Stage 2: The Greedy Selection of the Best Candidate

6.3 Selective Greedy Forwarding (SelG): Protocol Design

6.3.2 Stage 2: The Greedy Selection of the Best Candidate

the next subsection.

ST Ai. Assume Z(I_i^k) is an indicator variable such that,

Z(I_i^k) =











1 If I_i^k is currently being used for communication

0 Otherwise

According to the protocol interference model, only one interface can transmit among the interfaces that interfere with each other. Consider an interface I_i^k ∈Γ_i. Then either I_i^k can transmit or a set of interfaces from H(I_i^k) can transmit, where no two interfaces interfere with each other.

Theorem 6.2. For every interface I_i^k ∈Γ_i of ST A_i, following condition must hold true for minimum interference data communication,

Z(I_i^k) + X

I_j^m∈H(I_i^k)

Z(I_j^m)≤1.11364α (6.1)

where α is the beam-width of the interfaceI_i^k, expressed in radians.

Proof. The proof extends the concept provided in Lemma 1, given in [53], for protocol interference model in multi-interface mesh network. According to the protocol interference model, two interfaces interfere with each other, if they are at-most q×Rc distance apart.

Consider the following two cases,

Case I. AssumeI_i^ktransmits. In this case, none of the interfaces inH(I_i^k) can transmit.

Therefore,

Z(I_i^k) = 1 X

I_j^m∈H(I^k_i)

Z(I_j^m) = 0

This satisfies the inequality given in (6.1).

Case II. Assume I_i^k does not transmit. In this case, two interfaces from H(I_i^k) can transmit, provided that they are at-leastq×R_cdistance apart. Let us assume that there exists S number of interfaces in H(I_i^k), who are at least q ×Rc

distance apart. Considering the fact that all interfaces inH(I_i^k) must be within the disk centered at I_i^k and with radius q×R_c (based of the protocol model consideration, interference range is at-mostq×Rc), the problem of finding S can be reduced to the circle packing problem discussed in [218]. Let us consider

Interface I_i^k Beam direction for

MP_i qxR_c

qxR_c MP_u

MP_v 2

1.5xqxRc

Figure 6.2: Interference characterization

Fig. 6.2. If ST A_i does not use interface I_i^k, then the interfaces of ST A_u and ST Av can transmit simultaneously if they are at-least q×Rc distance apart.

Without considering the directionality of the interfaces, the problem of “finding maximum number of interfaces that can simultaneously transmit”, is similar to the problem of “finding maximum number of non-overlapping circles of radius (q×R_c)/2 that can be packed within a circle of radius 1.5×q×R_c”. From [218], the value is 7. Therefore, for uniform distribution of the directional beams of the interfaces,

S≤α

2π ×7≈1.11364α

whereαis the beam-width of the interfaceI_i^k, expressed in radians. As a result, Z(I_i^k) = 0

X

I_j^m∈H(I_i^k)

Z(I_j^m)≤1.11364α

This follows the theorem.

The interference characteristics for multi-interface mesh networks, as derived in equation (6.1), gives a theoretical upper bound for selecting maximum number of interfaces for simultaneous interference-free communications. The bound is free of the q value, which indicates that interference range does not affect interface selection, until the set of interfering interfaces (H(I_i^k)) are identified properly. IEEE 802.11sMCCA advertisement messageprovides the support for populating the set of interfering interfaces in every DTIM

interval. In practice interference-free interface selection can not be guaranteed because of the approximation in the protocol interference model². However, as shown in [214], with proper power adjustments, protocol interference model can capture interference with minimum error, and therefore, equation (6.1) can provide the bound for the minimum interference interface selection. Accordingly, equation (6.1) works as a constraint for the greedy selection to support the minimum interference mesh path selection. The next section provides the utility function for the greedy selection strategy.

Design of the Utility Function

The utility function is designed based on the effect of the current network dynamics (channel condition and interference) over the path metric information, to find out the set of PNCs. Let C^k_ij denote the link metric value between ST A_i and ST A_j through interface I_i^k ∈Γi, whenST Aj was selected as a PNC. Similarly assume thatP^k_ij denotes the path metric during the PNC computation. The utility function for the greedy selection of ST Aj, denoted as U(ST A_j, I_i^k) is expressed as follows;

U(ST A_j, I_i^k) =

























1 + v u u t

C^k_ij

2

−(^Ckij)²

C^k_ij2



×P^k_ij; ifC^k_ij <C^k_ij



1− v u u t

C^k_ij

2

−(^Ckij)²

C^k_ij2



×P^k_ij; ifC^k_ij >C^k_ij P^k_ij; ifC^k_ij =C^k_ij

(6.2)

The utility function estimates the path metric value based on the statistical dispersion in the current link metric value. A negative dispersion in the link metric value indicates possible cost reduction in the path metric value. Similarly, a positive dispersion in the link metric indicates cost inflation in the path metric. It can be noted that based on the stored information, the values ofC^k_ij andP^k_ij are obtained as follows;

C^k_ij =L_i[ST Aj, I_i^k] P^k_ij =Fi[ST Aj, I_i^k]

2Protocol interference model can not handle cumulative interference [214]. Therefore, the set of interfaces that introduce interferences due to cumulative power effect, can not be identified properly through MCCA broadcast messages. As a result, small error is observed in the set of interfering interfaces.

The utility function along with the interference constraint is used for the greedy selection ofhST A_j, I_i^ki as the next-hop forwarder, discussed next.

The Greedy Selection Strategy

The greedy selection is based on a constrained optimization problem, formulated as follows.

Problem 6.1.

min

hST Aj,I_i^ki

U(ST A_j, I_i^k) s.t. Z(I_i^k) + X

e⁰∈H(I_i^k)

Z(e⁰)≤1.11364α

ST A_j ∈Fi

I_i^k∈Γi

The constrained optimization given in Problem 6.1 returns the hST A_j, I_i^ki pair that provides minimum utility maintaining the interference constraint. The convergent time of the constrained optimization is analyzed using following theorem.

Theorem 6.3. Problem 6.1 can be solved within O(|Fi| × |Γ_i|) time complexity.

Proof. Let Y_i = hST A_j, I_i^ki represent the optimal solution of Problem 6.1 for ST A_i. Instead of allowing Y_i to take specific (MP, interface) value, let us define Y_i to be a vector in {O1,O2, ...,Ok}, where eachOj takes a possible solution, and is represented by an integer value from [0...k]. Let this integer is chosen as follows: consider an equilateral simplex Σ_k in R^k−1 with vertices b1, b2, ..., b_k. Let c_k = ^b1+b2+...+b_k ^k be the centroid of Σ_k, and let Oj = b_j−c_k for 1 ≤ j ≤k. Then problem 6.1 is represented as an integer semidefinite program with finite solution bound. From [219], the problem can be solved within O(k) time complexity by exploring all the elements in the vector {O1,O2, ...,Ok}.

In the present scenario,k=|Fi| × |Γ_i|. This follows the theorem.

Nevertheless in practice, the numeric value of the bound O(|Fi| × |Γ_i|) is very less (maximum within the order of two digits). So the running time for the problem is significantly low. The above selection procedure returns the next-hop candidate from the set of PNCs, and corresponding interface to be used, considering the link metric and its dispersion, as well as the interference constraint.

6.3.3 Extension of the SelG Protocol for Path Selection Between Two