference equation which can be iterated to find the approximate solution to equation (2):

This is called an explicit numerical scheme equation numbered 5 because the computation of at is completely determined by our computation of at.

### 2.1 Analysis of Numerical Error

Explicit numerical scheme used above is also called consistent because the finite difference approximations have a truncation error that approaches zero in the limit that,

Although this is a consistent method, we are still not guaranteed that iterating equation (5) will give a good approximation to the true solution of the diffusion equation (2). A numerical scheme is called convergent if the solution of the discretized equations (here, the solution of (5)) approaches the exact solution (here, the solution of (2)) in the limit that $\delta t \rightarrow 0$, $\delta x \rightarrow 0$.

For linear equations such as the diffusion equation, the issue of convergence is intimately related to the issue of stability of the numerical scheme (a scheme is called stable if it does not magnify errors that arise in the course of the calculation). Indeed, the Lax Equivalence Theorem says that for a properly posed initial value problem for a linear PDE, and a consistent finite difference approximation, stability is the necessary and sufficient condition for convergence. Moreover, it can be shown that the scheme given by (5) is only convergent when,

(6)

These issues are also quite important, but this is not the appropriate place to go into them in more detail. However, before moving on, let me emphasize that as the sizes $\delta t$and $\delta x$are made smaller, the truncation error of approximating the partial derivatives by finite differences decreases. However, for smaller sizes, more computations need to be done to get solutions for the same domain and total time, which leads to increased round off error. The total error as a function of these sizes is sketched in Figure 2.1

\begin{figure}\begin{center} \leavevmode \epsfbox{error.eps}\end{center}\end{figure}

Here it is assumed that the solution is calculated on the same domain and for the same total time.

### 2.2 Flowchart

XVI

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