Cda6530: Performance Models of Computers and Networks Chapter 4: Using Matlab for Performance Analysis and Simulation



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CDA6530: Performance Models of Computers and Networks Chapter 4: Using Matlab for Performance Analysis and Simulation

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Objective

  • Learn a useful tool for mathematical analysis and simulation
    • Interpreted language, easy to learn
  • Use it to facilitate our simulation projects
  • A good tool to plot simulation/experiment results figures for academic papers
    • More powerful than excel
    • Could directly create .eps for Latex

Introduction

  • MatLab : Matrix Laboratory
  • Numerical Computations with matrices
  • Why Matlab?
      • User Friendly (GUI)
      • Easy to work with
      • Powerful tools for complex mathematics
  • Matlab has extensive demo and tutorials to learn by yourself
    • Use help command

Matlab Software Access

  • all UCF in-campus computers have student-version Matlab installed
  • If you have no access to Matlab, you can use Octave, an open-source free software
    • http://www.gnu.org/software/octave/
  • The programming should be almost identical

Matrices in Matlab

  • To enter a matrix
          • 2 5 3
          • 6 4 1
  • >> A = [2 5 3; 6 4 1]
  • >> B = [1:1.5:6; 2 3 4 5]
  • >> for i=1:4
  • for j=1:3
  • C(i,j)=i*j;
  • end
  • end
  • >> D =[]; D=[D;5]; D=[D;6;7]
  • >> E = zeros(4, 5)

Basic Mathematical Operations

  • Remember that every variable can be a matrix!
  • Addition:
  • >> C = A + B
  • Subtraction:
  • >> D = A – B
  • Multiplication:
  • >> E = A * B (Matrix multiplication)
  • >> E = A .* B (Element wise multiplication, A and B same size)
  • Division:
  • Left Division and Right Division
  • >> F = A . / B (Element wise division)
  • >> F = A / B = A*inv(B) (A * inverse of B)
  • >> F = A . \ B (Element wise division)
  • >> F = A \ B=inv(A)*B (inverse of A * B)

Generating basic matrices

  • Matrix with ZEROS:
  • >> A = zeros(m, n)
  • Matrix with ONES:
  • >> B = ones(m, n)
  • IDENTITY Matrix:
  • >> I = eye(m, n)
  • m  Rows
  • n  Columns
  • zeros, ones, eye  Matlab functions

Obtain Information

  • Size(A): return [m n]
  • Length(A): length of a vector
    • Length(A) = max(size(A))
  • B = A(2:4,3:5)
    • B is the subset of A from row 2 to row 4, column 3 to column 5
  • A(:, 2)=[]
    • Delete second column

Basic Matrix Functions

  • Inv(A): inverse of A
  • Rank(A): rank of matrix A
  • A’: transpose of A
  • Det(A): determinant
  • V= eig(A): eigenvalue vector of A
    • [V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D

Random Number Generators

  • Rand(m,n): matrix with each entry ~ U(0,1)
    • You can use this for the programming project 1
  • Randn(m,n): standard normal distribution
    • You cannot use this in programming project 1
    • You must use the polar method I introduced!

Basic 2-D Figure Plot

  • Plot(X, Y):
    • Plots vector Y versus vector X
  • Hold: next plot action on the same figure
  • Title(‘title text here’)
  • Xlabel(‘…’), ylabel(‘…’)
  • Axis([XMIN XMAX YMIN YMAX])
  • Legend(‘…’)
  • Grid
  • Example demo

Elementary Math Function

  • Abs(), sign()
    • Sign(A) = A./abs(A)
  • Sin(), cos(), asin(), acos()
  • Exp(), log(), log10()
  • Ceil(), floor()
  • Sqrt()
  • Real(), imag()

Elementary Math Function

  • Vector operation:
    • Max(), min(): max/min element of a vector
    • Mean(), median()
    • Std(), var(): standard deviation and variance
    • Sum(), prod(): sum/product of elements
    • Sort(): sort in ascending order

Save/Load Data

  • Save fname
    • Save all workspace data into fname.mat
    • Save fname x y z
    • Save(fname): when fname is a variable
  • Load fname
    • Load fname x y
  • No error in data
  • You can run simulation intermittently
    • Save/load data between runs

Input/Output for Text Files

  • Input data file for further analysis in Matlab
    • Run simulation using C
      • matlab is slow in doing many loops
    • Use Matlab for post-data processing
      • Matrix calculation, utilize Matlab math functions
    • Simply use Matlab for figure ploting
      • Excel has constraint on data vector length (<300?)
  • Functions:
    • [A,B…]= Textread(fname, format)
      • Read formated data
    • Use fprintf(), fscanf() similar to C
      • Note that variables here can be vectors/matrices
      • Show examples here of writing data to text file

Advanced Graph

  • Subplot(m, n, p)
    • breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot, and returns the axis handle.
  • Semilogx(), semilogy(), loglog()

3-D plot

  • x=[0:10]; y=[0:10]; z=x’*y;
  • mesh(x,y,z); figure; surf(x,y,z);

M-file

  • Script or function
    • Scripts are m-files containing MATLAB statements
    • Functions are like any other m-file, but they accept arguments
    • It is always recommended to name function file the same as the function name
  • function A = changeSign(B)
  • % change sign for each element
  • [m,n] = size(B); A = zeros(m,n);
  • for i=1:m
  • for j=1:n
  • A(i,j)= -B(i,j);
  • end
  • end
  • return

Online Tutorials

  • Matlab itself contains many tutorials
  • Other online tutorials:
    • http://www.math.siu.edu/matlab/tutorials.html
    • http://www.cs.cmu.edu/~ggordon/780/lectures/matlab_tutorial.pdf
    • Google search “matlab tutorial ppt” to find a lot more
  • Example on Using Matlab for Markov Chain Steady State Calculation
  • Discrete-time Markov Chain transition matrix:
  • ¼ P = ¼ , ¼ [1 1 1… 1]T = 1
    • ¼ (P – I) = 0, But we cannot use it directly
    • Replace first column in (P-I) with [1 1..1]T to be A, then we can solve the linear equation set by ¼ = [1 0 0 … 0] A-1
  • Another way: P*P*P*P……
  • Tutorial on Matlab Simulink
  • Graphical programming language
  • Powerful modeling tool
    • Differential Equations
    • Physiological systems
    • Control systems
    • Transfer functions
  • M-file can call a simulink model
    • “sim fname”
    • Use current workspace variables
  • Simulation results can be saved to workspace variables
    • Thus can be process after simulink

Example: Internet Worm Propagation

Example 2: RC Circuit

  • Transfer function:

Save result to workspace variables

  • the save format is "structure with time".
  • Suppose the workspace variable is X_t.
  • Then:
    • X_t.time saves the simulation step times (vector)
    • X_t.signals.values saves the simulation results (vector).
  • plot(X_t.time, X_t.signals.values);
  • Variable step simulation or fixed step simulation:
    • "to workspace" use "-1" for sample time (inherited)
      • Then X_t.time has variable size
    • "to workspace" use "1" for sample time
      • Then each time tick has one result value

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