Decoupled Power Allocation Through Pricing on a CDMA Reverse Link Shared by Energy-Constrained and Energy-Sufficient Data Terminals

Abstract

We perform market-oriented management of the reverse link of a CDMA cell populated by data terminals, each with its own data rate, channel gain, willingness to pay (wtp), and link-layer configuration, and with energy supplies that are limited for some, and inexhaustible for others. For both types of energy budgets, appropriate performance indices are specified. Notably, our solution is “decoupled” in that a terminal can choose optimally, irrespective from choices made by the others, because it pays in proportion to its fraction of the total power at the receiver, which directly determines its signal-to-interference ratio (SIR), and hence its performance. By contrast, in other similarly-sounding schemes terminals’ optimal choices are interdependent, which leads to “games of strategy”, and their practical and theoretical complications. We study two situations: pricing for maximal (i) network revenue, and (ii) social benefit. The socially-optimal price is common to all terminals of a given energy class, and an energy-constrained terminal pays in proportion to the square of its power fraction. By contrast, the revenue-maximising network sets for each terminal an individual price that drives the terminal to the “revenue per Watt” maximiser. The network price is higher, and drives each terminal to consume less. Distinguishing features of our model are: (i) the simultaneous consideration of both limited and unlimited energy supplies, (ii) the performance metrics utilised (one for each type of energy supply), (iii) the generality of our physical model, which can lead to an optimal link-layer configuration, and (iv) our pricing of the received power fraction which yields a “decoupled” solution.

Appendix: Mathematical issues

In the analysis we assume that certain functions involving an S-curve or the derivative of an S-curve retain the S-shape, or the “single-peakedness” of the derivative (see Fig. 1).

Definition 9.1

$S:\Re_{+}\rightarrow[0,Y]$, is an S-curve with unique inflexion at x _f if (i) S(0) = 0, S is (ii) continuously differentiable, (iii) strictly increasing, (iv) convex over [0,x _f ) and concave over (x _f , ∞ ), and (v) surjective.

Remark A.1

In Definition A.1, S is strictly increasing and also surjective (for each y ∈ [0, Y] there is an $x\in\Re_{+}$ such that f(x) = y). Therefore, S must approach Y asymptotically as x goes to infinity (i. e. , this follows from the definition).

Definition 9.2

A function $h:\Re_{+}\rightarrow[0,Y]$ is single-peaked over $\Re_{+}$ if h is continuous, surjective and has a global maximum at X ∈ (0, ∞ ) (that is, h(X) = Y, $0\leq x_{1}<x_{2}\leq X\implies h(x_{2})>h(x_{1})$ and $X\leq x_{1}<x_{2}\implies h(x_{2})<h(x_{1})$).

Remark A.2

In Definition A.2, h is strictly increasing up to x = X and strictly decreasing thereafter. Since h is also surjective, it must approach 0 asymptotically as x goes to infinity.

Remark A.3

Definition A.2 is closely related to strict quasi-concavity. However, a strictly monotonic function satisfies strict quasi-concavity, but does not satisfy Definition A.2. For example, a function whose graph exhibits over the interval of interest the familiar “bell shape” of the Gaussian curve (as shown in Figs. 1 and 10, for example) is both strictly quasi-concave and single-peaked. On the other hand, the S-curve itself is strictly quasi-concave (since it is strictly increasing) but does not satisfy Definition A.2 (its “peak” occurs at infinity).

The specific assumptions are:

Assumption 1

If S satisfies Definition A.1, then the composite function s(z): = S(g(z)) with g(z) = Γz/(1 − z), Γ ≥ 1 and z ∈ [0,1) also satisfies Definition A.1.

Assumption 2

If S satisfies Definition A.1, then each of the following functions satisfies Definition A.2: (i)xS′(x), and for Γ ≥ 1(ii) (x/Γ + 1)²S′(x), (iii) x(x/Γ + 1)S′(x) and (iv) $x\mathcal{B}'(x)$ where $\mathcal{B}(x):=S(x)/x$ with $\mathcal{B}(0):=\lim_{x\downarrow0}\mathcal{B}(x)\equiv S'(0)$.

Remark A.4

By Lemma 4.2, if S satisfies Definition A.1 then $\mathcal{B}(x)$ satisfies Definition A.2 (i. e. , this is a proved statement, not an assumption).

Below we formally describe some more primitive technical properties for the concerned S-curve that lead to the assumptions above. These properties have the “single crossing” form; i.e., the value of certain function crosses the origin exactly once, a notion that has proved quite useful in certain contexts, such as economics and game theory [2]. In the present work, a strong version of this notion is formalised as follows:

Definition 9.3

A function $f:D\rightarrow\Re$ with $D\subset\Re$ satisfies the unique-crossing from above condition (UCC) over D′ ⊂ D if $\exists t_{0}\in D'$ such that f(t ₀) = 0 and $\forall t\in D'\quad t<t_{0}\implies f(t)>0\text{ and }t>t_{0}\implies f(t)<0$.

Lemma A.1

Consider the composite function S(g(z)), where $S:\Re_{+}\rightarrow[0,d]$, is an S-curve with inflexion at x _f , and $g:[a,b]\rightarrow\Re_{+}$, with 0 ≤ a < b, is a strictly increasing convex function such that lim_z→bg(z) = ∞. (i) If the function [g′(z)]²S′′(g(z)) + g′′(z)S′(g(z)) satisfies the UCC over [a,b], then: (ia) the composite function s(z): = S(g(z)) satisfies Definition A.1, and (ib) its inflexion abscissa z _f is such that g(z _f ) > x _f .

(ii) With g(z) = Γz/(1 − z), Γ ≥ 1 and $\mathcal{I}=[0,1)$ , conclusions (ia) and (ib) follow if the function (x + Γ)S′′(x) + 2S′(x) satisfies the UCC over the domain $\Re_{+}$ .

Proof

• (ia) We only show below that the composite function has, under the hypothesis, the curvature properties required by Definition A.1; that it also has the other properties can also be shown.

  The second derivative of S(g(z)) is [g′(z)]² S′′(g(z)) + g′′(z)S′(g(z)).

  g′′(z)S′(g(z)) is always positive because by hypothesis g′′ is positive (convexity).

  [g′(z)]² S′′(g(z)) has the sign of S′′; i.e. it is positive in [0,x _f ) and negative in (x _f , ∞ ).

  Thus, the composite function starts out convex (its second derivative starts out positive).

  If [g′(z)]² S′′(g(z)) + g′′(z)S′(g(z)) satisfies the UCC, then the composite function has exactly one inflexion point $z_{f}^{c}$.

  The fact that s asymptotically goes to d as z goes to b is immediate.

• (ib) $z_{f}^{c}$ must satisfy $g(z_{f}^{c})>x_{f}$ so that S′′(g(z _f )) be negative.

• (ii) If g(z) = Γz/(1 − z) then g′(z) = Γ/(1 − z)² and g′′(z) = 2Γ/(1 − z)³. Considering these expressions, the second derivative of the composite function becomes:

  $$ \frac{\Gamma^{2}}{(1-z)^{4}}S''(g(z))+\frac{2\Gamma}{(1-z)^{3}}S'(g(z))$$

  which has the same sign as ΓS′′(g(z)) + 2(1 − z)S′(g(z)).

  With x: = Γz/(1 − z), z = x/(x + Γ) and 1 − z = Γ/(x + Γ).

  Therefore, the second derivative has the sign of (x + Γ)S′′(x) + 2S′(x). The thesis follows from part (i) of this proof.□

The next result involves the bell shape exhibited by the graph of S′.

Lemma A.2

Consider the function h(t) = g(t)S′(t) where $S:\Re_{+}\rightarrow[0,d]$ is an S-curve with inflexion at t _f , and $g:\Re_{+}\rightarrow\Re_{+}$ is a strictly increasing continuously differentiable function. Furthermore, lim_t→ ∞h(t) = 0. If g′(t)S′(t) + g(t)S′′(t) satisfies the UCC at t₀ ∈ (0, ∞ ) then h satisfies Definition A.2, and has its maximal value at t₀ > t _f .

Proof

The derivative h′(t) = g′(t)S′(t) + g(t)S′′(t). The first term is always positive, and the second term has the sign of S′′ which is positive for t < t _f . Therefore, h starts out increasing.

If h′ satisfies the UCC at t ₀ then h reaches a global maximum at t ₀.

t₀ > t _f because h′(t₀) = 0 implies that g(t₀)S′′(t₀) < 0.□

Remark A.5

If in Lemma A.2 g(t) = t, then the function that must satisfy the UCC reduces to S′(t) + tS′′(t), or, equivalently, 1 + tS′′(t)/S′(t); that is, tS′′(t)/S′(t) must uniquely cross from above the horizontal line at ordinate negative 1. Division by S′(t) is possible because S′(t) > 0 ∀ t ∈ (0, ∞ ).