  We analyze the multihop delay of ad hoc cognitive radio networks, where the
transmission delay of each hop consists of the propagation delay and the
waiting time for the availability of the communication channel (i.e., the
occurrence of a spectrum opportunity at this hop). Using theories and
techniques from continuum percolation and ergodicity, we establish the scaling
law of the minimum multihop delay with respect to the source-destination
distance in cognitive radio networks. When the propagation delay is negligible,
we show the starkly different scaling behavior of the minimum multihop delay in
instantaneously connected networks as compared to networks that are only
intermittently connected due to scarcity of spectrum opportunities.
Specifically, if the network is instantaneously connected, the minimum multihop
delay is asymptotically independent of the distance; if the network is only
intermittently connected, the minimum multihop delay scales linearly with the
distance. When the propagation delay is nonnegligible but small, we show that
although the scaling order is always linear, the scaling rate for an
instantaneously connected network can be orders of magnitude smaller than the
one for an intermittently connected network.
