min-plus linear systems theory

Min-Plus Linear Systems TheoryMin-Plus Linear Systems Theory

System

(Classical) System 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 TheoryLinear Systems Theory

Linear Systems TheoryLinear Systems Theory

• Consider an arbitrary function

• Approximate by

Now we let

Linear Systems TheoryLinear Systems Theory

• The result of

“convolution”

System

(Classical) System Theory(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 SystemMin-Plus Linear System

Networkmin-plus Linear:Time invariant:

data

time t

A(t)

D(t)

delay

backlog

Consider arrival function:

.. and departure function:

Note:

Min-Plus Linear SystemMin-Plus Linear System

Min-Plus Linear SystemMin-Plus Linear System

• Consider an arbitrary function

• Approximate by

Now we let

Min-Plus Linear SystemMin-Plus Linear System

• The result of

“min-plus convolution

”

Min-Plus Linear SystemsMin-Plus Linear Systems

Network

• If input is burst function , output is the service curve

Min-Plus Linear SystemsMin-Plus Linear Systems

Network

• Departures can be calculated from arrivals and service curve:

“min-plus convolution”

System

Back to (Classical) SystemsBack to (Classical) Systems

• Now:

Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant

system

Time Shift System

eigenfunction

eigenfunction eigenvalue

System

Back to (Classical) SystemsBack to (Classical) Systems

• Solving:

• Gives:

eigenvalue

Fourier Transform

System

Now Min-Plus Systems againNow Min-Plus Systems again

• Now:

Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant

system

Time Shift System

eigenfunction

eigenfunctioneigenvalue

System

Back to (Classical) SystemsBack to (Classical) Systems

• Solving:

• Gives:

eigenvalue

Legendre Transform

TransformsTransforms

• 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