Min-Plus Linear Systems Theory Min-Plus Linear Systems Theory (Classical) System Theory Linear Time Invariant (LTI) Systems Linear:Time invariant: Consider an input signal:.. and its output at a system: Note: Linear Systems Theory Consider an arbitrary function Approximate by Now we let Linear Systems Theory The result of convolution (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 Min-Plus Linear System min-plus Linear:Time invariant: Consider arrival function:.. and departure function: Note: Min-Plus Linear System Consider an arbitrary function Approximate by Now we let Min-Plus Linear System The result of min-plus convolution Min-Plus Linear Systems If input is burst function, output is the service curve Min-Plus Linear Systems Departures can be calculated from arrivals and service curve: min-plus convolution Back to (Classical) Systems Now: Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system Time Shift System eigenfunction eigenvalue Back to (Classical) Systems Solving: Gives: eigenvalue Fourier Transform 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 Back to (Classical) Systems Solving: Gives: eigenvalue Legendre Transform 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