IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

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Joint Power Adaptation, Scheduling, and Routing for Ultra Wide Band Networks Arjunan Rajeswaran, Gyouhwan Kim, and Rohit Negi, Member, IEEE

Abstract— A general cross-layer optimization problem, to maximize network efficiency (min-max power) of ad-hoc networks, is formulated including power adaptation, scheduling and routing functionalities. The non-convexity of the link capacity, high dimensionality of multi-hop routing and inter-layer interactions among the protocol layers, renders the problem hard. Conversion to an equivalent form, results in two clearly separable subproblems, demonstrating the functionalities of the protocol layers. This decomposition allows the application of a simple shortest path based algorithm to the high-dimensional routing subproblem. Further, in the case of UWB networks, the non-convex scheduling & power adaptation sub-problem can be effectively approximated and solved by applying a novel quadratic lower bound to the link capacity function. Using these algorithmic solutions to these sub-problems, an interior point solver generating solutions to the joint UWB network problem is developed. The various simulation results demonstrate interesting characteristics of the optimal routing and scheduling solutions, and provide benchmarks for UWB network design. Comparison with prior information theoretic capacity results, validates the importance of this cross-layer optimization framework. Index Terms— Cross layer design, wireless, ad hoc network, ultra wide band, routing, scheduling, MAC.

I. I NTRODUCTION

W

IRELESS ad-hoc networks have been the subject of significant research interest, as a network model for various potential applications. Various physical layer and networking technologies, are under investigation to translate the salient features of ad-hoc networks into practical working systems. Recent changes in U.S. federal communications committee (FCC)’s regulations on UWB spectrum and a demand for higher data rates at short range, has fueled intense research and development efforts, to design and standardize commercial UWB radios [1], [2], [3]. UWB radios will be inexpensive and low power, making them ideal for ad hoc wireless applications [4], [5]. However, UWB radios possess properties (extremely large bandwidths and low power) that are drastically different from existing commercial radios (finite bandwidth and large Manuscript received September 30, 2005; revised April 14, 2006 and October 24, 2006; accepted December 4, 2006. The associate editor coordinating the review of this paper and approving it for publication was H. Li. This work was supported in part by the National Science Foundation under Career Award 0347455 and award CNS-0520153. This work was supported in part by Samsung Electronics. This work was presented in part at SPAWC 2005 and GLOBECOM 2004. A. Rajeswaran and G. Kim are pursuing their Ph.D.s at the department of Electrical and Computer Engineering, Carnegie Mellon University (e-mail: [email protected], [email protected]). R. Negi is an Associate Professor in the Department of Electrical and Computer Engineering, Carnegie Mellon University (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2007.05745.

power) and so, issues at all layers in ad-hoc networking merit an analytical revisit. Fundamentally, ad-hoc network design is burdened with the issues of power adaptation at the physical layer, scheduling at the link layer and routing (relaying of data packets) at the network layer. Scheduling is the process of sharing the common wireless medium between competing nodes in frequency and/or time domain. Power adaptation is the problem of choosing the power of transmission for each node in a given scheduled access to the medium. This scheduling & power adaptation problem, assuming a known routing solution, is known to be a hard non-convex optimization problem [6], due to the non-convex relation between link capacity and received desired and interference power, defined by the Shannon capacity formula [7]. At the network layer, the formal function of routing is to find the optimal path or paths, to connect source and destination nodes. A path is a sequence of wireless links and a wireless link is a transmission from a node to, potentially any other node. Thus, in wireless ad-hoc networks, the number of possible routing paths is exponential in the number of nodes and the formal routing problem, of choosing flows on these paths, is of high dimensionality [8]. Traditionally, power adaptation, scheduling and routing are treated independently, as separate functions of physical, link and network layers, respectively. Indeed, the concept of layered architecture has been used successfully in simplifying the design and implementation of wired communication networks. In wireless ad-hoc networks, however, all nodes use the shared wireless medium resulting in a strong interdependency between protocol layers. Therefore, in wireless ad-hoc networks, the power adaptation, scheduling and routing requires a joint consideration to optimize network performance. Interesting forays into this significant problem have been made with different simplifying assumptions [2], [9], [10]. In [2], link rate is assumed to be linear in the received signal to noise ratio (SNR), a simplification of the non-convex Shannon capacity. Based on this simplified physical layer and the assumed time-sharing link layer, it is proven that the maximum throughput (or log utility function) is achieved when each link operates at its maximum allowable power or remains silent. Thus, the scheduling problem consists of choosing a set of operational links in each time slot - a combinatorial problem. A heuristic that avoids strong interference, is the proposed solution. A fixed routing path, the Minimum Energy Path (MEP is the path with the minimum sum of inverse link gains), is chosen to carry all traffic. Thus, [2] provides a set of computationally efficient heuristics to solve the joint problem independently at the three layers.

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

A formal optimization method is applied to the scheduling and power adaptation problem in [9]. Similar to [2], link capacity is assumed to be linear in SNR. Using the assumed time-sharing link layer, it is proven that the duality gap is zero for the scheduling & power control problem. As in [2], these simplifications result in a combinatorial scheduling problem with a complexity of 2L , given L links. The routing is not formulated as a part of the optimization problem, but solved through an iterative procedure. The current optimal dual variables (i.e., sensitivities with respect to the link capacities) are chosen as link costs and a shortest path algorithm is applied to select a path. Traffic is incrementally added per iteration, to this selected path. Thus, [9] combines a routing heuristic with a solution to the simplified problem of lower layers. In [10], the functionalities of the three layers are formulated as a joint optimization problem with a generic concave objective function. Each link is assumed to operate at one of M predetermined rates. This practical discrete rate selection, is a simplification of the physical layer. The assumed link layer is a time-sharing MAC, resulting in a convex capacity region. The routing is modeled as a general multi-commodity flow problem. The dual problem to this convex formulation results in a separation; into a network sub-problem and a scheduling sub-problem. The network sub-problem is a convex problem which may be solved by any standard algorithm. The scheduling sub-problem is solvable under the particular structure of the capacity region and is a combinatorial issue of rate selection (complexity of M L ). The decomposition of layering functionalities (also observed in [11]), facilitates an efficient iterative solution. Thus, [10] provides an optimization tool to solve the joint problem with some restrictive assumptions. In this paper, a cross-layer optimization framework, developed in [8], is presented and validated with extensive simulation results. The joint problem of the physical, link and network protocol layers is first formulated with no simplifying assumptions on the nature of the solutions. The chosen objective function is min-max power so as to combine the important low power utilization and fairness across the equal capability nodes. The results may be converted to the max-min rate and so is this choice is a step towards the prevalent maxmin fair rate. Further, this cost function permits a comparison to prior information theoretic results, that are based on the max-min rate. However, due to a strict sense of fairness, i.e., maximizing the worst rate, the min-max power cost may result in lower efficiency as compared to other utility functions. The chosen cost function should also reflect topological variations, e.g., variable capabilities in nodes. Variations in the chosen metric may alter the value of the results but the qualitative results on scheduling and routing are expected to remain the same. At the physical layer, link rate is chosen to be the Shannon capacity function - the best per-link communication rate. This generic assumption results in a non-convex optimization problem. All possible pairs of transmitters and receivers are allowed as potential operational links, and simultaneous operation of several links is permitted at the link layer. General multi-hop/multi-path routing is considered at the network layer. Next, this non-convex problem is transformed into an equivalent linear program (LP), and its dual is developed.

The dual form shows a clear decomposition into two separate sub-problems - a routing sub-problem and an independent set sub-problem, demonstrating the functionality of the protocol layers. Utilizing this decomposed form, it is shown that the routing solution may be developed by incremental path selection, using a standard shortest path algorithm. The separation from the routing problem in the dual form, allows for an independent consideration of the non-convex scheduling & power adaptation sub-problem. This generic analysis is next applied to the important case of UWB networks. In this paper, the novel quadratic approximation to the UWB link capacity developed in [6], [12], is applied to solve the scheduling subproblem. Then, a Logarithmic Barrier Interior Point algorithm is developed to solve the cross layer optimization and shown to utilize the algorithmic solutions to the individual dual subproblems. In contrast to prior literature, this paper has no prior assumptions on the nature of the optimal solutions and so, the simulation results allows us to investigate characteristics of optimal solutions and network performance of UWB networks. The rest of the paper is organized as follows. In Section II, a general wireless ad-hoc network model is introduced. The joint optimization problem of the three protocol layers is formalized and its dual is investigated, in Section III. Subsequently, an iterative algorithm generating solutions to the joint problem is presented in Section IV. In Section V a limiting case (infinite bandwidth) is solved to provide a performance bound on the general case. Various simulation results and interesting observations are presented in Section IV and finally Section VII concludes the paper. II. W IRELESS AD - HOC N ETWORK M ODEL We begin by describing a general wireless ad-hoc network model and defining the functions of power adaption, scheduling and routing. Throughout this paper, (a)i and ai refer to the ith element of vector a, while (A)ij and aji refer to the (i, j)th element of the matrix A. Inequality ‘≥’ for vectors refers to element-wise inequality. ‘Â 0’ for matrices refers to positive-definiteness. A. Network elements Consider a wireless ad-hoc network with n nodes and S source-destination (S-D) pairs. Each node k, may transmit to n − 1 other nodes forming a wireless link and so there are a total L = n(n − 1) links. Further we define matrix D as (D)ki = 1 if node k is the source node of link i, else (D)ki = 0, to connect links with nodes. A path p, consists of a series of adjoining links and each S-D pair w, can communicate across multiple paths i.e., multiple-path and multi-hop communications are allowed. The mapping between paths and links may be represented by a binary matrix L such that (L)ip = 1 if the path p includes the link i (denoted as i ∈ Lp , where Lp is the set of links constituting the path p), and (L)ip = 0 otherwise. Similarly, F represents the mapping between paths and S-D pairs such that (F)wp = 1 if the path p connects the S-D pair w (denoted by p ∈ Pw , where Pw is the set of paths for S-D pair w). In this general ad-hoc network, the routing functionality is to choose the ‘path flow vector’ y, where yp is the flow (units of data rate) on path p, and hence determine the communication rate between S-D pairs.

RAJESWARAN et al.: JOINT POWER ADAPTATION, SCHEDULING AND ROUTING FOR ULTRA WIDE BAND NETWORKS

B. Physical layer model The total systemPbandwidth W is split into M sub-bands as M fW , f ≥ 0. Here, m=1 fm = 1, i.e., the sum of all elements of the ‘fractional bandwidth vector’, f , is unity. The bandwidth W may be written as W = W0 /x, where W0 represents a constant (unit) bandwidth, while the ‘bandwidth parameter’ x, defines the actual system bandwidth. This normalization facilitates parameterizing the effect of bandwidth (Section VI) in terms of x, denoting a UWB system with small x. The L × M matrix P is the ‘power assignment matrix’, where pm i is the transmit power of link i in sub-band m. The scheduling & power adaptation problem, thus consists of determining the transmit powers pm i of link i in sub-band m and the size of the sub-band fm . For each physical link, we assume capacity-achieving Gaussian channel codes and additive white Gaussian noise. Each link is assumed to provide a data rate equal to its Shannon capacity [7]. Thus, based on the scheduling & power adaptation choice, the rate bm i achieved by link i in sub-band m (of size fm W = fm Wx0 ) is à ! W0 pm i gii m b i = fm log 1 + , (1) P x fm N0 Wx0 + j6=i pm j gji where log(.) refers to the natural logarithm, and so, all capacities are in units of ‘nats’. N0 is the ambient Gaussian noise power spectral density (PSD) and gij ≥ 0 is an arbitrary gain from transmitting node of link i to receiving node of link j. In this paper, we use a simple distance-based gain model, i.e., 1/dα where d is the transmitter-receiver distance and α ≥ 1 is the path loss exponent. B denotes the ‘rate matrix’ corresponding to the power matrix P. Having described the elements of the physical, link and network layers, the joint problem is formalized. III. J OINT P OWER A DAPTATION , S CHEDULING AND ROUTING IN AD - HOC N ETWORKS This problem optimizes the min-max power (with uniform rate requirement) and so, the solution may be translated to the max-min rate (with equal power limit) by a search for the maximum permissable uniform rate, at which the min-max power reaches the power limit. A. Joint problem formulation Let rw be the required data rate for S-D pair w and r denote the ‘rate requirement vector’ for all S-D pairs. To minimize the ‘maximum transmission power’ P0 while achieving r, the joint power adaptation, scheduling and routing may be formalized as, Joint Nonlinear : subject to

min

M,P0 ,f ,y,P T

P0

e f = 1 , Fy = r ,

Be ≥ Ly , DPe ≤ P0 e .

(2) f ≥0, y≥0,

(3) (4) (5) (6)

The vector e = (1, 1, · · · 1)T is a column vector of the appropriate dimension. The equality constraint (3), represents the

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partition of the entire bandwidth W into M normalized subbands. The equality constraint (4), imposes the requirement that the chosen path flow vector y delivers the input traffic (with rate r). The inequality (5) ensures that link capacities (Be) support the total flow rate (Ly) due to all paths utilizing them. Thus the relation between link layer’s scheduling and network layer’s routing is defined by this constraint. Inequality (6), imposes the power constraint per node, by summing (by matrix D) the transmission powers of all links emanating from each node. Note that,mby considering fm as normalized time pi slots, pm i as energy, fm is the power in time slot m. Then, (1) m relates rate bi to power in time slot m and (6) constraints the total energy over unit time, i.e., average power. This variation allows Joint Nonlinear to model time-slotting with an average power constraint. Thus, solving Joint Nonlinear will yield the required optimal routing, scheduling and power adaptation solution through the optimal variables y, M, P, f . However, the non-convex relation between the elements of the rate matrix B and the power matrix P in (1), and the high dimensionality of path flow vector y, renders the problem intractable [13]. The problem is next analyzed to yield simpler formulations. Towards simplifying analysis, we first introduce the normalized variables, . . bm pm i qim = N0 Wi0 fm , cm i = f m W0 , . y . rw . 0 y¯p = Wp0 , r¯w = W , P¯0 = NP . 0 0 W0

(7) (8)

Here, qim , a normalized version of the transmit power vector pm i , is also the (transmit) SNR of link i in sub-band m (at x = 1). Similarly cm i , a normalized version of the link rate vector bm , is the corresponding spectral efficiency. y¯p , r¯w and i P¯0 are normalized versions of yp , rw and P0 , respectively. This normalization allows for a succinct representation while absorbing various constants. The problem Joint Nonlinear is a hard non-convex optimization problem and so we look for alternative representations that may aid the search for an intelligent solution. Hence, we transform Joint Nonlinear to a Linear Program (LP) in ¯ } by enumerating all the feasible (i.e., the variables {P0 , f , y q ≥ 0) infinite number of SNR vectors and collecting them as the columns of an infinite dimensional matrix Q. This is possible by considering all the countably infinite rational power vectors (a dense set in the reals). The corresponding spectral efficiency vectors c are collected into an infinite dimensional matrix C, while the infinite dimensional vector f denotes the series fi , i ² I. The infinite dimensional matrices Q, C and the vector f are notational tools to represent the convex combinations (Qf, Cf ) of the elements in the set of q ≥ 0 and the corresponding link capacities, respectively. Utilizing Q, C removes the variable M i.e., a solution with any number of bands is admissible. Similarly P is no more a variable since all possible q ( normalized p) are considered in the matrix Q. Thus, the non-convex relation between link rate and power no more exists.

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The transformed problem Primal LP is min P¯0

Primal LP : subject to

P¯0 ,f ,¯ y T

e f = F¯ y = Cf ≥ DQf ≤

1 , f ≥0, ¯r , ¯≥0, y L¯ y, P¯0 e .

(9) (10) (11) (12) (13)

The number of columns in L and F is the number of paths in the network. Thus, L and F consist of O(n!) columns, while Q and C consist of infinite columns. So, a direct solution of Primal LP is impossible. This infinite dimensional LP is simply a conceptual tool to convert the hardness of the nonconvex constraints (5) to a structured convex form. Therefore, we require an algorithm which intelligently produces the columns of L, F, Q and C that constitute an optimal solution. Towards such an algorithm, we next consider the dual of Primal LP. B. Dual problem analysis The variable f in Primal LP is infinite dimensional. Further, this link layer variable f is coupled with the network layer variable y ¯ (the path flow vector), in the constraint (12). Thus, the problem formulation Primal LP is both intractable and presents no intuition regarding the functionalities of the layers. The Lagrange dual problem is next analyzed to relax the coupling of primal variables in Primal LP towards separating the functionalities of the protocol layers. Also, we will show in Section IV that, the dual analysis is illustrative in a direct solution to Primal LP. Though we consider infinite dimensional variables the existence of a strictly feasible point and the finiteness of the cost function may be shown. These conditions are sufficient for strong duality [14] and so the dual function of Primal LP [13] is, min ( (λTc L)¯ y − (λTc C − λTq DQ)f ) , f ,¯ y

subject to

(14)

(10), (11), eT λq = 1 ,

where λq ≥ 0 and λc ≥ 0 are the dual variables corresponding to constraints (12) and (13), respectively. Due to the equality constraint (10), the minimization, (14), results in choosing the maximum element of λTc C − λTq DQ by setting the corresponding fm = 1. Similarly, (11) may be absorbed into the Lagrangian (14). The maximization of (14) yields the dual problem, Ã S ! X ∗ ∗ max r¯w Rw − I (15) Dual LP : λq ,λc ≥0, eT λq =1

where

w=1

R∗w = minp∈Pw (λTc L)p

(16)

I ∗ = maxm (λTc C − λTq DQ)m .

(17)

This decomposed dual form presents a mathematical view of the concept of ‘layering’ in networking. First, R∗w that selects a single path connecting a S-D pair, is a simpler version of the routing problem . By (16), the selected path has the

Fig. 1.

Least cost path problem with (λc )i as link cost.

minimum sum of (λc )i ’s among all p ∈ Pw , where Pw is the set of all paths connecting S-D pair w. As in Figure 1, each link has a corresponding (λc )i , since λc is the dual variable corresponding to the link rate constraints (12). λc is the sensitivity of the objective function value with respect to the corresponding link rate constraint and may be considered as a vector of link costs (a standard interpretation of the dual variables [9], [11], [13]). Thus (λTc L)p in (16) is the cost for path p. Therefore, R∗w is exactly a least cost path problem and any simple shortest path algorithm [15] may be applied. Next, sub-problem I ∗ , (17), is similar to the scheduling for a single sub-band. The vector q solving I ∗ maximizes the difference between ‘the weighted sum of the normalized link rates λTc c’ and ‘the weighted sum of transmission powers λTq Dq’, for a certain λc and λq . Transmission power is the expended resource (paid cost) to obtain the link rates. For a certain choice of dual variables, the optimal vector q achieves the most cost effective scheduling for a single sub-band. This is achieved by assigning the transmission powers for each link (zero power for a non-operational link) in the sub-band. Thus I ∗ solves the power adaptation problem by choice of transmission powers for each link. Scheduling is achieved by the choice of links that transmit simultaneously and so I ∗ is similar to a weighted maximal independent set problem [16]. The strong dependency between the protocol layers is demonstrated by the dual variables λc and λq . Routing and scheduling are coordinated through the dual variable λc , that affects the solutions to both R∗w and I ∗ . Scheduling and power adaptation interact in I ∗ through the dual variable λq , that may be considered to represent the cost of transmission power. Unfortunately, solving the scheduling sub-problem I ∗ is still hard due to the non-convex Shannon capacity formula relating q and c. However, the scheduling sub-problem I ∗ has been isolated through the dual decomposition. Thus effective link-level approximations can be applied to I ∗ . In fact, we can apply the approximation from our prior work on UWB scheduling [6], and extend the algorithmic procedure. In the next subsection, important results from [6] are reviewed and complexity issues in applying the result to the joint problem are discussed.

RAJESWARAN et al.: JOINT POWER ADAPTATION, SCHEDULING AND ROUTING FOR ULTRA WIDE BAND NETWORKS

C. UWB ad hoc scheduling UWB wireless networks have recently gained significance from a research as well as a practical perspective [4]. In this section, we consider the scheduling & power adaptation in UWB networks. As noted earlier, when W → ∞, the scheduling problem can be solved exactly (CDMA is optimal) [17], since the rate achieved is linear in the transmit power. Motivated by this result, we analyze the UWB physical layer towards a simplification of the dual sub-problem I ∗ . The key characteristic of a UWB physical layer is the low spectral efficiency, due to large W and the limited transmission power. FCC guidelines mandate that such a system have low spectral efficiency, to prevent an increase in the ambient noise power level. Hence, it is likely that Shannon capacity is ‘almost’ a linear function of power, with a small correction. This motivates the following approximation ˜bm i to the rate W0 , for large W = function bm (i.e., x ¿ 1). i x . pm i gii ˜bm = i N0 −x·

h i m X pm pm j gji i gii 1 pi gii + (18) N0 2f N W f N W | m{z0 0} j6=i m 0 0 {z } | L1 L2

˜bm i

Here, is the first-order Taylor series approximation to bm i m as a function of x about the point x = 0. Further, ˜bm i ≤ bi , m and so ˜bm i is a lower bound approximation to bi as shown in [6]. The bandwidth parameter x determines the accuracy of the approximation ˜bm the i . As x decreases (i.e., more UWB-like), pm m i gii ˜ lower bound bi approaches the upper bound N0 , and so, the approximation becomes accurate. There are two quadratic (in the power variables pm i ’s) terms which cause the lower bound to deviate from the linear upper bound, when x > 0. L1 is a ‘self-interference’ term that accounts for the concavity of the log(.) function. Secondly, the finite bandwidth implies that the system experiences co-channel interference, which is represented by L2 . The effect of both terms disappears when x → 0 (i.e., W → ∞). m m According to the approximation ˜bm i of bi , we redefine ci . Thus for UWB ad-hoc scheduling, . cm i = =

˜ bm i fm W0

qim gii − x · qim gii

³

1 m 2 qi gii

+

P

m j6=i qj gji

´ . (19)

This new definition of cm i is applied to the dual sub-problem I ∗ resulting in, I ∗ = max hT q − qT Aq , q ≥0

(20)

. . where A(i, j) = 12 x ((λc )i gii gji + (λc )j gjj gij ) and hi = (λc )i gii − (λq )k . k designates the index of transmitting node of link i. Now, the sub-problem I ∗ is a quadratic optimization problem, and although usually non-convex (A  0 in general), is simpler to solve than the general non-convex problem I ∗ in (15). Fortunately, the following result holds. Result 1: I ∗ may be obtained by the following procedure. 1) Choose a principal sub-matrix As  0 of A, corresponding to a subset of links s and the sub-vector hs of h. 2) If . 1 T −1 ∗ = A−1 s hs > 0, evaluate V al(s) = 4 hs As hs . 3) I

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max V al(s), achieved for index set s∗ , and q∗s∗ = 12 A−1 s∗ hs∗ . I ∗ is achieved by the optimal q∗ , which can be obtained by setting q∗ = q∗s∗ for indices s∗ and q∗ = 0 otherwise. (The proof is in [6]). Comments on the complexity: The procedure in Result 1 is much simpler than the exhaustive search for a general non-convex problem. However, since there are 2L = 2n(n−1) possible principal sub-matrices As , the procedure requires 2L checks for positive definiteness. For example, it requires 220 positive definiteness checks even for a small network with 5 nodes. The following result on the characteristic of the optimal solution q∗ allows further reduction of complexity. Result 2: In the optimal solutions to I ∗ , single links are the only type of communication allowed at link level, i.e., in each sub-band, every transmitting node transmits signal to only one receiver and also every receiving node receives from only a single transmitter. Proof: Firstly, it is shown below that in an optimal solution, each node transmits to at most one receiver in each sub-band. Consider links from a common transmitter to some N distinct receivers. We have gji = gii ∀ (i, j). This condition may be to the dual sub-problem I ∗ in (15). Papplied . N Define ³ S = x j=1 q´j and then, ci defined by (1), (7) is − x1 log

1+gii S−xqi gii 1+gii S

.

³ ´ Hence, I is max max λc T c − λq T Dq . For any fixed ∗

S ≥ 0q ≥ 0

value of S, say S = s, the spectral efficiency of link i, ci , is now a convex function of q and so is λc T c−λq T Dq, the inner objective function of I ∗ . Thus, I ∗ requires the maximization of a convex function over the simplex defined by S = s and q ≥ 0. Thus, the inner maximum will be achieved at one of the vertices of the simplex. This tells us that among all links emanating from a transmitting node, only one link must be active (qi > 0 and qj = 0 ∀j 6= i). This argument holds for the general case with multiple transmitting nodes, since irrespective of the ambient interference, ci is convex in q for a fixed S. Similarly, in the case of a common receiver, gji = gjj ∀ (i, j), and so in an optimal solution , each receiving node has only one transmitter per sub-band. In addition to Result 2, it is assumed that simultaneous transmitting and receiving is not possible, i.e., no mechanism for self-interference cancellation. Thus, the number of possible maximal independent set is reduced from 2n(n−1) Pn/2 n n−k to Ck Ck k!. This dramatic reduction of complexity k renders the joint problem tractable for medium sized networks with n ≤ 12. D. Heuristics for narrow band systems and large networks The quadratic approximation (18) may not be utilized for the analysis of narrow band systems (high SNR regime), due to its inaccuracy at large x. A computationally efficient heuristic is thereby developed to solve I ∗ for a narrow band scenario. This heuristic method, sequentially selects links that maximize the incremental contribution to the cost function at each stage [8]. The accuracy of this heuristic method was initially verified in sample UWB scenarios, through comparison with the optimal solution and subsequently applied to the narrow band scenario. The low complexity of this heuristic and its

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accuracy in the UWB case [8], motivates its application to networks with large number of nodes (∼ 50). Thus the heuristic is applied both in the narrow band case and for large networks. Next, using the LP formulation and the dual analysis results, an algorithm constructing the solution to Primal LP is developed. IV. A LGORITHM Primal LP may be solved by applying an interior point method similar to the method devised in [18]. Towards such a solution, a logarithmic barrier function [13] is defined as, N t X . Φt (P¯0 , α, ρ, π) = P¯0 − log(P¯0 − (π)k ) 2N k=1

L t X − log((ρ)i − (α)i ) . 2L i=1

(21)

Here, π = DQf , ρ = Cf and α = L¯ y are the feasible power vector, corresponding link capacity vector and link capacity requirement (imposed by the routing solution), respectively. The minimization of Φt (P¯0 , α, ρ, π) (a convex function) may be achieved by iterative updates of the variables along a descent direction while maintaining the feasibility of Primal LP [13]. The algorithm JOINT SOLVER, outlined below, relates the partial derivatives of Φt (with respect to fm and y¯p ) to the objective functions of dual sub-problems R∗w and I ∗ , using the appropriate dual variable estimates [13]. Subsequently, the solutions to the dual sub-problems in each iteration (a routing path per each S-D pair and a power vector for a sub-band), are utilized to obtain a descent direction. The solution to Primal LP is constructed by the multiple paths and power vectors obtained by solving R∗w and I ∗ iteratively. As noted, the solution to R∗w may be obtained by a shortest path algorithm (such as Dijkstra’s algorithm [15]). Also, I ∗ can be solved as in Subsection III-C or III-D. It is the particular form of the logarithmic potential function and linearity of the cost function that relates the descent direction and the dual sub-problems. The algorithm is not a primal-dual method. JOINT SOLVER provides a specified accuracy ² ∈ (0, 1), P¯0∗ such that the achieved min-max power is at most (1−²) , where P¯0∗ is the optimum value. In the following, a ← (a, a) denotes appending a to a, and A ← (A, a) denotes appending a to A. The dual cost is D(λq , λc ). Algorithm: JOINT SOLVER 1) Initialize all the variables with a feasible solution. D(λq ,λc ) 2) While maxk (π)k > (1−²) (till duality gap is met) do: t = 0 (compute unique a) Compute P¯0 satisfying ∂Φ ∂ P¯0 minimizer of Φt ). and (λc )i = (ρ)t/2L b) Set (λq )k = P¯0t/2N −(π)k i −(α)i (estimated feasible dual variables). c) Perform routing for all w sequentially. i) Find a shortest path pˆw by solving R∗w . y ¯w ← ((1 − δw )¯ yw , δw r¯w ). Here δw is the step size and y ¯w is the vector of non-zero y¯p ’s for S-D pair w. ii) Append the corresponding columns to F, L and update λc and α.

d) Perform scheduling & power adaptation. ˆ as the solution to I ∗ . f ← ((1−τ )f , τ ). i) Find q Here τ is the step size. ˆ ), C ← (C, c ˆ), update π, ρ. ii) Q ← (Q, q 3) Apply a Carath´eodory-theorem based algorithm to reduce the size of solution. The solution is a finite real set of vectors, i.e., a compact set. Hence by Carath´eodory theorem [7], there exists an equivalent optimal solution consisting of at most (L + S + 1) paths (dimensionality of the path flow L, F space) and (2L + 1) SNR vectors (dimensionality of the q, c space) [6], [8]. To reduce the dimensionality of the solution set to the Carath´eodory dimensionality, we apply an algorithm [19], that sequentially removes linearly dependent vectors. This results in a consistent succinct representation. The algorithm JOINT SOLVER, may be shown to strictly decrease the cost function at each stage, by the choice of descent direction (Step 2) and the convexity of the logarithmic barrier function. The descent direction is chosen to be along the coordinate with the steepest descent and this with the simplex constraint implies that the update (Step 2c,2d) strictly decreases the cost function. The combination of this sequential strict decrease and a lower bound on the cost function ensures the convergence of JOINT SOLVER. This is not sufficient to prove the convergence to the optimal or a bound on the number of iterations, i.e., the complexity of JOINT SOLVER. However, the stopping condition ensures that the duality gap at the converged point is within ². This proves the correctness of the simulation results presented in Section VI. V. L OWER B OUND OF M IN -M AX P OWER In the infinite bandwidth case (x = 0), each link requires minimum power for a certain fixed capacity. Thus, the minmax power at x = 0, provides a lower bound on the min-max power for the general case with x 6= 0. In this section, the joint problem is solved assuming infinite bandwidth, i.e., x = 0 and so, frequency assignment is not relevant. Thus, f is removed from Primal LP and Q is replaced with a single SNR column vector q. Also, the capacity of each link ci → gii qi , in the limit x → 0 (1). The resulting problem is min

P¯0 ,¯ y,q > 0

subject to

P¯0

(22)

(11) , Dq ≤ P¯0 e , diag(g)q ≥ L¯ y,

where g is the link gains. This LP may be easily solved resulting in the lower bound. VI. S IMULATION R ESULTS The presented simulation results demonstrate the effect of bandwidth on optimal routing and scheduling and the combined effect of node density and bandwidth on network performance. Prior to investigating solution characteristics, the accuracy of the quadratic approximation (18) to the log-capacity function (1) was measured in the test network shown in Figure 2. The 10 nodes forming 5 S-D pairs were distributed randomly (uniform distribution), in a unit area. Assuming equal rate

RAJESWARAN et al.: JOINT POWER ADAPTATION, SCHEDULING AND ROUTING FOR ULTRA WIDE BAND NETWORKS

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Fig. 2. Test network with 10 nodes and 5 S-D pairs; The squares denote source node and the circles denote the destination nodes. TABLE I ACCURACY OF THE QUADRATIC APPROXIMATION x m average of ˜bm i /bi

0.5 0.916

0.1 0.948

0.01 0.955

Fig. 3. Scheduling solutions with two different bandwidths; Equal required rate rw = 0.05W0 for all w, ² = 0.5. Each figure shows a typical schedule for a sub-band which has relatively large portion of the total bandwidth.

0

TABLE II

1(no approximation)

P ORTION OF BANDWIDTH ASSIGNED TO SCHEDULING PATTERNS

requirement, r¯w was set to be 0.05, the path loss exponent α = 3 and the approximation parameter ² = 0.2. Table I shows that the average error of the quadratic approximation (18) is less than 10% for the tested range of x, validating its accuracy. As a reference, assuming the 802.15.3a channel model and gii a link distance of 10m, x NP00W is −4.45 dB i.e., 0.36. At 0 such small SNR, our approximation (18) is accurate. Our normalized model allows the presentation of results in terms of the parameter x. At x = 0.1, the min-max power achieved by utilizing the quadratic approximation was 0.00512, while the lower-bound from Section V was 0.00444, demonstrating the network level accuracy. The heuristic method introduced in Subsection III-D is validated by the close agreement of the resulting min-max powers with the optimal algorithm (Subsection III-C). At x = 0.1, the min-max power obtained from the heuristic was 0.00506 representing only an insignificant error. Thus, for simulations with the large n ≥ 20 and/or narrowband networks (x ≥ 1), the heuristic in Subsection III-D was used. The joint problem was solved optimally at various bandwidths (0 ≥ x ≥ 25), for the network in Figure 2. The approximation ² = 0.5, i.e., a solution within 3dB of the optimum, was chosen. Figure 3 shows some typical scheduling patterns at two different bandwidth, i.e., a UWB (x = 0.01) case and a narrow band (x = 25) case. In [17], it was proven that the optimal scheduling solution in the ideal UWB system with infinite bandwidth is ‘simultaneous transmissions of all nodes’, i.e., CDMA in a common frequency band. As expected from the previous study, the scheduling solution of the practical UWB case with x = 0.01, shows that multiple links are scheduled together in each sub-band and the percentage of total bandwidth assigned to the sub-bands with 3 simultaneous transmitting links was 99% (see Table II). This high spatial reuse of frequency is possible because of the noise-limited nature of the UWB system. Contrastingly, in the

x

1link/sub-band

2link/sub-band

3link/sub-band

0.001 0.01 0.1 1 10 25

0.0049 0.0079 0.035 0.065 0.21 0.86

0.0011 0.0047 0.046 0.035 0.25 0.14

0.99 0.99 0.92 0.90 0.54 0

interference-limited narrow band case (x=25), the scheduler was conservative, and 86% of bandwidth was assigned to links scheduled individually, i.e., closer to a Frequency Division Multiple Access (FDMA). Thus, the scheduling solution transitions from a ‘CDMA like’ solution at large bandwidth (small x) to an ‘FDMA like’ solution at small bandwidth (large x). The effect of bandwidth is also visible in the optimal routing solution as in Figure 4. At x = 25, the limited system bandwidth, combined with the individual sub-band assignment (FDMA scheduling solution) allows very few operational links in the network. This results in routes with smaller number of links such as the direct path. As shown in Figure 4, for the shown S-D pairs, the direct path carries the majority of traffic flow. In the UWB case however, the routing solution consists of multi-hop paths with relatively large number of links. Towards optimizing the objective of min-max power, routing selects multi-hop paths constructed by power efficient short links to carry most of the traffic flow. The scheduling solution in the UWB case also provides such short links by the simultaneous scheduling of many links per sub-band (CDMA like solution in the noise-limited scenario). Thus, at x = 0.01 a significant amount of the traffic flow is carried on the MEP (Minimum Energy Path) [17]. The routing solution thus shifts traffic from the direct path to multi-hop paths (e.g., MEP) as the bandwidth increases. Further, note the similarity between the routing solutions at x = 0 and x = 0.01. This indicates that the infinite bandwidth routing solution (an easily obtained

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

Fig. 4. Routing solutions with two different bandwidths; Equal required rate rw = 0.05W0 for all w, ² = 0.5. For clarity of representation, main flows (with yp ≥ 0.1rw ) for only a single S-D pair, are displayed in each figure. The upper three figures show the routing solution for S-D pair 2 and the lower three show those for S-D pair 5. The thickness of a link represents the relative amount of data flow. −1

Required rate per S−D pair = 0.05W0

large power) and shows a decreasing uniform throughput capacity of Θ((n log n)−1/2 ) [20]. Whereas in [17], assuming an alternate UWB like communication model with limited transmission power but arbitrarily large bandwidth, a contrasting (increasing) network capacity result was derived. These two contrasting results hold for contrasting physical layers, at the extremes of very low bandwidth (narrow band) and arbitrarily large bandwidth (UWB) respectively. The upper bound of α−1 per node throughput for UWB networks is O((n log n) 2 ) (α > 2) [17], i.e., an increasing function of node density. Since capacity is a linear function of power at infinite bandwidth, for a certain fixed capacity, the lower bound of the required α−1 min-max power is Ω((n log n)− 2 ). As bandwidth increases (x → 0), the optimization result indeed approaches this information theoretic lower bound (solid line in Figure 5). Network performance at large x, indicates the opposite trend, as noted in [20]. Therefore, the optimization results support the theoretical result from [17], i.e., UWB networks have dramatically different characteristics. Consequently, in UWB regime, it may be feasible to realize dense ad-hoc networks.

10

min max power (Pmax/N0W0)

VII. C ONCLUSION

−2

10

−3

10

x = 10 x=3 x=1 x = 0.1 x=0 Info theoretic bound

−4

10

4

10 20 Node density (n)

30

40

50

Fig. 5. Effect of node density & bandwidth; Equal required rate rw = 0.05W0 for all w, ² = 0.5 and the number of S-D pairs is n/2.

solution as in Subsection V) may be a good approximation to the routing solution for practical UWB systems. In Figure 5, the min-max power is plotted as a function of node density. The equal input traffic rate of 0.05W0 is assumed and ² = 0.5. At large bandwidth (x = 0, x = 0.1), dense networks achieve low min-max power even with the increase in the total input traffic (linear in N ). Thus, when the bandwidth is large enough, dense networks increase network performance (decreasing min-max power or equivalently increasing uniform capacity). As in Figure 3, this trend is because power efficient short links are preferred at large bandwidth and dense networks provide shorter links. However, in the narrow band regime, network performance worsens with increasing node density. At x = 10, x = 3, the minmax power increases steeply with increasing node density at N > 10 and N > 30, respectively, indicating a limit on the supportable node density. Next, this performance is validated against known information-theoretic results. The well known Gupta-Kumar’s capacity result assumes a narrow band physical layer (limited bandwidth and arbitrarily

In this paper, we presented a general non-convex optimization framework for the joint wireless networking problem, including power adaptation, scheduling and routing functionalities. Analysis resulted in an illustrative dual decomposition that demonstrated a distinct mathematical representation of the routing and scheduling functionalities. Algorithmic solutions to these distinct sub-problems for a UWB network were integrated into an iterative algorithm to obtain the optimal solution. The simulation results showed the drastic variation of optimal routing and scheduling solutions with different bandwidth and node density. As the system becomes more UWB like (i.e., large bandwidth), scheduling solutions consist of increased simultaneous transmissions (CDMA like) while routing solutions selected multiple hops. Further, in UWB systems, densely located nodes improved network performance, providing an algorithmic validation of our prior information theoretic result in [17]. These extensive simulation results demonstrated the value of this general optimization theoretic approach. Future work will investigate computationally efficient and scalable distributed protocols. R EFERENCES [1] A. Rajeswaran, V. S. Somayazulu, J. R. Foerster, “Rake performance for a pulse based UWB system in a realistic UWB indoor channel,” in Proc. IEEE Int. Conf. on Commun. (ICC) 2003, pp. 2879-2883. [2] B. Radunovi´c and J. L. Boudec, “Optimal Power Control, Scheduling, and Routing in UWB Networks,” IEEE J. Sel. Areas Commun., vol. 22, pp. 1252-1270, Sept. 2004. [3] J. Foerster, E. Green, S. Somayazulu, and D. Leeper, “Ultrawide band technology for short or medium range wireless communications,” Intel Technology J., vol. 5, no. 2. Available at http://developer.intel.com/technology/itj/q22001/ [4] R. Fontana, A. Ameti, E. Richley, L. Beard, and D. Guy, “Recent advances in ultra wideband communications systems,” Digest of IEEE Conference on Ultra Wideband Systems and Technologies 2002, pp. 129133. [5] L. Krishnamurthy et al.,“Meeting the demands of the digital home with high-speed multi-hop wireless networks,” Intel Tech. J., vol. 6, pp. 57-68, 2002. [6] R. Negi and A. Rajeswaran, “Scheduling and Power Adaption for Networks in the Ultra Wide Band Regime,” in Proc. GLOBECOM ’04, pp. 139-145.

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[7] T. Cover and J. Thomas, Elements of Inform. Theory. John Wiley, 1991. [8] G. Kim, A. Rajeswaran, and R. Negi, “Joint power adaptation, scheduling and routing framework for wireless ad-hoc networks,” in Proc. SPAWC ’05, pp. 725-729. [9] R. Cruz and A. Santhanam, “Optimal routing, link control in multi-hop wireless networks,” in Proc. INFOCOM ’03, pp. 702-711. [10] M. Johansson and L. Xiao, “Cross-layer optimization of wireless networks using nonlinear column generation,” Tech. Report, RTI, Stockholm, Nov. 2003. [11] W. Yu and J. Yuan, “Joint source coding, routing and resource allocation for wireless sensor networks,” in Proc. ICC 2005, pp. 732-741. [12] A. Rajeswaran, G. Kim, and R. Negi, “A scheduling framework for UWB and cellular networks,” in Proc. Broadnet ’04, pp. 386-395. [13] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, 2004. [14] D. G. Luenberger, Optimization by Vector Space Methods. John Wiley & Sons, 1969. [15] D. Bertsekas and R. Gallager, Data Networks. 1992. [16] J. A. Bondy and U. Murthy, Graph Theory with Applications. Elsevier, 1976. [17] R. Negi and A. Rajeswaran, “Capacity of power constrained ad-hoc networks,” in Proc. INFOCOM ’04, pp. 443-453. [18] M. D. Grigoriaids and L. G. Khachiyan, “Coordination complexity of parallel price-directive decomposition,” Mathematics of Operations Research pp. 321-340, 1996. [19] D. G. Luenberger, Linear and Nonlinear Prog.. Addison-Wesley, 1989. [20] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Trans. Inf. Theory, vol. 46, pp. 388-404, March 2000.

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Arjunan Rajeswaran received his B.Tech. degree in Electrical Engineering from the Indian Institute of Technology, Bombay, India in 2001 and his M.S. degree in Electrical and Computer Engineering from Carnegie Mellon University in 2003. Since August 2003, he has been pursuing his doctoral research at Carnegie Mellon. His research interests are in the area of wireless networks. His focus is in the application of information and communication theoretic tools towards cross layer wireless network design. Arjunan received the best student paper award at IEEE/ACM Broadnets 2004. Gyouhwan Kim received his B.S. and M.S. degree in Electronic Engineering from Sogang University in Korea, in 1994 and 1996, respectively. Since 1996, he has been working in the CDMA cellular system development team in Samsung Electronics. Currently, he is also pursuing his Ph.D degree in the Department of Electrical and Computer Engineering at Carnegie Mellon University. His main research interests are wireless networks and communication theory.

Rohit Negi received the B.Tech. degree in Electrical Engineering from the Indian Institute of Technology, Bombay, India in 1995. He received the M.S. and Ph.D. degrees from Stanford University, CA, USA, in 1996 and 2000 respectively, both in Electrical Engineering. He has received the President of India Gold medal in 1995. Since 2000, he has been with the Electrical and Computer Engineering department at Carnegie Mellon University, Pittsburgh, PA, USA, where he is an Associate Professor. His research interests include signal processing, coding for communications systems, information theory, networking, cross-layer optimization and sensor networks.