Scalable Spectrum Allocation for Large Networks Based on Sparse Optimization

Transcription

1 Scaabe Spectrum ocation for Large Networks ased on Sparse Optimization innan Zhuang Modem R&D Lab Samsung Semiconductor, Inc. San Diego, C Dongning Guo, Ermin Wei, and Michae L. Honig Department of Eectrica Engineering and Computer Science Northwestern University Evanston, IL arxiv: v1 [cs.it] 19 Feb 2017 bstract Joint aocation of spectrum and user association is considered for a arge ceuar network. The objective is to optimize a network utiity function such as average deay given traffic statistics coected over a sow timescae. key chaenge is scaabiity: given n ccess Points (Ps), there are O(2 n ) ways in which the Ps can share the spectrum. The number of variabes is reduced from O(2 n ) to O(nk), where k is the number of users, by optimizing over oca overapping neighborhoods, defined by interference conditions, and by expoiting the existence of sparse soutions in which the spectrum is divided into k + 1 segments. We reformuate the probem by optimizing the assignment of subsets of active Ps to those segments. n 0 constraint enforces a one-to-one mapping of subsets to spectrum, and an iterative (reweighted 1) agorithm is used to find an approximate soution. Numerica resuts for a network with 100 Ps serving severa hundred users show the proposed method achieves a substantia increase in tota throughput reative to benchmark schemes. I. INTRODUCTION Heterogeneous ceuar networks with dense depoyment of access points (Ps) are anticipated to be a major component of 5G networks. chaenge with such dense depoyments is interference management. Coordinated radio resource aocation across mutipe ces is one approach for mitigating inter-ce interference. That incudes joint scheduing across mutipe ces over a fast timescae [1] [3] as we as the assignment of resources across ces over a sower timescae [4], [5]. Whereas the former approach requires instantaneous knowedge of channe gains, the atter approach reies on statistica knowedge of interference. We consider the joint aocation of spectrum and user association in a arge network with many Ps. The objective is to optimize a network utiity, such as average deay, given traffic statistics over a geographic region that change sowy reative to channe fading. Our approach buids on our prior work [4], [6] in which for a network of n Ps and k mobies (or User Equipments (UEs)), the spectrum is partitioned into 2 n patterns, corresponding to a possibe subsets of active Ps. The probem is to optimize the widths of spectrum segments, associated with the different patterns, aong with the user association under each pattern. This has been shown to provide significant performance improvement in throughput enhancement and deay reduction [4] [7]. The origina convex probem formuation in [4] is not usefu for arge networks because the number of patterns grows as O(2 n ). In prior work [6], we have reduced the number of variabes to O(n) by recognizing that each ink rate depends ony on oca patterns of active Ps. The probem can then be redefined over sets of overapping interference neighborhoods, associated with those oca patterns. Each P has its own interference neighborhood, which captures the interference from nearby Ps. The chaenge with the approach in [6] is to ensure that the spectrum assigned to each particuar P is consistent across the neighborhoods to which it beongs. To accompish that, the spectrum is discretized and a cooring agorithm is proposed to ensure that the oca patterns of active Ps are gobay consistent. Here we take a different approach to addressing the scaabiity probem. This is based on the fact that the soution is sparse, meaning that at most k + 1 out of the 2 n possibe patterns appear in the optima aocation for a genera network utiity function. Hence we reformuate the probem by dividing the spectrum into k + 1 segments, rather than 2 n, and attempt to identify the pattern that shoud be associated with each segment. This effectivey reverses the approach in [6], which attempts to assign a segment of spectrum to each pattern. In this reformuation, we initiay assume that any combination of patterns can be assigned to each of the k+1 segments. This probem is a convex reaxation of our origina probem. The one-to-one mapping of spectrum segments to patterns is then enforced with an 0 (cardinaity) constraint. n agorithm for finding an approximate soution to this probem is presented based on a reweighted 1 approximation of this constraint [8]. The approach to scaabiity presented here has the advantage of eiminating the combinatoria cooring probem that arises in [6]. Instead, a new 0 constraint is introduced. though this does not simpify the origina probem, it heps in finding an approximate soution, since the reweighted 1 approximations for the 0 constraints are known to perform we. Numerica resuts indicate that this method generay gives better performance for a fixed computationa compexity than the method in [6]. II. SYSTEM MODEL s in [6], we consider a network containing the set of Ps N = {1,, n} and the set of UEs K = {1,, k}. Each UE is actuay associated with a particuar ocation, and coud refer to a group of nearby mobies. The n Ps share W Hz of spectrum. For convenience, we normaize

2 P1 P2 P3 x 1 1 {1,3} x1 2 {1,3} x 3 1 {1,3} x3 2 {1,3} 0 y {1} y {1,2} y {1,2,3} y {1,3} y {2} y {2,3} y {3} 1 Fig. 1. n exampe off spectrum aocation among 3 Ps. W = 1. Each P can transmit on any part(s) of the spectrum, which is homogeneous (has the same distance attenuation). Ps sharing the same spectrum interfere at UEs within range of both transmitters. transmissions are assumed to be omni-directiona, athough a simiar set of probems can be reformuated with directiona transmissions. We define a subset of active (transmitting) Ps N as a pattern. There are 2 n patterns, and a particuar aocation of spectrum maps each pattern to a sice of spectrum. We denote an aocation as {y } N, where y denotes the amount of bandwidth assigned to pattern. n exampe with three Ps is depicted in Fig. 1. P 1 owns {1} excusivey; shares {1, 2} with P 2; shares {1, 3} with P 3; and shares {1, 2, 3} with both P 2 and P 3. User association is determined by how the spectrum assigned to a particuar P is aocated to different UEs. Specificay, denote the fraction of tota bandwidth used by P i to serve UE j under pattern as x i j, for i. UE j is then assigned to P i if x i j > 0 for some N. Since the tota bandwidth assigned to pattern is y, we have j K x i j y, N, i. (1) The tota bandwidth aocated to a patterns is then y = 1. (2) N Let denote the spectra efficiency of the ink from P i to UE j under pattern. This is measured over a sow timescae, and is therefore an average over short-term fading. The vaue of is therefore determined by the distance between P i and UE j, shadowing, and simiar ong-term characteristics of the interference inks from Ps in to UE ocation j. For concreteness we assume = W τ 1 {i } og i \{i} p i g i j p i g i j (3) + n j where 1 {i } = 1 if i and 0 otherwise, p i is the transmit power spectra density (PSD) at P i, g i j is the power gain of ink i j, and n j is the noise PSD at UE j. We assume fixed, fat transmit PSDs over the sow timescae considered. The factor W/τ, where τ is the average packet ength (bits), gives the units in packets/sec. The ink gain g i j incudes pathoss and shadowing effects. Ceary, = 0 if i, i.e., P i does not transmit on pattern. In practice, the spectra efficiencies can be measured as time-averaged channe gains. The tota rate received by UE j is therefore r j = xi j, j K. (4) N i III. PROLEM FORMULTION WITH GLOL PTTERNS The probem is to a network utiity function over the spectrum aocation and user association designated by x = ( x i j ) N,i,j K, y = (y ) N : r, x, y subject to x i j (P0a) 0, N, i, j K (P0b) and constraints (4), (1), and (2), where the network utiity function u depends on the service rates to a UEs. The optimization probem is convex if u is concave in r = [r 1,, r k ]. s in [6], we wi take u to be the average packet deay, given by = j λ j (r j λ j ) + (5) 1 where (x) equas 1 + x if x > 0 and + otherwise, and λ j is the Poisson packet arriva rate for UE j. This assumes exponentia packet engths and backogged interference [4]. Soving P0 becomes prohibitivey expensive as the network size grows, due to the inherit compexity from the 2 n goba patterns. However, a key property of soution(s) to (P0) is that at east one is sparse, i.e., contains at most k + 1 patterns. Proposition 1: ( [9]) P0 has a soution that divides the spectrum into at most k + 1 segments, i.e., { N y > 0} k + 1. (6) Furthermore, if u is eement-wise nondecreasing in r, k + 1 in (6) is reduced to k. Determining which k + 1 active patterns appear in a soution is then the key chaenge in soving P0. IV. REFORMULTION WITH SPRSITY CONSTRINTS. Loca Neighborhoods We first reduce the number of variabes in (P0) from O(2 n ) to O(n) by reformuating (P0) over interference neighborhoods based on oca patterns [6]. Due to pathoss, we assume interference vanishes beyond a certain distance. Let L N K denote the set of inks with nonzero gains. Hence each UE ony receives power from Ps within its neighborhood: j = {i (i j) L}. (7) From the P side, each P can ony transmit to a coection of UEs in a P neighborhood with positive rates: U i = {j (i j) L}. (8) We define the interference neighborhood for each P i as: N i = j Ui j, (9) i.e., N i incudes P i and a Ps that interfere with it.

3 N 2 = a b 1 N 1 = a 2 N 3 = b 3 U 2 U 1 a b U 3 Fig. 2. Neighborhoods in the case of three Ps and two UEs. n exampe with three Ps (denoted by {1, 2, 3}) and 2 UEs (denoted by {a, b}) is shown in Fig 2. The set of nonzero inks are L = {1 a, 2 a, 2 b, 3 b}. The P neighborhoods are U 1 = {a}, U 2 = {a, b}, and U 3 = {b}; and the UE neighborhoods are a = {1, 2} and b = {2, 3}. P 1 s interference neighborhood is N 1 = {1, 2}, as P 2 interferes with it at UE a. P 3 s interference neighborhood is N 3 = {2, 3}, as P 2 interferes with it at UE b. P 2 s interference neighborhood is N 2 = {1, 2, 3}, since P 1 and P 3 interfere with P 2 at UE a and UE b, respectivey. The spectra efficiency of ink i j ony depends on the oca patterns in j, according to the definition of UE neighborhoods in (7). Therefore, for any ink i j, the spectra efficiency under a goba pattern is equivaent to its intersection with UE j s neighborhood j : = si j j j K, i j, N. (10) We next express each rate in terms of oca patterns. We define bandwidth aocation variabes within a oca interference neighborhood N i as: = N : N i= x i j, i N, j U i, N i. (11) ecause a goba patterns sharing the same overap with N i contribute to the same oca pattern of P i, represents the bandwidth aocated to ink i j under oca pattern. Hence from P i s perspective, the bandwidth assigned to a oca pattern must be the sum of the bandwidths assigned to a goba patterns containing. The service rate of ink i j, as defined in (4), can then be cacuated as [6]: r j = j (12) i j where we have used (10) and (11), and the fact that ony Ps in j transmit to UE j with positive rate.. Sparse Optimization Motivated by Proposition 1, we reformuate (P0) by dividing the spectrum into k + 1 segments, and seek to assign a singe pattern to each segment. Proposition 1 impies that by optimizing this assignment we obtain a soution to P0. Let h denote the bandwidth of the th segment (to be optimized). We now associate a set of variabes z (for oca neighborhoods) and y for each segment. We can therefore rewrite the rate in (12) as r j = k+1 =1 i j j,, j K (13) where the first sum is over the k + 1 spectrum segments. part from the addition of segment index, the z variabe is the same oca bandwidth aocation variabe defined in (11). The amount of bandwidth assigned to oca pattern in P i s interference neighborhood N i within segment is y i, = j U i,, i N, N i. (14) The tota amount of spectrum assigned to N i satisfies y i, h, i. (15) We aso introduce the foowing consistency constraint to ensure that the amount of spectrum aocated to an P is consistent across any two neighborhoods N i and N m that contain it [6]. That is, for any nonempty C N i N m, y, i = y,. m (16) : N m=c N m: N i=c See, for exampe, N 1 and N 3 in Fig. 2. In interference custer N 1, P 2 transmits under pattern {2} and {1, 2}; in interference custer N 3, P 2 transmits under pattern {2} and {2, 3}. The tota bandwidth used by P 2 must be consistent across neighborhoods so that y 1 {2}, +y1 {1,2}, = y3 {2}, +y3 {2,3},, where {2} is the overapping pattern. This exampe can be extended to a set of Ps, which are members of two interference neighborhoods, giving (16). To ensure a one-to-one mapping of patterns to the k + 1 segments, we add the 0 -norm constraint y i, 0 1, i N, = 1,, k + 1 (17) where x 0 = 1 if x 0, and x 0 = 0 if x = 0. That is, we constrain each P to use at most one active pattern in each segment. We can now reformuate P0 in terms of the oca interference variabes z across the k + 1 spectrum segments: r, y, z (P1a) subject to, 0, (i j) L, N i, (P1b) h = 1, h 0, (P1c) =1,,k+1 and constraints (13)-(17) for each = 1,, k + 1. Theorem 1: P0 and P1 are equivaent (have the same soutions) given the oca neighborhood definitions (7)-(9). The proof is omitted due to imited space. Hence a soution to P1 aways satisfies Proposition 1. This reformuation eads to a computationay efficient approximation agorithm. V. ITERTIVE PPROXIMTION The number of variabes is reduced from O(2 n ) in P0 to O(nk) in P1. However, the 0 norm constraint makes the probem non-convex and difficut to sove. We appy the reweighted 1 approach, described in [8], to approximate the 0 constraint. Specificay, the 0 norm is approximated by the

4 weighted 1 norm, i w i x i, where w i is iterativey adapted, and taken to be inversey proportiona to x i computed at the preceding iteration. This has the effect of suppressing sma nonzero entries with arge weights. 1 The 1 approximation cannot be directy appied to P1 since the 0 norm constraints are couped through (17) and (P1c). Hence we present an iterative agorithm based on the 1 reweighted heuristic, where the weights depend on both the 1 norm and the bandwidth of each segment h. In each iteration of the agorithm, shown in gorithm 1, we sove P1, but with the 0 norm constraint (17) repaced by the weighted sum w,y i, i 1. (18) In gorithm 1, we refer to this reweighted 1 version of P1 as P2. gorithm 1 shows a random initiaization of the weights to introduce asymmetry in the first iteration. Otherwise, e.g., if a w, i s are initiaized to a constant, the soution stays symmetric over a segments, because the constraints and objectives are identica for each segment. symmetric soution, i.e., y,1 i = yi,2 =,, = yi,k, i N, N i, is generay not a soution to the origina probem. In each iteration, we sove P2 with the current weights w to obtain the current x, y, z and h = [h 1,, h k ]. Then the weights are updated as shown. The iterations terminate when the variabes converge or the maximum number of iterations t max is reached. The weight update in gorithm 1 is obtained by approximating the 0 norm with og(x+ɛ) for sma ɛ (see [8] and the references therein). dding ɛ in the denominator aows sma components to be propagated to the next iteration. Here the goa is to obtain a singe nonzero y, i = h, N i, so we take ɛ = αh. This is different from the fixed ɛ proposed in [8] in that αh changes with h in each iteration. That is, gorithm 1 simutaneousy searches for the optima reuse pattern and its associated bandwidth. It is possibe that gorithm 1 produces mutipe reuse patterns for a segment at termination. In those cases P i chooses the dominant pattern i = arg max Ni y, i. The goba reuse pattern assigned to segment is then given by = i:i i {i}. The spectra efficiencies for each segment are determined by the corresponding goba reuse pattern (set of interfering Ps): =. Given the assignment of j patterns to each spectrum segment, the widths of the segments, h = (h ) =1,,k+1 aong with the assignment of spectrum to mobies, x = ( ) j K,i j,=1,,k+1, can then be reoptimized by soving the convex probem: x, h subject to r j = k j=1 =1 (P3a) k, j K (P3b) i j h, j K, i j (P3c) 1 This agorithm has aso been used in [9] to sove an P activation probem. 0 and (P1c) for = 1,, k + 1. gorithm 1 Iterative re-weighted 1 approximation. INPUT: ( C ) j K,i j,c j, and (λ j ) j K. OUTPUT: The widths of k + 1 segments (h ) =1,,k+1, the k + 1 active patterns ( ) =1,,k+1, and the spectrum aocated to ink i j on segment, ( ) j K,i j,=1,,k+1 Initiaization: Choose w, i s randomy in (0,1). Set iteration counter t = 0. whie Variabes have not converged and t < t max do 1. Sove P2 with the current weights w. 2. Update w, i = 1 y, i +αh. 3. t = t + 1. end whie Post Processing: Determine the reuse patterns { } and spectra efficiencies { } across segments; sove P3. VI. NUMERICL RESULTS The simuation resuts assume one macro P is ocated at the center of the area, and the remaining n 1 pico Ps are randomy dropped around it. The k UEs are paced on a rectanguar attice. Link gains incude both pathoss and shadowing. dditiona simuation parameters are shown in the footnote. 2. Sma Network We first compare soutions to P0 and P1 for a sma network with n = 10 and k = 32. Since the number of variabes is reativey sma, we sove both versions of P0 with and without the oca neighborhood approximation using a standard convex optimization sover. The oca neighborhoods are constructed by incuding the strongest four Ps for each UE. The soution to P1 is obtained using gorithm 1. We compare those with fu spectrum reuse where each user is assigned to the P with the strongest signa (maxrsrp association), and aso the optima orthogona aocation, 3 i.e., ony {y {i} } i N are active. Fig. 3 shows deay versus traffic arriva rate for a schemes. The end of each curve represents the maximum arriva rate the scheme can support. The curves obtained by soving P0 with and without oca neighborhood approximation are very cose, which indicates considering the four strongest interferers is enough in such a sma network. The soution to P1 incurs sighty arger deay than the soution to P0. The jointy optimized spectrum aocations and user associations achieve substantia deay reduction as we as eight times throughput compared to fu frequency reuse with maxrsrp association. The optimized spectrum aocation and user associations obtained soving P1 are depicted in Fig. 4. The macro and pico Ps are represented by the bigger and smaer towers; each 2 The pathoss exponent is 3, standard deviation of shadow fading is 3, macro-transmit PSD is 5 µw/hz, pico transmit PSD is 1 µw/hz, noise PSD is 10 7 µw/hz, tota bandwidth is 20 MHz, and average packet ength is 1 Mb. 3 oth spectrum aocation and user association are optimized assuming each P excusivey occupies a fraction of the spectrum.

5 average deay per packet (seconds) fu spectrum reuse optima orthogona aocation soution to P0 without oca neighborhood approximation soution to P0 with oca neighborhood approximation soution to P1 using gorithm 1 average packet deay (seconds) fu spectrum aocation + maxrsrp association fu spectrum aocation + optimized association soution to P1 with 50 segments average packet arriva rate per user group (packets/second) Fig. 3. Deay versus traffic arriva rate curves for a sma network with n = 10 and k = Fig. 4. Exampe spectrum aocations and user associations from (P1). handset represents a UE. Each soid ine shows a connection between the corresponding P and UE. The grid for each UE shows the spectrum used to serve that UE. The traffic arriva rate (in (0, 100)) for each UE is shown under each grid. The user association in Fig. 4 is cose but not identica to that obtained by soving P0 (not shown) due to the approximation in gorithm 1, which expains the performance difference shown in Fig. 3.. Large Network Fig. 5 compares the performance of different aocation schemes in a arge network with n = 100 Ps and k = 200 UEs. To faciitate the simuations, we reduce the size of each UE neighborhood from four to three, i.e., each UE can ony be served by the three strongest Ps. We compare fu-spectrumreuse with maxrsrp association, fu-spectrum-reuse with optimized associations and the soution to P1 with 50 segments. The soution to P1 achieves 1.5 times the throughput of the fu-spectrum-reuse with optimized association. The soutions to P1 use no more than 23 active patterns (< 50 avaiabe segments). The throughput gain achieved by the soution to P average packet arriva rate (packets/second) Fig. 5. Deay versus traffic arriva rate curves for a arge network with n = 100 and k = 195. in this arge network is smaer than that in Section VI-. This is mainy because we ony consider the three strongest interferers, which compromises the benefits from interference management. VII. CONCLUSION n approach to joint aocation of spectrum with user association has been presented which expoits the sparsity of the optima soution. The proposed agorithm has been observed to achieve near-optima performance for sma to medium-size networks for which the optima soution can be computed, and provides substantia gains reative to fu frequency reuse. though not considered here, the formuation can be extended to accommodate spatia seectivity and different power eves. The performance-compexity tradeoff for such extensions is eft for future work. REFERENCES [1] W. Yu, T. Kwon, and C. Shin, Mutice coordination via joint scheduing, beamforming, and power spectrum adaptation, IEEE Trans. Wireess Commun., vo. 12, no. 7, pp. 1 14, [2] F. Wang, L. Song, Z. Han, Q. Zhao, and X. Wang, Joint scheduing and resource aocation for device-to-device underay communication, in Proc. Conf. Wireess Comm. and Networking, pp , IEEE, [3] M. Hong and Z.-Q. Luo, Distributed inear precoder optimization and base station seection for an upink heterogeneous network, IEEE Trans. Signa Process., vo. 61, pp , June [4]. Zhuang, D. Guo, and M. L. Honig, Traffic-driven spectrum aocation in heterogeneous networks, IEEE J. Se. reas Commun., vo. PP, no. 99, pp. 1 1, [5] Q. Kuang, Joint user association and reuse pattern seection in heterogeneous networks, in th Internationa Symposium on Wireess Communications Systems (ISWCS), pp , IEEE, [6]. Zhuang, D. Guo, E. Wei, and M. L. Honig, Scaabe spectrum aocation and user association in networks with many sma ces, submitted to IEEE Trans. Commun.. [7] Q. Kuang and W. Utschick, Energy management in heterogeneous networks with ce activation, user association, and interference coordination, IEEE Trans. Wireess Commun., vo. 15, June [8] E. J. Candes, M.. Wakin, and S. P. oyd, Enhancing sparsity by reweighted 1 minimization, Journa of Fourier anaysis and appications, vo. 14, no. 5-6, pp , [9]. Zhuang, D. Guo, and M. L. Honig, Energy-efficient ce activation, user association, and spectrum aocation in heterogeneous networks, IEEE J. Se. reas Commun., vo. PP, no. 99, pp. 1 1, 2016.