This work considers‘Protocol Interference Model’ [186,187] to capture interference among the contending interfaces. LetRcbe the communication range of an interface. According to the protocol interference model, two interfaces interfere with each other if their physical distance is less thanρ×Rc, where the value ofρdepends on the physical carrier sensing and the packet reception model. The supports of null steering [188] and capture effects [189] in modern wireless hardwares limit the interference such that in general,ρ≤2 [190]. Protocol interference model is an approximation of the physical interference model. However, it is distributed in nature, and can be used locally at every interface with the local neighborhood information [187]. In [191], the authors have shown that with proper parameter settings (such as the transmit power and the capture threshold), the protocol interference model can perform as good as the physical interference model. Further if ρ ≤ 2, the protocol interference model can be used correctly to capture the interference
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4 (b)
(c)
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3
4 2
1 (a) 3
Figure 3.2: (a) Same Receiver (b) Different Receiver, One-sided Interference (c) Different Receiver, Both-sided Interference
information with the two-hop broadcasting and the network overhearing.
A WMN can be represented as a directed graphGnet(Vnet,Enet), whereVnetis the set of directional interfaces, and Enet is the set of communication links between them. In case of directional antenna based communications, an edge e(Ii,Ij)∈Enet is used to connect two nodes, where {Ii,Ij} ∈Vnet, if the transmitter beam of Ii and the receiver beam of Ij are calibrated towards each other. In this case, Ii can directly communicate withIj. Definition 3.1. A Communication GraphGcom(Vcom,Ecom)⊆ Gnet(Vnet,Enet), where Vcom=Vnet and e(Ii,Ij)∈Ecom if Ii is communicating with Ij, denoted asIi → Ij. Definition 3.2. Given a network graph Gnet(Vnet,Enet) and the corresponding commu- nication graph Gcom(Vcom,Ecom), the Interference Graph Ginf(Vinf,Einf) for the net- work can be constructed as follows. Let Vinf =Vcom. There exist a directional edge from Ij ∈Vinf toIk∈Vinf if the transmissions of Ij interfere with the transmission of Ik.
It can be noted that interference is not symmetrical [192], i.e. Ij interfering with Ik
does not indicate thatIkis also interfering withIj. Further, the receiver forIj andIkmay be same or different. In Figure 3.2(a), interface 1 and interface 3 transmit to the same interface 2. Therefore interface 1 and interface 3 interfere with each other. There would be directed edges from 1 to 3 as well as 3 to 1 in the interference graph. In Figure 3.2(b), interface 1 and interface 4 transmit to different receivers, however, the transmissions of interface 1 affect transmissions of interface 4 only. So, there would be directed edge only from 1 to 4 in the corresponding interference graph. There would be no edge from 4 to 1.
Similarly in Figure 3.2(c), though interface 1 and interface 4 transmit to different receivers, their transmissions interfere with each other. Accordingly there would be directed edges from both 1 to 4 and 4 to 1 in the interference graph. This interface based interference graph is different from the link interference graph [193] used in the existing literatures, as
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(a) (b)
Figure 3.3: (a) Interference Graph (b) Channel Sharing Graph and Maximal Cliques discussed in Chapter 2, where the edges in a communication graph represent the vertices in the interference graph and there exists an edge if two links interfere with each other.
Interface based interference graphs are efficient for directional communications in a multi- interface directional mesh network because of the following reasons.
For multi-interface network, interference is interface specific rather than node specific. An interface can either be in transmit or receive mode. Only the interfaces that are in transmit mode, contribute to the network interference.
Interface based interference is easy to capture in the MCCA based mesh network.
Let every mesh STA broadcast the transmissions and the interference informations of its two-hop neighbors through the MCCAOP advertisement messages. As a result every mesh STA has three hop communication information for every interfaces, from where the local interference graph up to three hops can be constructed using the protocol interference model, through a method similar to [187]. Later it would be shown that three hop interference graph is sufficient to design an effective channel access mechanism.
Interface based interference model can easily capture the asymmetric nature of the interference from the directed interference graph.
Network interference is an important parameter that characterizes the maximum network capacity as well as the maximum throughput guarantee for the traffic flows. Let Ginf be the interference graph. The corresponding undirected graph ofGinf is constructed by removing the directionality of the edges. Then the corresponding graph is called Channel Sharing Graph, Gch. The reason behind removing the directionality of edges can be explained using an example. Let interfaceIj interferes with interface Ik, however, interface Ik does not interfere with Ij. This indicates that when Ik transmits, Ij should
Rc Rc
Figure 3.4: Worst case interference scenario
remain silent. Otherwise it would affect the transmissions of Ik. Similarly, when Ij
transmits, Ik should remain silent as it would affect its own transmissions. Let both the interfaces are of equal priority. Then the total channel share should be divided equally between the two interfaces to have a minimum interference channel access.
Figure 3.3(a) shows an interference graph, and Figure 3.3(b) shows the corresponding channel sharing graph by removing the directionality of the edges. The maximal cliques in a channel sharing graph are used to characterize the maximum throughput that an edge in a communication graph can achieve. A maximal clique in a channel sharing graph indicates that all the interfaces in that clique interfere with each other, and therefore the total channel share should be divided among all those interfaces. An important proposition from the channel sharing graph can be derived, stated as follows.
Proposition 3.1. Let C be a maximal clique in Gch. Then, X
Ij∈C
X
∀FIj
λ(FIj)
≤η (3.1)
where FIj is a traffic flow through the interface Ij, and λ(FIj) is the data rate for that flow. η is the maximum network capacity. Equation (3.1) is also called the ‘clique constraint’, and η is called the ‘clique capacity’. As an example, the total channel share for the interfaces 1, 2, 3 and 4 from Figure 3.3(b) should be less than the clique capacity.
The MCCA reservation procedure uses MCCAOP advertisement messages that broadcast two-hop interference information (both at the sender and at the receiver) to the neighborhood. Therefore, every mesh STA can find out the three hop interference graph using MCCAOP advertisement messages and the protocol interference model. Following theorem bounds the clique constraint in three hop neighborhood.
Theorem 3.1. Let C be a maximal clique in Gch. Then any two interfaces Ij ∈ C and Ik ∈C can be at maximum three-hop away.
Proof. It has been assumed that based on the protocol interference model, interference range is ρ×Rc, where ρ ≤2. Then for the interface based interference graph, the worst
case interference scenario is shown in Figure 3.4. Accordingly, the receiver of the interface Ik can be at most in two hop distance from Ij. This indicates thatIj and Ik are at most in three hop distance.
Following corollary can be derived directly from Theorem 3.1.
Corollary 3.1. Let Ij ∈Vcom. Then Ij can compute all the maximal cliques to which it belongs with three-hop neighbor information.
Proof. Let C be a maximal clique and Ij ∈ C. Then ∀Ik ∈ C;Ik 6= Ij, Ij and Ik are at most three hop distance away (according to Theorem 3.1). This indicates thatIj can computeC with three hop neighbor information.
Therefore, broadcasting two hop neighbor information with MCCAOP advertisement message helps the mesh STAs to characterize the interference constraints independently at every interface. This can capture the asymmetric behavior of the interference as well as can model the real-time interference scenario, where the interference range is more than the communication range. As discussed earlier, modern technologies, such as the adaptive power control [194], null steering [188] and the capture effect [189] are used to limit the interference range within twice the communication range. In this context, the proposed model can effectively characterize the interference in a multi-interface mesh network.
Once the interference information is characterized, it can be used to find out the required traffic demand at every interface. For this, a centralized optimization problem is formulated to find out the optimum traffic demand, as discussed in the following section.