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.
The basic idea of opportunistic spectrum access is to achieve spectrum efficiency and interoperability through a hierarchical access structure with primary and secondary users [1]. Secondary users, equipped with cognitive radios [2] capable of sensing and learning the communication environment, identify and exploit instantaneous and local spectrum opportunities without causing unacceptable interference to primary users [1].
In this paper, we focus on the connectivity and multihop delay of ad hoc cognitive radio networks. Due to the hierarchical structure of the spectrum sharing, these issues are fundamentally different from their counterparts in the conventional homogeneous networks. In particular, even in a static secondary network, the communication links are dynamic due to the spatial and temporal dynamics of the primary traffic. As a consequence, the connectivity of the secondary network depends not only on its own topological structure, but also on the topology, traffic pattern/load, and interference tolerance of the primary network. The multihop delay in the secondary network consists of not only the propagation delay but also the waiting time at each hop for the availability of the communication channel, i.e., the occurrence of a spectrum opportunity offered by the primary network. It is this interaction with the primary network that complicates the analysis of the connectivity and multihop delay of the secondary network.
Our technical approach rests on theories of continuum percolation and ergodicity by adopting a two-dimentional Poisson model for both the secondary and the primary networks. A disk model for signal propagation and interference is used as a starting point, which allows us to highlight the fundamental interactions between the primary and the secondary networks without delving into potentially intractable details.
We first analytically characterize the connectivity of the secondary network, where the connectivity is defined by the finiteness of the minimum multihop delay (MMD) between two randomly chosen secondary users. Specifically, the network is disconnected if the MMD between two randomly chosen secondary users is infinite almost surely (a.s.). The network is connected if the MMD between two randomly chosen secondary users is finite with a positive probability.
Under the Poisson model, the key parameter that characterizes the topological structure of the secondary network is the density λ S of the secondary users. For a given transmission power and interference tolerance of the primary network, the key parameter that characterizes the impact of the primary network is the density λ P T of the primary transmitters that represents the traffic load of the primary network. The connectivity of the secondary network can thus be characterized by a partition of the (λ S , λ P T ) plane as shown in Fig. 1. Specifically, we show that when the temporal dynamics of the primary traffic is sufficiently rich (for example, independent realizations of active primary transmitters and receivers across slots), whether the secondary network is connected depends solely on its own density λ S and is independent of the density λ P T of the primary transmitters. In other words, no matter how heavy the primary traffic is, the secondary network is connected, either instantaneously or intermittently, as long as its density λ S exceeds the critical density λ c of a homogeneous network (i.e., in the absence of the primary network). Note that when λ S > λ c , there is a.s. a unique infinite connected component in the secondary network formed by topological links connecting two secondary users within each other’s transmission range. We further show that for any two secondary users in this infinite topologically connected component, the MDD is finite a.s.
While the secondary network is connected and the MDD is finite with positive probability whenever there are sufficient topological links (i.e., λ S > λ c ), there may not be sufficient communication links to make the network instantaneously connected at any given time. The latter is determined by the traffic load of the primary network. As illustrated in Fig. 1, for any given density λ S of the secondary network with λ S > λ c , there exists a maximum density λ * P T (λ S ) of the primary transmitters beyond which the secondary network is only intermittently connected. When intermittently connected, the secondary network has no infinite connected component formed by communication links at any given time. Messages can only traverse the topological path connecting two secondary users by making stops in between to wait for spectrum opportunities.
It is thus natural to expect that the MDD will behave differently in an instantaneously connected secondary network as compared to an intermittently connected secondary network. Indeed, we show in this paper that the scaling behavior of the MDD with respect to the sourcedestination distance is starkly different depend
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