6.3 Selective Greedy Forwarding (SelG): Protocol Design
6.3.4 The Size of the Set of Potential Forwarders
The performance of the SelG protocol relies on the size of the initial set of PNCs. If the size is too small, then the behavior of the SelG protocol becomes similar to the proactive protocols. On the other hand, large number of entries in the set of PNCs may trap the protocol to the local best solutions (the local optimum based on the greedy selection), which is not the globally best solution, with minimum path metric value and interference constraints. Therefore, the size of the set of PNCs need to be chosen carefully for the best performance of the protocol.
In the proposed protocol, the size of the set of PNC is limited based on the dispersion in the path metric values received from the individual neighbors. AssumeST Aihas the set of PNCsFi. LetST Ai receives a PPREQ (or a PREQ in case of on-demand path selection between two mesh STAs) fromST Aj through the interface Iik ∈Γi. Then hST Aj, Iiki in included inFi, if and only if following condition is satisfied,
v u u u u t
Pkij 2
− Pdkij
2
dPkij2 ≤δ (6.4)
where,
dPkij = min{Pri`|hST A`, Iiri ∈Fi}
andδ is a constant, called the‘forwarder percentage’. The value ofδ depends on the path loss components, and can be decided by the service provider based on the deployment scenarios. The effect ofδ is analyzed in this chapter through simulation and experimental results.
6.3.5 Routing Efficiency of SelG
To ensure stable mesh paths, a path selection protocol should support following characteristics,
1. isotonic path selection, and 2. loop-free forwarding.
The concept of isotonicity is introduced in [220] and [221]. Assume two mesh paths P1 and P2 and two links L1 and L2. The isotonicity of a path selection (or routing) metric is defined as follows.
Z(L1+P1) <= Z(L1+P2) Z(P1+L2) <= Z(P2+L2) P2
P1
P1 P1
P2 P2
Z(P1) <= Z(P2)
A B
C’
L2
A B
C L1
A B
Figure 6.3: Explanation of Isotonicity
Definition 6.2. A path selection (or routing) decision metric Z(.) is called isotonic if Z(P1)≤ Z(P2)implies bothZ(L1⊕P1)≤ Z(L1⊕P2)andZ(P1⊕L2)≤ Z(P2⊕L2), where the operator ⊕ denotes addition of a link to a path.
The concept of isotonicity is explained using a figure, as shown in Fig. 6.3. Assume there exists two paths between the nodes A and B, P1 and P2 respectively. Further assume that the path metric value for P1 is less than the path metric value for P2. Let us add a link L1 from nodeC to nodeA. If isotonicity is satisfied, the path metric value from C toB through the path P1 should be less that the path metric value through the pathP2. Similarly, if the link is added to nodeB, the isotonicity of the path metric is to be satisfied also.
Theorem 6.4. The utility function, given in equation (6.2), for the decision making in the SelG protocol, supports isotonicity.
Proof. The airtime link metric value is additive in nature, and therefore it supports isotonicity. According to Algorithm 6.2, before broadcasting a PPREQ message, a mesh STA, ST Ai, updates the path metric value as follows,
Pkij∗= min{Pkij+Ckij|hST Aj, Iiki ∈Fi} (6.5) where Pkij∗ is the updated path metric value. Considering two sets A ⊆ R and B ⊆ R, whereR is the set of real numbers. if min{x|x∈A} ≤min{y|y∈B}, then for any ω∈R, following inequality always holds true.
min{x|x∈A}+ω≤min{y|y∈B}+ω
Based upon this inequality, it can be easily seen that Pkij∗ is isotonic. During greedy selection, let us assume the link dispersion factor be ρ. Considering Pkij∗ ≤Pwuv∗ for two candidate forwardershST Aj, Ijki andhST Av, Iuwi, following inequalities always hold true.
(1 +ρ)×Pkij∗≤(1 +ρ)×Pwuv∗ (6.6) (1−ρ)×Pkij∗≤(1−ρ)×Pwuv∗ (6.7) Therefore the utility function, as given in equation (6.2), also follows the isotonicity property.
The isotonicity property guarantees stable mesh paths in the network [220, 221], that is the mesh paths do not fall into the ping-pong effect, when the network and the channel condition is stable, and the best path at a moment can be selected unconditionally. The path is changed only when the network and the channel conditions are dispersed sufficiently to provide a better path. It can be noted that the interference constraint does not judge the quality of a path. It just gives a binary decision on whether to use a candidate or not, from the set of PNCs, such that the minimum interference mesh path selection is possible.
The candidate that gives the minimal utility value, according to the equation (6.2), is allowed to transmit only when the interference constraint is satisfied.
As discussed earlier, another important property to judge a mesh path selection protocol is to check whether loop-free property is satisfied. Following theorem shows that the proposed SelG protocol is loop-free.
Theorem 6.5. The greedy selection does not introduce a forwarding loop.
Proof. The method of contradiction is used to proof the theorem. For the notational shorthand, let us use only the next-hop mesh STA as the candidate from the set of PNCs, and imply that the corresponding interface is used. Let Pi and Ci denote the path metric and the link metric for the next hopST Ai respectively, with the implication of corresponding interfaces.
Let us assume that the greedy selection introduces a forwarding loop at ST Ai. Assume the forwarding loop be {ST Ai, ST Aj, ..., ST Ak, ST Ai}. Because of the additive and non-zero properties of the airtime metric value,
min{Pi+Ci}<min{Pj +Cj}< ...
<min{Pk+Ck}<min{Pi+Ci} (6.8)
∀i; min{Pi +Ci} ∈ R+, where R+ is the set of positive real numbers. Therefore, the inequality given in equation (6.8) is never possible. This contradicts with our assumption.
Hence, the greedy selection never introduces a forwarding loop.
6.4 Simulation Results
The proposed scheme is implemented and simulated using Qualnet Network simulator version 5.0.1 [71]. Qualnet supports IEEE 802.11s complaint HWMP protocol for the MAC layer path selection in a mesh network. The proposed SelG protocol is implemented as an extension of the HWMP protocol.