The number of possible states of the network is equal to the number of
ways in which P distinct objects (packets) can be arranged in 
slots (routing cells), one per slot. For a given
arrangement, the i-th packet can be going in one of two directions (from
to 
or vice versa). The total number of states,
, is therefore
Even for modestly sized networks 
can be very large; e.g. for an M = 16
grid with 100 packets, 
. For M = 100 and
P=6000, a case studied below, 
.
It is important to be aware that since this system operates
deterministically, its behaviour will always be periodic, with a
maximum possible period of 
. However, even for a 
network, 
is so large that it would take more than the lifetime of
the Universe to carry out a calculation in which even one complete
maximum period is executed, and so we can regard the number of states as
`effectively infinite' and the behaviour as `effectively aperiodic'. We also
tacitly assume that the period of any cycles that do occur is so long that
it does not affect any of our computer experiments.
In order to characterise the behaviour of the system, we have looked at a variety of state variables that could be used to define the state of the network as a function of discrete time (the number of token passes).
A vector 
can be used to represent the state of the entire
system on completion of the n-th token pass. This vector has 3P
discrete components; the 3k-2-th, 3k-1-th and 3k-th components
represent the x and y co-ordinates and the direction (from 
to 
or vice versa) of the k-th packet, 
. The two
spatial components range from 0 to M-1 and the direction component
takes on one of two values, 0 or 1. Note that this is not the most
compact representation of a given state (it does not reflect the fact
that each cell can only hold one packet at a time) but it has the
advantage of simplicity. It also has the necessary property that there
is a one to one mapping and invertible between states and vectors.
Scalar measures of the behaviour of the network include
, of all P packets on completion of the n-th
token pass
The results in this paper relate to 
as a function of n, since this is
convenient to calculate and is also a simple measure of the efficiency
of the system --- the higher the average age, the longer the packets
are taking to reach their destinations.