Here we present an example of bursty statistics from a different
class of VSS in the area of irregular traffic flows
on regular networks.
First we briefly recap the system
[3][4]. A square
grid of N by N cells is scanned
from the NW corner to the SE corner in a raster (or other sequence) by a
token, which alights on each cell in turn and enables development
to occur.
The grid is populated by packets numbering
a fraction f of the 
cells; the packets have set out
from transmitters on the W and S boundaries to make their way to
receivers at the E and N boundaries respectively. The fraction f may be
set by the program. When a receiver collects a packet it immediately
returns it towards the transmitter whence it came without removing
it from the grid. Thus f is a true constant of the motion and does not
change. Packets contain a record of their intended destination.
There can be only a single packet in a cell; thus a packet
cannot move into an occupied cell.
If the token alights on a cell containing a packet, the packet attempts to move in the direction of the receiver, or as close to that as possible according to a simple protocol (see Figure 13). If it cannot move in any direction it stays put. The token then passes to the next square. This is the only contingent protocol in the system, in the sense that what happens next is dependent on the arrangement of packets in the neighbourhood of the token cell, and also on the direction to the intended receiver. The protocol may easily be implemented in hardware by a VSS, as can be seen in the figure (Figure 13). Here, the individual cell sites contain the hardware switches as shown, and provide the protocol for onward routing of the itinerant packets. It would also be possible, at some increase in complexity, to regard the itinerant packets as having hardware attached which determines their onward paths. The concepts developed schematically in the figure provide, in our opinion, a firm demonstration that our data network is a true variable structure system, in exactly the same sense as in the other examples.
Figure 13: The local VSS for the network protocol
Here, the controlled switch selects each of the neighbouring cells in turn, in the order selected by the global variable which has been set by the programmer. On finding the first empty cell, the switch moves the packet into the empty cell leaving the source cell vacant. We have not gone into the precise details of the electronics needed to do this; nevertheless it is a true variable structure hardware arrangement.
So the network study is a simulation of another kind of
variable structure system.
The state vector, of dimension 
, consists of the
positions and attributes of the
pattern of populated
squares on the grid. It evolves with time,
and a single measure of the time development is taken to be
the average lifetime of all the packets on the net
since the last time they were reflected from a receiver.
It is this variable which is plotted on the vertical axes
in Figure 14, Figure 15, and Figure 18.
The packets have to move in one of the directions
set by the protocol, which is perfectly deterministic. However, the
system is sufficiently large [3] that individual global
(as opposed to local patterns in the region of the token)
patterns will most probably never reappear in reasonable amounts of runtime.
Thus the pattern and attributes of the occupied cells
governing the development of the state vector
between token passes almost certainly never repeats. It is truly variable
and chaotic, and is a direct consequence of the large number of possible
values of the
state vector. The variable structure is also represented by the global patterns
of packets on the net, affecting the development directly. The global
development is the aggregate of decisions set by the individual
VSS as described above.
Earlier papers [3][4] have investigated simple models of the network dynamics. Preliminary investigation of the phenomena reported there shows that traffic flows on the model network, whose structure is static and which supports constant traffic demand, displays bursty and intermittent behaviour for certain values of f; a further example of the statistically self-similar time series produced is shown in Figure 14.
Figure 14: Intermittent traffic blocking due to network dynamics
Another type of behaviour in this system is approximately periodic, showing cyclic oscillations between free flow and blocking behaviours. An example of the time series is shown below (Figure 15).
Figure 15: Cyclic traffic blocking on the network at a different load factor to that in
figure 14
Clusters of packets form and dissolve with time, resulting in a global variable structure of clustered packets which impedes the traffic flow. This is a kind of collective variable structure which has arisen out of the aggregate of the behaviour of all the individual controlled switch decisions described above for the individual token cells.
This is illustrated in Figure 16 and in Figure 17 which show, respectively, clustering (in a region of impeded flow) and less-impeded flow, with the points on the time series diagram in Figure 14 labelled with arrows.
Figure 16: Packet bunching impeding the traffic flow at a peak in figure 15
Figure 17: Less packet bunching at a point of easy data flow at a minimum in figure 15
An even more striking example of clumping and free flow is provided in the time series (Figure 18), together with the blocked packet distribution (Figure 19) and the subsequent unblocked packet distribution (Figure 20), for a 30 by 30 grid with 39 percent of the sites occupied by packets. The qualitative behaviour seen here is universal and does not depend on details of the size of the grid or the position of the receive and transmit sites. We always see critical levels of loading at which bursty behaviour or periodic behaviour sets in.
Figure 18: Packet blocking statistics, detail on a 30 by 30 grid with 39 percent loading
Figure 19: Packet bunching at a maximum latency (packet blocking) in figure 18.
Figure 20: The normal packet distribution for unblocked flow in figure 18.
It is interesting in this example that local VSS behaviour has resulted in a kind of stochastic variable structure across the network. We are observing collective behaviour of the dynamics resulting in global variable structure, that we did not set out with the intention of engineering when we wrote the protocol. Of course, there are many conditions for which the intermittent behaviour is not observed and the global VSS does not emerge; nevertheless the local implementation of the VSS always pertains.