1. Min-Plus Linear Systems Theory Min-Plus Linear Systems Theory

2. (Classical) System Theory Linear Time Invariant (LTI) Systems Linear:Time invariant:

3. Consider an input signal:.. and its output at a system: Note: Linear Systems Theory

4. Consider an arbitrary function Approximate by Now we let

5. Linear Systems Theory The result of “convolution”

6. (Classical) System Theory Linear Time Invariant (LTI) Systems If input is Dirac impulse, output is the system response Output can be calculated from input and system response: “convolution”

7. Min-Plus Linear System min-plus Linear:Time invariant:

8. Consider arrival function:.. and departure function: Note: Min-Plus Linear System

9. Consider an arbitrary function Approximate by Now we let

10. Min-Plus Linear System The result of “min-plus convolution”

11. Min-Plus Linear Systems If input is burst function, output is the service curve

12. Min-Plus Linear Systems Departures can be calculated from arrivals and service curve: “min-plus convolution”

13. Back to (Classical) Systems Now: Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system Time Shift System eigenfunction eigenvalue

14. Back to (Classical) Systems Solving: Gives: eigenvalue Fourier Transform

15. Now Min-Plus Systems again Now: Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system Time Shift System eigenfunction eigenvalue

16. Back to (Classical) Systems Solving: Gives: eigenvalue Legendre Transform

17. Transforms Classical LTI systems Fourier transform Min-plus linear systems Legendre transform Time domain Frequency domain Time domain Rate domain Properties: (1). If is convex: (2) If convex, then (3) Legendre transforms are always convex