Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom, Sweden Emai: mikaej@s.kth.se Information Systems Laboratory, Department of Eectrica Engineering Stanford University, Stanford, CA 90 90, USA Emai: {xiao, boyd}@stanford.edu Abstract The optima routing of data in a wireess network depends on the ink capacities, which, in turn, are determined by the aocation of transmit powers across the network. Thus, the optima network performance can ony be achieved by simutaneous optimization of routing and power aocation. In this paper, we study this joint optimization probem in CDMA data networks using convex optimization techniques. Athough ink capacity constraints of CDMA systems are not jointy convex in rates and powers, we show that coordinate projections or transformations aow the simutaneous routing and power aocation probem to be formuated as in systems with interference canceation or approximated by in systems without interference canceation a convex optimization probem which can be soved very efficienty. We aso propose a heuristic inkremova procedure based on the convex approximation to further improve the system performance. I. Introduction To make efficient the use of scarce radio resources, it becomes important not ony to optimize the operation of each ayer in wireess data networks, but aso to coordinate the operation of different ayers e.g., []. In particuar, the optima routing probem in the network ayer and resource aocation probem in the radio contro ayer are couped through the ink capacities, and the optima performance can ony be achieved by simutaneous optimization of routing and resource aocation. In a previous paper [], we formuated the simutaneous routing and resource aocation SRRA probem for wireess data networks. By assuming that the ink capacity is a concave and increasing function of the communications resources aocated the ink this assumption hods for TDMA and FDMA systems with orthogona channes the SRRA probem is a convex optimization probem over the network fow variabes and the communications variabes. We expoited the separabe structure of the SRRA probem via dua decomposition, and derived an efficient soution method which achieves the optima coordination of data routing and resource aocation. This research was sponsored in part by the Swedish Research Counci, AFOSR grant F90-0--0, NSF grant ECS-00700 and DARPA contract F-99-C-0 In this paper, we generaize the SRRA formuation to incude code-division mutipe access CDMA systems. In CDMA systems, the capacity of a ink depends not ony on the power aocated to itsef, but aso the powers aocated to other inks due to interferences. Moreover, the capacity constraints are not jointy convex in communication rates and power aocations, so a straightforward formuation of the SRRA probem is not a convex optimization probem and generay very hard to sove. We wi show, however, that by using a coordinate projection that ony considers the rate region the SRRA probem using Gaussian broadcast channes with superposition coding and interference canceation can be converted into an equivaent convex optimization probem. For practica CDMA systems without interference canceation, we suggest an approximate capacity formua for reativey high signa to interference and noise ratio SINR and show how a coordinate transform yieds a convex formuation. This approximate formuation is a restriction of the origina probem, and its soution is guaranteed to be feasibe for the origina non-convex formuation. We propose a heuristic ink-remova procedure to further improve the system performance based on this convex approximation. II. Formuation of the SRRA probem We briefy review the SRRA probem formuated in [] and generaize the mode to incude the CDMA case. A. Network fow mode Consider a connected communication network containing N nodes abeed n =,...,N and L directed inks abeed. The topoogy of the network is represented by a node-ink incidence matrix A R N L whose entry A n is associated with node n and ink via, if n is the start node of ink A n =, if n is the end node of ink 0, otherwise. 0-780-780-/0/$7.00 00 IEEE
We define On as the set of outgoing inks from node n. We use a muticommodity fow mode for the routing of data fows, with average data rates in bits per second. We identify the fows by their destinations, abeed d =,...,D, where D N. For each destination d, we define a source-sink vector s d R N,whosenth n d entry s n d denotes the non-negative amount of fow injected into the network at node n the source and destined for node d the sink, where s d d = n d sd n. We aso define x d R L + as the fow vector for destination d, whose component x d is the amount of fow on each ink and destined for node d. Lett and c be the tota traffic oad and the capacity of ink respectivey. Our network fow mode imposes the foowing group of constraints on the network fow variabes x, s and t: Ax d = s d, d =,...,D x d 0, s d d 0, d =,...,D t = d xd, t c, Here, means component-wise inequaity, and d means component-wise inequaity except for the dth component. The first set of constraints are the fow conservation aws for each destination, whie the ast set of constraints are capacity constraints for each ink. B. Communications mode In a wireess system, the capacities of individua inks channes depend on the media access scheme and the aocation of communications variabes, such as transmit powers, bandwidths or time-sot fractions, to the transmitters. We assume that the medium access, coding and moduation schemes are fixed, but that we can optimize over the communications variabes r. We use the foowing generic mode to reate the vector of tota traffic t and the vector of communications variabes r t c = φ r, Fr g, r 0 Here, the formua c = φ r describes the dependence of the capacity of ink on the communications variabes r =r,...,r L andr is the sub-vector of resources aocated to ink. The inear inequaity Fr g describes resource imits, such as the tota avaiabe transmit powers at each node. The constraint r 0 specifies that the communications variabes are non-negative. In this paper we consider two particuar casses of CDMA systems that we now describe in some detai. The Gaussian broadcast channe: The set of achievabe rates t i for an M-user Gaussian broadcast channe with noise powers σ σ... σ M in different inks and tota transmit power P tot assuming unit tota bandwidth is given by [, ] P i t i og + σ i + j<i P,i=,...,M j M i= P i P tot, P i 0, i=,...,m where P i is the transmit power aocated to ink i. Here the communications variabes are r =P,...,P M, and the constraints are exacty in the generic form. This mode can be used for inks starting from the same node, with superposition coding and interference canceation []. Interference-imited channes: Most CDMA systems in practice are designed without interference canceation. Let G R+ L L be the channe gain matrix across the whoe network, whose entry G ij is the power gain from the transmitter of ink j to the receiver of ink i. Ony the diagona terms G ii are desired, and the off-diagona terms G ij i j ead to interferences. Based on Shannon capacity, we have the ink capacity constraints t og + σ + j G jp j,. The power imits can be specified for each ink, P P,tot, or for each node shared by its outgoing inks P P n tot, n =,...,N. On Here the communications variabes are the powers P = P,...,P L, and the above constraints have the form. C. The generic SRRA formuation Consider the operation of a wireess data network described by the network fow mode and the communications mode and suppose that the objective is to minimize a convex cost function fx, s, t, r or maximize a concave utiity function. We have the foowing generic formuation of the SRRA probem: minimize fx, s, t, r subject to Ax d = s d, d =,...,D x d 0, s d d 0, d =,...,D t = d xd, t φ r, Fr g, r 0. Here the optimization variabes are the network fow variabes x, s, t and the communications variabes r. The SRRA probem is very genera and it incudes many important design probems for wireess data networks. For
exampe, the objective can be maximum tota utiity, minimum tota power or bandwidth, minimax power among the nodes, and minimax ink utiization []. For TDMA and FDMA systems, the capacity constraints usuay take the form t φ r the inks are orthogona and the functions φ are usuay concave and monotone increasing in r. These properties impy that the capacity constraints are jointy convex in t and r, hence the SRRA probem is a convex optimization probem, which can be soved gobay and efficienty by recenty deveoped interior-point methods e.g., []. More effective methods can be deveoped for soving the SRRA probem by expoiting its structure via dua decomposition see []. For the CDMA systems described in section II-B, however, the capacity constraints are not jointy convex in the rates t and vector of transmit powers P. This means that the direct formuation of the SRRA probem as in is not a convex optimization probem, and that it is generay very hard to find the goba optima soution. Nevertheess, we wi show in sections III and IV that coordinate projections or transformations aow us to approach the SRRA probem effectivey using convex optimization techniques. III. SRRA in CDMA networks using Gaussian broadcast channes In this section, we derive an equivaent convex formuation of the SRRA probem that uses Gaussian broadcast channe with interference-canceation at each node. A. An equivaent characterization of the rate region Athough the capacity constraints in for the Gaussian broadcast channe is not jointy convex in the rates t i and power vector P, it is a we-known fact that the achievabe rate region CP tot ={t, t 0} is a convex set. This is better seen from an equivaent characterization of the rate region [] CP tot ={t pt P tot,t 0} where the function p is defined as M pt = σ i σ i e i j M tj σ M 7 i= and σ 0 = 0. It is cear that the function p is convex, hence the rate region CP tot, the projection of the feasibe set of t, P ontothet-coordinates, is a convex set. Given any rate vector t CP tot, the corresponding power aocation is given by k P k = σ i σ i e i j<k tj e t k 8 i= for k =,...,M. Note that the proof of these resuts, given in appendix, reies on the fact that the channe gains in the mode are a equa. B. The convex formuation of the SRRA probem Consider a wireess data network where each node uses the Gaussian broadcast channe to send information over its outgoing inks. We assume that there is no interference among channes from different nodes they use disjoint frequency bands. We denote the oca tota traffic vector by t n = {t On} and define a convex function p n t n simiary as pt in 7 for each node n. Then we can formuate the convex SRRA probem minimize fx, s, t, r subject to A d x d = s d, d =,...,D x d 0, s d d 0, d =,...,D t = D d= xd, p n t n r n, n =,...,N r n P n tot, n =,...,N where the communications variabe r n is the tota transmit power used at node n. Introducing r n aows us to formuate the minimum tota power and minimax power SRRA probems. This convex optimization probem is equivaent to when the Gaussian broadcast channe mode is used at each node. After soving this convex probem, we can recover the power aocation P for each ink using 8. IV. SRRA in CDMA networks with interference-imited channes In this section, we derive a convex approximation for the SRRA probem with interference-imited channes when the SINRs are reativey high. We then give a heuristic ink-remova procedure to further improve the network performance based on the soution to the convex program. A. A convex approximation For CDMA systems with interference-imited channes described by, the SINRs are defined as γ = σ + j G,. jp j When the SINRs are reativey high e.g., γ or 0, we use the approximation og + γ og γ and re-write φ P og σ + j G jp j σ + j = og G jp j σ = og P + G j P j P. G G j Let Q = ogp i.e., P = e Q forand define σ ψ Q =φ P Q = og e Q + G j e Qj Q. G G j
Note that the functions ψ are concave in the variabe Q since og-sum-exp expressions are convex []. With the approximate capacity formua and the change of variabes, we can formuate the foowing SRRA probem minimize fx, s, t, r subject to A d x d = s d, d =,...,D x d 0, s d d 0, d =,...,D t = D d= xd, t ψ Q, On eq P n tot, n =,...,N where the ast constraint which is convex is in the new variabe Q. Here the capacity constraints t ψ Q are jointy convex in t and Q. This impies that 9 is a convex optimization probem, which can be soved gobay and efficienty. We have the foowing remarks: Note that og γ og + γ, i.e., weusedanunderestimate for the ink capacity. This means that the soution to the restricted convex program 9 is aways feasibe to the origina probem. The average throughput of many channes with biterror-rate constraints e.g., variabe-rate M-QAM in additive Gaussian noise or Rayeigh fading environment can be we approximated e.g., [7, 8] by og + kγ og k + og γ 9 provided that kγ is reativey high. We can formuate the SRRA probem with these channes simiary. The change of variabes that we have used is weknown in geometric programming see, e.g., [] and provides an expicit ink between our work and the optimization of communication systems with quaity of service constraints considered in [9, 0]. B. A heuristic ink-remova procedure Any soution to 9 must satisfy γ for a inks, since we require 0 t og γ. The inks with γ = have zero capacity, but are aocated nonzero powers. We can safey remove these inks, and sti guarantee that the soution is feasibe for the SRRA probem with the new reduced network topoogy. If we sove the SRRA probem for the new topoogy, the objective can ony be improved eading to arger tota utiity or ess tota power. Hence, we propose to use the foowing ink-remova procedure given a network topoogy and the SRRA probem repeat sove the convex SRRA formuation 9 remove inks whose SINRs equa one capacity zero unti no inks were removed a SINRs greater than one This procedure usuay takes very few iterations to stop. In many cases we can continue to remove inks with very Fig.. Fig.. 7 8 8 7 9 9 0 A network with nodes and 0 directed inks. Fig.. Initia routing soution. Routing soution after the ink-remova procedure. sma SINRs, even if they are greater than one. This may resut in sparser routing pattern and higher SINRs for the remaining inks, but feasibiity is no onger guaranteed and the objective may be degraded. V. A numerica exampe Consider the network shown in figure, which has nodes and 0 directed inks ink abes are next to the arrows. Each node has tota power P n tot = to be shared by its outgoing inks. A the receivers have the same noise power σ = 0.00. We use the interference-imited channe mode. A diagona entries of the channe gain matrix G are set to one, and the off-diagona entries are generated randomy with a uniform distribution on [0, 0.0]. There are two data fows, one form node to node and the other form node to node, to be supported by the network. The goa is to maximize the sum throughput of the two fows s + s. Foowing the ink-remova procedure in section IV-B, we soved the probem in three steps: a Sove the convex SRRA probem 9 with 0 inks. The soution is shown in figure and tabe I. The tota throughput is.89. b Remove the 8 inks with SINRs equa to one, and sove 9 with the new network topoogy having inks. The routing soution is roughy the same as before see figure, and the tota throughput is.0. c Further remove ink and 0 which have very ow SINRs, and sove 9 again for the resuting network with 0 inks. The routing soution, shown in figure, achieves a tota throughput of.00. 0
ink SINR power aocation a b c a b c... 0.7 0.7 0.7...9 0.07 0.07 0.0 7. 7.8. 0. 0. 0..8 7.. 0.0 0.08 0.8 0. 0.. 0.9 0.7 0.0 9. 0. 0. 0.7 0. 0. 7.9 9. 9. 0.97 0.0 0.7 8 9. 0. 0. 0.9 0. 0. 9 0. 0.. 0.07 0.07 0.089 0 0...0 0.7 0.0 0..0 0 0 0.0 0 0.. 0 0.0 0.0 0.0 0 0 0.009 0 0.0 0 0 0.0 0 0.0 0 0 0.00 0 0.0 0 0 0.0 0 0 7.0 0 0 0.0 0 0 8.0 0 0 0.0 0 0 9.0 0 0 0.0 0 0 0.8.8 0 0.0 0.0 0 TABLE I SINR and power aocation. The SINRs are a stricty greater than one after ony one ink-remova step b, with a sight improvement in the objective vaue. Athough the further removing of inks with sma SINRs in step c degrades the objective a itte bit, it eads to a sparser routing pattern cf. figure and. In a three cases, the sum throughputs are very cose. VI. Concusions We have generaized the formuation of the SRRA probem in wireess data networks to incude CDMA systems. Athough the capacity constraints for CDMA systems are not jointy convex in rates and powers, we have shown that coordinate projection and transformation techniques aow the SRRA probem to be formuated as with interference canceation or approximated by without interference canceation a convex optimization probem that can be soved very efficienty. We have aso proposed a heuristic ink-remova procedure based on the convex approximation to further improve the system performance. Distributed agorithms have been deveoped for power contro probems in CDMA systems to achieve maxmin or specified SINRs e.g., [,, ]. It woud be interesting to expore if these coud be used to deveop distributed agorithms for soving the SRRA probem in the dua decomposition framework [], where distributed power contro is done based on the pricing of the ink capacities. Appendix The foowing derivation detais the outine in []. To derive an equivaent characterization of the achievabe rate region for the Gaussian broadcast channe, we sove the P i s in terms of the rate vector t. We can rewrite the equaities in into j i P j = e ti j<i P j + e ti σ i σ i, i =,...,M. For a given k, we mutipy the ith equaity by e i<j k tj for i k. Then adding them together yieds j k P j = k i= σ i σ i e i j k tj σ k. where σ 0 = 0. We define the functions p k t = j k P j = k i= σ i σ i e i j k tj σ k for k =,...,M. Note that p k t is non-negative and non-decreasing in k, since p t =σ e t 0and p k t p k t = k i= σ i σ i e i j<k tj e t k 0 for a k>. Hence, given any t 0, provided that pt p M t = M i= σ i σ i e i j M tj σ M P tot, there must exist an aocation of powers P k such that P k = p k t p k t 0, k =,...,M where p 0 t =0and M i= P k P tot, i.e., t ies in the capacity region. Conversey, any rate vector satisfying satisfy pt P tot, and the corresponding power aocation is given by P k = p k t p k t, i.e., equation 8. References [] N. Bambos. Toward power-sensitive network architectures in wireess communications: Concepts, issues, and design aspects. IEEE Pers. Commun., :0 9, 998. [] L. Xiao, M. Johansson, and S. P. Boyd. Simutaneous routing and resource aocation via dua decomposition. Submitted, Juy 00. http://www.stanford.edu/ boyd/srra.htm [] T.M.Cover.Broadcastchannes.IEEE Trans. Inform. Theory, 8:, 97. [] P. P. Bergmans. Random coding theorem for broadcast channes with degraded components. IEEE Trans. Inform. Theory, 9:97 07, 97. [] S. P. Boyd and L. Vandenberghe. Course reader for EE: Introduction to Convex Optimization with Engineering Appications. Stanford University, 998. [] D. N. Tse. Optima power aocation over parae Gaussian broadcast channes. Unpubished, a short summary pubished in Proc. of Int. Symp. Inform. Theory, Um, Germany, 997. [7] X. Qiu and K. Chawa. On the performance of adaptive moduation in ceuar systems. IEEE Trans. Commun., 7:88 89, June 999. [8] A. J. Godsmith and S.-G. Chua. Variabe-rate variabepower M-QAM for fading channes. IEEE Trans. Commun., 0:8 0, October 997. [9] S. Kandukuri and S. P. Boyd. Optima power contro in interference imited fading wireess channes with outage-probabiity specifications. IEEE Trans. Wireess Commun., :, 00. [0] D. Juian, M. Chiang, D. O Nei, and S. P. Boyd. QoS and fairness constrained convex optimization of resource aocation for wireess ceuar and ad hoc networks. In Proc. IEEE IN- FOCOM 0, 00. [] J. Zander. Distributed cochanne interference contro in ceuar radio systems. IEEE Trans. Veh. Techno., :0, 99. [] G. J. Foschini and Z. Mijanic. A simpe distributed autonomous power contro agorithm and its convergence. IEEE Trans. Veh. Techno., :, 99. [] R. D. Yates. A framework for upink power contro in ceuar radio systems. IEEE J. Seect. Areas Commun., 7: 7, 99.