A routing protocol using a reliable and high-throughput path metric for multi-hop multi-rate ad hoc networks

Abstract

In this paper, a high-throughput routing protocol for multi-rate ad hoc networks using lower layer information is proposed. By choosing the route with the minimum value of the proposed “Route Assessment Index” metric which has the form of entropy function, the selected route is ensured to have high throughput and link reliability among route candidates. Link bottleneck is avoided in the chosen route; hence, the packet drop rate due to buffer overflow is alleviated. Furthermore, an effective route discovery strategy is also introduced along with new routing metric. The correctness of the proposal is proven, and the simulation results show that our new metric provides an accurate and efficient method for evaluating and selecting the best route in multi-rate ad hoc networks.

Appendix: Properties of entropy function applied for the route assessment

Let X be a discrete random variable with alphabet χ and probability mass function $p(x) = \Pr \{ X = x\} ,x \in \chi$. The discrete random variables have a finite number of possible values (x ₁, x ₂, ..., x _n ) with probabilities (p ₁, p ₂, ..., p _n ), respectively, such that p _i  ≥ 0,i = 1,2, ...,n and $\sum\nolimits_{i = 1}^n {{p_i} = 1}$. The entropy of that set is defined as

$$ \label{equ_pro1_app1} H({p_1},{p_2},\ldots,{p_n}) = \sum\limits_{i = 1}^n {{p_i}} \log \frac{1}{{{p_i}}} = - \sum\limits_{i = 1}^n {{p_i}} \log {p_i} $$

(11)

The function H(p ₁,p ₂,...,p _n ) (or H(P)) is a nonnegative, continuous, and symmetric function [7]. In the paper, the observing sets are the set of the coefficient {α _i } for route candidates instead of {p _i }.

Property 1

(Maximality)

The entropy H(p ₁,p ₂,...,p _n ) is maximum when all the probabilities are equal.

$$ \label{equ_pro2_app1} H(P) \!=\! H({p_1},{p_2},\ldots,{p_n}) \le H\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right) \!=\! \log n, $$

(12)

with equality if and only if p _i  = 1/n, ∀ i = 1,2,...,n.

Proof

The proof is straightforward following the information inequality theorem in [7] as follows: The relative entropy of two distributed function satisfies

$$ \label{equ_pro3_app1} D(p\parallel q) = \sum\limits_{x \in \chi } {p(x)\log \frac{{p(x)}}{{q(x)}}} \ge 0{\rm{ }} $$

(13)

with equality if and only if p(x) = q(x) for all x. Also, from Eq. 11, we have

$$ H\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right) = - \sum\limits_{i = 1}^n {\frac{1}{n}} \log \frac{1}{n} = \log n $$

Applying the Eq. 13 for two distributed function H(P) and $H(\frac{1}{n},\frac{1}{n},\ldots, \frac{1}{n})$, we have

$$ D\left(H(P)\parallel H\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right)\right) = \log n - H(P) \ge 0 $$

Therefore, H(P) ≤ logn , the maximality properties is proven. □

It also can be shown that in the general case, for two sets (p ₁,p ₂,...,p _n ) and (q ₁,q ₂,...,q _n ) that satisfy q ₁ ≥ q ₂ ≥ ... ≥ q _n , also p ₁ = q ₁ + Δ _u , p ₂ = q ₂ − Δ _u , and p _i  = q _i , ∀ i = 3,4...,n, we have:

$$ \label{equ_pro4_app1} H(p) \le H(q) $$

(14)

To prove this property, we have a lemma as follows:

Lemma 2

(Recursive property) For n ≥ 3, we have

$$\begin{array}{rll} H\left( P \right) &=& H\left( {{p_1} + {p_2},{p_3},\ldots,{p_n}} \right) \\ && + \left( {{p_1} + {p_2}} \right)H\left( {\frac{{{p_1}}}{{{p_1} + {p_2}}},\frac{{{p_2}}}{{{p_1} + {p_2}}}} \right) \end{array}$$

(15)

Proof

The proof is straightforward and had been shown in [7]. □

Using Lemma 2, we have

$$\begin{array}{rll}H\left( P \right) &=& H\left( {{p_1} + {p_2},{p_3},\ldots,{p_n}} \right) \\&& + \left( {{p_1} + {p_2}} \right)H\left( {\frac{{{p_1}}}{{{p_1} + {p_2}}},\frac{{{p_2}}}{{{p_1} + {p_2}}}} \right) \\ &=& H\left( {{q_1} + {q_2},{q_3},\ldots,{q_n}} \right) \\&&+ \left( {{q_1} + {q_2}} \right)H\left( {\frac{{{q_1} + {\Delta _u}}}{{{q_1} + {q_2}}},\frac{{{q_2} - {\Delta _u}}}{{{q_1} + {q_2}}}} \right), \end{array}$$

(16)

and

$$\begin{array} {rll}H\left( Q \right) &=& H\left( {{q_1} + {q_2},{q_3},\dots,{q_n}} \right) \\&& + \left( {{q_1} + {q_2}} \right)H\left( {\frac{{{q_1}}}{{{q_1} + {q_2}}},\frac{{{q_2}}}{{{q_1} + {q_2}}}} \right) \end{array}$$

(17)

Therefore, from Eqs. 16 and 17, we need to prove that $H\left( {\frac{{{q_1} + {\Delta _u}}}{{{q_1} + {q_2}}},\frac{{{q_2} - {\Delta _u}}}{{{q_1} + {q_2}}}} \right) \le H\left( {\frac{{{q_1}}}{{{q_1} + {q_2}}},\frac{{{q_2}}}{{{q_1} + {q_2}}}} \right)$.

Denote $x \buildrel \Delta \over = \frac{{{q_1}}}{{{q_1} + {q_2}}}$, then x ≥ 0.5 since q ₁ ≥ q ₂. We have

$$\begin{array}{lll}&&{\kern-7pt}H\left( {\frac{{{q_1}}}{{{q_1} + {q_2}}},\frac{{{q_2}}}{{{q_1} + {q_2}}}} \right)\nonumber\\ &&= H\left( {x,1 - x} \right) \\ &&= - x\log \left( x \right) - \left( {1 - x} \right)\log \left( {1 - x} \right) \\&&\buildrel \Delta \over = f\left( x \right) \end{array}$$

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f(x) is a decreasing function for 0.5 ≤ x ≤ 1, and then $f\left( x \right) \ge f\left( {x + \frac{{{\Delta _u}}}{{{q_1} + {q_2}}}} \right) = H\left( {\frac{{{q_1} + {\Delta _u}}}{{{q_1} + {q_2}}},\frac{{{q_2} - {\Delta _u}}}{{{q_1} + {q_2}}}} \right)$. Hence, the general case is proven. It means that for two sets with the same number of elements in each set, the set with more resemble elements will have the higher entropy outcome.

Property 2

(Uniform Distribution)

Suppose we have $\phi (n) = H(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}),n \ge 2,n \in \textbf{N}$, then $\frac{{\phi (n)}}{n} \ge \frac{{\phi (n + 1)}}{{n + 1}}$.

Proof

The proof is straightforward using the result of Property 1. We have φ(n) = logn and φ(n + 1) = log(n + 1). Also, $\mathop {\lim }\limits_{n \to \infty } \frac{{\log n}}{n} = 0$. Therefore, the entropy function will reduce when the number of elements in the set increases (n large) and the theorem is proven. It means that for sets with the different number of homogeneous elements, the higher number of elements a set has, the lower entropy outcome per element that set gets. □