Periodic Systems:

Periodic Example:

This is an example of a periodic system. Two nodes just start sending traffic between themselves and the system effectively runs in an infinite loop, never converging. Systems that meet this criteria are periodic.

Example of a periodic system:

0.0 1.0 0.0

0.5 0.0 0.5

0.0 1.0 0.0

Finding the Periodic System Steady-State:

When this system is multiplied by itself, it cycles between two different matrices, representing the transition probabilities from each node. The cycles prevent the matrix from converging.

Cycle #1:

0.5 0.0 0.5

0.0 1.0 0.0

0.5 0.0 0.5

Cycle #2:

0.0 1.0 0.0

0.5 0.0 0.5

0.0 1.0 0.0

Feedback Loops:

The cycles can be broken by adding a small feedback loop into the system. A small percentage of the time and node can go to itself.

0.001 0.999 0.0

0.4995 0.001 0.4995

0.0 0.999 0.001

The steady state of the system can now be calculated:

These solutions match the algebraic solutions for the equations exactly. However, adding the feedback loops can significantly increase the number of iterations required to converge on the steady state. It is apparently unknown what combinations of feedback loops produce the minimum error and fastest convergence.

0.25 0.5 0.25

0.25 0.5 0.25

0.25 0.5 0.25