Spatial Reuse in Dense Wireless Areas: A Cross-layer Optimization Approach via ADMM

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Spatia Reuse in Dense Wireess Areas: A Cross-ayer Optimization Approach via ADMM Haeh Tabrizi, Member, IEEE, Borja Peeato, Member, IEEE, Gonaz Farhadi, Member, IEEE, John M. Cioffi, Feow, IEEE, and Ghadah Adabbagh, Member, IEEE Abstract This paper introduces an efficient method for communication resource use in dense wireess areas where a nodes must communicate with a common destination node. The proposed method groups nodes based on their distance from the destination and creates a structured muti-hop configuration in which each group can reay its neighbor s data. The arge number of active radio nodes and the common direction of communication toward a singe destination are expoited to reuse the imited spectrum resources in spatiay separated groups. Spectrum aocation constraints among groups are then embedded in a joint routing and resource aocation framework to optimize the route and amount of resources aocated to each node. The soution to this probem uses coordination among the ower-ayers of the wireess-network protoco stack to outperform conventiona approaches where these ayers are decouped. Furthermore, the structure of this probem is expoited to obtain a semi-distributed optimization agorithm based on the aternating direction method of mutipiers ADMM) where each node can optimize its resources independenty based on oca channe information. Index Terms Aternating direction method of mutipiers ADMM), crossayer optimization, dynamic resource aocation, routing. I. INTRODUCTION With the remarkabe growth of wireess technoogy and advancement of mobie devices, the demand for more efficient use of imited radio resources is increasing rapidy. Optimizing performance through sophisticated physica-ayer techniques is no onger enough, so studies have considered coordinating mutipe users through scheduing and admission contro, as we as joint operation of the physica and network ayers in wireess networks. Lin et a. summarize such cross-ayer optimization techniques in [1]. This paper studies efficient resource use in dense wireess areas where a nodes communicate with a singe destination node. Dense wireess areas are ocations popuated with many radio devices that must communicate over a shared spectrum. Some exampes incude concert has and stadiums popuated H. Tabrizi is with the Department of Eectrica Engineering, Stanford University, Stanford, CA, USA, e-mai: htabrizi@stanford.edu) B. Peeato is with the Department of Eectrica and Computer Engineering, Purdue University, West Lafayette, IN, USA, e-mai: bpeeato@purdue.edu). J. Cioffi is with the Department of Eectrica Engineering, Stanford University, Stanford, CA, USA, and Department of Computer Science, King Abduaziz University, Jeddah, Saudi Arabia, e-mai: cioffi@stanford.edu). G. Farhadi is with Fujitsu Labs of America, Sunnyvae, CA, USA, e-mai: gfarhadi@us.fujitsu.com). G. Adabbagh is with the Department of Computer Science, KAU, Jeddah, Saudi Arabia, e-mai: gadabbagh@kau.edu.sa) This paper has been submitted in part to IEEE goba communications conference, Atanta, GA, December D wireess node destination node wireess ink D spectrum reuse Fig. 1: Mutipe hops and reuse between hops. with user handhed devices, which must a communicate with a common access point or base-station BS). Home Area Networks HAN) are another exampe of dense wireess areas, particuary when such residentia oca area networks may eventuay connect hundreds of digita devices within a home in the so-caed Internet of things. Figure 1 suggests a configuration that can increase spectrum efficiency by reaying data over mutipe hops rather than a singe hop or direct communication). The node configuration motivates a hierarchica muti-hop architecture, where spectrum can be reused among the hops as we as among ces [2, Ch. 15]. Furthermore, the presence of a arge number of nodes in a dense wireess area offers many possibe routes that the proposed muti-hop configuration can optimize. This approach does not require additiona infrastructure, but the nodes act as reays for their neighbor s data. In this scenario a nodes communicate with a fixed destination node, hence the reuse pattern and fow of data can be structured. The nodes that are in cose proximity of each other are grouped together to form an FDMA frequency division mutipe access) system where each node operates on different spectra. Each such group of nodes can reay data for neighboring groups 1 aong an optima routing path determined by the capacity of the inks between nodes. The capacity of each ink is determined by the power and bandwidth aocated to that ink, which in turn depend on the tota spectrum aocated to each group. Finding the optima network configuration therefore requires a joint optimization considering a constraints. Joint optimization and coordination of the different network ayers is not a new concept. It has been used in many different appications in wireess and wireine networks [4], 1 A configurabe MAC e.g. patform proposed in [3]) may be used to enabe a different channe access method random access, FDMA, etc.) for different nodes depending on their congestion eve aong the path.

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2 [5]. Simutaneous routing and resource aocation methods for orthogona mutipe access systems have been proposed by [6] FDMA and TDMA) and [7] CDMA). Both approaches jointy optimize network-ayer routing and physica-ayer resource aocation through a convex formuation of the probem. A joint scheduing, routing, and resource aocation for mutihop ceuar networks with reay stations is proposed by [8]. In this case, the genera probem is non-convex, and hence an iterative agorithm is used to coordinate the ayers. Beyond resource aocation and reay seection, [9] jointy optimizes the reay strategy that shoud be used between each source and destination pair based on a cross-ayer design. This paper focuses on an extended version of the simutaneous routing and resource aocation SRRA) framework proposed by [6], incuding additiona constraints on the tota avaiabe spectrum resources and spectrum reuse among the predetermined user groups. The joint optimization of routing and goba resource aocation in a dense area is a non-convex probem that can ony be soved exacty through exhaustive search methods, such as branch and bound. Unfortunatey, these methods are often sow and have exponentia worstcase performance [10]. Hence, we propose simpifying the probem into a convex form that can be soved using ADMM Aternating Direction Method of Mutipiers), which has O1/k) convergence rate [11]. The predetermined user groups are seected based on each node s reative distance to the destination and a subset of a possibe inks connecting nodes from one group to a neighboring group are seected as candidate data fow inks in the optimization probem. The goa is to achieve jointy optimized routing, goba spectrum assignment, and oca resource aocation for the seected inks through convex optimization. Distributed impementations of network utiity maximization probems often ead to simper subprobems, each optimizing a subset of the decision variabes based on oca information. Paomar et a. [12] propose a systematic framework to expoit decomposition structures that ead to different distributed agorithms. A very popuar approach is using dua decomposition to generate a highy distributed optimization probem [6], but some studies seek faster convergence using Newton s method [13]. Certainy, there is a tradeoff between the amount of message-passing among computing entities, convergence speed, and computationa compexity of each distributed agorithm. This paper expoits the structure in the proposed joint optimization framework to generate two different distributed agorithms: The first agorithm centraized) separates the probem into two subprobems, aowing decouped optimization of network-ayer and physica-ayer variabes. The second agorithm semi-distributed) empoys a active nodes as computing resources that perform parae optimization based on oca channe information. The proposed decompositions use the aternating direction method of mutipiers ADMM) [11] agorithm, which is more numericay stabe than conventiona decomposition methods. For exampe, ADMM does not require strict convexity of the objective, but the dua decomposition method proposed in [6] does. ADMM has been widey empoyed in producing distributed agorithms. Some recent exampes incude mutice coordinated beamforming [14] and distributed mode predictive contro [15]. The main contributions of this paper are: 1) designing a hierarchica structure of nodes that aows resource reuse among spatiay separated nodes, 2) integrating goba resource aocation and reuse in a cross-ayer optimization framework of routing and resource aocation, and 3) deveoping a semidistributed optimization agorithm based on ADMM where each node optimizes the resources based on its oca channe information. The rest of this paper is organized as foows. Section II describes the system mode, consisting of the network topoogy and the physica ayer constraints in a dense wireess area. Section III formuates the cross-ayer optimization probem for the previous mode in a convex form. Section IV proposes a centraized and a distributed agorithm to sove this probem, both based on ADMM. It aso provides a numerica exampe to iustrate the performance and convergence of the agorithm for a sma network. Finay, Section V presents more detaied simuations evauating the performance of the agorithm on arger networks and Section VI concudes and summarizes the paper. II. PROBLEM FORMULATION The system mode and the proposed hierarchica node configuration dictate a set of system constraints that need to be satisfied whie optimizing network performance. This section formuates and motivates these constraints. A. Network Fow Mode We consider N active users/nodes randomy distributed in a given area that need to communicate simutaneousy with a common destination node, such as a ceuar base station BS). This paper investigates upink transmission, but downink communication can be formuated in a simiar manner with minor modifications. A standard directed graph is used to represent the network topoogy. The graph is assumed to be connected, i.e., there is a route between every node and the BS. The nodes are abeed as n N = {1,2,...,N}, with node 1 representing the BS. A nodes can transmit, receive and reay data over the existing inks. Transmission and reception occur on disjoint spectra. A ink exists between two nodes i and j if direct communication among the two is possibe. With a tota of L inks in the network, denote the set of a inks by L = {1,2,...,L}, where each ink is identified with an integer vaue between 1 and L. The network topoogy is represented by a ink-node incidence matrix A R N L, whose entries determine a ink s source and destination nodes by +1 and 1, respectivey. For exampe, for the configuration shown in Figure 2, where N = 4 and L = ) 3, the ink-incidence matrix is given by: A = Each coumn of A corresponds to a ink and each row corresponds to a node. The first coumn indicates that ink 1 goes from node 2 to node 1. The set of outgoing inks from node n are denoted by On) and the set of incoming inks are denoted by In).

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS Fig. 2: Simpe node and ink exampe. A muti-commodity fow mode [16] that satisfies fow conservation at each node is considered. For the upink case considered here, x R L identifies the fow on each ink data rate) that is destined to node 1 BS). Let A n denote the eement on the n-th row and -th coumn of A, and et r R N be a vector of data traffic injected in the network at each node. Then Ax = r, where r n = A nx, denotes the amount of data traffic injected into the network at node n. Positive components wi indicate fow sources, whie negative ones wi indicate fow sinks. For the exampe in Figure 2, the fow conservation aw indicates that r 1 = x 1, r 2 = x 1 x 2 x 3, r 3 = x 2, and r 4 = x 3. Furthermore, the amount of fow on each ink x ) is constrained by that ink s physica ayer capacity c : x c, L. B. Physica Layer Mode An FDMA channe access method is considered, where each node s outgoing inks are assigned disjoint frequency bands. The received signa is corrupted by additive white Gaussian noise with power spectra density σ and the ink capacity is given by a concave and monotone increasing function of its aocated bandwidth w and power p, as foows: c = w og p q w σ 3 4 ), L. Parameter q is the channe gain from the source to the destination of ink. The tota amount of power aocated to a node n is constrained to be smaer than a pre-fixed vaue P max,n, hence p P max,n, n N. On) The tota amount of bandwidth aocated to node n is denoted by w = v n, n N. On) If necessary, after obtaining the continuous bandwidth variabesw, they can be quantized based on the underying system moduation approach. For exampe, if the underying system is an OFDMA system, the bandwidth vaues can be rounded down to an integer number of subcarrier bandwidths and if the underying system is an LTE system, it can be rounded to the cosest integer number of resource bocks. C. Group Assignment The objective of the proposed muti-hop configuration is to reuse as much of spectrum as possibe. Hence, direct inks between far nodes and the BS are divided into mutipe hops that operate on different spectrum bands, aowing reuse Figure 1). To this end, nodes are grouped based on their distance to the BS as shown in Figure 3: nodes ocated between radia distances d g 1 and d g from the BS beong in group G g, and a nodes within distance d 1 of the destination beong to groupg 1. The coverage area is thus partitioned intom disjoint sections such that G g Gg =, G g, g,g, and each node beongs to a group: M g=1 G g = N. In genera, the coser a ink to the BS, the arger the fow it wi be carrying. The inks that terminate at the BS carry the argest tota fow. The inks that are physicay farther away from the BS, carry smaer fows Figure 3). Let f denote the frequency reuse factor, such that every f + 1)-th group uses the same spectrum set f > 2 to avoid the hidden node probem). As such, the maximum avaiabe spectrum W max is divided among groups 1 through f: f W g = W max, g=1 wherew g denotes the amount of bandwidth aocated to group g. The rest of the groups f+1 throughm) reuse the spectrum aocated to groups 1 through f: W g = W g 1) mod f)+1, g > f. For exampe, if f = 3, the tota bandwidth is divided between groups 1, 2 and 3, whie groups 4, 7, 10,... use the same spectrum as group1; groups5,8,11,... use the same spectrum as group 2; and groups 6,9,12,... use the same as group 3. D. Routing Link Assignments In order to optimize the routing from each node to the BS, an initia set of possibe inks shoud be determined. This subsection describes a method, simiar to that in [17], for seecting this initia poo of inks or refining a pre-defined set of inks if desired). The method assumes that the BS knows the ocation of each node and uses reative distances between them to seect the initia poo of inks. If this assumption does not hod, the initia poo of inks coud incude a those between nodes in consecutive groups. Regardess of how the initia poo of inks is seected, resource and fow aocation are based on channe information. The proposed method wi incude a ink from node i to node j in the initia poo if: 1) j G g 1, where i G g. 2) the anguar distance between nodes i and j is ess than a pre-fixed θ, as shown in Figure 3. 3) the distance between nodes i and j is ess than a prefixed d th. The first condition enforces a singe hop between consecutive groups to aow spectrum reuse. If i G g and j G h with h < g 1, then the ink from i to j traverses more than one group spectrum region) and defeats the purpose of reuse. On the other hand, if h = g then i and j beong to the same group creating an extra hop within the group, and if h > g, then j is farther away from the BS than i aready is. In both cases, j has no reaying benefit for i.

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4 d... dg g-1 d i G g j d 1 G g-1 θ Increase in fow D Fig. 3: Link assignment conditions. The second and third conditions ensure that the reay nodej is not too far away from the source node i. Parametersd th and θ contro the number of outgoing inks from a node and aow seecting a reasonabe non-empty) set of possibe routes. The maximum transmission power per unit bandwidth per ink is imited to avoid interference between groups operating on the same spectrum. This maximum transmission power depends on the radia distances d g, g = {1,2,...,M} defining the groups and on the frequency reuse factor f. If every group has the same radia distance d, the minimum possibe distance between the transmitter of one ink and a receiver suffering its interference is equa to d int = f 2) d. The received power per unit bandwidth has to be ess than a fraction α of the noise power for interference to be negigibe, hence the constraint on the transmission power is: p w αn 0 K 0 d int ) a, where a is the pathoss exponent, 0 is the reference distance 2, N 0 is the noise power spectra density at the receiver σ ), and K is the attenuation factor based on the simpified pathoss mode [2]. This work denotes the bandwidth coefficient by ) a ) 1 0 γ = αn 0 K, d int such that the per ink transmission power constraint becomes p w γ. III. JOINT ROUTING, GLOBAL AND LOCAL RESOURCE ALLOCATION Based on the network conditions, such as node density and ocations, the user groups and possibe routing inks are determined. Given these, the routing and resource aocation probem can be formuated as a convex optimization probem with the objective of maximizing a concave utiity function. To maintain fairness among a users, the utiity function was chosen to be the minimum rate among a users n N\{1}). Ony upink communication is considered here, so the set of a transmitters a nodes other than the BS) is Ñ = N \{1}. For each ink {1,...,L} in the initia poo, we define the system variabes p and w, respectivey denoting the amount of power and spectrum aocated to the ink, and x, denoting 2 This is a conservative bound, using the shortest possibe distance between interfering nodes. Resuts coud be improved using the actua distance for each node, if static and known. the amount of fow on the ink destined to the BS. W g, g {1,...,M} wi denote the amount of spectrum aocated to each group. The probem then becomes: p,w,x,r n,w g) minr n 1) n Ñ A n x = r n, n N 1a) x w og 1+ p ) q, L 1b) w N 0 p w γ, L 1c) p P max, n Ñ 1d) On) n G g On) f W g = W max g=1 w W g, g {1,2,...,M} W g = W g 1) mod f)+1, g > f x 0, p 0, w 0, L. 1e) 1f) 1g) 1h) Constraints 1a) through 1c) are as expained in Section II. Constraint 1d) is obtained by setting the maximum transmission power of a users equa to P max. Constraints 1e) through 1g) identify the spectrum reuse constraints introduced in Section II. The variabes W g and r n are auxiiary variabes introduced to make the probem formuation easier to understand. The vaue for r n can be derived from x using equations 1a). Simiary, it is easy to prove that constraint 1e) wi be tight in the optima soution, so the vaue for W g can aso be uniquey derived from w. Consequenty, both r n and W g appear between brackets in the ist of optimization variabes. This notation for auxiiary variabes continues throughout the rest of the paper. The constraints in 1) define a convex set and the objective function is concave. Hence, the probem is convex and can be soved gobay and efficienty by interior-point methods [18]. However, there might be mutipe soutions with the same objective vaue. Adding a sma negative factor of power sum, ǫ p, to the objective woud guarantee that a soution with sma power vaues is seected among the optima set. This penaty term has negigibe effect on the resuting minimum rate vaue r n. The conventiona direct-mode resource-aocation probem can be derived as a specific scenario of the muti-hop optimization probem 1). In this formuation, there is ony one group of nodes, and the ony possibe route from each node to the destination node is a direct ink between the node and the destination. In this case, the probem can be rewritten as: p,v,r n) min n Ñ r n 2) 1+ p ) nq n, n v n N Ñ 0 0 r n v n og 0 p n P max, n Ñ

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 5 v n = W max, n Ñ v n 0, n Ñ, where v n and p n are equivaent to the origina parameters w and p, respectivey, since there is ony one outgoing ink from each node. IV. DISTRIBUTED OPTIMIZATION Dua decomposition is an od method for soving a convex optimization probem in a semi-) decentraized manner 3 [19]. Generay, if the objective function is separabe in its variabes, the probem can be spit into smaer subprobems. These subprobems can then be soved iterativey in parae based on dua ascent or descent). However, this method requires some hard assumptions such as strict convexity and finiteness of the objective function; otherwise it can resut in numerica instabiity. Augmented Lagrangian methods such as the method of mutipiers achieve better convergence resuts by adding a penaty parameter to the dua ascent objective; however, the penaty parameter is not separabe when the origina probem is. The aternating direction method of mutipiers ADMM), on the other hand, is an agorithm that combines the decomposabiity property of dua ascent with the robust convergence properties of the method of mutipiers [11]. The objective function in probem 1) is not stricty convex so dua decomposition cannot be appied directy. A weighted quadratic reguarization term woud make the objective stricty convex, but finding the appropriate weights is a chaenging probem. If the weights are too sma, dua decomposition wi sti suffer from numerica stabiity, whie excessive weights woud invove an optimaity cost. ADMM is in genera more numericay stabe and faster in convergence than the conventiona dua decomposition method [14]. This paper thus proposes ADMM to impement distributed cross-ayer optimization agorithms for spatia reuse. A. Disjoint Network-Layer and Physica-Layer Optimization One common method for deveoping efficient agorithms in cross-ayer optimization probems is to decoupe the networkayer and physica-ayer probems [6], [12]. This method divides the probem into two simper subprobems, the optimization of network-ayer and physica-ayer variabes, which can be soved separatey and updated iterativey by exchanging messages between the two ayers. This section takes a simiar approach in decouping the probems through ADMM. At each iteration k Z, the network ayer and physica ayer subprobems are soved in parae and based on the resuts, the dua variabes are updated. The network-ayer parameters x and the physica-ayer parameters w and p are couped by constraint 1b) in the above centraized optimization probem. To decompose the probem, a new auxiiary variabe t R L and L equaity 3 Contrary to popuar beief, dua decomposition does not provide a fuy distributed soution. There sti needs to be a centra entity gathering information from a nodes. constraints t = x, are introduced. ADMM decoupes constraints t = x, through a dua variabe u R L, and decomposes probem 1) into two subprobems corresponding to the network and physica ayers. The augmented Lagrangian for the extended probem with penaty parameter ρ > 0 and dua variabe y is L = minr n ) y T x t) ρ n Ñ 2 x t 2 2, constraints 1a)-1h), but it is often more convenient to scae the dua variabe y and combine the inear and quadratic terms, appying ADMM to L = minr n ) ρ n Ñ 2 x t+u 2 2, whereu = ρ 1 y and sti constraints 1a)-1h). Both formuations are equivaent, as shown in [11] section 3.1.1). The network-ayer optimization probem is then: x 0,r n) and the physica ayer probem is: p,w,t 0,W g) minr n ) ρ n Ñ 2 x t+u 2 2 3) A n x = r n, n N, ρ 2 x t+u 2 2 4) t w og 1+ p ) q, L w N 0 1c) 1g). Let superscripts denote iteration indices, omitted in the above subprobems for carity. At iteration k = 1, the network ayer parameters x 1) are obtained from 3) using arbitrary initia vaues for the dua variabesu 0) and auxiiary variabes t 0). The physica ayer parameters p 1), w 1) and t 1) are then obtained from 4) using the same u 0) and the obtained vaues for x 1). Finay, the dua variabe u 1) is updated based on u 0), t 1), and x 1) as foows: u k+1) = u k) +x k+1) t k+1). 5) ADMM has guaranteed convergence for any ρ > 0 but in practice the speed of convergence can change depending on its vaue. Unfortunatey, there is no way to know in advance which vaue wi yied the fastest convergence. Convergence resuts of this agorithm are investigated in Section IV-D through an exampe. B. Semi-Distributed Device Optimization Agorithm The authors are not aware of any fuy distributed agorithm that guarantees convergence to the goba optimum using ony oca coordination among nodes. In any case, such an agorithm woud probaby require a very arge number of iterations, since the constraints in probem 1) are not oca. Information such as data rate from each node needs to reach every potentia reay down the chain, then trave back and forth unti they converge on a soution. Instead, a semi-distributed agorithm is proposed where each node independenty optimizes its own resources based on oca channe information and imited

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 6 communication with a centra entity such as the BS). The agorithm uses ADMM to decompose probem 1) into an independent subprobem for each node and a shared dua update. One major benefit of the semi-distributed agorithm is that the agorithm can adapt to changing oca channe conditions during the iterations of soving the goba probem. In the centraized approach, however, the fina system parameters are obtained based on the initia channe information sent to the centra unit 4. In the semi-distributed probem there are two sets of couping variabes: the per-ink fows x and the per-node bandwidthsv n. To decoupe the per-node optimization subprobems from the goba probem, auxiiary sets of variabes t, L and b n, n Ñ are introduced aong with the foowing equaity constraints: t = x, L b n = v n, n Ñ. 6a) 6b) Even though it appears that the probem compexity increases with the increased number of constraints and variabes, it wi be shown that such decouping generates simper convex subprobems. The set of origina variabes, ψ R L+N 1, and the set of new variabes, z R L+N 1, are defined as: ψ = [x 1,...,x L,v 2,...,v N ],z = [t 1,...,t L,b 2,...,b N ]. The two variabes v 1 and b 1, which correspond to the destination node bandwidth, are eiminated. The variabe ξ R L+N 1 represents the set of dua variabes ξ = u,y), such that u R L corresponds to the routing variabes and y R N 1 corresponds to the bandwidth variabes. The Lagrangian then becomes L = minr n ) ρ n Ñ 2 ψ z +ξ 2 2, constraints 1a)-1h). With this notation and using ADMM, probem 1) is decomposed into two subprobems. The first subprobem optimizes routing and the amount of bandwidth aocated to each user: ψ=[x,v] 0,r n,w g) minr n ) ρ n Ñ 2 ψ z +ξ 2 2 7) A n x = r n, n N n G g v n W g, g {1,2,...,M} 1f) 1g). The second subprobem determines variabesz, per ink power p, and bandwidth w, : p,w,z=[t,b] 0 ρ 2 ψ z +ξ 2 2 8) t w og 1+ p ) q, L w N 0 w = b n, n Ñ On) 4 The addition energy consumption for message-passing in the semi-distributed approach and the performance gain that can be obtained by updating oca channe information are very system dependent and are out of the scope of this work. 1c) 1d). Subprobems 7) and 8) are then soved iterativey with dua updates: ξ k+1) = ξ k) +ψ k+1) z k+1). 9) As demanded, probem 8) can now be decomposed into N 1 separate subprobems, such that node n soves the foowing probem based on its oca channe information q, On): p,w,t, On),b n ρ 2 On) x t +u ) ρ 2 v n b n +y n 2 2 t w og 1+ p ) q, On) w N 0 10a) p P max 10b) On) p w γ, On) w = b n On) p,w,t 0, On). Furthermore, subprobem 7) is separabe into bandwidth v and fow variabes x. The objective in 7) can be rewritten as min n Ñ r n) ρ 2 x t + u 2 2 ρ 2 v b + y 2 2. Hence this subprobem can further be decomposed into routing and spectrum assignment probems. The routing subprobem is: x 0,r n) minr n ) ρ n Ñ 2 x t+u ) A n x = r n, n N. The spectrum assignment subprobem is simpy the Eucidean projection onto a constrained set: v 0,W g) ρ 2 v b+y ) v n W g, g {1,2,...,M} n G g 1f) 1g). Hence probems 11) and 12) can be soved in parae and even by two independent entities. The dua variabes u and y can aso be updated separatey as: u k+1) = u k) +t k+1) x k+1) 13) y k+1) = y k) +b k+1) v k+1). 14) The prima residuas corresponding to ink fows are h k) 1 = x k) t k) and the prima residuas corresponding to bandwidth are h k) 2 = v k) b k). The dua residuas are then defined as s k) 1 = ρt k) t k 1) ) and s k) 2 = ρb k) b k 1) ). Various criteria can be considered as stopping points for the ADMM method. According to [11], one possibiity coud be: h k) 1 2 ǫ p 1, hk) 2 2 ǫ p 2, 15) s k) 1 2 ǫ d 1, sk) 2 2 ǫ d 2 for sma predefined ǫ p 1, ǫp 2, ǫd 1, and ǫd 2 vaues.

7 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 7 C. Impementation This section focuses on the impementation of the semidistributed device optimization agorithm introduced above and investigates the required message passing between different entities. A the information regarding channes, network topoogy, and other such infrastructure can be communicated to the nodes by the centra unit base-station) through contro channes. In ceuar networks, signaing for ink estabishment can be done via sight modifications to the standard radio resource contro RRC) connection reconfiguration procedure [20]. This procedure can incude signaing to the sink nodes of each hop indicating their roe as we as providing a ist of source nodes that the sink node is providing wireess broadband service to. Based on the optimization subprobems deveoped in the previous section, the entity soving probem 11) which can be the BS) provides each user n with the fows x, On) that it shoud carry on each of its outgoing inks and the tota amount of bandwidth v n that it can use. Given v n and x, each node cacuates p,w, and t, On) by soving optimization probem 10). Each node n then broadcasts t, On) and b n = On) w. The prima variabes x and v can be cacuated at a singe centra entity in parae or by two independent entities. Denote by CU 1, the centra unit that coects t vaues from a nodes and soves probem 11) to obtain x. A second centra unit CU 2) is in charge of group bandwidths: it gathers per user cacuated bandwidths b n and soves 12) to obtain v. After each iteration, the dua variabes u and y are updated. The dua variabe updates can be performed either at the centra units or at each node, resuting in different message passing structures. Figure 4 shows the message-passing when the updates are performed at the CU s, and the corresponding distributed agorithm is summarized in Agorithm 1. Figure 4 shows the variabes that are updated by each entity and the reference number of the probem soved to obtain the corresponding variabe. In this configuration, based on initiaized vaues of u 0) and t 0), CU 1 soves optimization probem 11) to obtain x 1). Then the sum of the dua variabe and the updated prima variabe u 0) +x 1) ) is broadcast. Simiary, based on initiaized vaues of y 0) and b 0), CU 2 soves optimization probem 12) to obtain v 1). Then y 0) + v 1) ) is broadcast. Each node n captures its required information y n 1) +v n 1) ) and u 0) +x 1) ), On), cacuates b 1) n and t 1), On) by soving probem 10), and broadcasts the updated variabes. During the next iteration, CU 1 cacuates u 1), and CU 2 cacuates y 1) based on equations 13) and 14), respectivey. In this manner, at each iteration, a the dua and prima parameters are updated. Each update step in Agorithm 1 is foowed by broadcasting the updated parameters. The dua variabe update is performed at the CU s, because there are generay more computing resources at these ocations as compared to individua nodes. According to probem 10), the ony information that each node requires from the CU s is the sum of x and u for a its outgoing inks and the sum of v n and y n. Hence in order to reduce the amount of message-passing between the nodes and CU 1 x + u, 11, 13) t, On) 10) Node n CU 2 v n +y n, n 12, 14) b n 10) Fig. 4: 3-eve optimization and message passing among different entities. Agorithm 1 Proposed distributed cross-ayer optimization. initiaize u, y n, t, b n, and n 1: repeat 2: perform in parae: CU 1: givenu and t, updatex 11) and broadcast x +u,. CU 2: given y n and b n, update v n 12) and broadcast v n +y n, n. 3: Each node n: given x +u, On), v n +y n and oca channe information, update t, On) and b n 10) and broadcast resuts. 4: CU 1 and2: update corresponding dua variabesu, and y n, n 9). 5: unti stopping criterion 15) met. CU s, the sum of the dua and prima variabes x +u, and w n +y n, n are broadcast by the CU s instead of the separate variabe vaues Figure 4). After convergence, the optimum routing variabes x and physica-ayer resources p and w are obtained. In summary, the semi-distributed optimization agorithm works as foows: Each node n gathers and stores the vaues of x +u, On), v n +y n. Then it performs an Eucidean projection onto a set of constraints 10), which has very sma compexity, obtaining t, On), and b n. Each node then broadcasts its updated resuts. Two centra units, CU 1 and CU 2, that can be the same or independent entities, gather the updated vaues of t and b n, respectivey. CU 1 and CU 2 then sove 11) and 12), respectivey, in parae. These probems aso have very sma compexity. The resuts of such optimizations are x, L, and v n, n Ñ. CU 1 and CU 2 then update the dua variabes u and y, respectivey, whie storing a oca copy of each variabe for the next dua variabe update. In theory, the proof of convergence for ADMM requires that a nodes update t, On) and b n in step-3 before the agorithm can proceed to step-4. In most practica cases, however, the agorithm sti converges even if some nodes occasionay fai to perform the update. A good practica approach is to aow partia updates ony a subset of nodes update their oca variabes) during the first few iterations, and impose that a nodes perform the update in subsequent ones. This approach guarantees convergence to the same soution, whie aowing faster execution of the agorithm in cases where

8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 8 meters meters meters meters Fig. 5: Top: simpe node configuration N = 12), bottom: resuting optimum routing configuration. some nodes are significanty sower than others. Wei and Ozdagar proposed in [21] an asynchronous distributed version of ADMM that converges in O1/k), but it assumes that a entities send updates infinitey often and ony guarantees convergence with probabiity 1, unike reguar ADMM. The number of iterations required with this asynchronous version can aso be significanty arger than with reguar ADMM. The compexity of Agorithm 1 scaes gracefuy to arge number of nodes N. Step-2 consists of soving a simpe quadratic program 11) and performing an Eucidean projection onto the convex set specified in 12). Both of them can be soved using interior point methods with ON) fops per iteration. Step-3 is performed in parae by the nodes, so its compexity does not grow. An interior point method at node n woud use O On) ) fops per iteration. Finay, Step-4 computes a inear combination of optimized variabes, which again is a very simpe operation with compexityon). The amount of data broadcast by the nodes aso increases ineary with the number of nodes and inks. In ow-power arge scae networks such as wireess sensor networks WSN), the message-passing overhead can be comparabe to the actua data size. The proposed agorithm can then demoish the owpower objective of WSNs; however, it can prove beneficia in dense areas popuated with devices that have higher energy resources. D. A Numerica Exampe Figure 5 iustrates the agorithm operation over a circuar ceuar area of radius 280m. A tota of 11 nodes with destination N = 12) are randomy distributed in this area, creating 6 groups separated by d = 40m except d 1 = 2d). We assume that a tota bandwidth of W max = 10MHz is avaiabe with a carrier frequency of 800Mhz. The receiver noise power spectra density isn 0 = W/MHz. The reference distance 0 in constraint 1c) is set to 1 and the pathoss exponent is a = 4. Each user s maximum transmission power is set to P max = 0.5W. Setting the anguar threshod θ = 10 o and the distance threshod d th = 1.5d in the ink assignment step, a tota of L = 15 initia inks are generated, as shown in the top pot of Figure 5. CVX, a package for specifying and soving convex programs [22], is used to sove the centraized optimization ρt k t k 1 ) 2 x k t k dua residua prima residua iteration k Fig. 6: Norms of prima and dua residuas versus iteration for ayered optimization probem. max min rate Kbps) 250 r* iteration k Fig. 7: Max-min rate convergence for ayered optimization probem. probem 1), resuting in a max-min rate of 208Kbps. For the disjoint network ayer and physica ayer optimization probems, the penaty parameter is set to ρ = 1 for variations refer to [11, Ch 3.4]) and routing parameterst 0) are initiaized to 1. Soving the individua subprobems 3) and 4) iterativey via ADMM resuts in the same max-min rate of 208Kbps, as shown in Figure 7. The norms of the prima and dua residuas at each iteration are depicted in Figure 6. The resuts show that the centraized agorithm converges in about 15 iterations. Before impementing the subprobems 10), 11), and 12) of the semi-distributed agorithm in Section IV-B, it is important to investigate their nature. Optimization probem 10) is performed by a the nodes in parae) and has a quadratic objective and inear constraints other than constraint 10a). By approximating the og function with a piecewise inear function, probem 10) can be converted to a simpe Quadratic Programming QP) probem with a reativey sma number of variabes. Then CVXGEN [23], which can generate fast custom code for sma QP-representabe convex optimization probems, can be used to sove the probem. A sequentia convex programming approach [24] is used to convert subprobem 10) into a QP: the capacity term in 10a), cw,p ) = w og1 + p q w N 0 ), is approximated by } a piece-wise inear function: cβ) = min a1,a 2) Σ{ a T 1 β +a 2, where β = p,w ) and Σ represents a set of panes tangent to cβ). For simpicity, Σ in this case consists of ony 10 panes. Consequenty, constraint 10a) is repaced with the foowing inear constraint: t min a 1,a 2) Σ { a T 1 β +a 2 }, On) This approximation can be repeatedy re-adjusted during the execution of the agorithm to choose panes that are tangent

9 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 9 ρt k t k 1 ) 2 ρb k b k 1 ) dua residuas iteration k x k t k 2 v k b k prima residuas iteration k Fig. 8: Norms of prima and dua residuas versus iteration for the semi-distributed optimization probem. max min rate kbps) 250 r* iteration k max min rate kbps) iteration k Fig. 9: Max-min rate convergence for the semi-distributed optimization probem for a time-invariant channe eft) and N 0 changed after 50 iterations right). at the current iterate, and hence the approximation error is progressivey reduced. Eventuay, when a point is reached where panes are tangent at the iterate points and the iterates remain constant reached an equiibrium point), it is ensured that the optima soution with zero approximation error is achieved. The number of variabes in this subprobem depends on the number of outgoing inks from each node. Each node s unknown parameters are t, p and w, and hence if node n has L n outgoing inks, it wi have 3L n variabes. Since the nodes in Figure 5 ony have 1 or 2 outgoing inks, each node has a 3 or 6 variabe QP probem to sove. Subprobem 12) is readiy in QP format and subprobem 11) can be easiy converted to a QP by repacing it with its epigraph form. The probem then becomes x 0,ν) ν ρ 2 x t+u ) A n x = ν, n Ñ, Both subprobems 11) and 12) with L = 15 and N = 12 variabes, respectivey, can now be impemented using CVX- GEN, as we. Setting ρ = 0.5 and initiaizing the fows t 0) and bandwidths w 0) to 1, a max-min rate of 208Kbps is obtained through the above approach. The prima and dua residua norms at each iteration are represented in Figure 8, and Figure 9 eft) represents the max-min rate convergence resuts. Simuations using CVXGEN on an Inte core i7, 2.7 GHz processor) took an average of 0.327ms to sove probem 11) and 0.132ms to sove probem 12). Probem 10), soved in parae by a nodes, took 0.145ms in average. In this scenario, the semi-distributed agorithm requires approximatey 50 iterations to converge. Ignoring the time required for message passing max{0.132, 0.327}) = 23.6ms are required on average to obtain the optima routing and resource aocation soution. Ignoring the inks that have negigibe fows e.g. ess than 10Kbps in this case), the bottom pot of Figure 5 is obtained, where Link 4 has been eiminated due to carrying ony 8Kbps. If desired, the resource aocation can then be re-optimized based on the updated inks. The number of iterations and time required for the agorithm to converge can be reduced significanty through warm start initiaization techniques [11, ch.4.3]. The rapid convergence of ADMM when the initia point is cose to the optima soution aso aows the semi-distributed impementation to adapt to changing network conditions. Figure 9 right) shows the convergence of the max-min rate when each node s receiver noise power spectra density N 0 is scaed by a random factor between 1 2 and 5 2 after the first 50 iterations. The nodes do not need to notify the CUs of this change, they just use the new vaue when soving their subprobems. ADMM took 50 iterations to converge from the initia conditions, but ony 12 more iterations to adapt to the new vaues for N 0. V. PERFORMANCE EVALUATION This section evauates the performance of the proposed hierarchica node configuration and cross-ayer optimization agorithms by using the direct communication in 2) as the base ine for comparison. Simuations consider a singe sector of a ceuar network. However, the resuts can be easiy extended to the entire ceuar area, aowing spectrum to be reused among different sectors as we as among the groups within a sector. A. Simuation Setup An instance of the simuation scenario is shown in Figure 10. It consists of a singe sector from a 6-sector circuar ceuar network with radius 210 meters. A tota of 44 users N = 45 with destination node) are randomy distributed within this sector and the common destination node is ocated at 0,0). The sector is sub-divided such that the radia distance between consecutive groups is d = 30m. Since a nodes within group 1 communicate directy with the node at 0,0), the radia distance to the boundary of group 1 d 1 ) is set to doube the reguar distance d. Such vaue keeps the average ink engths within group 1 simiar to other inks in the network. According to the routing-ink assignment poicy, with a distance threshod of 1.5d and anguar threshod of 15 0, a tota of 86 inks were generated for the network in Figure 10. The number of hops can be adjusted by varying d. For ow rate requirements, as the number of hops increases, the tota required transmission power from a source to destination decreases. However, there is a imit on the number of hops, since there must be at east one node per group to take advantage of the frequency reuse. Furthermore, the reduction in transmission power with increasing number of hops comes at the expense of increasing the number of nodes cooperating to transmit data and a arger routing tabe.

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 10 meters meters Fig. 10: Nodes distributed randomy N = 45) with initia set of possibe inks L = 86) tota transmission power dbm) muti hop f=3 muti hop f=4 5 muti hop no reuse direct max power per user dbm) Fig. 12: Tota network transmission power vs. per user maximum transmission power N = 45). max of min rate Kbps) muti hop f=3 muti hop f=4 muti hop no reuse direct max power per user dbm) Fig. 11: Per user rate vs. per user maximum transmission power. The tota avaiabe bandwidth was assumed to be W max = 10MHz with carrier frequency of 800Mhz. The receiver power spectra density is N 0 = W per 1MHz bandwidth. The reference distance 0 in constraint 1c) is set to 1 and the pathoss exponent a = 4. Four different methods are compared in terms of their average performance: muti-hop configurations with frequency reuse of f = 3 and f = 4, muti-hop configuration without reuse f = ), and direct communication 2). The resuts were averaged over 1000 randomy generated networks, constrained to have at east one node in each group. A negative factor of power sum, ǫ p, with ǫ = 10 6 is used in the simuations, as described in Section III. B. Resuts and Discussion The maximum transmission power per device P max is varied between 0dBm and 30dBm and the minimum rate among a users is potted in Figure 11. Furthermore, the tota transmission power empoyed by the network for each P max is represented in Figure 12. It is evident from these figures that the performance of muti-hop configurations with reuse can be significanty imited because of transmission power imits that avoid interference. To sove convex probems 1) and 2), CVX, a package for specifying and soving convex programs [22] is used. The minimum rate and tota power varied significanty among the 1000 networks that were averaged, but the reative performance of each scheme remained fairy constant. For exampe, when the maximum power per user is imited to P max = 20dBm, the standard deviation in the minimum transmission rate for the muti-hop scheme was 8.6 Kbps when f = 3, 45 Kbps when f = 4, and 47 Kbps with no reuse, but the rates with the atter two were neary identica and better than with the f = 3, for a networks. Simiary, the tota transmission power had a standard deviation of 0.5 dbm for f = 3, 2 dbm for f = 4, and 1.8 dbm without frequency reuse, but the reative positions of the curves remained constant. The standard deviations with the direct transmission scheme were negigibe in a cases. Probem 1) contains two constraints on transmission power: constraint 1c), which imits the transmission power on each ink and constraint 1d), which imits per node transmission power P max i.e., the tota power a node transmits over a its outgoing inks). This impies that when P max is arge, the transmission powers are imited by constraint 1c) and when P max is sma, constraint 1d) dominates. Figure 12 shows the P max threshod vaues for frequency reuse factors of f = 3, 15dBm) and f = 4 25dBm), after which increasing P max does not change the tota network transmission power and hence does not increase the achievabe rate. Figure 12 aso shows that tota network transmission power increases ineary withp max in direct mode where constraint 1c) does not appy. At maximum transmission power of 0dBm, as Figure 11 depicts, muti-hopping provides a factor of 10 increase in data rate reative to direct communication, whie approximatey 10 times ess transmission power is required Figure 12). However, this performance gain decreases as P max increases. As theoretica studies of spatia reuse and muti-hopping [25] suggest, using direct communication between source and destination is preferred over spatia reuse and muti-hopping when arge transmission powers are aowed. However, for imited power sources, muti-hopping and spatia reuse perform better than direct mode. At moderatey ow transmission powers the configurations with reuse factors of 3 and 4 obtain the same data rate as that

11 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 11 without reuse, but the tota network transmission power for each case is different. In genera, for the same data rate, the required transmission power increases with the reuse factor. When P max = 10dBm, the tota transmission power required by the muti-hop scheme without reuse is 2.7 times the amount of power required when f = 3 and 1.5 times the amount of power required in the f = 4 case, despite a three provide the same minimum rate. The above resuts, obtained through joint optimization of routing and resource aocation, often require data to fow from a source node through mutipe routes to the destination node. The number of used routes is determined in part by the initia poo of inks seected during the ink assignment step of Section II-D. One method for reducing the number of used routes, abeit with a oss in optimaity, is to restrict the number of outgoing inks from each node by reducing d th and θ. At the expense of further reducing network performance, a ess compex approach is to enforce a singe route from each source to the destination. This can be done by eiminating routing from the joint optimization framework and simpy aowing each node to connect to the cosest neighbor in the group above it. Such approach even though simper to impement, performs significanty worse than cross-ayer optimization, because the nodes that are ocated near the boundary of the groups wi be congested with incoming traffic. Another aternative woud be to sove the probem with mutipe routes and have each node use a its resources for the most promising outgoing ink. The performance oss with each of these approaches depends on many network parameters such as the number of nodes present and their configuration. VI. SUMMARY The demand for better spectrum use grows dramaticay as the wireess technoogy and mobie devices progress rapidy. This demand is more stressed in areas where radio devices are densey packed and need to communicate with a common destination node. This paper proposes empoying these radio devices as both reays and computing resources for better system performance. In this method, each device is used to reay other user s data and perform parae optimization of goba resource aocation. Simuation resuts show that the proposed method can boost each user s rate by a factor of 10 whie using ower transmission power. However this gain is at the expense of higher system compexity. It is shown that muti-hop communication with reuse has better performance when transmission power is imited by device power and not interference. By further increasing spectrum reuse, the avaiabe bandwidth increases, but at the expense of tightening transmission power imits to mitigate co-channe interference. Pathoss can be further reduced and transmission power owered by increasing the number of hops between each node and the destination, but again at the expense of higher system compexity with arger number of hops. ACKNOWLEDGMENT The authors woud ike to thank Professor Stephen Boyd for his vauabe discussions. This paper was funded by the Deanship of Scientific Research DSR), King Abduaziz University, under grant No HiCi). The authors, therefore, acknowedge with thanks DSR technica and financia support. REFERENCES [1] X. Lin, N. Shroff, and R. Srikant, A tutoria on cross-ayer optimization in wireess networks, Seected Areas in Communications, IEEE Journa on, vo. 24, no. 8, pp , [2] A. Godsmith, Wireess Communications. New York, NY, USA: Cambridge University Press, [3] C. D. A. S. Michae Neufed, Jeff Fified and D. Grunwad, Softmac fexibe wireess research patform, Fourth Workshop on Hot Topics in Networks HotNets-IV), [4] T. S. R. Shakkottai, Sanjay and P. C. 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