The third example shows that the natural strategy of assigning priorities to the packets at random is not effective either.
Proof: As in Example 1, the network consists of many copies of a
subnetwork. Each subnetwork is constructed so that 
. A
subnetwork consists of a linear chain of length d, with loops of
length 
between adjacent nodes (see Figure 10).
The packets are broken into 
groups numbered 0 through
of 
packets each. The packets in group i use
the linear chain for 
steps and then use 
loops
as their path. As in the previous example, we assume that queues have
unlimited capacity and that at each step a node can send a single
packet.
If the random ranks are assigned so that the packets in group i have
smaller ranks than the packets in groups with larger numbers, then the
packets in group i delay the packets in group i+1 by 
steps. Thus the last packet experiences an 
delay.
Once again the ranks of the packets must have a specific order, which
can be shown to happen with high probability given enough copies of
the subnetwork. As in Observation 4.1, it is not hard to
show this requires 
.
