Loss-based proportional fairness in multihop wireless networks

Abstract

Proportional fairness is a widely accepted form of allocating transmission resources in communication systems. For wired networks, the combination of a simple probabilistic packet marking strategy together with a scheduling algorithm aware of two packet classes can meet a given proportional vector of n loss probabilities, to an arbitrary degree of approximation, as long as the packet loss gap between the two basic classes is sufficiently large. In contrast, for wireless networks, proportional fairness is a challenging problem because of random channel variations and contention for transmitting. In this paper, we show that under the physical model, i.e., when receivers regard collisions and interference as noise, the same packet marking strategy at the network layer can also yield proportional differentiation and nearly optimal throughput. Thus, random access or interference due to incoherent transmissions do not impair the feasibility of engineering a prescribed end-to-end loss-based proportional fairness vector. We consider explicitly multihop transmission and the cases of Markovian traffic with a two-priority scheduler, as well as orthogonal modulation with power splitting. In both cases, it is shown that sharp differentiation in loss probabilities at the link layer is achievable without the need to coordinate locally the transmission of frames or packets among neighboring nodes. Given this, a novel distributed procedure to adapt the marking probabilities so as to attain exact fairness is also developed. Numerical experiments are used to validate the design.

Appendices

Appendix 1: Proof of Theorem 2

The proof of the theorem is based on certain results about nonnegative matrices and its eigenvalues. We begin by recalling the relevant properties. A real matrix is called essentially nonnegative if its off-diagonal elements are nonnegative. The maximal real eigenvalue of an essentially nonnegative, irreducible matrix is increasing when any of its elements increases [10]. Moreover, the maximal real eigenvalue of A + D, where A is any irreducible and essentially nonnegative matrix and D is diagonal with not all elements identical, is a strictly convex function of D [6]. Now, let Q be the infinitesimal generator of an irreducible Markov process, and let $\Uplambda$ be the matrix of instantaneous arrival rates associated with that process. Solving the eigenvalue problem

$$\eta {\bf b} (\Uplambda - c I) = {\bf b} Q$$

(21)

is equivalent to solving

$$-\eta g(\eta) {\bf b} = {\bf b} (Q - \eta \Uplambda)$$

(22)

if the service rate c is regarded as a function of the eigenvalue η. We adopt in the following lemma the proper point of view for our purpose: assume η is constant and interpret g(η) as a function of the matrix $\Uplambda$ (or, more precisely, as a function of the diagonal elements in $\Uplambda$).

Lemma 7

The maximal real eigenvalue g(η) of the matrix $\Uplambda - \frac{1}{\eta} Q$ is increasing when any element of $\Uplambda$ increases.

Proof

The steps of the proof mimic those in [8]. The matrix $Q(\eta) = Q - \eta \Uplambda$ is essentially nonnegative. Since η < 0, increasing one of the elements in the diagonal matrix $\Uplambda,$ say λ _i , has the effect of increasing the same element in Q(η), while the others remain unchanged. So the maximal real eigenvalue of $Q(\eta),\;\hbox{mre}(Q(\eta)),$ increases and is a convex function of λ _i , too. To see this, pick two points a ¹ _i and a ² _i . Put $\Uplambda_1 = \Uplambda + (a_i^1 - \lambda_i) E$ and, similarly, $\Uplambda_2 = \Uplambda + (a_i^2 - \lambda_i) E,$ where E is a diagonal matrix with E _ii  = 1 and zero otherwise. By the matrix-convexity of $\hbox{mre}(\cdot),$ using $A_1 = Q - \eta \Uplambda_1$ and $D = \eta(\Uplambda_1 - \Uplambda_2)$ we can write

$$\begin{aligned} \hbox{mre} \left((1 - h) A_1 + h (A_1 + D)\right) &= \hbox{mre}\left(Q - \eta ((1 - h) \Uplambda_1 + h \Uplambda_2)\right) < (1 - h) \hbox{mre}(A_1) + h \hbox{mre}(A_2) \\ &= (1 - h) \hbox{mre} ( Q - \eta \Uplambda_1) + h \hbox{mre}(Q - \eta \Uplambda_2). \end{aligned}$$

(23)

The second and the last terms establish the strict convexity in a _i . The last step of the proof begins noticing that

$$g(\eta) = -\frac{\hbox{mre}\left(Q(\eta)\right)}{\eta}.$$

(24)

Using the monotonicity of $\hbox{mre}(\cdot),$ it is straightforward to deduce that

$$\frac{\partial \hbox{mre}}{\partial \lambda_i} = -\eta \frac{\partial g(\eta)}{\partial \lambda_i} > 0$$

(25)

because η is a constant independent of λ _i . Since η < 0, the maximal real eigenvalue g(η) is increasing in λ _i . Finally, taking the derivative of (25) and recalling the convexity of the maximal real eigenvalue, the inequality

$$\frac{\partial^2 \hbox{mre}}{\partial \lambda_i^2} = -\eta \frac{\partial^2 g(\eta)}{\partial \lambda_i^2} > 0$$

(26)

shows that g(η) is a convex function of λ _i . □

By application of the chain rule, the inverse function of g(η) is monotonically increasing in λ _i and strictly concave.

Appendix 2: Proof of Lemma 3

A direct calculation suffices. Under the stated assumptions, the first derivative of

$$h(x) = \frac{1 - f(x)}{1 - g(x)}$$

is

$$h^\prime(x) = \frac{f^\prime(x) g(x) - f(x) g^\prime(x) + g^\prime(x) - f^\prime(x)}{\left(1-g(x)\right)^2},$$

(27)

and the first derivative of f(x)/g(x) is

$$\frac{f^\prime(x) g(x) - f(x) g^\prime(x)}{g^2(x)}.$$

If $(f(x)/g(x))^\prime$ is asymptotically decreasing, then the numerator $f^\prime(x) g(x) - f(x) g^\prime(x) < 0.$ Since f(x) and g(x) tend to 0 as $x \to \infty.$ for sufficiently large x the difference $g^\prime(x) - f^\prime(x) < \epsilon,$ for any $\epsilon > 0.$ Using these in (27) shows that $h^\prime(x) < 0$ for sufficiently large x.