Here we present an example of bursty statistics from a different class of complex system in the area of irregular traffic flows on regular networks. Earlier papers[3][4] have investigated simple models of network dynamics. Further investigation of the phenomena reported there show that traffic flows on the model network, whose structure is static and which supports constant traffic demand, can display bursty and intermittent behaviour; an example of the time series produced is shown below (Figure 9).
Figure 9: Intermittent traffic blocking due to network dynamics
Another type of behaviour in this system shows approximately periodic, cyclic, oscillations between free flow and blocking behaviours. An example of the time series is shown below (Figure 10).
Figure 10: Cyclic traffic blocking
In the cases considered above, the role of the controlled switch is taken by clusters of packets which form and dissolve with time, resulting in a variable structure of clustered packets which impedes the traffic flow. This is illustrated in Figure 11 and in Figure 12 which show, respectively, clustering (in a region of impeded flow) and less-impeded flow, with the points on the time series diagram in Figure 10 labelled with arrows.
Figure 11: Packet bunching impeding the traffic flow
Figure 12: Less bunching at a point of easy data flow
For completeness, we now show the time series (Figure 13), the blocked packet distribution (Figure 14) and the subsequent unblocked packet distribution (Figure 15) 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 13: Packet blocking statistics, detail on a 30 by 30 grid with 39 percent loading
Figure 14: Packet bunching blocks the flow
Figure 15: Free flow has re-emerged, no packet blocking occurs
In a sense, there is no direct implementation of the controlled switch, but just as in the case of the electronic circuits considered above, the dynamics are isomorphic with those of a system containing deliberately-embedded controlled switching.