Fractional Power Control for Decentralized Wireless Networks

1 Fractional Po wer Control for Decentralized W ireless Networks Nihar Jindal, Ste ven W eber , Jef frey G. Andrews Abstract W e consider a new approach to po wer control in decentralized wireless networks, termed fractional power control (FPC). T ransmission power is chosen as the current channel quality raised to an exponent − s , where s is a constant between 0 and 1. The choices s = 1 and s = 0 correspond to the familiar cases of channel in version and constant power transmission, respectiv ely . Choosing s ∈ (0 , 1) allows all intermediate policies between these two extremes to be e valuated, and we see that usually neither extreme is ideal. W e derive closed-form approximations for the outage probability relati ve to a target SINR in a decentralized (ad hoc or unlicensed) network as well as for the resulting transmission capacity , which is the number of users/m 2 that can achie ve this SINR on average. Using these approximations, which are quite accurate o ver typical system parameter values, we prove that using an exponent of s ∗ = 1 2 minimizes the outage probability , meaning that the in verse square root of the channel strength is a sensible transmit power scaling for networks with a relati vely low density of interferers. W e also show numerically that this choice of s is rob ust to a wide range of variations in the netw ork parameters. Intuitively , s ∗ = 1 2 balances between helping disadv antaged users while making sure they do not flood the network with interference. I . I N T RO D U C T I O N Po wer control is a fundamental adaptation mechanism in wireless networks, and is used to at least some extent in virtually all terrestrial wireless systems. For a single user fading channel in which the objectiv e is to maximize expected rate, it is optimal to increase transmission power (and rate) as a function of the instantaneous channel quality according to the well-known waterfilling policy [2]. On the other hand, if the objecti ve is to consistently achie ve a target rate (or SNR), then the po wer should be adjusted so that this target lev el is exactly met. Such an objecti ve is philosophically the opposite of waterfilling, since po wer is inv ersely related to the instantaneous channel quality: we call this c hannel in version . Although suboptimal from an information theory point of view , some channel in version is used in many modern wireless systems to adapt to the extreme dynamic range (often > 50 dB due to path loss differences as well as multipath fading) that those systems experience, to provide a baseline user experience ov er a long-term time-scale. A. Backgr ound and Motivation for F ractional P ower Contr ol In a multi-user network in which users mutually interfere, po wer control can be used to adjust transmit power le vels so that all users simultaneously can achiev e their target SINR le vels. The Foschini-Miljanic algorithm is an iterati ve, distrib uted power control method that performs this task assuming that each recei ver tracks its instantaneous SINR and feeds back power adjustments to its transmitter [3]. Considerable work has deeply explored the properties of these algorithms, including developing a framew ork that describes all power control problems of this type [4], as well as studying the feasibility and implementation of such algorithms [5], [6], including with varying channels [7]; see the recent monographs [8][9] for e xcellent surveys of the v ast body of literature. This body of w ork, while in many respects quite general, has been primarily focused on the cellular wireless communications architecture, particularly in which all users ha ve a common recei ver (i.e., the uplink). More recently , there has been considerable interest in power control for decentralized wireless networks, such as unlicensed spectrum access and ad hoc networks [10], [11], [12], [13], [14], [15]. A key distinguishing trait of a decentralized network is that users transmit to distinct receiv ers in the same geographic area, which causes the power control properties to change considerably . The contact author N. Jindal (nihar@umn.edu) is with the University of Minnesota, S. W eber is with Drex el Univ ersity , J. Andre ws is with the Uni versity of T exas at Austin. This research was supported by NSF grant no. 0634763 (Jindal), no. 0635003 (W eber), nos. 0634979 and 0643508 (Andre ws), and the D ARP A IT -MANET program, grant no. W911NF-07-1-0028 (all authors). An early , shorter version of this work appeared at Allerton 2007 [1]. Manuscript date: May 29, 2018. 2 In this paper , we explore the optimal power control policy for a multi-user decentralized wireless network with mutually interfering users and a common tar get SINR. W e do not consider iterativ e algorithms and their con ver gence. Rather , motiv ated by the poor performance of channel inv ersion in decentralized networks [16], we de velop a ne w transmit po wer policy called fractional power contr ol , which is neither channel in version nor fix ed transmit power , but rather a trade-of f between them. Motiv ated by a recent Motorola proposal [17] for fairness in cellular networks, we consider a policy where if H is the channel po wer between the transmitter and receiv er , a transmission po wer of H − s is used, where s is chosen in [0 , 1] . Clearly , s = 0 implies constant transmit power , whereas s = 1 is channel inv ersion. The natural question then is: what is an appropriate choice of s ? W e presume that s is decided of fline and that all users in the network utilize the same s . B. T echnical Appr oach W e consider a spatially distributed (decentralized) network, representing either a wireless ad hoc network or unlicensed spectrum usage by many nodes (e.g., Wi-Fi or spectrum sharing systems). W e consider a network that has the following key characteristics. • Each transmitter communicates with a single receiv er that is a distance d meters aw ay . • Channel attenuation is determined by path loss (with exponent α ) and a (flat) fading value H . • Each transmitter knows the channel power to its intended receiver , b ut has no knowledge about other trans- missions. • All multi-user interference is treated as noise. • T ransmitters do not schedule their transmissions based on their channel conditions or the activities of other nodes. • T ransmitter node locations are modeled by a homogeneous spatial (2-D) Poisson process. These modeling assumptions are made to simplify the analysis, b ut in general reasonably model a decentralized wireless network with random transmitter locations, and limited feedback mechanisms. In particular , the above assumptions refer to the situation where a connection has been established between a transmitter and receiv er , in which case the channel po wer can be learned quickly either through reciprocity or a fe w bits of feedback. It is not howe ver as easy to learn the interference level since it may change suddenly as interferers turn on and off or physically mov e (and reciprocity does not help). The fixed transmit distance assumption is admittedly somewhat artificial, but is significantly easier to handle analytically , and has been shown to preserve the inte grity of conclusions e ven with random transmit distances. For example, [16], [18] prove that picking the source-destination distance d from an arbitrary random distribution reduces the transmission capacity by a constant factor of E [ d 2 ] / ( E [ d ]) 2 ≥ 1 . Therefore, although fixed distance d can be considered best-case as far as the numerical value of transmission capacity , this constant factor will not change fractional po wer control’ s relativ e effect on the transmission capacity , which is the subject of this paper . C. Contributions and Or ganization The contributions of the paper are the suggestion of fractional po wer control for wireless networks and the deri vation of the optimum po wer control exponent s ∗ = 1 2 . The e xponent s = 1 2 is sho wn to be optimal for an approximation to the outage probability/transmission that is valid for relatively lo w density networks that are primarily interference-limited (i.e., the effect of thermal noise is not overly large); if the relativ e density or the ef fect of noise is large, then our numerical results show that no power control ( s = 0 ) is generally preferred. In the relativ ely large parameter space where our primary approximation is valid, fractional po wer control with the choice s ∗ = 1 2 is shown to greatly increase the transmission capacity of a 1-hop ad hoc network for small path loss exponents (as α → 2 ), with more modest gains for higher attenuation channels. The results open a number of possible a venues for future work in the area of power control, and considering the prev alence of power control in practice, carry several design implications. The remainder of the paper is organized as follows. Section II provides background material on the system model, and ke y prior results on transm ission capacity that are utilized in this paper . Section III holds the main results, namely Theorem 3 which gi ves the outage probability and transmission capacity achie ved by fractional po wer control, and Theorem 4 which determines the optimum po wer control exponent s ∗ for the outage probability approximation. Section IV provides numerical plots that explore the numerically computed optimal s ∗ , which provides insight on 3 ho w to choose s in a real wireless network. Section V suggests possible extensions and applications of fractional po wer control, while Section VI concludes the paper . I I . P R E L I M I NA R I E S A. System Model W e consider a set of transmitting nodes at an arbitrary snapshot in time with locations specified by a homogeneous Poisson point process (PPP), Π( λ ) , of intensity λ on the infinite two-dimensional plane, R 2 . W e consider a reference transmitter-recei ver pair , where the reference recei ver , assigned index 0 , is located without loss of generality , at the origin. Let X i denote the distance of the i -th transmitting node to the reference receiv er . Each transmitter has an associated receiv er that is assumed to be located a fixed distance d meters aw ay . Let H i 0 denote the (random) distance–independent fading coefficient for the channel separating transmitter i and the reference recei ver at the origin; let H ii denote the (random) distance–independent fading coefficient for the channel separating transmitter i from its intended recei ver . W e assume that all the H ij are i.i.d. (including i = j ), which implies that no source- destination (S-D) pair has both a transmitter and recei ver that are very close (less than a wav elength) to one another , which is reasonable. Received power is modelled by the product of transmission power , pathloss (with e xponent α > 2 ), and a fading coef ficient. Therefore, the (random) SINR at the reference receiv er is: SINR 0 = P 0 H 00 d − α P i ∈ Π( λ ) P i H i 0 X − α i + η , (1) where η is the noise power . Recall our assumption that transmitters hav e kno wledge of the channel condition, H ii , connecting it with its intended receiv er . By exploiting this knowledge, the transmission power , P i , may depend upon the channel, H ii . If Gaussian signaling is used, the corresponding achiev able rate (per unit bandwidth) is log 2 (1 + SINR 0 ) . The Poisson model requires that nodes decide to transmit independently , which corresponds in the abo ve model to slotted ALOHA [19]. A good scheduling algorithm by definition introduces correlation into the set of transmitting nodes, which is therefore not well modeled by a homogeneous PPP . W e discuss the implications of scheduling later in the paper . B. T ransmission Capacity In the outage-based transmission capaci ty frame work, an outage occurs whene ver the SINR falls below a prescribed threshold β , or equiv alently whenever the instantaneous mutual information falls belo w log 2 (1 + β ) . Therefore, the system-wide outage probability is q ( λ ) = P (SINR 0 < β ) (2) Because (2) is computed over the distribution of transmitter positions as well as the iid fading coef ficients (and consequently transmission po wers), it corresponds to fading that occurs on a time-scale that is comparable or slower than the packet duration (if (2) is to correspond roughly to the packet error rate). The outage probability is clearly a continuous increasing function of the intensity λ . Define λ (  ) as the maximum intensity of attempted transmissions such that the outage probability is no larger than  , i.e., λ (  ) is the unique solution of q ( λ ) =  . The transmission capacity is then defined as c (  ) = λ (  )(1 −  ) b , which is the maximum density of successful transmissions times the spectral efficienc y b of each transmission. In other words, transmission capacity is area spectral efficiency subject to an outage constraint. For the sake of clarity , we define the constants δ = 2 /α < 1 and SNR = pd − α η . Now consider a path-loss only en vironment ( H i 0 = 1 for all i ) with constant transmission power ( P i = p for all i ). The main result of [18] is gi ven in the following theorem. Theor em 1 ([18]): Pure pathloss. Consider a network wher e the SINR at the r efer ence r eceiver is given by (2) with H i 0 = 1 and P i = p for all i . Then the following expr essions give bounds on the outage pr obability and transmission attempt intensity for λ,  small: q pl ( λ ) ≥ q pl l ( λ ) = 1 − exp ( − λπ d 2  1 β − 1 SNR  − δ ) , (3) λ pl (  ) ≤ λ pl u (  ) = − log(1 −  ) 1 π d 2  1 β − 1 SNR  δ . (4) 4 Here pl denotes pathloss. The transmission attempt intensity upper bound, λ pl u (  ) , is obtained by solving q pl l ( λ ) =  for λ . These bounds are shown to be approximations for small λ,  respectiv ely , which is the usual regime of interest. Note also that − log(1 −  ) =  + O (  2 ) , which implies that transmission density is approximately linear with the desired outage lev el,  , for small outages. The following corollary illustrates the simplification of the above results when the noise may be ignored. Cor ollary 1: When η = 0 the expr essions in Theor em 1 simplify to: q pl ( λ ) ≥ q pl l ( λ ) = 1 − exp n − λπ d 2 β δ o , (5) λ pl (  ) ≤ λ pl u (  ) = − log(1 −  ) 1 π d 2 β δ . (6) I I I . F R AC T I O N A L P O W E R C O N T R O L The goal of the paper is to determine the ef fect that fractional power control has on the outage probability lo wer bound in (3) and hence the transmission capacity upper bound in (4). W e first revie w the key prior result that we will use, then deri ve the maximum transmission densities λ for different power control policies. W e conclude the section by finding the optimal power control exponent s . A. T ransmission capacity under constant power and channel in version In this subsection we restrict our attention to two well-known po wer control strategies: constant transmit power (or no po wer control) and channel inv ersion. Under constant po wer, P i = p for all i for some common po wer le vel p . Under channel in version, P i = p E [ H − 1 ] H − 1 ii for all i . This means that the received signal power is P i H ii d − α = p E [ H − 1 ] d − α , which is constant for all i . That is, channel inv ersion compensates for the random channel fluctuations between each transmitter and its intended recei ver . Moreover , the expected transmission power is E [ P i ] = p , so that the constant power and channel in version schemes use the same expected power . W e would like to emphasize the distribution of H is arbitrary and can be adapted in principle to any relev ant fading or compound shado wing-fading model. For some possible distributions (such as Rayleigh fading, i.e. H ∼ exp(1) ), the value E [ H − 1 ] may be undefined, strictly speaking. In practice, the transmit po wer is finite and so P i = p E [ H − 1 ] H − 1 ii is finite. The value E [ H − 1 ] is simply a normalizing factor and can be interpreted mathematically to mean that H → min( H , δ ) for an arbitrarily small δ . Such a definition would not affect the results in the paper . A main result of [16] extended to include thermal noise is giv en in the following theorem, with a general proof that will apply to all three cases of interest: constant power , channel in version and fractional po wer control. Note that cp and ci are used to denote constant po wer and channel in version, respectiv ely . Theor em 2: Constant power . Consider a network wher e the SINR at the refer ence r eceiver is given by (2) with P i = p for all i . Then the following expr essions give good appr oximations of the outage pr obability and transmission attempt intensity for λ,  small. q cp ( λ ) ≥ q cp l ( λ ) = 1 − P  H 00 ≥ β SNR  E " exp ( − λπ d 2 E [ H δ ]  H 00 β − 1 SNR  − δ )    H 00 ≥ β SNR # ≈ ˜ q cp l ( λ ) = 1 − P  H 00 ≥ β SNR  exp ( − λπ d 2 E [ H δ ] E "  H 00 β − 1 SNR  − δ    H 00 ≥ β SNR #) λ cp (  ) ≈ ˜ λ cp (  ) = − log   1 −  P  H 00 ≥ β SNR    1 π d 2 1 E [ H δ ] E "  H 00 β − 1 SNR  − δ    H 00 ≥ β SNR # − 1 . (7) Channel in version. Consider the same network with P i = p E [ H − 1 ] H − 1 ii for all i . Then the following expr essions give tight bounds on the outage pr obability and transmission attempt intensity for λ,  small: q ci ( λ ) ≥ q ci l ( λ ) = 1 − exp ( − λπ d 2 E [ H δ ] E [ H − δ ]  1 β − E [ H − 1 ] SNR  − δ ) (8) λ ci (  ) ≤ λ ci u (  ) = − log(1 −  ) 1 π d 2 1 E [ H δ ] E [ H − δ ]  1 β − E [ H − 1 ] SNR  δ . (9) 5 Pr oof: The SINR at the reference receiv er for a generic power vector { P i } is SINR 0 = P 0 H 00 d − α P i ∈ Π( λ ) P i H i 0 X − α i + η , (10) and the corresponding outage probability is q ( λ ) = P (SINR 0 < β ) = P P 0 H 00 d − α P i ∈ Π( λ ) P i H i 0 X − α i + η < β ! . (11) Rearranging yields: q ( λ ) = P   X i ∈ Π( λ ) P i H i 0 X − α i ≥ P 0 H 00 d − α β − η   . (12) Note that outage is certain when P 0 H 00 < η β d α . Conditioning on P 0 H 00 and using f ( · ) to denote the density of P 0 H 00 yields: q ( λ ) = P ( P 0 H 00 ≤ η β d α ) + Z ∞ η β d α P   X i ∈ Π( λ ) P i H i 0 X − α i ≥ p 0 h 00 β d α − η    P 0 H 00 = p 0 h 00   f ( p 0 h 00 )d( p 0 h 00 ) . (13) Recall the generic lower bound from [16]: if Π( λ ) = { ( X i , Z i ) } is a homogeneous marked Poisson point process with points { X i } of intensity λ and iid marks { Z i } independent of the { X i } , then P   X i ∈ Π( λ ) Z i X − α i > y   ≥ 1 − exp n − π λ E [ Z δ ] y − δ o , (14) Applying here with Z i = P i H i 0 and y = p 0 h 00 β d α − η : q ( λ ) ≥ P ( P 0 H 00 ≤ η β d α ) + Z ∞ η β d α 1 − exp ( − π λ E [( P i H i 0 ) δ ]  p 0 h 00 β d α − η  − δ )! f ( p 0 h 00 )d( p 0 h 00 ) = 1 − Z ∞ η β d α exp ( − π λ E [( P i H i 0 ) δ ]  p 0 h 00 β d α − η  − δ ) f ( p 0 h 00 )d( p 0 h 00 ) = 1 − P ( P 0 H 00 ≥ η β d α ) E " exp ( − λπ d 2 E [( P i H i 0 ) δ ]  P 0 H 00 β − η d − α  − δ )    P 0 H 00 ≥ η β d α # . (15) The Jensen approximation for this quantity is: q ( λ ) ≈ 1 − P ( P 0 H 00 ≥ η β d α ) exp ( − λπ d 2 E [( P i H i 0 ) δ ] E "  P 0 H 00 β − η d − α  − δ    P 0 H 00 ≥ η β d α #) . (16) For constant po wer we substitute P i H i 0 = pH i 0 (for all i ) into (15) and (16) and manipulate to get the expressions for q cp l ( λ ) and ˜ q cp l ( λ ) in (7). T o obtain ˜ λ cp (  ) , we solve ˜ q cp l ( λ ) =  for λ . For channel in version, P 0 H 00 = p E [ H − 1 ] while for i 6 = 0 we have P i H i 0 = p E [ H − 1 ] H i 0 H ii . Plugging into (15) and using the fact that H ii and H i 0 are i.i.d. yields (8), and (9) is simply the in verse of (8). Note that channel inv ersion only makes sense when SNR E [ H − 1 ] = pd − α η E [ H − 1 ] , the ef fective interference-free SNR after taking into account the po wer cost of in version, is lar ger than the SINR threshold β . The validity of the outage lo wer bound/density upper bound as well as of the Jensen’ s approximation are ev aluated in the numerical and simulation results in Section IV. When the thermal noise can be ignored, these results simplify to the expressions given in the following corollary: 6 Cor ollary 2: When η = 0 the expr essions in Theor em 2 simplify to: q cp ( λ ) ≥ q cp l ( λ ) = 1 − E h exp n − λπ d 2 β δ E h H δ i H − δ 00 oi ≈ ˜ q cp l ( λ ) = 1 − exp n − λπ d 2 β δ E h H δ i E h H − δ io , q ci ( λ ) ≥ q ci l ( λ ) = 1 − exp n − λπ d 2 β δ E h H δ i E h H − δ io , λ cp (  ) ≈ ˜ λ cp (  ) = − log(1 −  ) 1 π d 2 β δ 1 E [ H δ ] E [ H − δ ] , λ ci (  ) ≤ λ ci u (  ) = − log(1 −  ) 1 π d 2 β δ 1 E [ H δ ] E [ H − δ ] . (17) Note that these expressions match Theorem 3 and Corollary 3 of the SIR-analysis performed in [16]. In the absence of noise the constant po wer outage probability approximation equals the channel in version outage probability lo wer bound: ˜ q cp l ( λ ) = q ci l ( λ ) . As a result, the constant power transmission attempt intensity approximation equals the channel in version transmission attempt intensity upper bound: ˜ λ cp (  ) = λ ci u (  ) . Comparing ˜ λ cp (  ) = λ ci u (  ) in (17) with λ pl u (  ) in (6) it is evident that the impact of fading on the transmission capacity is measured by the loss factor , L cp = L ci , defined as L cp = L ci = 1 E [ H δ ] E [ H − δ ] < 1 . (18) The inequality is obtained by applying Jensen’ s inequality to the con vex function 1 /x and the random v ariable H δ . If constant power is used, the E [ H − δ ] term is due to fading of the desired signal while the E [ H δ ] term is due to fading of the interfering links. Fading of the interfering signal has a positive effect while fading of the desired signal has a negati ve ef fect. If channel inv ersion is performed the E [ H − δ ] term is due to each interfering transmitter using po wer proportional to H − 1 ii . When the path loss exponent, α , is close to 2 then δ = 2 /α is close to one, so the term E [ H − δ ] is nearly equal to the expectation of the in verse of the fading, which can be extremely large for se vere fading distributions such as Rayleigh. As a less sev ere example, α = 3 , the loss factor for Rayleigh fading is L cp = L ci = 0 . 41 . B. T ransmission capacity under fractional power contr ol In this section we generalize the results of Theorem 2 by introducing fractional power control (FPC) with parameter s ∈ [0 , 1] . Under FPC the transmission power is set to P i = p E [ H − s ] H − s ii for each i . The receiv ed po wer at recei ver i is then P i H ii d − α = p E [ H − s ] H 1 − s ii d − α , which depends upon i aside from s = 1 . The expected transmission power is p , ensuring a fair comparison with the results in Theorems 1 and 2. Note that constant po wer corresponds to s = 0 and channel inv ersion corresponds to s = 1 . The following theorem giv es good approximations on the outage probability and maximum allow able transmission intensity under FPC. Theor em 3: Fractional power control. Consider a network where the SINR at the r efer ence r eceiver is given by (2) with P i = p E [ H − s ] H − s ii for all i , for some s ∈ [0 , 1] . Then the following expr essions give good appr oximations of the outage pr obability and maximum transmission attempt intensity for λ,  small q fpc ( λ ) ≥ q fpc l ( λ ) = 1 − P ( H 00 ≥ κ ( s )) × E " exp ( − λπ d 2 E [ H − sδ ] E [ H δ ]  H 1 − s 00 β − E [ H − s ] SNR  − δ )    H 00 ≥ κ ( s ) # ≈ ˜ q fpc l ( λ ) = 1 − P ( H 00 ≥ κ ( s )) × exp ( − λπ d 2 E [ H − sδ ] E [ H δ ] E "  H 1 − s 00 β − E [ H − s ] SNR  − δ    H 00 ≥ κ ( s ) #) λ fpc (  ) ≈ ˜ λ fpc (  ) = − log  1 −  P ( H 00 ≥ κ ( s ))  1 π d 2 1 E [ H − sδ ] E [ H δ ] × E "  H 1 − s 00 β − E [ H − s ] SNR  − δ    H 00 ≥ κ ( s ) #! − 1 7 where κ ( s ) =  β SNR E [ H − s ]  1 1 − s . Pr oof: Under FPC, the transmit power for each user is constructed as P i = p E [ H − s ] H − s ii . Substituting this v alue into the proof for Theorem 2 immediately gi ves the expression for q fpc l ( λ ) . Again, the transmission attempt intensity approximation is obtained by solving ˜ q l ( λ ) =  for λ . As with Theorem 2, the approximation q fpc l ( λ ) ≈ ˜ q fpc l ( λ ) is accurate when the exponential term in q fpc l ( λ ) is approximately linear in its argument and thus Jensen’ s is tight. In other words, this approximation utilizes the fact that e − x is nearly linear for small x . Looking at the expression for q fpc l ( λ ) we see that this reasonable when the r elative density λπ d 2 is small. If this is not true then the approximation ˜ q fpc l ( λ ) is not sufficiently accurate, as will be further seen in the numerical results presented in Section IV. The FPC transmission attempt intensity approximation, ˜ λ fpc (  ) , is obtained by solving ˜ q fpc l ( λ ) =  for λ . The following corollary illustrates the simplification of the abo ve results when the noise may be ignored. Cor ollary 3: When η = 0 the expr essions in Theor em 3 simplify to: q fpc ( λ ) ≥ q fpc l ( λ ) = 1 − E h exp n − λπ d 2 β δ E h H δ i E h H − sδ i H − (1 − s ) δ 00 oi ≈ ˜ q fpc l ( λ ) = 1 − exp n − λπ d 2 β δ E h H δ i E h H − sδ i E h H − (1 − s ) δ io , λ fpc (  ) ≈ ˜ λ fpc (  ) = − log(1 −  ) 1 π d 2 β δ 1 E [ H δ ] E [ H − sδ ] E  H − (1 − s ) δ  . (19) The loss factor for FPC, L fpc , is the reduction in the transmission capacity approximation relativ e to the pure pathloss case: L fpc ( s ) = 1 E [ H δ ] E [ H − sδ ] E  H − (1 − s ) δ  . (20) Clearly , the loss factor L fpc for FPC depends on the design choice of the exponent s . C. Optimal F ractional P ower Contr ol Exponent Fractional po wer control represents a balance between the extremes of no po wer control and channel in version. The mathematical ef fect of fractional po wer control is to replace the E [ H − δ ] term with E [ H − sδ ] E [ H − (1 − s ) δ ] . This is because the signal fading is softened by the power control exponent − s so that it results in a leading term of H − (1 − s ) (rather than H − 1 ) in the numerator of the SINR expression, and ultimately to the E [ H − (1 − s ) δ ] term. The interference po wer is also softened by the fractional power control and leads to the E [ H − sδ ] term. The key question of course lies in determining the optimal power control exponent. Although it does not seem possible to deriv e an analytical e xpression for the e xponent that minimizes the general expression for q fpc l ( λ ) gi ven in Theorem 3, we can find the exponent that minimizes the outage probability approximation in the case of no noise. Theor em 4: In the absence of noise ( η = 0 ), the fractional power contr ol outage pr obability appr oximation, ˜ q fpc l ( λ ) , is minimized for s = 1 2 . Hence, the fractional power contr ol transmission attempt intensity appr oximation, ˜ λ fpc (  ) is also maximized for s = 1 2 . Pr oof: Because the outage probability/transmission density approximations depend on the exponent s only through the quantity E  H − sδ  E  H − (1 − s ) δ  , it is suf ficient to show that E  H − sδ  E  H − (1 − s ) δ  is minimized at s = 1 2 . T o do this, we use the follo wing general result, which we prove in the Appendix. For any non-ne gati ve random v ariable X , the function h ( s ) = E  X − s  E  X s − 1  , (21) is con vex in s for s ∈ R with a unique minimum at s = 1 2 . Applying this result to random variable X = H δ gi ves the desired result. The theorem shows that transmission density is maximized, or equiv alently , outage probability is minimized, by balancing the positi ve and negati ve effects of power control, which are reduction of signal fading and increasing interference, respectively . Using an exponent greater than 1 2 o ver -compensates for signal fading and leads to interference le vels that are too high, while using an exponent smaller than 1 2 leads to small interference levels but an u nder-compensation for signal fading. Note that because the ke y expression E  H − sδ  E  H − (1 − s ) δ  is con vex, the loss relativ e to using s = 1 2 increases monotonically both as s → 0 and s → 1 . 8 One can certainly en vision “fractional” po wer control schemes that go ev en further . For example, s > 1 corresponds to “super” channel inv ersion, in which bad channels take resources from good channels ev en more so than in normal channel inv ersion. Not surprisingly , this is not a wise policy . Less obviously , s < 0 corresponds to what is sometimes called “greedy” optimization, in which good channels are giv en more resources at the further expense of poor channels. W aterfilling is an example of a greedy optimization procedure. But, since E  H − sδ  E  H − (1 − s ) δ  monotonically increases as s decreases, it is clear that greedy power allocations of any type are worse than ev en constant transmit power under the SINR-target set up. The numerical results in the ne xt section sho w that FPC is very beneficial relati ve to constant transmit po wer or channel in version. Ho wever , fading has a deleterious effect relati ve to no fading even if the optimal exponent is used. T o see this, note that x − 1 2 is a conv ex function and therefore Jensen’ s yields E [ X − 1 2 ] ≥ ( E [ X ]) − 1 2 for any non-negati ve random variable X . Applying this to X = H δ we get  E h H − δ 2 i 2 ≥  E [ H δ ]  − 1 , which implies L fpc (1 / 2) = 1 E [ H δ ]  E h H − δ 2 i 2 ≤ 1 . Therefore, fractional PC cannot fully ov ercome fading, but it is definitely a better po wer control policy than constant po wer transmission or traditional power control (channel in version). I V . N U M E R I C A L R E S U L T S A N D D I S C U S S I O N In this section, the implications of fractional power control are illustrated through numerical plots and analytical discussion. The tightness of the bounds will be considered as a function of the system parameters, and the choice of a robust FPC exponent s will be proposed. As default parameters, the simulations assume α = 3 , β = 1 (0 dB) , d = 10m , SNR = pd − α η = 100 (20 dB) , λ = 0 . 0001 users m 2 . (22) Furthermore, Rayleigh fading is assumed for the numerical results. A. Effect of F ading The benefit of fractional po wer control can be quickly illustrated in Rayleigh fading, in which case the channel po wer H is exponentially distributed and the moment generating function is therefore E [ H t ] = Γ(1 + t ) , (23) where Γ( · ) is the standard gamma function. If fractional power control is used, the transmission capacity loss due to fading is L fpc = 1 E [ H δ ] E [ H − sδ ] E  H (1 − s ) δ  = 1 Γ(1 + δ ) · Γ(1 − sδ ) · Γ(1 − (1 − s ) δ ) (24) In Fig. 1 this loss factor ( L ) is plotted as a function of s for path loss exponents α = { 2 . 1 , 3 , 4 } . Notice that for each value of α the maximum takes place at s = 1 2 , and that the cost of not using fractional po wer control is highest for small path loss exponents because Γ(1 + x ) goes to infinity quite steeply as x → − 1 . This plot implies that in severe fading channels, the gain from FPC can be quite significant. It should be noted that the expression in (24) is for the case of no thermal noise ( η = 0 ). In this case the po wer cost of FPC completely v anishes, because the same po wer normalization (by E [ H − s ] ) is performed by each transmitting node and therefore this normalization cancels in the SIR expression. On the other hand, this power cost does not vanish if the noise is strictly positi ve and can potentially be quite significant, particularly if SNR is not large. A simple application of Jensen’ s shows that the po wer normalization factor E [ H − s ] is an increasing function of the exponent s for any distribution on H . For the particular case of Rayleigh fading this normalization factor is Γ(1 − s ) which makes it prohibiti vely expensi ve to choose s very close to one; indeed, the choice s = 1 requires infinite po wer and thus is not feasible. On the other hand, note that Γ( . 5) is approximately 2 . 5 dB and thus the cost of a moderate exponent is not so large. When the interference-free SNR is reasonably large, this normalization factor is relativ ely negligible and the effect of FPC is well approximated by (24). 9 B. T ightness of Bounds There are two principle approximations made in attaining the expressions for outage probability and transmission capacity in Theorem 3. First, the inequality is due to considering only dominant interferers; that is, an interferer whose channel to the desired receiv er is strong enough to cause outage even without any other interferers present. This is a lower bound on outage since it ignores non-dominant interferers, but nev ertheless has been seen to be quite accurate in our prior work [18], [20], [16]. Second, Jensen’ s inequality is used to bound E [exp( X )] ≥ exp( E [ X ]) in the opposite direction, so this results in an approximation to the outage probability rather than a lo wer bound; numerical results confirm that this approximation is in fact not a lower bound in general. Therefore, we consider the three rele vant quantities: (1) the actual outage probability q fpc ( λ ) , which is determined via Monte-Carlo simulation and does not depend on any bounds or approximations, (2) a numerical computation of the outage probability lo wer bound q fpc l ( λ ) , and (3) the approximation to the outage probability ˜ q fpc l ( λ ) reached by applying Jensen’ s inequality to q fpc l ( λ ) . Note that because of the two opposing bounds (one lower and one upper), we cannot say a priori that method (2) will produce more accurate expressions than method (3). The tightness of the bounds is explored in Figs. 2 - 5. Consider first Fig. 2 for the default parameters giv en above. W e can see that the lo wer bound and the Jensen approximation both reasonably approximate the simulation results, and the approximation winds up serving as a lo wer bound as well. The Jensen’ s approximation is very accurate for large v alues of s (i.e., closer to channel in version), and while looser for smaller values of s , this “error” actually mov es the Jensen’ s approximation closer to the actual (simulated) outage probability . The Jensen’ s approximation approaches the lo wer bound as s → 1 because the random variable H (1 − s ) δ approaches a constant, where Jensen’ s inequality tri vially holds with equality (see, e.g., (19)). Changing the path loss exponent α , the SNR, the tar get SINR β , or the density λ can hav e a significant ef fect on the bounds, as we will see. W ith the important exception of high density networks, the approximations are seen to be reasonably accurate for reasonable parameter values. Path loss. In Fig. 3, the bounds are gi ven for α = 2 . 2 and α = 5 , which correspond to much weaker and much stronger attenuation than the (more likely) default case of α = 3 . For weaker attenuation, we can see that the lo wer bound holds the right shape but is less accurate, while the Jensen’ s approximation becomes very loose when the FPC exponent s is small. For path loss exponents near 2 , the dominant interferer approximation is weakened because the attenuation of non-dominant interferers is less drastic. On the other hand, both the lower bound and Jensen’ s approximation are very accurate in strong attenuation en vironments as seen in the α = 5 plot. This is because the dominant interferer approximation is very reasonable in such cases. SNR. The behavior of the bounds also v aries as the background noise lev el changes, as shown in Fig. 4. When the SNR is 10 dB, the bounds are quite tight. Howe ver , the behavior of outage probability as a function of s is quite different from the default case in Fig. 2: outage probability decreases slowly as s is increased, and a rather sharp jump is seen as s approaches one. When the interference-free SNR is only moderately larger than the target SINR (in this case there is a 10 dB dif ference between SNR and β ), a significant portion of outages occur because the signal po wer is so small that the interfer ence-fr ee recei ved SNR falls below the target β ; this probability is captured by the P ( H 00 ≥ κ ( s )) terms in Theorem 3. On the other hand, if SNR is much larger than the target β , outages are almost always due to a combination of signal fading and large interference po wer rather than to signal fading alone (i.e., P ( H 00 ≥ κ ( s )) is insignificant compared to the total outage probability). When outages caused purely by signal fading are significant, the dependence on the exponent s is significantly reduced. Furthermore, the po wer cost of FPC becomes much more significant when the gap between SNR and β is reduced; this explains the sharp increase in outage as s approaches one. When SNR = 30 dB, the behavior is quite similar to the 20 dB case because at this point the gap between SNR and β is so large that thermal noise can effecti vely be neglected. T arget SINR. A default SINR of β = 1 was chosen, which corresponds roughly to a spectral efficienc y of 1 bps/Hz with strong coding, and lies between the lo w and high SINR regimes. Exploring an order of magnitude abov e and below the default in Fig. 5, we see that for β = 0 . 1 the bounds are highly accurate, and sho w that s ∗ = 1 2 is a good choice. For this choice of parameters there is a 30 dB gap between SNR and β and thus thermal noise is essentially ne gligible. On the other hand, if β = 10 the bounds are still reasonable, but the outage behavior is very similar to the earlier case where SNR = 10 dB and β = 0 dB because there is again only a 10 dB gap between SNR and β . Despite the qualitativ e and quantitativ e dif ferences for low SNR and high target SINR from the default values, it is interesting to note that in both cases s = 1 2 is still a robust choice for the FPC exponent. Density . The default value of λ = 0 . 0001 corresponds to a some what lo w density network because the expected 10 distance to the nearest interferer is approximately 50 m, while the TX-RX distance is d = 10 m. In Fig. 6 we explore a density an order of magnitude lo wer and higher than the def ault value. When the network is e ven sparser , the bounds are extremely accurate and we see that s ∗ = 1 2 is a near-optimal choice. Ho wever , the behavior with s is very different in a dense network where λ = . 001 and the nearest interferer is approximately 17 m away . In such a network we see that the nearest neighbor bound is quite loose because a substantial fraction of outages are caused by the summation of non-dominant interferers, as intuitiv ely e xpected for a dense network. Although the bound is loose, it does capture the fact that outage increases with the exponent s . On the other hand, the Jensen approximation is loose and does not correctly capture the relationship between s and outage. The approximation is based on the fact that the function e − x is approximately linear for small x . The quantity x is proportional to π λd 2 , which is large when the network is dense relativ e to TX-RX distance d , and thus this approximation is not v alid for relati vely dense networks. C. Choosing the FPC exponent s Determining the optimum choice of FPC exponent s is a key interest of this paper . As seen in Sect. III-C, s ∗ = 1 2 is optimal for the Jensen’ s approximation and with no noise, both of which are questionable assumptions in many regimes of interest. In Figs. 7 – 10, we plot the truly optimal choice of s ∗ for the default parameters, while v arying α , SNR, β , and λ , respecti vely . That is, the value of s that minimizes the true outage probability is determined for each set of parameters. The FPC exponents s l (∆) and s u (∆) are also plotted, which provide ∆ % error below and abov e the optimum outage probability . For the plots, we let ∆ = 1 and ∆ = 10 . The key findings are: (1) In the pathloss ( α ) plot, s ∗ = 1 2 is a very robust choice for all attenuation regimes; (2) For SNR, s ∗ = 1 2 is only robust at high SNR, and at low SNR constant transmit power is preferable; (3) For target SINR β , s ∗ = 1 2 is robust at low and moderate SINR targets (i.e. low to moderate data rates), but for high SINR tar gets constant transmit power is preferred; (4) For density λ , s ∗ = 1 2 is rob ust at lo w densities, but constant transmit po wer is preferred at high densities. The explanation for findings (2) and (3) is due to the dependence of outage behavior on the difference between SNR and β . As seen earlier , thermal noise is essentially negligible when this gap is larger than approximately 20 dB. As a result, it is reasonable that the exponent shown to be optimal for noise-free networks ( s = 1 2 ) would be near-optimal for networks with very low lev els of thermal noise. On the other hand, outage probability behaves quite dif ferently when SNR is only slightly larger than β . In this case, power is very v aluable and it is not worth incurring the normalization cost of FPC and thus v ery small FPC exponents are optimal. Intuitiv ely , achieving high data rates in moderate SNR or moderate data rates in lo w SNR are dif ficult objectives in a decentralized network. The low SNR case is somewhat anomalous, since the SNR is close to the target SINR, so almost no interference can be tolerated. Similarly , to meet a high SINR constraint in a random network of reasonable density , the outage probability must be quite high, so this too may not be particularly meaningful. T o explain (4), recall that the Jensen-based approximation to outage probability is not accurate for dense networks and the plot shows that constant po wer ( s = 0 ) is preferred at high densities. 1 Fractional po wer control softens signal fading at the expense of more harmful interference po wer , and this turns out to be a good tradeoff in relativ ely sparse networks. In dense networks, howe ver , there generally are a large number of nearby interferers and as a result the benefit of reducing the ef fect of signal fading (by increasing exponent s ) is overwhelmed by the cost of more harmful interference po wer . Note that this is consistent with results on channel in version ( s = 1 ) in [16], where s = 0 and s = 1 are seen to be essentially equiv alent at low densities (as expected by the Jensen approximation) but in version is inferior at high densities. V . P O S S I B L E A R E A S F O R F U T U R E S T U DY Gi ven the historically very high level of interest in the subject of power control for wireless systems, this new approach for po wer control opens man y ne w questions. It appears that FPC has potential for many applications due 1 Based on the figure it may appear that choosing s < 0 , which means users with good channels transmit with additional power , outperforms constant power transmission. Howev er , numerical results (not shown here) indicate that this provides a benefit only at extremely high densities for which outage probability is unreasonably lar ge. Intuitively , a user with a poor channel in a dense network is extremely unlikely to be able to successfully communicate and global performance is improved by ha ving such a user not even attempt to transmit, as done in the threshold-based policy studied in [16]. 11 to its inherent simplicity , requirement for only simple pairwise feedback, and possible a priori design of the FPC parameter s . Some areas that we recommend for future study include the following. How does FPC perform in cellular systems? . Cellular systems in this case are harder to analyze than ad hoc networks, because the base stations (receiv ers) are located on a regular grid and thus the tractability of the spatial Poisson model cannot be exploited. On the other hand, FPC may be e ven more helpful in centralized systems. Note that some numerical results for cellular systems are gi ven in reference [17], but no analysis is provided. Can FPC be optimized f or spectral efficiency? . In this paper we hav e focused on outage relativ e to an SINR constraint as being the metric. Other metrics can be considered, for example maximizing the a verage spectral ef ficiency , i.e. max E [log 2 (1 + SINR)] , which could potentially result in optimal exponents s < 0 , which is conceptually similar to waterfilling. What is the effect of scheduling on FPC? If scheduling is used, then how should power le vels between a transmitter and recei ver be set? Will s = 1 2 still be optimal? W ill the gain be increased or reduced? W e conjecture that the gain from FPC will be smaller but non-zero for most any sensible scheduling policy , as the effect of interference in version is softened. Can FPC be used to improv e iterative power control? At each step of the Foschini-Miljanic algorithm (as well as most of its variants), transmitters adjust their power in a manner similar to channel-in version, i.e., each transmitter fully compensates for the current SINR. While this works well when the target SINR’ s are feasible, it does not necessarily work well when it is not possible to satisfy all users’ SINR requirements. In such a setting, it may be preferable to perform partial compensation for the current SINR le vel during each iteration. For example, if a link with a 10 dB tar get is currently experiencing an SINR of 0 dB, rather than increasing its transmit po wer by 10 dB to fully compensate for this gap (as in the Foschini-Miljanic algorithm), an FPC-motiv ated iterative policy might only boost power by 5 dB (e.g., adjust po wer in linear units according to the square root of the gap). V I . C O N C L U S I O N S This paper has applied fractional po wer control as a general approach to pairwise po wer control in decentralized (e.g. ad hoc or spectrum sharing) networks. Using two approximations, we hav e shown that a fractional po wer control exponent of s ∗ = 1 2 is optimal in terms of outage probability and transmission capacity , in contrast to constant transmit power ( s = 0 ) or channel in version ( s = 1 ) in networks with a relatively low density of transmitters and lo w noise le vels. This implies that there is an optimal balance between compensating for fades in the desired signal and amplifying interference. W e saw that a gain on the order of 50% or larger (relati ve to no power control or channel in version) might be typical for fractional power control in a typical wireless channel. A P P E N D I X W e prov e that for any non-negati ve random variable X , the function h ( s ) = E  X − s  E  X s − 1  , (25) is con vex in s for s ∈ R with a unique minimum at s = 1 2 . In order to sho w h ( s ) is con vex, we show h is log-con vex and use the fact that a log-con vex function is con ve x. W e define H ( s ) = log h ( s ) = log  E  X − s  E  X s − 1  , (26) and recall H ¨ older’ s inequality: E [ X Y ] ≤ ( E [ X p ]) 1 p ( E [ Y q ]) 1 q , 1 p + 1 q = 1 . (27) The function H ( s ) is con vex if H ( λs 1 + (1 − λ ) s 2 ) ≤ λH ( s 1 ) + (1 − λ ) H ( s 2 ) for all s 1 , s 2 and all λ ∈ [0 , 1] . Using H ¨ older’ s with p = 1 λ and q = 1 1 − λ we ha ve: H ( λs 1 + (1 − λ ) s 2 ) = log  E h X − ( λs 1 +(1 − λ ) s 2 ) i E h X ( λs 1 +(1 − λ ) s 2 ) − 1 i = log  E h X − λs 1 X (1 − λ ) s 2 i E h X λ ( s 1 − 1) X (1 − λ )( s 2 − 1) i ≤ log  E  X − s 1  λ E [ X s 2 ] 1 − λ E  X s 1 − 1  λ E  X s 2 − 1  1 − λ  = λ log  E  X − s 1  E  X s 1 − 1  + (1 − λ ) log  E [ X s 2 ] E  X s 2 − 1  = λH ( s 1 ) + (1 − λ ) H ( s 2 ) . (28) 12 This implies H ( s ) is con vex, which further implies con vexity of h ( s ) . 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Muhlethaler , “ An Aloha protocol for multihop mobile wireless networks, ” IEEE T rans. on Info. Theory , pp. 421–36, Feb. 2006. [20] S. W eber , J. G. Andrews, X. Y ang, and G. de V eciana, “T ransmission capacity of wireless ad hoc networks with successive interference cancellation, ” IEEE T rans. on Info. Theory , vol. 53, no. 8, pp. 2799–2814, Aug. 2007. 13 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Power Control Exponent (s) Multiplicative Effect of Fading α =4 α =3 α =2.1 Fig. 1. The loss factor L vs. s for Rayleigh f ading. Note that L cp and L ci are the left edge and right edge of the plot, respecti vely . 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) simulation lower bound Jensen approx Fig. 2. The outage probability (simulated, lo wer bound, and Jensen’ s approximation) vs. FPC e xponent s for the default parameters. 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) α = 2.2 simulation lower bound Jensen approx 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) α = 5.0 simulation lower bound Jensen approx Fig. 3. The outage probability (simulated, lower bound, and Jensen’ s approximation) vs. FPC exponent s for α = 2 . 2 (left) and α = 5 (right). 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) SNR = 10 dB simulation lower bound Jensen approx 0.05 0.055 0.06 0.065 0.07 0.075 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) SNR = 30 dB simulation lower bound Jensen approx Fig. 4. The outage probability (simulated, lower bound, and Jensen’ s approximation) vs. FPC exponent s for SNR = 10 dB (left) and SNR = 30 dB (right). 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) β = -10 dB simulation lower bound Jensen approx 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability (q) Fractional power control parameter (s) β = 10 dB simulation lower bound Jensen approx Fig. 5. The outage probability (simulated, lower bound, and Jensen’ s approximation) vs. FPC exponent s for β = − 10 dB (left) and β = 10 dB (right). 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability Fractional power control parameter (s) λ = 0.00001 simulation lower bound Jensen approx 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage probability Fractional power control parameter (s) λ = 0.001 simulation lower bound Jensen approx Fig. 6. The outage probability (simulated, lower bound, and Jensen’ s approximation) vs. FPC exponent s for λ = 0 . 00001 (left) and λ = 0 . 001 (right). 15 0 0.2 0.4 0.6 0.8 1 2 2.5 3 3.5 4 4.5 5 Fractional power control parameter (s) Pathloss attenuation constant ( α ) sl,su ( Δ = 10%) sl,su ( Δ = 1%) s opt Fig. 7. The optimal choice of FPC exponent s vs. PL exponent α , with ± 1 % and ± 10 % selections for s . 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 Fractional power control parameter (s) Transmitter signal to noise ratio (SNR, dB) sl,su ( Δ = 10%) sl,su ( Δ = 1%) s opt Fig. 8. The optimal choice of FPC exponent s vs. transmitter SNR = ρ η , with ± 1 % and ± 10 % selections for s . 0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 5 10 15 20 Fractional power control parameter (s) Required receiver SINR ( β , dB) sl,su ( Δ = 10%) sl,su ( Δ = 1%) s opt Fig. 9. The optimal choice of FPC exponent s vs. SINR constraint β , with ± 1 % and ± 10 % selections for s . 16 0 0.2 0.4 0.6 0.8 1 -50 -45 -40 -35 -30 -25 -20 Fractional power control parameter (s) Spatial density of interferers ( λ , dB) sl,su ( Δ = 10%) sl,su ( Δ = 1%) s opt Fig. 10. The optimal choice of FPC exponent s vs. density λ , with ± 1 % and ± 10 % selections for s .