Total optimisation of a media-streaming wirelessterminal: Energy-efficient link adaptation under

higher-layer criteriaVirgilio Rodriguez

Communication Networks (ComNets)RWTH Aachen

Aachen, Germanyemail: rodriguez@comnets.rwth-aachen.de

Abstract—It has long been recognised that a wireless commu-nication system can be more efficient if link-layer parameterssuch as modulation order, symbol rate and packet size, are(adaptively) optimised. A common optimising criterion is to max-imise spectral efficiency (bits per second per Hertz (bps/Hertz))subject to a very low bit-error constraint. But a packet-orientedcriterion for link adaptation seems more appropriate for practicalcommunication networks fitted with strong error detection and aselective packet re-transmission mechanism. In recent work, weperformed link optimisation for maximal bits per second or bitsper Joule for data (delay-tolerant) traffic. In the present work, weextend our previous analysis to consider a (delay-sensitive) mediastreaming application (e.g. music, or video), which introducesyet another degree of freedom: the number of bits allocated toa second of the media signal (encoding rate). We consider anyarbitrary combination of link-layer parameters, with the onlyrestriction that the corresponding packet-success rate function(PSRF) is an S-curve. Under a very general optimisation criteria,we obtain a robust result for a media-streaming terminal withan adjustable symbol rate: the key is a tangent line from (0,0)to a scaled graph of the PSRF, the steeper the line (greaterslope) the better the configuration. Under additional reasonableassumptions, at the optimum, both throughput per unit SNRand “fidelity” per unit rate are maximal, which is intuitivelyappealing.

I. INTRODUCTION

The importance of (adaptively) optimising the link layerconfiguration of a wireless communication systems has longbeen recognised. The adaptation to a time-varying channel ofsuch parameters as constellation size, symbol rate and codingcan result in increased efficiency. In particular, modulationadaptation has received significant attention in the literature[1],[2], [3], [4]. In these and most studies, the focus is the symbol:for example, to choose bits per symbol in order to maximisebps/Hertz (spectral efficiency), while holding the bit error rateunder a specified (very low) level. More recently, it has beenrecognised that packet-oriented (OSI layer-2) link adaptationis more appropriate under certain interesting scenarios [5].

In many practical communication networks, medium-access-control (MAC) packets are “guaranteed” in the sensethat binary data is packetised, strong error-detecting codes(e.g., cyclic-redundancy codes (CRC)) are added to each datapacket, and an automated repeat-request (ARQ) mechanism is

installed. Sufficiently long CRC codes (e.g., 16 bits) performextremely well in practise, and the feedback ARQ channel canbe designed for very high reliability. Under these conditions,the error rate at the input of the error detector can berelatively high, because a packet received in error is simplyretransmitted. Therefore, the common criterion of maximisingspectral efficiency subject to an extremely low bit-error con-straint does not seem appropriate for the ARQ-based system.The engineer should focus instead on the post-retransmissionperformance (e.g., “net” throughput or “goodput”). Alongthose lines, [5] seeks to (adaptively) configure the link layerparameters for maximal “goodput”. Following a related butdifferent approach, [6] reports adaptive rate and power policiesfor network utility maximisation (NUM) — under an averagetransmitted power constraint — that are significantly differentfrom spectral-efficiency policies.

While following the rationale and motivation of [5], [7]employs a different methodology: it observes that the impactof any given combination of link configuration parameters isfully captured by the packet-success rate function (PSRF) ofthe symbol SNR, and postulates that the graph of the PSRF isan “S-curve”– a very mild assumption because, as shown byFig. 3, the S-shape is very versatile, including as special casescurves that are sufficiently close to many common shapes.Then, analytical geometry leads [7] to conclude that a set ofpossible link configurations can be ranked by the slope of atangent line from the origin to the graph of a scaled version ofthe PSRF: the steeper the tangent (greater slope) the better theconfiguration. However, the optimisation criterion utilised by[7] for an energy-limited delay-tolerant (data) terminal, totaltransferred information bits over a battery charge (bits perJoule), is not appropriate for media streaming applications. Forthese applications, the desired media signal is reconstructedperiodically by converting the received information bits intoa corresponding segment of media (e.g., music or video)for immediate user perception. If the number of bits perreconstructed media segment is very low, the correspondingperceptual quality may be unacceptably low, in which casethe system will perform poorly, even if the total number oftransferred bits per Joule is high. A different criterion is needed

for a media streaming terminal. Presently, we extend [7] tothe media streaming case, with a focus on video. Anotherpredecessor work is [8], in which transmission power is theonly link parameter considered.

Below, we start by describing the video streaming process,and the data packet communication model. Then we discussoptimisation criteria appropriate for link layer configurationfor an energy-limited video-streaming terminal. Subsequently,we provide the core of our analysis and main results with afocus on the energy-limited terminal that can adjust its symbolrate. Then, we illustrate and discuss our results. An appendixprovides some additional technical details.

II. SYSTEM MODEL

A. Video streaming and reconstruction

The process that gives rise to a video signal is as follows.A video camera takes a sequence of still pictures. Eachstill picture (“frame”) is an image represented by a grid of“pixels”, with each pixel value represented by several bits(a matrix). When these images are shown in sequence at asufficiently high rate (e.g., 30 frames per second), the humaneye cannot perceive the transition between frames, whichcreates the illusion of motion. If each frame is a sufficientlyrich representation of reality (a very large number of pixelseach encoded through many bits), the human brain perceives“realistic” video.

In addition to efficient image encoding per frame, a goodvideo coder can exploit the similarities between successiveframes. However, even efficiently encoded, “full quality” videomay be too resource expensive (especially for wireless trans-mission). Fortunately, several degrees of freedom exist to saveresources at the expense of perceptual quality: pixels per frame(image size), bits per pixel, and frames per second. Generally,the perceptual quality of the reconstructed video segment (sayone second long) will be an increasing function of the numberof bits allocated to the segment (“rate”). In fact, some videocoders represent a segment of video through a bit stream that,ideally, can be truncated at any point and decoded, yieldinga continuum of perceptual quality levels, depending on thetruncation point (rate)[9].

Fig. 1 shows schematically a system engaged in videostreaming over a wireless channel. We assume that an idealscalable video encoder is used, although this is not strictlynecessary. Each T -second of raw video is encoded as a Y -bit file, which is broken up into packets. Error-control bitsare added to each packet (error-control system not shown).These packets enter a large buffer prior to transmission.Packets are transmitted at the symbol rate of Rs with b bitsper symbol. Packets received in error are retransmitted untilreceived correctly (further detail below). Correctly receivedpackets are placed in a large buffer. Every Δ ≤ T seconds theinformation bits brought by the correct packets are decoded toreproduce the corresponding video segment, which is viewedby an end-user. If only the first y bits of the original Y -bit encoded file are available, the reproduced video segmentwill have “perceptual quality” (i.e., the segment quality as

VIDEO DECODER

Y bits perT secs of video

LARGE

BUFFER

LARGE

BUFFER

VIDEO ENCODER

Figure 1. High-level view of wireless video streaming

0 i K0

k

1

S(x)/x

S(x)

xS’(x)

Figure 2. An S-curve, S(x), xS′(x) (solid bell curve), and the tangenu (tangentline from origin). The inflexion (and hence the peak of S′(x) which is notshown but is similar to xS′(x)) occurs at x = i, and the genu (knee) at (K,k).S(x)/x (dash, scaled) is maximised at x = K. The interpretation of x and S(x)varies depending on the context.

perceived by the end user) q(y), where q is some increasingfunction.

B. Physical layer model

∙ N0 is the average Gaussian noise spectral density∙ E is the energy budget, when applicable∙ P is the power constraint, if any∙ H is the channel gain, and h := H/N0

∙ p = HP is the received power∙ Rs is the symbol rate∙ b bits per symbol∙ σs is the signal-to-noise ratio (SNR) per symbol∙ L-bit packets carrying L−C information bits are used.∙ For a given combination of the relevant parameters,a,

F(x;a) is the packet-success rate function (PSRF), whichgives the probability of correct reception of a data packetas a function of the per-symbol SNR, x.

∙ For some technical reasons, f (x;a) := F(x;a)−F(0;a)replaces F [10]. For any a, f is assumed to satisfyDefinition A.1; that is, its graph as function of the persymbol signal-to-noise ratio (SNR) has the S-shape shownin Fig. 2, which is a very mild assumption (see Fig. 3).The relevant parameters depend on specific characteristicsof the communication system, and technical assumptionsmade by the analyst. Further detail– a very mild assump-tion because, as shown by Fig. 3, the S-shape is verygeneral, including curves that are sufficiently close to

Figure 3. The S shape is very versatile : Besides the S-shape proper (U3), italso covers the “ramp” (U2), the (“mostly”) concave (U1), and the (“mostly”)convex curve (U4). The “ramp” of an S-curve (see U3) can approximate a lineover a range of interest (axes intentionally left unmarked).

many common shapes. Further detail in the appendix.

For convenience, x is used as a generic function argument.

C. Information transferred over a period of interest

The total number of information bits that a terminal operat-ing with packet-success rate f (⋅;a) — where a includes packetlength L, C overhead bits per packet, and b bits per symbol— and SNR held at x can transfer over time period τ is givenby

τL−C

LbRs f (x;a) (1)

III. LINK ADAPTATION CRITERIA FOR MEDIA TRAFFIC

We focus on a terminal with a long video stream to transfer,but could apply the analysis to other media signals (such asmusic).

If the terminal of interest has reliable information aboutfuture channel states it could take it into account. In practise,such information is rarely available. Thus the terminal optto manage its resources as if the present channel state werepermanent (that is, it is the most reliable estimate of the futurestate). As channel state information is updated periodically,the terminal re-calculates its optimum operating parameters,which may lead to a change in its link configuration.

The terminal’s performance is determined by a functionu which is increasing in each of two arguments: (i) thetotal number of transferred media “segments”, ν, and (ii) the“perceptual quality” of each segment, q. This simply meansthat for a given number of segments, the terminal prefersmore perceptual quality per segment; and for a given levelof perceptual quality, q, the terminal prefers more segments,which is eminently reasonable. For technical reasons, we alsoassume that for any q, u(⋅,q) is bounded (thus a terminal thatviews a very large number of segments of given q will stillhave a finite utility). In our analysis below we do not assumeany more about u, unless explicitly stated.

For simplicity, we take a “segment” of video to last exactlyone second (T = 1 in Fig. 1). Thus, a terminal with energy

budget E while operating at power P will transfer a total ofE/P segments by the time its energy runs out. With powerP, and noise-normalised channel gain h the terminal willtransfer y information bits per second (more precisely y(hP)),on average, which will result in a segment of perceptual qualityq(y), where q is strictly increasing in y. In this case, theterminal’s performance will be

u(E/P,q(y)) (2)

This is the index that the energy-limited terminal wishes tomaximise.

It is useful to think that each segment is ideally scalably-encoded [9], so that whatever the number y of transferredinformation bits corresponding to that segment, the decoderat the receiver can produce a corresponding video segment ofperceptual quality q(y).

It is not unusual for some studies to assume that a mediaterminal demands an inflexible level of perceptual qualitywhich is presumably exogenous (i.e., determined by forcesoutside the analysis) and often expressed as a hard constrainton the maximal level of “distortion” which the terminaltolerates. This in turns induces a hard constraint in the “rate”that must be delivered (y herein).

However, such a hard constraint on perceptual quality israrely appropriate, and even less so for a terminal operatingover a wireless channel, and with limited energy budget.Experience shows that video streams can be usable at a verywide range of perceptual quality levels. For example, sportsevent video streams in actual use in the Internet today areoffered in a wide range that starts near 100 Kbps and extendsto several Mbps. While the corresponding level of perceptualquality for the lowest rate is clearly far from ideal, it iscertainly usable, which is why it is offered, and could representthe rational choice under severe resource constraint. Thus, itseems more appropriate to assume that the “rate” of the videois one more variable to be chosen optimally from a wide range,considering channel conditions and resource availability. Thechoice may well be constrained from below, but we do notexplicitly consider such constraint herein.

IV. ENERGY-EFFICIENT LINK CONFIGURATION FOR MEDIA

STREAMING

Lemma IV.1. For a given link configuration, a, a terminalwith adjustable symbol rate, energy budget E, power limit Pand a long sequence of media segments to stream, allocatesits resources seeking to maximise

maxx≥0,0≤y≤Y (x;a)

u

(Ehy

B(x;a),q(y))

(3)

where x denotes the terminal’s per-symbol signal-to-noiseratio, y denotes the number of transferred information bitsper 1-second segment, q(y) gives the corresponding per-ceptual quality of the decoded media segment, S(x;a) =((L − C)/L)b f (x;a), B(x;a) := S(x;a)/x with B(0;a) :=limx↓0 B(x;a) ≡ S′(0;a) and Y (x;a) = hPB(x;a). S satisfiesDefinition A.1.

Proof: If the terminal operates with constant power P >0, it can transfer ν = E/P 1-sec media segments. Thus, theterminal should maximise u(E/P,q(y)) (Eq. (2)). If it also setsits symbol rate to Rs, it achieves the per-symbol signal-to-noiseratio (SNR) of x = hP/Rs.

By (1), per time unit the terminal transfers

y =L−C

LbRs f

(hPRs

;a)= hP

L−CL

bf (x;a)

x(4)

= hPS(x;a)/x with S(x;a) = ((L−C)/L)b f (x;a).Thus, y = hPB(x;a).The condition B(0;a) := limx↓0 B(x;a)≡ S′(0;a) is given to

specify the value of B at the origin.For a given link configuration, b(L−C)/L is a constant.

Thus, S satisfies Definition A.1 because f does.For P > 0, (4) can be re-written as 1/P = (h/y)B(x;a)Thus, the terminal must maximise u((Eh/y)B(x;a),q(y)).With an adjustable symbol rate, the terminal can freely

choose any x> 0, since for given P it can always set Rs = hP/xto achieve the desired x. However, by (4), for given x, thelargest y that can be achieved is Y (x;a) := hPB(x;a) (withpower at the maximal level). Thus the terminal can also freelychoose any positive y such that y ≤ Y (x;a).

Remark IV.1. One way to reduce the symbol rate from amaximal rate of Rs to Rs/K is by transmitting K identicalsymbols each at the maximal rate Rs. These K symbols are,at the receiver, “coherently accumulated” which increases theratio of the energy per symbol to the noise spectral density[11, section II.B].

Lemma IV.2. For the terminal of Lemma IV.1, the optimalper-symbol SNR x∗ is the maximiser of B(x;a).

Proof: The terminal wants to maximiseu((Eh/y)B(x;a),q(y)). Operating at the x that maximisesB(x;a) is advantageous for two reasons: (i) it evidentlyincreases (Eh/y)B(x;a), the first argument of u, andtherefore performance, since u is increasing in each argumentand (ii) it also increases the largest possible y; i.e., it widensthe interval from which y can be chosen (the largest y is notnecessarily optimal, but to have a wider interval from whichto choose y cannot possibly hurt).

Lemma IV.3. Let S : ℜ+ → [0,d] satisfy Definition A.1with inflexion at x f . Let B(x;a) := S(x;a)/x with B(0;a) :=limx↓0 B(x;a)≡ S′(0;a). (i) There is a unique tangent line fromthe origin to S(x), denoted as c∗x and called the tangenu,with tangency point, genu, (x∗,S(x∗)), where x∗ > x f . (ii) Bis strictly quasi-concave, and its unique maximiser in [0,X ] ismin(x∗,X) .

Proof: See [12].

Remark IV.2. Figure 2 shows the tangenu (tangent from(0,0)) and genu (tangency point) for the S-curve S(x) =[1− exp(−x/2)/2]80 − 2−80, as well as graphs of xS′(x) andS(x)/x.

Theorem IV.1. Consider a set of I link layer configurationseach identified by a combination of parameters ai; that is,a1, . . . ,aI. With each i associate the function S(x;ai) = b(L−C) f (⋅;ai)/L and denote the abscissa of its genu as x∗i . Theconsidered terminal should choose the configuration j∗ definedby S(x∗j∗ ;aj∗)/x∗j∗ = maxi{S(x∗i ;ai)/x∗i }; that is, S(x;aj∗) hasthe steepest tangenu among the functions S(x;ai).

Proof: By Lemma IV.2, for any ai, the terminal operatesat the SNR that maximise S(x;ai)/x.

By Lemma IV.3, the maximum occurs at the genu (tangencypoint) of S(x;ai).

Then, for any value of y that the terminal ultimately chooses,its performance with link configuration i will be given byu((Eh/y)(S(x∗i ;ai)/x∗i ) ,q(y)). Since u is increasing in eachargument, the optimal configuration is evidently the one withthe highest ratio S(x∗i ;ai)/x∗i (which is the slope of its tangenu).

Remark IV.3. Theorem IV.1 indicates that the media-streamingterminal can sort a set of possible link configurations. Hence,there is an energy-ideal configuration, and it can be identifiedgraphically because the tangenu of the associated S-curve (amultiple of the corresponding PSRF) is the steepest (has thegreatest slope) among those considered (see Fig. 4). If theterminal has an inflexible symbol rate and limited power itmay be unable to reach the corresponding SNR value (at thetangency point), x∗i . In this case, its best policy is unclear.

Corollary IV.1. If S(x∗i ;ai)/x∗i ≥ S(x∗j ;aj)/x∗j and x∗i ≤ x∗jthen ai dominates aj in the sense that ai provides superiorperformance, and ai’s optimal operating point is reachablewhenever aj

′s optimal operating point is.

Proof: Follows directly (see also Remark IV.3).

Remark IV.4. Corollary IV.1 indicates that certain link para-meter combinations can be permanently eliminated, becausewhenever their implied optimal operating point can be reached,there is at least one combination of parameters that performsbetter and whose optimal operation point can also be reached(compare S2 to S3 in Fig. 4). Anyhow, the terminal that hasa flexible symbol rate can always reach the optimal operatingpoint of the ideal configuration, and therefore has no reasonto choose any other configuration. (See also Remark IV.1).

V. EFFICIENT LEVEL OF PERCEPTUAL QUALITY

Theorem V.1. For the given configuration, let c∗ :=S(x∗;a)/x∗ (slope of the tangenu), e∗ := hEc∗ and Y ∗ := hPc∗.Then the terminal chooses y∗ to satisfy

max0≤y≤Y ∗

u

(e∗

y,q(y)

)

Proof: Follows from Lemma IV.1, Lemma IV.2, andLemma IV.3 (direct substitution into (3)).

Corollary V.1. Suppose u is multiplicative in the sense thatu(a,b) ≡ u(ta,b/t) ∀t > 0. Suppose q is continuous, and

00

SNR

perf

orm

ance

S3

S1

T2

T3

S2

S2(x)

T1

x12

x2

xx

3x1

Figure 4. The steeper the tangent the better the configuration: S2 wins.Configuration 1 may be usable if x2 is unreachable. But config. 3 is eliminated,because whenever x3 is feasible so is x2, and S2 performs better [7].

limy↓0 q(y)/y exists. Then, let ρ be defined by

ρ(y) =

{q(y)

y y > 0

limy↓0q(y)

y y = 0

The maximiser y∗ of ρ(y) over the closed and bounded set[0,Y ∗] exists, and the terminal’s optimal power level is P∗ =y∗/(c∗h).

Proof: If u is multiplicative, u(e∗/y,q(y))≡ u(e∗,q(y)/y)(multiply first argument by y > 0 and divide the second bysame y). Thus, the terminal can maximise u(e∗,q(y)/y), withthe first argument now a constant. Since u is increasing inthe second argument, the maximiser y∗ of ρ(y) over [0,Y ∗]also maximises u. The optimal power level P∗ is the one thatproduces y∗, that is, hP∗c∗ = y∗.

Remark V.1. The simplest example of a multiplicative per-formance function is (E/P) ⋅ q(y) . In this case, the terminalmaximises the weighted number of the segments it gets totransfer, where the weight is the perceptual quality of eachsegment, hence, E/P segments times q(y).

Remark V.2. For several reasons, one may hypothesise thatthe function that yields perceptual quality in terms of the rate,y, has also the shape shown in Fig. 3; that is, q(y) satisfiesDefinition A.1 (for some arguments, see [8], [13]).

Corollary V.2. If u is multiplicative, and q(y) satisfies Defin-ition A.1, then the optimal y∗ equals min(yk,Y ∗) where yk isthe value of y at the tangency point of the unique tangent linefrom the origin to the graph of the function q.

Proof: Follows directly from Lemma IV.3 and CorollaryV.1.

VI. DISCUSSION

In [7], we followed [5]’s lesson that packet-oriented linkadaptation — as opposed to symbol/ spectral-efficiency fo-

cused adaptation — is more appropriate for complex multi-layered networks. On the basis of analytical geometry, [7]provides a simple and very general result: among a set oflink parameter combinations, the “best” combination is theone whose scaled PSRF has the steepest tangent from (0,0).However, [7] considered only delay-tolerant (data) traffic.

The present work has extended [7] to consider a media-streaming wireless terminal with a limited energy supply.In addition to its link parameters, the media terminal canalso choose optimally the number of information bits thatare on average available to the receiver to reconstruct onesecond of media. This rate is, in the scientific literature, oftenassumed fixed (e.g., a maximal acceptable distortion) anddetermined by forces outside the analysis. However, at presentit is common for a given video programme to be offeredfor Internet streaming in a wide range of “rates”, starting atfairly low values. Well-performing “scalable” video coders canfurther facilitate offering a given content at multiple rates. Thislends support to our approach of considering media “rate” asa variable to be chosen judiciously given channel state andresource constraints.

Under a very general performance criterion, the energy-limited media terminal should utilise the same link config-uration identified by [7] for the delay-tolerant terminal. Thisassumes that the media terminal can reduce its symbol rate asnecessary to achieve the optimal SNR associated with the idealconfiguration, even under unfavourable channel conditions.Symbol-rate adaptation to a continuum of values is signific-antly more challenging than “modulation order” adaptation[5], but adapting the symbol rate in discrete “steps” (half-rate,quarter rate, etc.) — as described for example in [11] — isrelatively simple, and would be sufficient for our purposes. Animportant implication of our results is that if even the simplestform of symbol-rate variation is available, then this rate isthe only link parameter to be adapted to the channel, becausethen the terminal can always reach the optimal SNR of the“best” overall link configuration (i.e., the optimal combinationof modulation family and order, coding, packet length, etcis held constant). The message to engineers and scientists isclear: symbol rate adaptation should receive special attention.

Figure 4 illustrates the procedure to find an optimal linkconfiguration. The curves in Fig. 4 may arise when an optimalM value for the M-QAM modulation family is sought, as inthe literature that follows [1]. The combination of parameterscorresponding to S2 is best. Further discussion is found in [7].

The above is true as long as the terminal prefers moresegments for a given perceptual quality, and more perceptualquality for given number of segments, which should be thecase for any reasonable situation. To describe the optimal“rate” we need to assume a bit more about the terminal per-formance index. If this index is multiplicative (e.g. increasingin the product of the number of transferred segments by theaverage quality level per segment) then the optimal “rate”maximises perceptual quality per unit of rate (q(y)/y), whichis eminently reasonable.

We do not assume much about the function q(y): it must be

continuous and increasing (unlike the distortion-rate functionwhich is decreasing and convex). In fact, q depends on thespecific media content, as well as the “observer”, and henceit could not be determined without well-conceived experi-mentation with human subjects. However, the same reasonsthat make the S-curve a very good model for the packet-success rate function also makes it a good model for q.Assuming that q is an S-curve practically implies that it is asmooth, increasing, and bounded function, which is certainlyreasonable for a function that yields the perceptual quality ofa segment of media, in particular video, as a function of itsencoding rate (see again Fig. 3). If q is in fact an S-curve (orat least can be approximated as such), and the terminal’s indexis multiplicative, we can further say that the optimal “rate”,y∗, occurs at the tangency point of a line from the origin tothe S-curve q. In this case, if several encoders are available fora given media content, the one whose associated quality-rateS-curve has the “steepest” tangent from the origin would bethe best.

Characterising this terminal’s optimal adaptation policywhen it cannot reach the optimal operating point of the“best” configuration because its symbol rate is fixed deservesadditional attention. Likewise, the procedure to identify thebest link configuration can and should be applied “off line” togenerate appropriate tabular information, which can then beloaded in the terminal’s memory. Developing such tables forspecific communication systems/standards is a possible avenueof further research. It is important to keep in mind that whilethe total number of link parameter combinations could be verylarge if all degrees of freedom are fully exploited, the pro-cedure (i) need not be applied repetitively while the terminaloperates, and (ii) is easily parallelisable, and hence doablethrough parallel or “grid” computing. Each combination ofparameters produces exactly one S-curve, which has exactlyone tangent from (0,0), which has exactly one slope. Thus,a large number of parameter combinations could be dividedamong many computing devices, so that each independentlyobtains the slopes for a number of parameter combinations.Ultimately, the overall largest of the computed slopes leadto the best of the available configurations. Finally, it may beinteresting to embedded our analysis into a network modelsuch as [6]’s.

ACKNOWLEDGEMENT

The Deutsche Forschungsgemeinschaft (DFG) has, throughthe UMIC project, provided partial financial support. RWTH’sCommunication Networks Institute (ComNets) has also con-tributed.

APPENDIX

Definition A.1. S : ℜ+ → [0,Y ], is an S-curve with uniqueinflexion at x f if (i) S(0) = 0, S is (ii) continuously differ-entiable, (iii) strictly increasing, (iv) convex over [0,x f ) andconcave over (x f ,∞), and (v) surjective (see Fig. 3).

Remark A.1. In Definition A.1, S is strictly increasing andalso surjective (for each y ∈ [0,Y ] there is an x ∈ ℜ+ such thatS(x) = y). Therefore, S must approach Y asymptotically as xgoes to infinity (i.e., this follows from the definition).

Definition A.2. A function B : ℜ+ → [0,Y ] is single-peakedover ℜ+ if B is continuous, surjective and has a globalmaximum at X ∈ℜ+ (that is, B(X) =Y , 0≤ x1 < x2 ≤ X =⇒B(x2)> B(x1) and X ≤ x1 < x2 =⇒ B(x2)< B(x1)).

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