Hard Fairness Versus Proportional Fairness in Wireless Communications: The Multiple-Cell Case

Hard F airness V ersus Proportional F airness in W ireless Communications: The Multiple-Cell Case Daeyoung Park, Member , IEEE and Giuseppe Caire, F ellow , IEEE Abstract W e consider the uplink of a cellular communication system with K users per cell and infinite base stations equally spaced on a line. The system is con ventional, i.e., it does not make use of joint cell-site processing. A hard f airness (HF) system serves all users with the same rate in any channel state. In contrast, a system based on proportional fairness serves the users with variable instantaneous rates depending on their channel state. W e compare these two options in terms of the system spectral efficienc y C (bit/s/Hz) versus E b / N 0 . Proportional fair scheduling (PFS) performs generally better than the more restrictiv e HF system in the regime of lo w to moderate SNR, b ut for high SNR an optimized HF system achiev es throughput comparable to that of PFS system for finite K . The hard-f airness system is interference limited. W e characterize this limit and validate a commonly used simplified model that treats outer cell interference power as proportional to the in-cell total power and we analytically characterize the proportionality constant. In contrast, the spectral efficienc y of PFS can gro w unbounded for K → ∞ thanks to the multiuser diversity effect. W e also show that partial frequenc y/time reuse can mitigate the throughput penalty of the HF system, especially at high SNR. Index T erms Delay-limited capacity , partial reuse transmission, proportional fair scheduling. D. P ark is with the School of Information and Communication Engineering, Inha Univ ersity , Incheon, 402-751 Korea. E-mail: dpark@ieee.org. G. Caire is with the Ming Hsieh Department of Electrical Engineering, Univ ersity of Southern California, Los Angeles, CA 90089 USA. E-mail: caire@usc.edu. This work was supported by the Korea Research Foundation Grant funded by the K orean Gov ernment (MOEHRD) (KRF- 2006-352-D00141). 1 I . I N T RO D U C T I O N Consider a wireless cellular system with K user terminals (UTs) per cell where all users share the same bandwidth and Base Stations (BSs) are arranged on a uniform grid on a line (see Fig. 1). This model was pioneered by W yner in [1] under a very simplified channel gain assumption, where the path gain to the closest BS is 1, the path gain to adjacent BSs is α and it is zero elsewhere. W yner considered optimal joint processing of all base stations. Later , Shamai and W yner [2] considered a similar model with frequency flat fading and more con ventional decoding schemes, ranging from the standard separated base station processing to some forms of limited cooperation . A very large literature followed and extended these works in v arious ways (see for example [3], [4]). In this paper we focus on the uplink of a con ventional system , such that each BS decodes only the users in its own cell and treats inter-cell interference as noise. W e extend the model in two directions: 1) we consider realistic propagation channels determined by a position-dependent path loss, and by a slo wly time-varying frequency-selecti ve fading channel; 2) we compare the optimal isolated cell delay-limited scheme [5] with the Proportional Fair Scheduling (PFS) scheme [6], [7]. In a delay-limited scheme each user transmits at a fixed rate in each fading block, and the system uses power control in order to cope with the time-v arying channel conditions [5]. A delay-limited system achie ves “hard fairness” (HF), in the sense that each user transmits at its o wn desired rate, independently of the fading channel realization. On the other hand, generally higher throughput can be achiev ed by relaxing the fixed rate per slot constraint. Under v ariable rate allocation, the sum throughput is maximized by letting only the user with the best channel transmit in each slot [8]. Howe v er , if users are af fected by dif ferent distance-dependent path losses that change on a time-scale much slower than the small scale fading, this strategy may result in a very unfair resource allocation. In this case, PFS achie ves a desirable tradeoff, by maximizing the geometric mean of the long-term av erage throughputs of the users [7]. HF and PFS hav e been compared in terms of system throughput versus E b / N 0 in [9] for the single-cell case. This comparison is indeed rele vant: HF models how voice-based systems work today and how Quality-of-Service guaranteed systems will work in the future (each user makes a rate request and the system struggles to accommodate it). In contrast, PFS is being implemented 2 in the so-called EV -DO 3rd generation systems [6] in order to take advantage of delay-tolerant data traf fic. Hence, a meaningful question is: what system capacity loss is to be expected by imposing har d-fairness? In this paper we address this question by extending the results of [9] to the multicell case with con ventional decoding (i.e., without joint processing of the BSs). I I . S Y S T E M M O D E L Each cell experiences interference from the signals transmitted by UTs in other cells. Frequency- selecti vity is modeled by considering M parallel frequency-flat subchannels. Roughly speaking, we may identify M with the number of fading coherence bandwidths in the system signal bandwidth [10], [11], [12]. The receiv ed signal at BS n in subchannel m is giv en by r m ( n ) = ∞ X j = −∞ K X k =1 h m k ( n, j ) x m k ( j ) + z m ( n ) (1) where h m k ( n, j ) denotes the m -th subchannel gain from user k at cell j to cell n , and x m k ( j ) denotes the signal of m -th subchannel transmitted by user k in cell j , and z m ( n ) is an additive white Gaussian noise with variance N 0 . The channel (power) gain is giv en by g m k ( n, j ) = | h m k ( n, j ) | 2 and the transmit po wer of a user is giv en by E [ | x m k ( j ) | 2 ] = E m k ( j ) . W e model the channel gain as the product of two terms, g m k ( n, j ) = s k ( n, j ) f m k ( n, j ) , where s k ( n, j ) denotes a frequency-flat path gain that depends on the distance between the BS and the UT , and f m k ( n, j ) is a “small-scale” fading term that depends on local scattering en vironment around user terminal [12]. These two components are mutually independent as they are due to dif ferent propagation ef fects. P ath loss takes the expression s k ( n, j ) = d k ( n, j ) − α , where d k ( n, j ) denotes the the distance from base station n to user k in cell j and α is the path loss exponent. W e assume that UTs are not located in a forbidden region at distance less than δ from the BS so that the path loss does not div erge. When the users are uniformly distributed in each cell, the cdf of s ≡ s k ( n, n ) is giv en by F s ( x ) =          0 , x < r − α 1 − x − 1 α − δ r − δ , r − α ≤ x < δ − α 1 , x ≥ δ − α (2) where the cell radius is r and the minimum distance between BSs is D = 2 r . The distance d k ( n, j ) , n 6 = j , is gi ven by d k ( n, j ) = | n − j | D − d k ( j, j ) or d k ( n, j ) = | n − j | D + d k ( j, j ) , 3 depending on the location of user k in cell j as shown in Fig. 2. Consequently , the path loss s k ( n, j ) can be expressed as s k ( n, j ) = θ k ( n, j )  | n − j | D − s k ( j, j ) − 1 α  − α + (1 − θ k ( n, j ))  | n − j | D + s k ( j, j ) − 1 α  − α , (3) where θ k ( n, j ) takes on values 0 or 1 with equal probability since the UTs are distributed uniformly in each cell. W e assume that the path losses change in time on a very slo w scale, and can be considered as random, but constant, over the whole duration of transmission. In contrast, the small-scale fading changes relativ ely rapidly , e ven for moderately mobile users [10]. W e assume Rayleigh block-fading, changing in an ergodic stationary manner from block to block, i.i.d., on the M subchannels, F f ( x ) = 1 − e − x . As in [2], we assume that all users send independently generated Gaussian random codes. Let R m k ( n ) denote the rate per symbol allocated by user k in cell n on subchannel m . When outer- cell interference is akin Gaussian noise, the uplink capacity region of cell n for fixed channel gains is giv en by the set of inequalities X k ∈ S R m k ( n ) ≤ log  1 + P k ∈ S g m k ( n, n ) E m k ( n ) N 0 + I m ( n )  (4) for all S ⊆ { 1 , 2 , · · · , K } , where the interference at cell n on subchannel m is giv en by I m ( n ) = X j 6 = n K X k =1 g m k ( n, j ) E m k ( j ) . (5) The capacity region of the M parallel channel case can be achiev ed by splitting each user infor- mation into M parallel streams and sending the independent codewords ov er parallel channels. The aggregate rate of user k at cell n is giv en by R k ( n ) = M X m =1 R m k ( n ) , k = 1 , 2 , · · · , K . (6) A. Delay-Limited Systems In a delay-limited system, the rates R k ( n ) are fixed a priori , and the system allocates the transmit energies in order such that the rate K -tuple ( R 1 ( n ) , · · · , R K ( n )) is inside the achie vable region in each fading block [5], [9]. W e define the system E b / N 0 under a coding strategy that supports user rates ( R 1 ( n ) , · · · , R K ( n )) as  E b N 0  sys = 1 N 0 Γ K X k =1 M X m =1 E m k ( n ) , (7) 4 where the total number of bits per cell, Γ , is giv en by Γ = P K k =1 R k ( n ) . 1 The system spectral ef ficiency C is giv en by C = Γ M and it is expressed in bits per second per hertz (bit/s/Hz) or , equi valently , in bits per dimension. For giv en user rates ( R 1 ( n ) , · · · , R K ( n )) , we allocate the partial rates R m k ( n ) under the constraints (6) in order to minimize ( E b / N 0 ) sys . As a subproblem for this optimization problem, we first consider the m -th subchannel energy allocation assuming that the partial rates and the interference I m ( n ) are giv en. Thanks to the fact that the recei ved ener gy region is a contra- polymatrid [5], the optimal energy allocation is giv en explicitly as E m π m k ( n ) ( n ) = N 0 + I m ( n ) g m π m k ( n ) ( n, n ) " exp X i ≤ k R m π m i ( n ) ( n ) ! − exp X ii ν m π m l ( n ) ( n ) d m π m l ( n ) ( n ) exp Γ K X l 0 "! 17?@AB3CBAA3+D;!"0 17?@AB3CBAA3+D;$"0 EFA87GAB3CBAA3+D;!"0 EFA87GAB3CBAA3+D;$"0 Fig. 3. Spectral ef ficiency versus system E b / N 0 for the optimal delay-limited systems for K = ∞ . The channel parameters are M = 10 , the path loss exponent α = 2 , the cell size D = 2 , and the forbidden region δ = 0 . 01 . ! !"# $ $"# % %"# & &"# ' ! !"% !"' !"( !") $ $"% $"' $"( $") % *+,-./010234 ! " 5%6+75%6+ # 5!"!$6+85$! 7 ! " , $ , " 6$0%49 $ , " 6&0%44 ! %7 ! " $ , " 6$4 Fig. 4. Power fraction β and its upper and lower bounds for the optimal delay-limited systems for K = ∞ . The channel parameters are path loss exponent α = 2 , the cell size D = 2 , and the forbidden region δ = 0 . 01 . 24 D=2 cell -1 cell 0 cell 1 r 0 r 0 r=1 inner zone outer zone r 0 r 0 (a) D=2 cell -1 cell 0 cell 1 r 0 r=1 r=1 r 0 inner zone outer zone (b) Fig. 5. Cellular model for partial reuse transmission in which users located in shaded areas are only allowed to transmit signals in (a) phase 1 and (b) phase 2. ! !"# !"$ !"% !"& ' ! '$ ! '# ! '! ! & ! % ! $ ! # ! # $ ()! ()' ()# ()* ()$ + ! ,- . /0 ! 1 232 4,561 ! )#748)#74 " )!"!'749)'! Fig. 6. System E b / N 0 versus r 0 for the delay-limited system in the partial reuse transmission scheme for K = ∞ . The channel parameters are the path loss exponent α = 2 , the cell size D = 2 , the forbidden re gion δ = 0 . 01 , and M = 10 . 25 ! !" ! # " # !" !# " ! $ % & # ' ( ) * !" +, - ./ " 0 121 3+450 63+-78.1.9:0 ! ;$<3=;$<3 " ;">"!<3?;!" @ABB38CDE1F7117GE3+17EHBI3JIBB0 @ABB38CDE1F7117GE3+FAB87KBI3JIBB0 87FI3CIA1I3$38CDE1F7117GE3+FAB87KBI3JIBB0 GK87FDB3KDC87DB3CIA1I38CDE1F7117GE3+FAB87KBI3JIBB0 Fig. 7. Spectral efficiency versus system E b / N 0 for the delay-limited system in the optimal partial reuse transmission scheme for K = ∞ . The channel parameters are the path loss exponent α = 2 , the cell size D = 2 , the forbidden region δ = 0 . 01 , and M = 10 . ! !" ! !# ! $" ! $# ! " # " $# $" !# # $ ! % & " ' () * +, # - ./. 0(12- 30(*4.+56- ! 7!8097!80 " 7#:#$ ;44<=0*>?@1 ABC?DEFB>@ G>H<=0*>?@1 I7!# I7$# Fig. 8. Spectral efficienc y lower and upper bounds versus system E b / N 0 for the proportional fair scheduling for K = 10 and K = 20 . The channel parameters are path loss e xponent α = 2 , the cell size D = 2 , and the forbidden region δ = 0 . 01 .