Joint Scheduling and Power-Allocation for Interference Management in Wireless Networks

Transcription

1 Jont Schedulng and Power-Allocaton for Interference Management n Wreless Networks Xn Lu *, Edwn K. P. Chong, and Ness B. Shroff * * School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN Emal: {xnlu, shroff}@ecn.purdue.edu Dept. of Electrcal and Computer Engneerng Colorado State Unversty Fort Collns, CO Emal: echong@engr.colostate.edu Abstract Interference management s crucal n wreless communcaton systems because nterference ultmately lmts the system capacty. Opportunstc schedulng and power allocaton are effectve nterference management mechansms. In ths paper, we present jont schedulng and power-allocaton schemes to allevate ntercell nterference. Frst, we study the problem wth the objectve to mnmze the average transmsson power, and thus nterference to other cells, whle mantanng the requred data-rate for each user wthn the cell. Then we study the problem to maxmze the net utlty, de ned as the dfference between the value of throughput and the cost of power consumpton, wth the same data-rate requrements. We establsh the optmalty of our jont schedulng and power-allocaton schemes for both problems n the paper. I. INTRODUCTION Wreless spectrum ef cency s becomng ncreasngly mportant because of the expandng demand for wreless servces, especally for wde-band data-communcaton servces. Interference management s a crucal component of ef cent spectrum utlzaton n wreless systems. Power allocaton and opportunstc schedulng are effectve mechansms for nterference management and ef cent spectrum utlzaton. Power allocaton has been well studed and wdely used n wreless systems to mantan desred lnk qualty, allevate nterference to others, and mnmze power consumpton [5]. In ths paper, we ntroduce the dea of combnng opportunstc schedulng wth power allocaton. Because users experence tme-varyng and locaton-dependent channel condtons n wreless envronments, we can schedule users opportunstcally so that a user can explot more of ts good channel condtons and avod (as much as possble bad tmes, at least for applcatons (e.g., data servce that are not tme-crtcal. Hence, jont schedulng and Ths research s supported n part by the Natonal Scence Foundaton through grants NCR , ANI , ANI, ANI , and ECS. power-allocaton scheme should be able to further mprove the spectrum ef cency and decrease power consumpton. In ths paper, we study the ntercell-nterference-allevaton problem usng jont schedulng and power-allocaton mechansms. We study two dfferent versons of ths problem. In the rst problem, the objectve s to mnmze the total transmsson power, and thus nterference to other cells, subject to a mnmum-data-rate requrement for each user wthn the cell. In the second problem, the objectve s to maxmze the system net utlty, whch s de ned as the value of the throughput mnus the power cost, wth the same mnmum-data-rate constrants. In both problems, we have jont schedulng and power-allocaton decsons. II. SYSTEM MODEL We consder a cell n a tme-slotted system;.e., tme s the resource to be shared among users, and users transmt one at a tme. Examples of such a system ncludes TDMA systems and tme-slotted CDMA systems (e.g., IS-856/Qualcomm HDR system. We consder the downlnk of a cell, and use a stochastc model to capture the tme-varyng channel condton of a user. To elaborate, let α k be a random varable representng the receved SIR for user at tme k gven that the transmsson power s 1. For smplcty of the analyss, we assume that the stochastc process {α k, k = 1, 2, } s statonary and ergodc. Hence, we drop the tme ndex k. Let α = {α 1,, α N }, where N s the number of users n the system. Bascally, α ndcates the channel condtons of users at a generc tme-slot. Let f (c be the requred SIR for relable transmsson at datarate c for user, whch s an ncreasng functon of c, and f (0 = 0. Dfferent users may have dfferent forms of f (c. Gven α, whch ndcates the channel condton, the mnmum requred transmsson power to support a data-rate c for user s f (c/α. Let P max be the maxmum transmsson power,

2 whch represents the restrcton on the transmsson power of a practcal system. Let R denote the requred average data-rate for user. There are two components n a jont schedulng and powerallocaton scheme: a schedulng polcy that decdes whch user to use the tme-slot and a power allocaton polcy that decdes the transmsson power of the selected user (and thus ts correspondng data-rate. Let Q be a schedulng polcy; Q decdes whch user should transmt at a generc tme-slot, gven the channel condtons. In general, Q( α {1,, N, Null}. If Q( α =, = 1,..., N, then user s scheduled to transmt. If Q( α = Null, then no user s scheduled to transmt. Ths may occur f all users experence relatvely bad channel condtons. Let p( be the power allocaton polcy, 0 p( α P max. If user s selected to transmt and ts transmsson power s p( α, then c (p = f 1 (α p s ts achevable data-rate. In summary, a polcy for a jont schedulng and power-allocaton scheme s gven n the form of {Q, p}. III. MINIMIZING TRANSMISSION POWER Frst, we study the problem where the objectve s to mnmze the overall transmsson power whle mantanng the requred data-rate for each user wthn the cell. To acheve ths goal, we need to decde whch user should be scheduled at a generc tme-slot and what should be ts transmsson power. Let P (Q, p be the overall transmsson power of the system under polcy {Q, p}: P (Q, p = E ( p( α1 {Q( α=}. The problem that we are nterested n can be formally stated as: mnmze Q,p P (Q, p subject to 0 p( α P max, E ( c (p1 {Q( α=} = R, = 1,, N. (1 Our objectve s to mnmze the overall transmsson power P (Q, p under two sets of constrants. The rst constrant, 0 p( α P max, ndcates the maxmum-transmssonpower restrcton of the system. The second constrant, E(c (p1 {Q( α=} = R, s the mnmum-data-rate constrant, where E(c (p1 {Q( α=} s the average data-rate of user gven {Q, p}. Note that we could have wrtten the second constrant n the more general (nequalty form: E(c (p1 {Q( α=} R. However, because our objectve s to mnmze the transmsson power, a soluton to (1 s certanly a soluton to the problem wth the more general nequalty constrants. Hence, wthout loss of generalty, we study the problem wth the equalty constrants, as de ned n (1. Next, we present our soluton to the jont schedulng and power-allocaton problem de ned n (1. Let L( λ = We de ne E ( p( α1 {Q( α=} ( ( λ E c (p1 {Q( α=} R = E (p( α λ c (p 1 {Q( α=} + λ R. l ( λ, α, p = p( α λ c(p p ( λ, α = argmn l ( λ, α, p 0pP max l ( λ, α = l ( λ, α, p ( λ, α. Note that we have l ( λ, α 0 because l ( λ, α, 0 = 0. Proposton 1: Suppose there exsts λ such that E ( c (p 1 {Q ( α=} = R, = 1,, N, where Q ( α s de ned as Q ( α = argmn l ( λ, α. (2 Then, {Q, p } s an optmal soluton to the problem de ned n (1. The above proposton s vald for all f s that are ncreasng functons wth f (0 = 0. Further, f f s a strctly convex functon, then p has a closed-form expresson: 0 f f p ( (0 > λ α λ, α = f (f 1 (λ α α f f (0 λ α f (C, P max f f (C < λ α where C = f 1 (α P max s the maxmum data-rate of user gven P max and the channel condton. From Proposton 1, we observe that a user s chosen to transmt when t s a relatvely-best user. User s relatvelybest f l ( λ, α mn j l j ( λ, α (and hence has the same form as the opportunstc scheduler n [2]. Moreover, the transmsson power of the selected user s the power that mnmzes l ( λ, α, p. We can thnk of λ as the unt reward (n terms of power/data-rate to compensate power consumpton. It controls the value of transmsson power, and n turn the data-rate. The fact that l ( λ, α 0 ndcates that the transmsson power (p ( λ, α should be no greater than the reward (λ c (p of the user for transmttng at data-rate c (p. Also note that the resultng data-rate of a user s an ncreasng functon of ts unt reward λ. Ths property enables us to obtan λ teratvely n the mplementaton.

3 Proof of Proposton 1: Suppose a polcy {Q, p} sats es the maxmum transmsson power constrant and the data-rate constrant. We wll show that P (Q, p P (Q, p. Because {Q, p} sats es the rate constrant, we have P (Q, p = P (Q, p We have λ ( E(c (p1 {Q( α=} R = E (p( α λ c (p 1 {Q( α=} + λ R. l ( λ, α, p ( λ, α l ( λ, α, p, 0 p P max by the de nton of p ( λ, α. If Q Null, then (p( α λ c (p 1 {Q( α=} l ( λ, α, p ( λ, α1 {Q( α=} l ( λ, α, p ( λ, α1 {Q ( α=}, where the last nequalty s because of the de nton of Q. If Q = Null, because l ( λ, α, p ( λ, α 0, we have IV. MAXIMIZING NET UTILITY Now we study a jont schedulng and power-allocaton problem n a dfferent scenaro. In [2], we consder a schedulng problem that maxmzes the system throughput subject to each user s mnmum data-rate requrement. Because a user s throughput s an ncreasng functon of ts transmsson power and the objectve s to maxmze the throughput, t s obvous that the base staton should always transmt wth ts maxmum power. However, because transmsson power causes nterference to other cells n wreless systems, we need to take the power consumpton nto account as well. To acheve ths goal, we ntroduce the noton of net utlty, whch s de ned as the dfference between the value of the throughput and the cost of the power consumpton [4]. Let g (p be the power cost of user ; then, c (p g (p s de ned as the net utlty of user. As before, let Q be the schedulng polcy and p be the power polcy. Let T (Q, p be the average net utlty of user gven the jont polcy {Q, p}: T (Q, p = E ( (c (p g (p1 {Q( α=} C (Q, p = E ( c (p1 {Q( α=} T (Q, p = T (Q, p. We formulate the problem as: maxmze Q,p subject to T (Q, p 0 p( α P max C (Q, p R, = 1,, N (3 In other words, the objectve s to maxmze the net utlty gven the maxmum power constrant and data-rate requrement constrants. We de ne (p( α λ c (p 1 {Q( α=} = 0 l ( λ, α, p ( λ, α1 {Q ( α=}. Hence, P (Q, p E l ( λ, α, p ( λ, α1 {Q ( α=} + λ R = P (Q, p. Let Q ( α be de ned as where η sats es: 1 mn,,n η = 1 b ( η, α, p = c (pη g (p p ( η, α = max b ( η, α, p 0pP max b ( η, α = b ( η, α, p ( η, α. Q ( α = argmax b ( η, α, (4 2 C (Q, p R for all 3 For all, f C (Q, p > R, then η = 1. Proposton 2: {Q, p } s an optmal soluton to the problem de ned n (3. Proof: Suppose {Q, p} sats es the maxmum transmsson power constrant and the data-rate constrant. We wll show that

4 T (Q, p T (Q, p. We have T (Q, p T (Q, p + Further, = (η 1(C (Q, p R E ( (c (pη g (p1 {Q( α=} (η 1R. (c (pη g (p1 {Q( α=} (c (p η g (p 1 {Q( α=} (c (p η g (p 1 {Q ( α=}. The rst nequalty s due to the de nton of p, and the second to the de nton of Q. Hence, T (Q, p = E ( (c (p η g (p 1 {Q ( α,p =} (η 1R T (Q, p + (η 1 (C (Q, p R = T (Q, p, whch completes the proof. Note that the solutons of the two problems have certan smlartes. Both polces choose the relatvely-best user to transmt and the optmal transmsson power maxmzes/mnmzes b /l. Here, a user s relatvely-best f b ( η, α max j b j ( η, α. The problem studed n [2] can be consdered as a specal case of the problem de ned n (3 wth g (p 0. In other words, f we do not penalze transmsson power at all, then the base staton always transmts wth ts maxmum power P max. In ths case, the jont schedulng and power-allocaton degenerates to a pure schedulng problem. V. NUMERICAL RESULTS We only present numercal results of our jont schedulng and power-allocaton scheme that mnmzes the total transmsson power n ths secton. In the followng, we rst descrbe bre y our smulaton model of a cellular system, and then we show the smulaton results for the jont polcy usng the cellular model. Rate (kbps SIR (db A. Cellular Model TABLE I ACHIEVABLE DATA-RATE VS. SIR In our smulaton, we consder a mult-cell system consstng of a center hexagonal cell surrounded by hexagonal cells of the same sze. The base staton s at the center of each cell, and smple omn-drectonal antennas are used by mobles and base statons. The frequency reuse factor s 3, and the co-channel nterference from the sx rst-rng neghborng cells are taken nto account. We assume that each cell has a xed number of frequency bands, and focus on one frequency band that s shared by 10 users n the central cell. The users have exponentally dstrbuted on and off perods. Users move wth random speed and drecton n the cell. They perceve tme-varyng and locaton-dependent channel gans. The channel gans of the users are mutually ndependent random processes determned by the sum of two terms: one due to path (dstance loss and the other to shadowng. We adopt the path-loss model (Lee s model and the slow log-normal shadowng model n [3]. Recall that f(c s the requred SIR for relable transmsson at data-rate c. In practce, f s usually not a contnuous functon because the system only supports dscrete data-rates by adaptng dfferent codng rates and modulaton schemes. Hence, n the smulaton, we assume that there are ten dscrete data-rates avalable and the correspondng SIR requrements are lsted n Table I. The data s smlar to the result presented n [1]. B. Numercal Results In the followng, we show smulaton results for the jont schedulng and power-allocaton scheme that mnmzes the overall transmsson power wth mnmum-data-rate guarantees. Frst, we set λ 0. Then the system performs the followng procedure at every tme-slot. Bascally, the users measure ther channel condtons, and then send the nformaton to the base staton ( α k. The base staton decdes the user to be scheduled and ts transmsson power usng the jont schedulng and power-allocaton polcy by substtutng λ k nto (2. Then the base staton updates the parameter used n the schedulng polcy by λ k+1 = λ k + a k (R R k, where a k s the step sze (a k = 0.001, Rk s the estmated average data-rate of user at tme k, R s the requred datarate, and R = 3 (kbps. The above parameter tunng scheme s smlar to the one n [2]. For the purpose of comparson, we also smulate a roundrobn polcy. In the round-robn schedulng polcy, actve users follow a predetermned order. When a user s turn comes, f ts

5 average transmsson rate s lower than ts requred rate, then we let the user transmt. Its transmsson power equals the mnmum power requred to support the hghest data-rate achevable to the user gven ts channel condton. Otherwse, f the use does not qualfy to transmt, then we go to the next actve user untl we nd a user to transmt n the current tme-slot or all actve users have been exhausted. We compare the average transmsson power of our polcy wth that of round-robn. The smulaton s run for 100,000 tme-slots. In the smulaton, both polces can mantan the requred data-rates of users. Fgure 1 shows the average transmsson power of each user usng our polcy and usng roundrobn. The x-axs s the user s ID. The y-axs s the average power consumpton, whch s the amount of transmsson power consumed by a user dvded by ts throughput. The rst bar s the average power consumpton of a user n the round-robn scheme and the second bar s that n our scheme. Note that the power consumpton of each user n our scheme s unanmously lower than that n round-robn. The total transmsson power of round-robn s 33% hgher than that of our polcy. VI. CONCLUSIONS Opportunstc schedulng explots tme-varyng channel condtons to mprove spectrum ef cency, provdng an addtonal degree of freedom to the system. Its merts also le n ts ablty to work n conjuncton wth other resource management mechansms. In ths paper, we study jont schedulng and powerallocaton schemes to allevate ntercell nterference. We study two problems wth dfferent but related objectves. The rst objectve s to mnmze the total transmsson power and the second objectve s to maxmze the net utlty, both under mnmum-data-rate constrants. We provde optmal solutons to the studed problems and use smulaton results to evaluate the power savngs compared wth a round-robn scheme. Further research ncludes the study of the system behavor when Power Consumpton Fg RR PC User ID The average transmsson power of round-robn and our polcy. all cells mplement the same schedulng and power-allocaton algorthms. REFERENCES [1] A. Furuskar, M. Frodgh, H. Olofsson, and J. Skold. System performance of EDGE, a proposal for enhanced data rates n exstng dgtal cellular systems. In Proceedngs of 48th IEEE Vehcular Technology Conference, volume 2. IEEE, [2] X. Lu, E. Chong, and N. Shroff. Transmsson schedulng for ef cent wreless resource utlzaton wth mnmum-utlty guarantees. In IEEE Vehcular Technology Conference, Atlantc Cty, October [3] G. Stuber. Prncples of Moble Communcaton. Kluwer Academc Publshers, [4] M. Xao, N. Shroff, and E. Chong. Utlty-based power control n cellular wreless systems. In Proceedngs of IEEE INFOCOM 2001, volume 1, [5] R. Yates. A framework for uplnk power control n cellular rado systems. IEEE Journal on Selected Areas n Communcatons, 43(7: , 1995.