Objective: Use a system of ODE to model data degradation in a digital storage system.

Description: This assignment has two parts: theoretical and practical. You will build a system of ODE that models the loss or corruption of data in a digital storage system, as follows:

x 1 = a11x1 + + a1nxn + f1

x 2 = a21x1 + + a2nxn + f2

. . . . . . . . . . . . . . . . . . . . . . . .

x m = am1x1 + + amnxn + fm

You can assume that the functions aij(t) and fj(t) are given, while the functions x1(t), , xn(t) are the unknowns. Remember that the rate of change is given by:

rate of change = input rate output rate.

Consider several factors that affect degradation, for example, flipping bits, dispersions of electric charger, leaks due to defective insulation, physical media decomposition, and other. You decide which factors to include and their values. Then, you will implement a computer simulation to visualize the data degradation.

Part 1:

Consider the three processors shown in the figure. Let us suppose, for the sake of discussion that processor A contains 100 Mbytes of data in which 25 Mbytes is used for I/O and 75 Mbytes for internal housekeeping. Suppose processor B and C also contains 100 Mbytes of data (75 MB for housekeeping and 25 MB for I/O). Data is pumped into and out of the processors as indicated in the figure. The I/O plus the Data exchanged between the three processors and the I/O data propagated to processor C and going downstream to the network are well processed.

Construct a mathematical model that describe the number of I/O data x1 (t),x2 (t) and x3 (t) of network data in processors A, B and C , respectively, at time t .

Put the mathematical model (which is a system of 1st order differential equations) into the form: x=Atxt+ft, with ft=0.

Find the eigenvalues of matrix A.

Use formal mathematical rigor when writing and explaining your model, the system of ODE, and the quantification of the outcome.

Part 2:

Two Processors A and B have each 100 Mbytes of memory. Initially, 100 Mbytes of data + I/O is stored in processor A and 100 Mbytes of data + I/O is stored in processor B . The system is closed in that data + I/O is pumped only between the processors, as shown in the Figure.

Use the information given in the figure to construct a mathematical model for the number of I/O data x1tand x2t at time t in processors A and B , respectively.

Write the system of linear differential equations in mathematical form x=Atxt+ft, with ft=0.

Calculate eAt from matrix A found in the system of 1st order linear differential equations,

Find the solution to the initial value problem x0=1, x0=-1 by matrix method.

100 MB

B I/O + data

3 MB/sec

A 100 MB

I/O + data

2 MB/sec

Choose the appropriate software packages to generate graphs for the solution in (d) or animate the data, as best suited for your project.