Mobile Netw Appl (2011) 16:640–660 DOI 10.1007/s11036-011-0300-z Decoupled Power Allocation Through Pricing on a CDMA Reverse Link Shared by Energy-Constrained and Energy-Sufficient Data Terminals Virgilio Rodriguez · Friedrich Jondral · Rudolf Mathar Published online: 30 March 2011 © Springer Science+Business Media, LLC 2011 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 similarlysounding 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 V. Rodriguez (B) Institute for Communication Networks (ComNets), RWTH, Aachen, Germany e-mail: rodriguez@comnets.rwth-aachen.de F. Jondral Institut für Nachrichtentechnik, Karlsruher Institut für Technologie, Karlsruhe, Germany e-mail: friedrich.jondral@kit.edu R. Mathar Institute for Theoretical Information Technology, RWTH, Aachen, Germany e-mail: mathar@ti.rwth-aachen.de 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 linklayer configuration, and (iv) our pricing of the received power fraction which yields a “decoupled” solution. Keywords power control · pricing · game theory · microeconomics · CDMA · revenue maximisation 1 Introduction In telecommunication networks, pricing is a critical tool to both generate revenues, and induce efficient resource use. Herein, we propose and analyse a technicaleconomic scheme for the 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 possibly individual link-layer configuration). Some terminals are battery-powered while others have boundless energy, and we specify pertinent utility functions for each type. The present paper builds upon [19], where the network sets prices that maximise its revenues, and [18], where a “social planner” uses pricing to drive the terminals to a socially-optimal allocation. For reverse link CDMA power allocation, many useful price-based algorithms have been reported [1, 6, 10, 22, 23, 25]. Previous schemes rely on per-Watt pricing. However, a terminal’s performance does not solely depend on the amount of power it buys. It also depends on Mobile Netw Appl (2011) 16:640–660 the power choices made by the others. Thus, terminals choices are interdependent, which leads to a “game of strategy”, a model in which several “selfish” agents make interdependent choices [7, 12]. Games engender both theoretical and practical problems. The typical solution concept is the Nash equilibrium (NE), which (i) may not exist, (ii) if existing, may not be unique, and (iii) is in general inefficient [5]. And even if a unique NE can be proved to exist, it may be unclear: (a) how will the players reach the NE, and (b) after how many “iterations”. The facts that terminals frequently enter and exit the network at arbitrary times, and that they are “anonymous” to each other, further complicate matters. Additionally, if perWatt pricing is implemented as a “true” billing scheme in a commercial network, it may face substantial consumer resistance, because in the common economic scenario with which the customer is familiar, his/her utility depends solely on how s/he spends her/his own income among a number of “goods”, irrespectively of how others spend theirs. Below we provide a “decoupled” solution. A terminal pays according to its fraction of the total power at the receiver. Because this fraction directly determines a terminal’s signal-to-interference ratio (SIR), and hence its performance, for a given price, a terminal can make an optimal choice without worrying about the choices made by others. A decoupled solution is superior for both technological and marketing reasons: it shifts complexity from the terminal to the base station, it avoids Nash-equilibrium existence/convergence issues, and places the user in the familiar territory in which s/he “controls her/his own destiny”. As a base line for performance evaluation, we do analyse a “game” in which each terminal chooses its power without economic incentives or a central controller. By also analysing the conditions yielding a “socially optimal” allocation, we conclude that our pricing scheme can, with relatively minor modifications, maximise “socially benefit”, instead of network profit. Additional distinguishing features of our model are: (i) the simultaneous consideration of both limited and unlimited energy budgets as in real networks, where battery-powered terminals coexist with those powered from a vehicle or the power-grid (including fixed wireless local loop), (ii) the performance metrics utilised (one for each type of energy budget), and (iii) the generality of our physical model (each terminal may have its own data rate, channel gain, willingness to pay (wtp), and even link-layer configuration). Below, we first discuss the physical model, and the terminals’ technical-economic rationale. Then, we address feasibility issues, and identify a terminal’s power 641 fraction as key. We then analyse how each terminal reacts under power-fraction pricing. Subsequently, we describe how the network can set prices to maximise its revenue. At that point, we shift to the zero-price game. Then, we describe the “socially optimal” allocation. Several numerical examples and figures are given throughout. In a final section, we summarise and discuss key contributions. 2 Generalities 2.1 Physical model N terminals upload data to a CDMA base station (BS). The sub-index i identifies a terminal. z or x may be used √ as generic function arguments (e.g., x is a concave graph). – – – – – – – p0 is the average Gaussian noise power Ei is the energy budget P̂i is the power constraint hi is the channel gain pi = hi Pi is the received power For convenience, we let p̄ := N  Yi =  (1) pi i=1 j=i p j + p0 is i’s interference An S−curve and its 1st and 2nd derivatives 1 0 0 a c Fig. 1 An S-curve and scaled versions of the graphs of its 1st (dash-dot) and 2nd (dash) derivatives. The inflexion occurs at x = a 642 Mobile Netw Appl (2011) 16:640–660 Table 1 Notation summary Power Ratios p0 : noise P̂i : constraint pi := hi Pi : received  Yi := p0 + j=i p j N p̄ := i=1 pi Ŵi := W/Ri : Spreading gain κi := pi /Yi (carrier-to-interference) σi := Ŵi κi : (signal-to-interference) πi := κi /(1 + κi ) (power fraction)  π̄ := N j=1 π j – – – – – – – – – κi = pi /Yi is the carrier-to-interference ratio (CIR) W is the available bandwidth, assumed equal to the common chip rate Ri is the data rate. Ŵi = W/Ri is the spreading gain σi = Ŵi κi is the signal-to-interference-plus-noise ratio (SIR) Mi -bit packets carrying Li < Mi information bits are used. Fi is the packet-success rate function (PSRF) giving the probability of correct reception of a data packet as a function of the SIR. For some technical reasons, fi (x) := Fi (x) − Fi (0) replaces Fi . Its graph is assumed to have the Sshape shown in Fig. 1. Our analysis does not rely on any specific PSRF. Furthermore, as described in the Appendix, fi satisfies the technical Assumptions 1–2 that guarantee that each of certain graphs retains a desired shape (“S” or “bell”) (Table 1). 2.2 Terminals’ general objective Two categories of data terminals are of interest: (i) energy-constrained (battery powered), and timedriven, referred as e-terminals and t-terminals, below. An e-terminal focuses on the total number of information bits transferred with its total energy budget. The t-terminal focuses on the number of information bits transferred over a reference time period (such as the time unit). The model below works for both categories. The utility function has the “quasi-linear” form [24, Ch. 10]: vi Bi + yi where: – – – (i) vi is the “willingness to pay” (wtp) (the monetary value to the terminal of one information bit successfully transferred), (ii) Bi is the (average) number of information bits the terminal has successfully transferred within a reference length of time, say τ , and (iii) yi is the amount of money the terminal has left after any charges and rewards are computed. – Bi generally depends on a vector of “resources” z. When the terminal must pay ci (z), it chooses z to maximise vi Bi (z) + [Di − ci (z)]. Di is the terminal’s monetary budget, which limits its total expenditure and, if “large”, need not be considered, in which case the terminal maximises benefit minus cost: (2) vi Bi (z) − ci (z) 3 Feasibility of power ratios 3.1 SIR/CIR feasibility Definition 3.1 Let πi be defined by πi = σi κi ≡ 1 + κi Ŵi + σi (3) Proposition 3.1 πi equals i’s fraction of the total power at the receiver (including noise), that is, pi κi = N 1 + κi j=1 p j + p0 (4)  Proof With Yi := Nj=1 p j − pi + p0 (the interference experienced by terminal i), and κi = pi /Yi (the CIR), Eq. 3 can be written as: pi pi pi /Yi ≡ ≡ N pi /Yi + 1 pi + Yi j=1 p j + p0 (5) ⊔ ⊓ Proposition 3.2 There is a one-to-one correspondence between κi and πi given by κi = κ(πi ), where κ(z) := z for z ∈ [0, 1) 1−z (6) κ is strictly increasing and convex. Proof πi = 1 1 κi =⇒ = +1 1 + κi πi κi =⇒ κi = πi 1 ≡ 1/πi − 1 1 − πi κ(z) = z/(1 − z) =⇒ ∀ z ∈ [0, 1) κ ′ (z) = (1 − z)−2 > ⊔ ⊓ 0 and κ ′′ (z) = 2(1 − z)−3 > 0 (see Fig. 2). Consider a system of N equations of the form pi N j=1 j=i p j + p0 = κi (7) Mobile Netw Appl (2011) 16:640–660 643 15 Proof Perform direct substitution into Eq. 7 with p0 = 0 and pi = kπi where k > 0: pi N 10 j=1 z/(1−z) j=i kπi = N p j k j=1 π j − kπi = πi 1 − πi By Proposition 3.2, the last expression equals κi . 5 ⊔ ⊓ 3.2 Power limits and the “ruling terminal” 0 0 0.2 0.4 0.6 0.8 1 z Fig. 2 Graph of κ(z) = z/(1 − z), the CIR as a function of the terminal’s fraction of the total receiver power Theorem 3.1 If π̄ := N  (8) πj < 1 j=1 pi πi = p0 1 − π̄ (9) Proof Condition (8) implies that Eq. 9 is positive. Replacing Eq. 9 into Eq. 7 yields: pi j=1 j=i = = p j + p0 p0 Lemma 3.1 Let k be a ruling terminal. If πk π̄ ≤ 1 − hk ( P̂k / p0 ) (11) (12) then the power levels given by Eq. 9 obey the applicable power constraints (the system is feasible). Proof By definition of k, Eq. 12 implies that then the system def ined by Eq. 7 has a positive solution given by N Definition 3.2 Terminal k is a ruling terminal if πk πi ≥ ∀i ∈ {1, . . . , N} hk P̂k hi P̂i N j=1 p0 πi /(1 − π̄) π j/(1 − π̄ ) − p0 πi /(1 − π̄ ) + p0 πi πi = π̄ − πi + 1 − π̄ 1 − πi By Proposition 3.2, the last expression equals κi . ⊔ ⊓ π̄ ≤ 1 − πi hi ( P̂i / p0 ) (13) ∀i Since Eq. 12 =⇒ Eq. 8, Eq. 9 has a positive solution: pi∗ = p0 πi /(1 − π̄ ). From Eq. 13, πi πi p0 ≤ hi P̂i 1 − π̄ (14) Thus, Eq. 12 implies that pi∗ ≤ hi P̂i ∀i. ⊔ ⊓ hi ( P̂i / p0 ) ≤ 1 − π̄ =⇒ Remark 3.1 πi represents terminal i’s desired quality of service (QoS), and hi P̂i the highest power level that i can get to the receiver; thus the ratio πi /(hi P̂i ) yields i’s desired QoS per unit of power available to i. The terminal with the highest such ratio wants, loosely speaking, “the most for the least”, and hence is in the worst situation to achieve its QoS. If the ruling terminal can achieve its desired QoS, all others can as well. Corollary 3.1 If p0 ≈ 0 (cell is “interference limited”), and N  j=1 4 Terminal’s choice under pricing πj = 1 (10) then any power vector proportional to the vector (π1 , · · · , π N ) is a solution to the system def ined by Eq. 7. 4.1 Pie division through pricing πi is terminal i’s fraction of the total power at the receiver. Thus, the network can view power allocation as a “pie division” problem: how big the “slice” 644 Mobile Netw Appl (2011) 16:640–660 to be allocated to a given terminal. The case of the interference-limited cell is the easiest to visualise: the “pie size” is necessarily 1 (Eq. 10), and the “slices” immediately indicate the received power levels. For a given value of πi , the terminal can obtain directly the corresponding CIR (Eq. 6), and hence its performance. Thus, if the resource manager sets a price ci at which terminal i can “buy” a “slice” πi , for the given ci , i can choose optimally πi , irrespective of the choices made by others. In principle, the terminal’s individual choices may exceed “capacity”, but this may be avoided if the price is “right”, or the network may select an “optimal” subset of terminals to be served. In the general case, a terminal needs to know the total received power p̄ + p0 in order to calculate its emission power, for its chosen power fraction. However, the analysis below shows that, for a given price, the total received power will not affect the terminal’s optimal fraction. 4.2 General technical results We first provide some technical results that are relevant to various parts of this work. Others more specific results will be inserted below where appropriate. Lemma 4.1 Let h : [d, e] → ℜ+ be a continuous strictly quasi-concave function h with maximal value Y at X where d < X < e, that is, d ≤ x1 < x2 ≤ X =⇒ h(x2 ) > h(x1 ) and X ≤ x1 < x2 ≤ e =⇒ h(x1 ) > h(x2 ) (Fig. 3). Suppose h(d) ≥ h(e). Let S := {x : h(x) = c} (the set of solutions to h(x) = c). ⎧ ⎪ ∅ if c > Y or c < h(e) ⎪ ⎪ ⎪ ⎨{X} if c = Y S= ⎪ {a, b } with a < X < b if h(d) < c < Y ⎪ ⎪ ⎪ ⎩{b } with X < b h(e) < c < h(d) Furthermore, a < x < b =⇒ h(x) > c. Proof The first two cases are immediate, by definition. The next two cases follow from the fact that h is strictly increasing in (d, X) and strictly decreasing in (X, e). ⊔ ⊓ Lemma 4.2 Suppose S : ℜ+ → [0, d] is an S-curve (Def inition A.1) with inf lexion at z f . Let B(z) := S(z)/z with B(0) := limz↓0 B(z) ≡ S′ (0). Then, (i) there is a unique tangent line from the origin to S(z), denoted as c∗ z and called the tangenu, with tangency point, genu, (z∗ , S(z∗ )), where z∗ > z f . (ii) B is strictly quasiconcave, and its unique maximiser in the interval [0, Z ] is min(z∗ , Z ). Proof See [16]. ⊔ ⊓ Remark 4.1 Figures 4 and 5 show the tangenu (tangent from (0,0)) and genu (tangency point) for some S-curves. Figure 6 shows the shape of the graph S(z)/z. 0.97 4 16 0.86 32 64 Packet success rate Y c 0 0 0.15 0.26 0.43 0.78 1 Power Fraction at receiver d a X b e Fig. 3 Graph of a single-peaked function h with maximal value Y at X. a and b are the two solutions to h(x) = c, where h(d) < c<Y Fig. 4 The composite function f (Ŵκ(z)) = f (x), with f the PSRF, Ŵ the spreading gain, and κ(z) = z/(1 − z), plays a key role, especially its tangenu (tangent to (0,0)) and genu (tangency  80 point). f (x) = 1 − exp (−x/2) /2 − 2−80 is displayed. The value of Ŵ ∈ {4, 16, 32, 64} is indicated near the genu Mobile Netw Appl (2011) 16:640–660 645 Then, the terminal should choose the value of πi that maximises benefit minus cost, that is: vi Li Ri fi (Ŵi κ(πi )) − ci πi Mi (16) fi is the (slightly modified) packet-success-rate function, whose graph as a function of its argument, the SIR, is an S-curve. fi (Ŵi κ(z)) is a composite function of fi and κ, with independent variable z and parameter Ŵi . The graph of fi (Ŵi κ(z)) inherits the S-shape of fi (see Fig. 4 and the Appendix). Thus, the terminal maximises an expression of the form S(z) − cz, where S is some S-curve, a problem whose solution is given by Theorem 4.1: Fig. 5 Maximising S(z) − cz subject to z ≤ Z , where S(z) is an “S-curve”. If c > c∗ or c = c′ and Z < a then z = 0 is optimal. Otherwise min(Z , z′ ) is the maximiser. At z′ , the curve’s tangent (short blue line) is parallel to the cost line c′ z 4.3 Choice by a time-driven terminal 4.3.1 Analysis in term of the power fraction As discussed in Section 2.2, a t-terminal wishes to maximise its performance over some pre-specified length of time, say the time unit, considering both its benefit and its cost. For a given πi , the terminal’s cost is ci πi , and (with κ(z) given by Eq. 6) the average number of information bits transferred over a time unit is Bi : Li Ri fi (Ŵi κ(πi )) Mi (15) c2 c* Money c1 0 a z* b Resource Fig. 6 For c ≤ c∗ the e-terminal chooses z∗ ; else z = 0 is optimal Theorem 4.1 Consider the problem of maximising S(z) − cz subject to 0 ≤ z ≤ Z where c is a positive number and S is an S-curve (Def inition A.1) with inf lexion at z f , and satisfying c∗ z∗ = S(z∗ ). (i) If c > c∗ then z = 0 is the maximiser. (ii) For c < c∗ , the equation S′ (z) = c has exactly one solution, zc , that is greater than z f . (iii) If zc ≤ Z then zc is the maximiser. If zc > Z then the maximiser is either Z , if S(Z ) ≥ cZ , or zero, otherwise. Proof (i) If c > c∗ then ∀z > 0, cz > S(z) =⇒ S(z) − cz < 0. Thus, z = 0 is the maximiser. (ii) If c < c∗ then a maximiser of S(z) − cz that is inside (0,Z) must satisfy the first-order necessary optimising condition: S′ (z) = c. Because S is an S-curve, its derivative is strictly increasing between zero and z f and strictly decreasing for z > z f (as shown in Fig. 1). If S′ (0) ≤ c < c∗ , then by Lemma 4.1, S′ (z) = c has 2 solutions zo , zc such that z0 < z f < zc . If c < S′ (0) (as is c′′ in Fig. 5) then, also by Lemma 4.1, zc > z f is the only solution to S′ (z) = c. (iii) At zc , S′ is decreasing (S′′ is negative) which implies that zc is, if feasible, a maximiser. By contrast, at z0 , S′ is increasing, therefore z0 is a minimiser. If zc < Z , zc is the only feasible point that satisfies the necessary and sufficient conditions for a maximiser, and therefore it is the global maximiser. If zc > Z , then one of the extreme points is the maximiser: 0 or Z . If S(Z ) − cZ > 0 (e. g. Z > a in Fig. 5) then Z is the maximiser. Otherwise zero maximises S(z) − cz. ⊔ ⊓ Remark 4.2 The case c = c∗ , Z ≥ z∗ is, in principle, indeterminate, because for z = z∗ , c∗ z∗ = S(z∗ ) which implies that the objective function evaluates to zero, 646 Mobile Netw Appl (2011) 16:640–660 just as it does for z = 0. Thus, in this case, either 0 or z∗ are optimal. However, we shall assume that in any such situation the decision-maker prefers the positive value (i.e., to use the system) over zero (i.e., to remain inactive). If the x2 is feasible, Eq. 18 is satisfied, and revenue reduces to: x + 1 S′ (x) (19) x Ŵ Under suitable assumptions, the graph of Eq. 19 has the bell shape shown in Fig. 6. 4.3.2 Analysis in terms of the SIR 4.4 Choice by an energy-limited terminal One can perform an analysis similar to that of Section 4.3.1 in terms of the SIR, x. In this case, the benefit depends directly on xi , and no composite function is needed. However, the cost is now a nonlinear function of xi , namely ci xi /(xi + Ŵi ) where Ŵi is the spreading gain. Considering Eqs. 15 and 16, the terminal chooses xi to maximise xi Li Ri fi (xi ) − ci vi Mi xi + Ŵi (17) Thus, the terminal maximises an expression of the form S(x) − c(x) subject to 0 ≤ x ≤ X, where S is some S-curve and c(x) := cx/(x + Ŵ) with c and Ŵ positive real numbers. Below we sketch a solution for this problem that follows closely the proof of Theorem 4.1. Some technical details are ignored. An interior maximiser must satisfy S′ (x) = c′ (x) = cŴ/(x + Ŵ)2 , or Ŵ x +1 Ŵ 2 S′ (x) = c (18) By solving Eq. 18 together with the intersection condition cx/(x + Ŵ) = S(x) one obtains c∗ such that c∗ x/(x + Ŵ) is tangent to S(x) at x∗ . It is clear that c > c∗ implies ∀x > 0, S(x) − c(x) < 0. Thus, x = 0 is the maximiser. For c < c∗ , it is also clear that if X is less than the abscissa of the point where the cost curve first intersects S(x) (e.g., X < a2 for c = c2 in Fig. 6), ∀x ∈ [0, X], S(x) − c(x) < 0. Then, x = 0 is again the maximiser. For c < c∗ and X > a2 , let h(x) := Ŵ(x/ Ŵ + 1)2 S′ (x) (left side of Eq. 18). By Assumption 2 in the Appendix, h retains the single-peaked shape of S′ (see Fig. 1). Let h(x̂) be h’s maximal value. By Lemma 4.1, as long as c < h(x̂), Eq. 18 has two solutions, x1 and x2 such that x1 < x̂ < x2 , and h(x) > c ∀x ∈ (x1 , x2 ). Thus over (x1 , x2 ) the derivative of the objective function S′ (x) − cŴ/(x + Ŵ)2 is positive, which implies that the largest feasible value in (x1 , x2 ) is best. Hence, the maximiser is min(x2 , X). The resulting payment by the terminal (network’s revenue) at the chosen operating point is cx/(x + Ŵ). We first establish some additional technical results about general maximisation problems involving an Scurve. Subsequently we explain how these results relate to the e-terminal problem. 4.4.1 Relevant technical results Theorem 4.2 Suppose S is an S-curve (Def inition A.1) with inf lexion at z f , such that c∗ z∗ = S(z∗ ). Let B(z) := S(z)/z with B(0) := limz↓0 B(z) ≡ S′ (0). Consider the problem of maximising B(z) − c subject to 0 ≤ z ≤ Z where c > 0 (see Fig. 7). If (ia) c > c∗ or (ib) c ≤ c∗ , z∗ > Z and B(Z ) < c, then the maximiser is zero. (ii) If c ≤ c∗ and z∗ ≤ Z then z∗ is optimal. (iii) If c ≤ c∗ , z∗ > Z and B(Z ) ≥ c, then z = Z is optimal. Proof Since c is a constant independent of z, the z > 0 that maximises B(z) − c is the same that maximises B(z). By Lemma 4.2 min(z∗ , Z ) maximises B(z). Fig. 7 Terminal chooses SIR x that maximises benefit minus cost: S(x) − cx/(x + Ŵ); e.g., for c = c2 it chooses x2 where the derivative of the S-curve equals that of the cost-curve. The largest c for which the terminal will operate is c∗ . The bell-shaped curve corresponds to the terminal’s payment (network’s revenue) as a function of its choice, over an appropriate range Mobile Netw Appl (2011) 16:640–660 647 By definition of z∗ , S(z∗ )/z∗ − c ≡ c∗ − c. Therefore, c > c∗ implies that z = 0 maximises B(z) − c, while c ≤ c∗ and z∗ ≤ Z implies that z∗ is optimal. If c ≤ c∗ but z∗ > Z then the highest feasible value for the objective function is S(Z )/Z − c. Therefore, if c ≤ c∗ , z∗ > Z and S(Z )/Z ≥ c, then z = Z is optimal; while c ≤ c∗ , z∗ > Z and S(Z )/Z < c, implies that z = 0 is optimal. ⊔ ⊓ Theorem 4.3 Suppose S is an S-curve (Def inition A.1) with inf lexion at z f , such that c∗ z∗ = S(z∗ ). Let B(z) := S(z)/z with B(0) := limz↓0 B(z) ≡ S′ (0). By Lemma 4.2 B is strictly quasi-concave with a global maximum at z∗ (i.e., it is single-peaked). Further assume that, as shown in Fig. 8, B(z) starts out convex and has a single inf lexion point between 0 and z∗ , at zl . (i) Then, there is a unique line from the origin, denoted as cx z, that is tangent to the graph of B (Fig. 8) at zx < z∗ . Consider the problem of maximising B(z) − cz subject to 0 ≤ z ≤ Z with c > 0. (ii) If c > cx , then z = 0 is the maximiser. (ii) For c < c∗ , the equation B ′ (z) = c has exactly one solution, z M , that is greater than zl . (iii) If z M ≤ Z then z M is the maximiser. If z M > Z then the maximiser is either Z , if B(Z ) ≥ cZ , or zero, otherwise. (a) Proof (i) By hypothesis, between 0 and z∗ , B(z) has the Sshape, and the development in [16] implies the existence and uniqueness of the tangent cx z. The reminder of the proof is very similar to that of Theorem 4.1. (ii) If c > cx , the line cz lies entirely over the curve except at the origin, thus, B(z) − cz < 0 for any z > 0, and z = 0 is the maximiser. (iii) For 0 < c < cx , a maximiser in (0, Z ) must satisfy B ′ (z) = c. Any solution of B ′ (z) = c must be less than z∗ because by strict quasi-concavity, B ′ (z) ≤ 0 for z ≥ z∗ . Also, since B(z) has, by hypothesis, an S-shape over (0, z∗ ), B ′ is itself single-peaked over (0, z∗ ), therefore by Lemma 4.1, B ′ (z) = c has at least one solution, z M , to the right of zl (where its peak is), and B ′′ (z M ) < 0. If z M ≤ Z , then z M is optimal, since it is feasible, and it is the only interior point that satisfies the necessary and sufficient condition for a maximiser. If z M > Z , then one of the extreme points is the maximiser. If B(Z ) > cZ > 0 (e.g., Z > a in Fig. 8a), then Z is the maximiser. Otherwise zero maximises B(z) − cz. ⊔ ⊓ (b) Fig. 8 Maximising B(z) − cz subject to 0 ≤ z ≤ Z Remark 4.3 The case c = cx is indeterminate, but we assume that zx is chosen (see Remark 4.2). Figure 8 illustrates the solution (z M is denoted as M). If c > cx , the cost line cz lies entirely over the curve except at the origin, and z = 0 is optimal. At the other extreme, if 648 Mobile Netw Appl (2011) 16:640–660 c ≈ 0, the maximiser is ≈ z∗ . If c < cx , at the maximiser z M a tangent to the curve is parallel to the cost line. 4.4.2 E-Terminal’s choice with linear pricing For an energy-limited terminal, the natural period of interest is battery life. For any given p̄ the transmission power corresponding to πi is: Pi = pi / hi ≡ πi ( p̄ + p0 )/ hi (20) Ei hi ci πi p̄ + p0 (21) Now, πi does not drop out (compare to Eq. 22). The terminal chooses πi to maximise total benefit minus total cost: With energy Ei , battery life is Ti = Ei /Pi ≡ Ei hi πi ( p̄ + p0 ) By Eq. 21, the terminal’s total cost over the period of interest is given by ci πi Ti ≡ ci Ei hi p̄ + p0 (22) Notice that πi drops out of the total cost expression. The terminal’s benefit is vi Bi , with Bi the total (average) number of (energy-earned) information bits over the period Ti : Li Ri fi (Ŵi κ(πi ))Ti Bi (πi ) = Mi (23) The terminal chooses πi to maximise utility (total benefit minus total cost): Ei hi p̄ + p0 Li fi (Ŵi κ(πi )) vi Ri − ci Mi πi operating point. This is intuitively unappealing, and in fact undesirable, for reasons given below. In this section we consider a cost function of the form cz2 (quadratic pricing). With energy Ei , battery life Ti is Ei hi /(πi ( p̄ + p0 )) (Eq. 21). Total cost is now ci πi2 Ti , or (24) ci is a known constant. The composite function fi (Ŵi κ(z)) retains the S-shape of fi (see Fig. 4 and the Appendix). Thus, the terminal must maximise an expression of the form S(z)/z − c, a problem whose solution is given by Theorem 4.2. Remark 4.4 Equation 24 indicates that the the total received power, p̄ + p0 does affect the terminal’s utility (benefit minus cost). But it appears as a factor of both the benefit and the cost; in the end it does not affect the terminal’s optimal choice. Whatever the total received power turns out to be, the terminal still wants the quantity in brackets to be as large as possible. Thus, it chooses the πi that maximises the bracketed expression in Eq. 24. 4.4.3 E-terminal’s choice with quadratic pricing As discussed in Section 4.4.2, with a linear price the independent variable drops out of the e-terminal’s total cost expression. Thus, the price determines whether or not this terminal operates, but it does not affect its Ei hi p̄ + p0 fi (Ŵi κ(πi )) Li vi Ri − ci πi Mi πi (25) (26) As before, fi (Ŵi κ(z))/z is the familiar ratio of an Scurve to its argument. The solution to the e-terminal’s maximisation problem with quadratic pricing follows Theorem 4.3 (Section 4.4.1). See also Remark 4.4. 5 Revenue-maximising prices 5.1 Optimal single-terminal pricing 5.1.1 Pricing for a time-driven terminal The analysis of Section 4.3 shows that as the price grows, the terminal consumes less; i.e., chooses smaller values of the resource. Since the network is interested in revenue (the product of the quantity purchased by the price), it is unclear what is the network’s “best” price. To determine this, the network must observe how its revenue varies as a function of the price. First, suppose that the resource constraint is “large”, Z ≫ z∗ . By Theorem 4.1 and as illustrated by Fig. 9a, for a given ck < c∗ , the terminal chooses a value zk that satisfies S′ (zk ) = ck . Then, the resulting network’s revenue is ck zk ≡ zk S′ (zk ). Thus, the network’s revenue follows the curve zS′ (z). The curve zS′ (z) has a single peak, say at z R (Assumption 2 in the Appendix). In principle, the network would like to set a price such that the terminal chooses z R . But it is shown below that c R > c∗ which implies that c R z R > S(z R ). Thus, c R would be “too high” for the terminal, which would choose zero instead, yielding no revenue for the network. Lemma 5.1 The maximiser z R of zS′ (z) satisf ies z f < z R ≤ z∗ where z f and z∗ denote respectively the inf lexion point and the genu of the Benef it S-curve. Mobile Netw Appl (2011) 16:640–660 649 maximises the network’s revenue when the resource constraint is Z is (i) c∗ if Z ≥ z∗ or (ii) S(Z )/Z if z∗ > Z . Proof (i) By Lemma 5.1, with Z > z∗ , it is the decreasing side of the revenue curve zS′ (z) that intersects S(z) at z∗ . Thus, z > z∗ =⇒ zS′ (z) < z∗ S′ (z∗ ). Therefore, the network sets the price equal to c∗ and the terminal chooses z∗ . Then, the network’s revenue equals c∗ z∗ ≡ z∗ S′ (z∗ ) = S(z∗ ) = L v R f (Ŵκ(z∗ )) M (a) (27) (ii) By Theorem 4.1, if c > S(Z )/Z then cz > S(z) for z ∈ (0, Z ), and the terminal chooses z = 0 which produces no revenue for the network. Thus S(Z )/Z is the largest price for which the terminal will operate. If the network choose a lower price, the terminal cannot choose z > Z , yielding a lower revenue for the network. Therefore the network sets the price equal to S(Z )/Z , the terminal chooses z = Z , and the network receives a revenue equal to S(Z ). ⊔ ⊓ 5.1.2 Pricing for an energy-constrained terminal (b) Fig. 9 Pricing involves both a terminal’s side and a network’s side. Once the network understands how the terminal reacts to a price level (a), it can set the price that maximises its revenue (b) Proof At z R the derivative of zS′ (z) vanishes; that is, S′ (z R ) + zS′′ (z R ) = 0. S′ (z R ) is necessarily positive, thus S′′ (z R ) < 0 which implies z R > z f . If the increasing (rising) side of zS′ (z) intersected S(z), its falling side would intersect S(z) again. But by Lemma 4.2, the equation zS′ (z) = S(z) has only one positive solution, z∗ . Thus, it is the falling side of the graph zS′ (z) that intersects S(z). Therefore, z R ≤ z∗ . ⊔ ⊓ Theorem 5.1 Suppose that a t-terminal’s benef it from utilising a power fraction z is given by the S-curve S(z), which satisf ies c∗ z∗ = S(z∗ ). Then, the price that Linear pricing By Theorem 4.2 and the discussion in Section 4.4, under linear pricing, the e-terminal will either decline to operate, or operate at the point that maximises “benefit per Watt” (the maximiser of fi (Ŵi κ(z))/z). The specific value of ci plays a role indirectly, because it can make the cost exceed the benefit. It makes sense for the network to set the highest value of ci that is acceptable to the terminal (see Eq. 24). That is: ci∗ = Li fi (Ŵi κ(z∗ )) vi Ri Mi z∗ (28) At such level, the terminal’s benefit equals its cost. The total revenue provided by this terminal during the life of its battery equals (see Eq. 22): Ei hi ∗ c p̄ + p0 i (29) What the network receives from this terminal per time unit equals: ci∗ z∗ = Li vi Ri fi (Ŵi κ(z∗ )) Mi (30) Quadratic pricing The development here follows closely that of Section 5.1.1. From Section 4.4.3, and Theorem 4.3, if the available resource Z > z∗ , and with B(z) = S(z)/z the terminal’s choice satisfies B ′ (z) = c, 650 Mobile Netw Appl (2011) 16:640–660 and the network revenue is cz = zB ′ (z). By part (iv) of Assumption 2 from the Appendix, zB ′ (z) retains the single-peaked shape (see Fig. 10). Proceeding along those lines, one can prove the following theorem: Theorem 5.2 Suppose that an e-terminal’s benef it from utilising a power fraction z is given by B(z), as def ined in Theorem 4.3. Then, the price that maximises the network’s revenue when the resource constraint is Z is (i) cx if Z ≥ zx or (ii) B(Z )/Z if zx > Z . Remark 5.1 The argument to prove Theorem 5.2 is virtually identical to the proof of Theorem 5.1. The key observation is that, as pointed out in the proof of Theorem 4.3, only those values of z satisfying 0 < z ≤ z∗ play any role, and over this range B(z) (its “rising side”) has—by assumption—the S-shape (see Fig. 8). 5.2 Serving many terminals We assume that the network can set an individual price per terminal, and in principle treat each terminal independently, following Section 5.1. However, the sum of the individually chosen πi∗ may violate Eq. 12. From all the sets of terminals that satisfy Eq. 12, the network must choose the “best” set. This problem follows the pattern of the well-known “knapsack problem”. 5.2.1 Which terminals to serve? The (fractional) knapsack problem There is a finite set of items, each characterised by a “weight” and a “value”. One seeks the combination of items that maximises the sum of the values, without exceeding a total weight constraint. The problem is in general NPhard [13]. However, if one can include in the knapsack any desired fraction of any item, the problem admits a very simple and intuitive solution. Items are sorted by their “value to weight” ratio, and whole items are inserted in order. When no space is left for another whole item, the pertinent fraction of the next item is added to completely fill the knapsack [4]. In our problem, serving “a fraction” of a terminal is to admit it with a lowered πi than it wants. However, the analogy is imperfect, because the “value” of the terminal is not linear with its “slice”, πi . Thus, the fractional knapsack solution yields a suboptimal choice in our case (which we neglect below). “Benef it per Watt” priority A terminal’s “weight” should be (a function of) its service “slice”, πi , which is itself proportional to the terminal’s received “Wattage” (Eq. 9). The obvious “value” measure (from the network’s viewpoint) is revenue contribution, but over which period (time unit or battery life)?. The time unit is a natural choice for t-terminals. It turns out that it makes sense for the network to consider, for value-toweight purposes, an e-terminal’s revenue per second contribution. By doing so, the network measures both categories of terminal with the same yard stick. Furthermore, an e-terminal whose battery charge runs out is likely replaced by a new terminal which (statistically) has similar properties to the departing one. Thus, the network may as well focus on revenue per second. Equations 27 and 30 are equivalent. Thus, the value/weight ratio for terminal i, while operating with a power fraction of πi > 0 and paying the network an amount that equals the terminal’s benefit, can be expressed as vi R̂i /πi , with R̂i := Li fi (Ŵi κ(πi ))Ri Mi (31) Given the preceding pricing analysis, at the operating point, πi should either be (i) z∗ , the value at the genu of fi (Ŵi κ(z)), or, if such value is “too high” for some reason, (ii) the highest reachable z, provided that at such z the terminal’s cost does not exceed its benefit. Optimal physical layer conf iguration Notice that (Li /Mi )(W/ Ŵi ) fi (Ŵi κ(z∗ ))/z∗ is determined by the physical-layer configuration (modulation, coding, data rate). If several such configurations are available, the network should impose the one that offers the largest: Fig. 10 With B(z) := S(z)/z, peakedness property zB ′ (z) retains the single- Li fi (Ŵi κ(z∗ )) Mi Ŵi z∗ (32) Mobile Netw Appl (2011) 16:640–660 651 because it leads to greater “benefit per Watt” when the terminal operates optimally. Thus, two terminals that have a common spreading gain (or data rate), should have a common PSRF, fi . This line of reasoning is explored further in a more general context in [20]. 5.2.2 Power limited cell The key is Section 3.2. If A are the indices of a certain set of terminals, they can occupy the cell each with a power fraction πi , if the total fractional “slice” allocated to them satisfies a condition similar to Eq. 12: with k ∈ A, and πk / p̄k ≥ πi / p̂i ∀i ∈ A,  πk (33) πi ≤ 1 − p̂k / p0 i∈A Terminal k is the “ruling terminal” of the set A. Notice that πk appears on the left side of constraint 33. Thus, with terminal k active, the total fractional “slice” left for possible companions is 1 − πk − πk p̂k / p0 (34) The largest achievable value for πk occurs when terminal k is alone in the cell, and equals p̂k /( p̂k + p0 ). A terminal’s “best subjects” Let J(1), J(2), . . ., J(N) be indices such that π J(1) π J(2) π J(N) ≥ ≥ ··· ≥ p̂ J(1) p̂ J(2) p̂ J(N) Thus, when all terminals are active, terminal J(1) is the ruling terminal. When J(1) is not active, terminal J(2) becomes ruling, and so on. Evidently, terminal J(N) “rules”, only when no one else is active. The network is interested in identifying the “best subjects” of terminal J(m) (m < N), defined as the set of terminals that produces the most revenue (while satisfying the pertinent feasibility condition) when J(m) is the ruling terminal, that is, with terminals J(1), . . . , J(m − 1) not active. One can identify this set, by applying the knapsack solution described in Section 5.2.1, with “knapsack capacity” given by Eq. 34, that is, 1 − π J(m) (1 + p0 / p̂ J(m) ). Let A J(m) be the set that contains (the indices of) terminal J(m) and its “best subjects” (with A J(N) := {J(N)}). The idea is to find A J(1) and compute and store the combined revenue that J(1) together with its “best subjects” produce. Subsequently, find A J(2) , and compute and store the combined revenue produced by the terminals in A J(2) . Then, proceed analogously with respect to J(3), J(4), and so on. Finally, from the previously obtained sets, choose the one that produces the most revenue, overall. Table 2 Key parameters for each terminal i vi Ŵi πi∗ g(πi∗ ) R̂i ri p̂i 1 2 3 4 5 6 7 3.5 4.0 3.1 3.5 3.7 3.5 3.0 32 64 32 64 128 128 64 0.26 0.15 0.26 0.15 0.08 0.08 0.15 0.88 0.86 0.88 0.86 0.85 0.85 0.86 3.52 1.72 3.52 1.72 0.85 0.85 1.72 46.6 46.3 41.3 40.5 39.7 37.6 34.7 1 5 4 3 1 4 4 Numerical illustrations Table 2 provides the key parameters for 7 terminals. Power limits are given as multiples of p0 . The common PSRF is that of Fig. 4, and units are such that (Li /Mi )Ri = 128/ Ŵi . ri stands for “value to weight” ratio. The service SIR’s (Ŵi κ(πi∗ )) are 11.5, 11.1 and 10.9 for spreading gains 32, 64, and 128 respectively, and g(πi ) := f (Ŵi κ(πi∗ ), the packet success rate. Table 3 applies the solution procedure of Section 5.2.2 to these terminals. The first column has the indices of the terminals sorted in order of descending value/weight ratio (ri in Table 2). The top row has the “slice” that can be allocated to all terminals when the terminal whose index is directly below “rules” (d j := π ∗j / p̂ j). The second row has the terminals’ indices sorted in descending order, from left to right, by ε j (terminals 7, 2, and 6 are not shown for reasons Table 3 A set of the terminals is chosen for service. 0.74 0.92 0.93 0.95 1 − dj πi v̂i [0] 0.26 12.3 1 1 0.15 6.9 1 [1] [0] 0.26 10.6 0,40 1 1 [1] 0.15 6.0 5 0 [1] [0] [0] 0.08 3.1 6 0 1 1 1 0.08 2.9 7 0 1 1 1 0.15 5.1 0.74 0.87 0.79 0.52 ∑ πi 30.1 35.0 31.9 21.0 ∑ v̂i i 1 5 3 4 1 [1] [0] [0] 2 1 1 3 1 4 652 explained below). Thus, terminal 1 has the greatest “value to weight” ratio, but coincidentally, it also has the highest π ∗j / p̂ j ratio (it is in the “worst situation”), which implies that terminal 1 “rules” when all are active. With terminal 1 absent, terminal 5 “rules”; and with 1 & 5 off, terminal 3 “rules”, and so on. The last two columns have respectively πi and v̂i := vi R̂i (the individual revenue contribution). A “1” (resp. “0”) in the position (i, j), (row, column), means that when terminal j “rules”, terminal i is (resp. is not) among the “best subjects” of j. A number between 0 and 1 at such position indicates that when j “rules”, terminal i is “fractionally served”. The brackets denote mandatory (from the algorithm viewpoint) inclusions or exclusions. Thus, when terminal 1 “rules”, terminal 2 and 3 are “fully” served, and terminal 4 is also served but with less than half of its desired “slice”. In this case, the sum of the slices is 0.74 and the total revenue brought by these terminals is 30.1. When terminal 5 “rules” (next column), terminal 1 is turned off by construction, and each of the other terminals can be “fully” served. The sum of the slices is 0.87 and the total revenue is 35.0. At this point the algorithm can be stopped. The group “ruled” by terminal 5 is chosen, because it produces more revenue than terminal 1 and its best subjects. Thus, even though terminal 1 itself produces more revenue (in absolute and relative terms) than any other terminal acting alone, it is the only terminal left out because of its “bad situation”. The final two columns (3 and 4 rules, respectively) are provided as illustration, but not needed. 5.2.3 Analysis of the algorithm The core algorithm that finds A J(m) and computes revenue utilises the simple knapsack solution of Section 5.2.1, and is applied at most N − 1 times (A J(N) = {J(N)} by definition). Additionally, it becomes progressively simpler, because as m grows to N, fewer terminals need to be considered. E.g., J(N) and J(N − 1) are the only potential members of A J(N−2) . Furthermore, it may be unnecessary to run the core procedure N − 1 times. The process can be stopped with terminal J(m), if each one of terminals J(m + 1) to J(N) can join J(m), each operating at its optimal power fraction and satisfying constraint 33 with k = J(m). Then, from the sets A J(1) , . . . , A J(m) , the one that produces the most revenue is chosen. For instance, if J(1) can be joined by all other terminals (each operating at πi∗ and satisfying Eq. 12) then, evidently, that is the best the network can do, and the process can be stopped right there. Mobile Netw Appl (2011) 16:640–660 Finally, this procedure is performed at the energy and processing-power rich base station. Thus, computational complexity does not seem to be a problem. 6 Game without direct cost For performance comparison purposes, we discuss below a situation in which each terminal can choose its power without a direct cost. The network may still charge the terminal some conventional subscription fee. But if there is no connection between what the terminal pays and its resource usage, it will use resources as if they were “free”. This scenario is best modelled as a “game of strategy”. Because there is neither pricing nor a central controller, this situation can be thought of a “worst case scenario”, which provides a “base line” for performance evaluation. 6.1 Game of strategy In a game of strategy, each of several “selfish” agents chooses a “strategy” in order to maximise its own “payoff”, which depends on the strategies chosen by all players. Herein, the strategy is a transmission power level, which increases the payoff of the transmitting terminal, but degrades the performance of all others (for introductory treatments, see [7, 12]). The key solution concept is the Nash Equilibrium (NE); i.e., a strategy per player (power level) such that no player would be better off by unilaterally changing strategy. The key to identifying the NE(s) is to first characterise each player’s “best response”. 6.2 A terminal’s best response In our game, the “best response” of terminal i is the power level terminal i chooses given a vector representing one power level for each of the other terminals. In fact, terminal i only needs the sum of the components of such vector, which determines its level of interference, Yi . 6.2.1 T-terminal best response Since the benefit of a time-driven terminal is strictly increasing in its SIR, it is self-evident that, without cost considerations, its (selfish) best response to any level of interference or noise is to transmit at maximal power. The resulting received power can be added into the noise term. Mobile Netw Appl (2011) 16:640–660 653 6.2.2 “Statutory battery” discipline A situation in which each t-terminal operates continuously at its maximal power is intuitively undesirable. Since power-pricing is now unavailable (by construction), a reasonable policy is to endow each terminal with a (virtual) “statutory battery”, i.e., a limit on the total amount of energy that it can emit over a specified time period (comparable to a typical battery life). The base station can tell the power level received from a given terminal, its channel gain, the times it is active, and, hence, its total energy emitted. Under this policy, the t-terminal maximises its total benefit by the time its energy emission allowance is exhausted (see Section 2.2); that is, the t-terminal behaves as an e-terminal. 6.2.3 E-terminal best response In any case, we can think that all players are eterminals, experiencing a fixed level of “noise” p0 (which may include power received from t-terminals that are not “virtual battery” powered, if any). From Section 4.4, for a given Yi and with σi := Ŵi hi Pi /Yi (the SIR), a terminal with energy budget Ei chooses its transmission power, Pi , to maximise Ei vi Ri fi (Ŵ i hi Pi /Yi )/Pi , or hi Li fi (Ŵi hi Pi /Yi ) fi (x) Ei vi W ∝ Yi Mi Ŵi hi Pi /Yi x (35) Thus, the terminal’s best response is to set the power level so that its SIR maximises fi (x)/x. By Lemma 4.2, this ratio is single-peaked, and its unique maximiser, x∗ , can be easily identified at the genu of the graph of the PSRF. If x∗ cannot be reached because of power limitations, the terminal operates at its power limit. All terminals with a common PSRF will aim at the same SIR (even if each has its own willingness to pay, data rate, channel state, packet size, and energy budget). If several link-layer configurations are available, the configuration whose PSRF has the highest ratio f (x∗ )/x∗ should be chosen (see [20]). We assume below a common PSRF, thus, all terminals aim at the same SIR, x∗ . 6.3 Equilibrium of the game 6.3.1 Symmetric equilibrium From Section 3.2, it is feasible for each of N terminals to experience SIR x∗ if and only if condition 12 is  satisfied; that is, πi ≤ 1 − πk /( p̂k / p0 ), where πi = x∗ /(x∗ + Ŵi ) ≡ (1 + Ŵi /x∗ )−1 , and k such that πk / p̂k ≥ πi / p̂i ∀i ∈ {1, . . . , N} (k is the “ruling terminal”). Thus, if condition 12 holds, each terminal can reach x∗ with the feasible power level given Eq. 9: h j P j = 1− π j p0 N n=1 πn ≡ p0 (1 + Ŵ̄ j)−1 N 1 − n=1 (1 + Ŵ̄n )−1 (36) Since each terminal operates at its best-response SIR (for any level of interference), Eq. 36 describes a Nash equilibrium in closed form (with Ŵ̄ j := Ŵ j/x∗ , each quantity in Eq. 36 is presumed known). Of course, condition 12 may not be satisfied. 6.3.2 Nash equilibrium in the general case The general procedure to find the Nash equilibrium of this game is reminiscent of Section 5.2.2. Each terminal must obey inequality 14, which can be written (with π̄ =  πi , and p̂i := hi P̂i ) as πi / p̂i ≤ (1 − π̄ )/ p0 . The right side is the same for all terminals, thus the terminal with the greatest ratio πi / p̂i has the most difficulty to meet this common constraint, therefore it is in the “worst situation” (or it is the “weakest”). If this terminal “gives up” on obtaining the optimal SIR, and simply operates at its maximal power, then constraint 14 no longer counts for this terminal, since it no longer aims for the optimal SIR (or any specific one). Then, the terminal with the second highest πi / p̂i can be thought of being in the worst situation (among those still aiming for the optimal SIR). Notice that when the data rate is common, Ŵi = Ŵ and hence πi = π , then, the terminal with the smallest p̂i is in the worst situation. But a terminal with a large spreading gain, Ŵi , could have the smallest p̂i and not be in the worst situation because it may achieve the desired SIR with a very small percentage of the receiver power (πi = 1/(1 + Ŵi /x∗ )). Then, to find the Nash equilibrium in the general case, first sort the terminals by their πi / p̂i ratios, and re-label the terminals such that π1 / p̂1 ≤ · · · ≤ π N / p̂ N . First check whether the symmetric NE (Section 6.3.1) exists, and if yes, stop. Otherwise, (i) find the terminal in the worst situation and assume it operates at its maximal power level, (ii) add its received power to the noise term, and then (iii) check whether the remaining N − 1 terminals can achieve the desired SIR under this new scenario. If the answer is yes, stop. Otherwise, proceed recursively with two terminals maxed out. In the second iteration (if necessary), to check whether terminals 1, . . . , N − 1 can reach the desired SIR with terminal N maxed out, the condi N−1 tion is i=1 πi ≤ 1 − π N−1 /( p̂ N−1 / p0(1) ) where p0(1) := 654 Mobile Netw Appl (2011) 16:640–660 p0 + p̂ N . At the (m + 1)th iteration, to check whether terminals 1, . . . , N − m can reach the desired SIR, while each of terminals N − m + 1, . . . , N operates at its maximal transmission power, the condition becomes operating with SIR xi◦ , the terminal successfully uploads (Li /Mi )Ri fi (xi◦ ) information bits per second, yielding a benefit of N−m  6.3.4 Numerical illustration i=1 πi ≤ 1 − π N−m p̂ N−m / p0(m) (37)  with p0(m) := p0 + Nj=N−m+1 p̂ j. If constraint 37 is satisfied, the procedure stops after m + 1 iterations. Otherwise, it continues recursively. In general, the game ends with N∗ terminals achieving x∗ (N∗ = 0 is possible, of course), and each of the remaining N − N∗ operating at its maximal power level, and experiencing the resulting SIR. It is clear that an x∗ -achieving terminal has no incentive to unilaterally change its power level. It is also clear that each of the remaining terminals would gain by unilaterally raising its power level, but cannot do so, because it is maxed out already. Thus, the procedure indeed stops at a Nash equilibrium. 6.3.3 Benef it at equilibrium Once the equilibrium is found, one can apply Eq. 35 to calculate a terminal’s total benefit while operating at equilibrium. With xi◦ and pi◦ denoting respectively the  ◦ equilibrium SIR and received power, p̄ = p0 + pi , and assuming a common energy budget, E, Eq. 35 yields: f (x◦ ) vi hi f (x◦ ) vi hi L EW ◦i := K ◦i M xi Yi xi p̄ + p0 − pi◦ (38) The constant K (measured in Joules) just gathers a product of parameters, and could be set to 1 by choosing convenient units of measurement. For an x∗ achieving terminal, xi◦ = x∗ and pi◦ is given by Eq. 9. If these terminal’s data rates are of the same order of magnitude, so will be the equilibrium power levels, and thus the denominator of Eq. 38 will be approximately constant for all these terminals. Then, the benefit of an x∗ -achieving terminals will be, with convenient units, ≈ vi hi . For terminals not achieving x∗ , pi◦ = p̂i and xi◦ = Ŵi p̂i /( p̄ + p0 − p̂i ). Equation 38 yields a terminal’s benefit over its “battery life”. This is of interest to the network, because what the terminal is willing to pay under any “conventional” fee structure is, ultimately, limited by what the terminal’s “gets out of” the system. However, for reasons discussed in Section 5.2.1, the network may be more interested in benefit per time unit. While vi (Li /Mi )Ri f (xi◦ ) (39) In Table 4, we apply the procedure of Section 6.3.2, for the terminals of Table 2, assuming they are all energy-constrained. The common PSRF is that of Fig. 4. The maximiser of f (x)/x is 10.75, hence, πi = 10.75/(10.75 + Ŵi ). With the original indices, the πi / p̂i ratios are, respectively, 0.25, 0.03, 0.06, 0.05, 0.08, 0.02 and 0.04. This yields the sorting [ 6 2 7 4 3 5 1 ]. The first column of Table 4 has the terminals sorted by their “situations”, from “worst” to “best”. Each has been re-labelled (original indices shown in parenthesis). We use the notation of condition 37 to explain the next columns. The 2nd column corre sponds to p0(m) = p0 + Nj=N−m+1 p̂ j, the third column has the ratio p̂ N−m / p0(m) , and the fourth column has π N−m p0(m) / p̂ N−m . The fifth and sixth columns have, respectively, the left and right side of condition 37. The first time the left side is less than or equal to the right side, a NE has been found. Thus, this example ends at the 2nd iteration, with only the terminal 7(1) operating at maximal power, and all others enjoying their chosen SIR σ ∗ . The row corresponding to 5(3) is given as illustration, but not needed. The equilibrium power levels can be obtained explicitly through Eq. 9 (with p0 replaced by p0(1) = p0 + p̂7 = 2). 1 − π̄ = 0.16, thus hi Pi = 2(πi /0.16). E.g., hi Pi = 7/4, for terminals 4, 3(7), and 2. Then, the equilibrium benefits can be calculated through Eq. 38. Table 4 Finding a Nash equilibrium i p+0 p̂w p+ 0 πw p̂w ∑z 1− d πi p̂i 7(1) 1 0.25 0.25 1.09 0.75 0.25 1 6(5) 2 0.50 0.15 0.84 0.85 0.08 1 5(3) 3 1.33 0.19 0.76 0,.81 0.25 4 4(4) 7 - - - 0.14 3 3(7) - - - - 0,.14 4 2(2) - - - - 0.14 5 1(6) - - - - 0.08 4 Mobile Netw Appl (2011) 16:640–660 7 Performance experiments Below we use the no-direct-cost game as a base line for performance. The key performance index is the total benefit derived by the terminals (which equals the network’s revenue under the direct pricing scheme). Benefit is normalised by dividing it by “representative benefit”, which is obtained assuming that (i) the number of terminals is the largest that can be served at the optimal SIR, (ii) when all have the average spreading gain, (iii) are uniformly distributed along the cell radius (1 Km), and (iv) have the average willingness to pay. Figure 11 shows how performance changes with some measures of “congestion”. The pricing scheme (a) 655 performs only marginally better than the game when the number of terminals is “small” (meaning that all can reach the optimal SIR in the game). However, as the number of terminals grow, the performance gap increases. This is reasonable because under pricing, the network always chooses to serve the “best” (revenue/Watt) users. Thus, having many users to choose from is actually good for the network. The game, however, always “serves” everybody (at equilibrium all terminals transmit). With many terminals operating at full power, most get very poor performance. However the game performance does not “collapse” quickly, because each of many terminals does transmit and “gets something”, and there are always a few of them that are very close to the base station, and get reasonable performance, which helps the game total. Thus, the specific “gain” of the pricing scheme depends on how “busy” the cell is. For instance, Fig. 11a indicates that with the “correct” arrival rate (that equal to the number of terminals that can be served at the optimal operating point), pricing outperforms the game significantly, but modestly (8 to 7). If the arrival rate drops to 60% of this level, the performance gap is only a few percentage points. Pricing outperforms the game about 4 to 3 (33%) when the arrival rate is 40% above the “right level”. Figure 12 describes how the “spread” of the willingness-to-pay values (“social classes”) affects performance. The equally-likely wtp are the components of 4v/s, where v = [1 a a2 a3 ] and s is the sum of v’s components (e.g., with a=1/2, v = [1 1/2 1/4 1/8] and s = 15/8, but with a = 1 vi = 1 ∀i). With this choice, the expected value of the wtp vector (4v/s) is 1, for (b) Fig. 11 Performance versus “congestion” under power-share pricing (solid) and at the equilibrium of a no-direct-cost game Fig. 12 The greater the inequality (lower a), the better the performance of the pricing scheme. But the game performance is unaffected 656 Mobile Netw Appl (2011) 16:640–660 all a. With the arrival rate fixed at the “right level”, the pricing scheme advantage grows with “inequality” (doubles with a = 1/2), which is reasonable because the network considers a users’ wtp to decide whom to serve, but the game is blind to monetary considerations. 8 Socially optimal allocation 8.1 Power limits of data terminals 8.4 Numerical illustration 8.4.1 Finding the socially optimal price One can solve the system of (nonlinear) Eq. 43 by algebraic or numerical means. But Fig. 13 provides greater insight (recall Figs 9b and 8). The planner sweeps a price line, from vertical to horizontal, until it finds the optimal slope. Any price greater than c∗1 (a line to the left of c∗1 z (red, dash)) is “too high”. When the price To simplify the exposition, we assume that there is a power-constrained media terminal with a service level agreement that specifies its spreading gain Ŵ M and SIR, σ M , and denote the total “slice” available for data terminals as 1 − d.  In principle, any combination of πi satisfying π̄ = πi ≤ 1 − d can be assigned to the data terminals. The largest possible value of pi occurs when πi = π̄ = 1 − d (i is the only active data terminal), which yields pi = p0 (1 − d)/d (Eq. 9). We assume below that for all i, p̂i / p0 ≥ (1 − d)/d. Therefore, each data terminal can reach its resulting power level. 8.2 Objective function With Vi (πi ) denoting the appropriate benefit function (depends on the terminal’s energy class), a reasonable criterion for a social planner is to solve maximise: subject to, N Vi (πi ) (40) πi ≤ 1 − d (41) πi ≥ 0 (42) i=1 N i=1 (a) The necessary optimising conditions are [9, 11]: Vi′ (πi ) − μ0 ≤ 0 with equality for πi > 0 (43) 8.3 Social optimum through pricing From Eq. 43, any terminal receiving a positive share of the power must satisfy Vi′ (πi ) = μ0 . From Section 4.3, and Fig. 9b, a t-terminal can satisfy Eq. 43, and hence reach a “socially optimal” allocation, provided that a common price is set, which coincides with μ0 . Per Section 4.4.2 and Fig. 7, in order for the eterminal’s behaviour to lead to Eq. 43, quadratic pricing must be used. The corresponding analysis is in Section 4.4.3. (b) Fig. 13 Bell and S curves are benefit graphs. A terminal’s optimal allocation is identified by a short tangent, parallel to the solid blue line representing the socially optimal price, for the given resource constraint. All can be served when total resource is 0.84 (a), but terminal 5 is left out when the resource drops to 0.54 Mobile Netw Appl (2011) 16:640–660 falls to c∗1 , terminal 1 chooses to operate, and as the price continues to drop, more terminals become active and/or those already active increase their purchase. The planner stops when the sum of “slices” equals 1 − d. 8.4.2 Operating point: planner’s versus network’s As shown by Fig. 13, the first terminals to become active are precisely those with the “steepest” tangenu, that is, those with the highest “value to weight” ratio, which is precisely the criterion used by the (monopolistic) network. However, the network chooses an individual price per terminal such that each operates at the genu (“knee”), where each maximises “benefit per Watt”. The planner chooses a common price, and each active terminal ends up paying less that it would under the network’s price. But at the lower planner’s price, each active terminal consumes more, which may reduce the total number of terminals that can receive service. 9 Discussion In the reverse link of a CDMA network with N terminals, each can receive service quality (SIR) σi only if  πi ≤ 1 − d where πi = σi /(σi + Ŵi ) (Ŵi is the spreading gain and 0 < d < 1). πi also equals i’s share of the total power at the receiver. Thus, the uplink management can be approached as a “pie division” problem, where the total receiver power is the pie and πi is the fractional “slice” assigned to i. We have proposed, analysed, and evaluated a technical-economic scheme based on the key variable chosen by nature: πi . Each data terminal has its own data rate, channel gain, willingness to pay (wtp), and link-layer configuration. Some have limited energy (e-terminals), but others not (t-terminals) and we have specified appropriate performance metrics for both types. The receiver power fraction, πi , immediately determines the carrier-tointerference ratio, κi = πi /(1 − πi ), which directly leads to the SIR, σi = Ŵi κi . Thus, given a price on πi , the terminal can individually make an optimal choice irrespective of choices made by others. This is a major advantage of our proposal. The network ultimately chooses for each terminal an individual price that forces it to operate where “revenue per Watt” is highest. Of course, the sum of the “slices” ordered by the terminals may exceed resource availability. Then, the network follows a “knapsack” approach [13] to find, among all sets of terminals that satisfy the constraint, the revenue maximiser. Thus, our proposal simplifies the terminal’s choice at the 657 expense of (reasonably) complicating the network’s (a favourable trade-off, given the energy and/or computational limitations of a terminal). As a base-line for performance evaluation, a “game” in which each terminal chooses its power to maximise its own performance without a direct cost is solved “from first principles”. A “player” with unlimited energy will evidently always set its power to the maximal level, raising the “noise floor” of all. However, energy-conservation can be induced through a limit on total energy spent over an appropriate period of time (“statutory battery”). The existence of a Nash equilibrium is proved “constructively”, by showing it in closed form. The pricing scheme always outperforms the game, and the performance gap grows with the number of terminals in the system, and also tends to increase with “social inequality”. Moreover, a limited study of “social benefit” maximisation indicates that our proposal can achieve the social optimum, with a price common to all terminals, and with each energyconstrained terminal paying in proportion to the square of its power fraction. Thus, our analysis can equally lead to a “true” economic pricing scheme for network profit, or may be employed as a benefit-maximising algorithmic metaphor along the lines of [3]. We have captured the critical packet-success-rate function (PSRF) through an S-curve of unspecified algebraic form, and approach from [16, 17], which have been found useful in numerous scientific contributions, such as [14, 15]. This makes our analysis relevant to a wide variety of physical layer configurations, and in fact allows us to optimally (re)configure the link layer, as discussed in greater detail in [20]. For each of the three environments considered (network pricing, game, and “social” pricing) complete numerical examples have been given in tabular and/or graphical form. However, this work has an analytical core of three “general” maximisation problems: (1) B(x), (2) S(x) − cx and (3) B(x) − cx (S(x) is an Scurve and B(x) = S(x)/x, a single-peaked curve). The 1st is key when an e-terminal faces linear or zero price (Fig. 7), and the 2nd is fundamental for the t-terminal’s analysis (Fig. 9). The third arises when an e-terminal faces quadratic pricing, a requirement of the “social optimum” (Fig. 8). We have characterised the solutions through the geometrical properties of the graphs, under the presumption that “all we know” is that S is an Scurve, and that it satisfies some additional technical conditions—discussed in the Appendix—which guarantee that the shapes of certain graphs are as assumed (S(x)/x has been proved to be “single-peaked” in [16, 17]). 658 The present work has no doubt limitations. As in [22] and many related works, we have only considered one cell (inter-cell interference can be added to noise). Furthermore, as commonly done in the power control literature, channel gains have been assumed fixed. This does not mean that our analysis only applies when users are immobile and channel are stable, but rather that there is a separation of timescale between power updates and changes in propagation conditions [8, Section 2.6.2]. In a practical system, channel state information is updated periodically. After each update, our analysis may lead to a revised resource allocation. We believe the single-terminal pricing analysis to be rigorous. However, the extension to a multi-terminal scenario through the knapsack approach should be viewed as an analytically-supported heuristic. Likewise, our solution to the interesting and challenging social planner problem is partial. And we have neglected internetwork price competition. These limitations suggest avenues for future work. Along these lines, in [21] we have applied a similar approach to fourth-generation cellular networks, by proposing an auction procedure to allocate frequency sub-channels combined with pricing for power allocation. We remain optimistic that, despite these limitations, (i) the advantages offered by our proposal, (ii) the generality of our model, and (iii) the innovative methodology we have utilised combine to produce a useful addition to existing scientific literature. Acknowledgements Most of the basic research reported herein was performed at the Universität Karlsruhe (now Karlsruher Institut für Technologie) with the financial support of the European Commission through the PULSERS II project. Important additional work was done at RWTH Aachen with the support of the Deutsche Forschungsgemeinschaft (DFG), through the UMIC project. The preparation of the final revised manuscript was supported by the Communication Networks (ComNets) Chair at RWTH Aachen. 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 : ℜ+ → [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. Mobile Netw Appl (2011) 16:640–660 Remark A.1 In Definition A.1, S is strictly increasing and also surjective (for each y ∈ [0, Y] there is an x ∈ ℜ+ 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 : ℜ+ → [0, Y] is singlepeaked over ℜ+ if h is continuous, surjective and has a global maximum at X ∈ (0, ∞) (that is, h(X) = Y, 0 ≤ x1 < x2 ≤ X =⇒ h(x2 ) > h(x1 ) and X ≤ x1 < x2 =⇒ h(x2 ) < h(x1 )). 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 satisf ies Def inition A.1, then the composite function s(z) := S(g(z)) with g(z) = Ŵz/(1 − z), Ŵ ≥ 1 and z ∈ [0, 1) also satisf ies Def inition A.1. Assumption 2 If S satisf ies Def inition A.1, then each of the following functions satisf ies Def inition A.2: (i)xS′ (x), and for Ŵ ≥ 1(ii) (x/ Ŵ + 1)2 S′ (x), (iii) x(x/ Ŵ + 1)S′ (x) and (iv) xB ′ (x) where B(x) := S(x)/x with B(0) := limx↓0 B(x) ≡ S′ (0). Remark A.4 By Lemma 4.2, if S satisfies Definition A.1 then 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 Mobile Netw Appl (2011) 16:640–660 game theory [2]. In the present work, a strong version of this notion is formalised as follows: Definition 9.3 A function f : D → ℜ with D ⊂ ℜ satisfies the unique-crossing from above condition (UCC) over D′ ⊂ D if ∃t0 ∈ D′ such that f (t0 ) = 0 and ∀t ∈ D′ t < t0 =⇒ f (t) > 0 and t > t0 =⇒ f (t) < 0. Lemma A.1 Consider the composite function S(g(z)), where S : ℜ+ → [0, d], is an S-curve with inf lexion at x f , and g : [a, b ] → ℜ+ , with 0 ≤ a < b , is a strictly increasing convex function such that limz→b g(z) = ∞. (i) If the function [g′ (z)]2 S′′ (g(z)) + g′′ (z)S′ (g(z)) satisf ies the UCC over [a, b ], then: (ia) the composite function s(z) := S(g(z)) satisf ies Def inition A.1, and (ib) its inf lexion abscissa z f is such that g(z f ) > x f . (ii) With g(z) = Ŵz/(1 − z), Ŵ ≥ 1 and I = [0, 1), conclusions (ia) and (ib) follow if the function (x + Ŵ)S′′ (x) + 2S′ (x) satisf ies the UCC over the domain ℜ+ . 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 zcf . The fact that s asymptotically goes to d as z goes to b is immediate. (ib) zcf must satisfy g(zcf ) > x f so that S′′ (g(z f )) be negative. (ii) If g(z) = Ŵz/(1 − z) then g′ (z) = Ŵ/(1 − z)2 and g′′ (z) = 2Ŵ/(1 − z)3 . Considering these expressions, the second derivative of the composite function becomes: 2Ŵ Ŵ2 S′′ (g(z)) + S′ (g(z)) (1 − z)4 (1 − z)3 which has the same sign as ŴS′′ (g(z)) + 2(1 − z)S′ (g(z)). With x := Ŵz/(1 − z), z = x/(x + Ŵ) and 1 − z = Ŵ/(x + Ŵ). 659 Therefore, the second derivative has the sign of (x + Ŵ)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 : ℜ+ → [0, d] is an S-curve with inf lexion at t f , and g : ℜ+ → ℜ+ is a strictly increasing continuously dif ferentiable function. 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