An overview of local capacity in wireless networks

Abstract

This article introduces a metric for performance evaluation of medium access schemes in wireless ad hoc networks known as local capacity. Although deriving the end-to-end capacity of wireless ad hoc networks is a difficult problem, the local capacity framework allows us to quantify the average information rate received by a receiver node randomly located in the network. In this article, the basic network model and analytical tools are first discussed and applied to a simple network to derive the local capacity of various medium access schemes. Our goal is to identify the most optimal scheme and also to see how does it compare with more practical medium access schemes. We analyzed grid pattern schemes where simultaneous transmitters are positioned in a regular grid pattern, ALOHA schemes where simultaneous transmitters are dispatched according to a uniform Poisson distribution and exclusion schemes where simultaneous transmitters are dispatched according to an exclusion rule such as node coloring and carrier sense schemes. Our analysis shows that local capacity is optimal when simultaneous transmitters are positioned in a grid pattern based on equilateral triangles and our results show that this optimal local capacity is at most double the local capacity of ALOHA based scheme. Our results also show that node coloring and carrier sense schemes approach the optimal local capacity by an almost negligible difference. At the end, we also discuss the shortcomings in our model as well as future research directions.

Appendix

1.1 A.1 Locating the starting point z on the closed curve bounding the reception area

The SIR S _i (z) at point z should be greater or at least equal to β. We assume that it is equal to β. Therefore, at point z, we have

$$\frac{|z-z_{i}|^{-\alpha}}{\underset{j\neq i}{\sum}|z-z_{j}|^{-\alpha }}=\beta. $$

Our aim is to find the coordinates of point z which satisfy the above relation. To simplify computation of point z on the closed curve, bounding reception area of transmitter i located at z _i =(x _i ,y _i ), its y coordinate can be fixed such that z=(x,y)=(x,y _i ). This reduces the above equation to

$$ \frac{|x-x_{i}|}{\underset{j\neq i}{\sum}|z-z_{j}|}-\beta=0. $$

(10)

Equation (10) is a function of variable x and can be solved using Newton’s Method.

Remark

Newton’s Method: Given a function f(x) and its derivative f′(x), begin with a first guess x ₀. Provided the function is reasonably well-behaved, a better approximation x ₁ is $:=x_{0}-\frac{f(x_{0})}{f'(x_{0})}$. The process is repeated until a sufficiently accurate value is reached:

$$ x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}. $$

(11)

From (10)

$$f(x)=\frac{|x-x_{i}|^{-\alpha}}{\underset{j\neq i}{\sum }|z-z_{j}|^{-\alpha}}-\beta. $$

We assume that $g=r_{i}^{-\alpha}$ and $h=\underset{j\neq i}{\sum }r_{j}^{-\alpha}$ where r _i =|x−x _i | and r _j =|z−z _j |.

Newton’s Method requires the derivative of f(x) which is computed as below.

$$\begin{aligned} &{\frac{d}{dx}g=-\alpha\frac{x-x_{i}}{r_{i}^{\alpha+2}}}\\ &{\frac{d}{dx}h=\underset{j\neq i}{\sum}-\alpha\frac {z-z_{j}}{r_{j}^{\alpha+2}}} \end{aligned}$$

The first derivative of the function f(x) is,

$$\begin{aligned} f'(x) & =\frac{1}{h^{2}}\biggl(h\frac{d}{dx}g-g \frac{d}{dx}h\biggr) \\ & =\frac{ ((\underset{j\neq i}{\sum}r_{j}^{-\alpha})(-\alpha\frac {x-x_{i}}{r_{i}^{\alpha+2}})-(r_{i}^{-\alpha})(-\alpha\frac {z-z_{j}}{r_{j}^{\alpha+2}}) )}{\frac{1}{ [\underset{j\neq i}{\sum}r_{j}^{-\alpha} ]}} \end{aligned}$$

Newton’s Method also requires first approximation of the root, x ₀. An approximate value, closer to the actual root, can significantly reduce the number of iterations in Newton’s Method.

In all three types of grid networks, the transmitter closest to i, hereafter referred to as j, lies at distance d and hence can give the best estimate x ₀. For first approximation x ₀, we can ignore all other transmitters in the network. In this case,

$$\begin{aligned} &\frac{|z-z_{i}|^{-\alpha}}{|z-z_{j}|^{-\alpha}} \geq\beta \\ &\frac{r_{i}^{-\alpha}}{r_{j}^{-\alpha}} \geq\beta:i\neq j, \end{aligned}$$

where r _i =|z−z _i | and r _j =|z−z _j | and we get

$$\frac{r_{j}}{r_{i}}\leq(\beta)^{\frac{1}{\alpha}} $$

The location of transmitters i and j and point z in the plane form three corners of a triangle with angle θ equal to 0 in case of square and triangular grid and π/6 radians in case of hexagonal grid layout. Figure 7 shows the location of transmitters z _i and z _j , point z and distances r _i , r _j and d. Using above relation between r _i and r _j and the Law of Cosines, we get the solution of r _i as

$$r_{i}=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}, $$

where $A=1-K^{\frac{2}{\alpha}}$, B=−2.d.cos(θ) and C=d ² where d is the distance between transmitters i and j and is a known parameter of the grid layout.

Fig. 7

Geometric representation of finding the first approximation of point z=(x ₀,y _i )

Remark

Select positive value of r _i as the solution of the above quadratic equation. Using x ₀=x _i +r _i as the first approximate solution in Newton’s Method (11), and after a few iterations, we can get a sufficiently accurate value x _n+1 which will be the x coordinate of the point z.

The coordinates of point z will be: (x _n+1,y _i ).