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Mobile Networks and Applications 11, 9–20, 2006 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. DOI: 10.1007/s11036-005-4457-1

A Scheduling Framework for UWB & Cellular Networks* ARJUNAN RAJESWARAN, GYOUHWAN KIM and ROHIT NEGI Department of Electrical and Computer Engineering, Carnegie Mellon University Published online: 9 December 2005

Adstract. The max-min fair scheduling problem in wireless ad-hoc networks is a non-convex optimization problem. A general framework is presented for this optimization problem and analyzed to obtain a dual problem, which involves solving a series of optimization sub-problems. In the limit of infinite bandwidth (W → ∞), the scheduling solution reduces to simultaneous transmission (spread spectrum) on all links (Negi and Rajeswaran, INFOCOM ’04 (March 2004)). This motivates the analysis of the scheduling problem in the Ultra Wide Band (UWB) regime (W  1, but finite), a model for certain practical radios. A quadratic (in 1/W) lower bound to the single link capacity function is developed, which simplifies the dual sub-problem to a quadratic optimization (Negi and Rajeswaran, GLOBECOM ’04, (Dec. 2004)). The solution to this sub-problem is then obtained under both total power and power spectral density constraints. This solution is utilized to iteratively construct the schedule (sub-band sizes) and power allocation, thus optimally solving the UWB max-min fair scheduling problem, to within any desired precision. Simulations on medium sized networks demonstrate the excellent performance of this scheme. A cellular architecture (not necessarily UWB) may also be considered in this framework. It is proved that Frequency Division Multiple Access is the optimal scheduling for a multi-band cellular architecture. Keywords: wireless communications, ad-hoc network, Ultra Wide Band, cross layer design, scheduling, MAC

1. Introduction Wireless communication networks consist of nodes that communicate with each other over a wireless channel. ‘Infrastructure wireless networks’, such as cellular networks, are widely prevalent. They typically consist of a wired infrastructure of controllers (base stations), with nodes connected to the controllers over wireless links. Other ‘infrastructureless’ networks, such as ad-hoc networks, consist of purely wireless links. Ad-hoc wireless networks allow speedy deployment, low cost and low maintenance, which lend towards applications such as sensor networks, personal area networks (PAN’s) and military battle-field communications. Wireless network design deals with the issues of scheduling at the link layer and relaying of data packets (routing) at the network layer. At the link layer, the resources to be allocated are access to the wireless medium and the power of transmission. In this paper, we consider frequency sub-bands to be the resource to be allocated, although our method applies equally well to time-slotted systems. Thus, we call the ‘frequency assignment problem’, a ‘scheduling problem’ in this paper. Medium Access Control (MAC) is the process of scheduling the shared wireless channel between competing nodes. Power allocation is the problem of choosing the power of transmission for each node in a given scheduled access to the channel. In this paper, a ‘CDMA-like’ (Code Division Multiple Access) scheme will refer to a schedule that uses very few sub-bands (several transmissions in the same * This

work was supported in part by the National Science Foundation under Career award 0347455.

sub-band). Conversely, a ‘FDMA-like’ (Frequency Division Multiple Access) scheme will refer to a schedule that uses several sub-bands (few transmissions in the same sub-band). The variety of topologies possible for ad-hoc networks makes the scheduling problem substantially harder, compared to infrastructure networks. In the simplest case (assuming a constant per-link capacity and a pairwise interference model between nodes), the MAC problem is equivalent to a graph coloring problem [14], an NP-hard problem [3]. Interference mitigation techniques and distributed MAC protocols towards solving the scheduling problem have been considered in [1,11] among others. MAC schemes have also been designed to meet different performance criterion [7,17]. However, the generic scheduling problem of multi-band power assignment remains unsolved. In this paper, we develop a general framework for the max-min scheduling problem in static wireless networks. This framework is presented formally as a non-convex optimization problem. This non-convex problem is converted into an equivalent linear programming problem and its dual developed. Applying a recent result of optimization theory [9] to the dual problem, a scheduling algorithm is devised. This algorithm yields the optimal solution to the general multi-band power assignment problem by iteratively solving the sub-problem of the dual problem. The value and flexibility of this analytical framework is demonstrated through important examples. The application of this scheduling framework to UltraWide Band (UWB) networks was introduced in [15]. UWB networks are receiving considerable research and development attention due to both the demand for higher data rates at

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short distances and the changes in Part 15 of FCC regulation. The extremely large bandwidths (W  1), and low power of UWB radios make them drastically different from existing commercial radios (finite bandwidth and large power). Under this very different physical layer communication model, the scheduling problem at the link layer requires analysis from a new perspective. Interestingly, in the limit of infinite bandwidth (W→∞), the optimal scheduling scheme is a CDMA scheme, as shown in [13]. Motivated by the CDMA solution in the limiting case (W→∞) the scheduling problem was studied in the Ultra Wide Band (W  1) regime with a per node power constraint [15]. We showed that the solution resulted in a simplification of the general non-convex scheduling problem and developed an algorithm to generate the solution iteratively. Here, we further develop the work in [15] by considering the scheduling problem under a Power Spectral Density (PSD) limit. Application of this different physical layer constraint results in a simple algorithmic variation demonstrating the utility of the framework. Next, the case of cellular networks is considered as a particular simplified topology of an ad-hoc network. Scheduling in cellular networks is a well researched topic [10]. Existing cellular systems use schemes such as ‘Time Division/Frequency division’ (e.g. GSM networks) or ‘Code division’ (e.g. IS-95 systems) for MAC. Power control in cellular systems has been considered from different perspectives such as target SNR’s and Quality Of Service metrics [6,19]. Schemes for power control have also been incorporated in standards [5]. The cellular structure renders the uplink scheduling problem simpler than the general ad-hoc wireless network scheduling problem. This simplified solution is demonstrated effectively through the dual optimization framework. The rest of the paper is organized as follows. In Section 2, the scheduling problem for a general static ad-hoc network is introduced and formalized as a non-convex optimization problem. The dual problem is developed and its structure noted to be a series of optimization sub-problems. Assuming a solution to the dual optimization sub-problem, an iterative algorithm that optimally constructs the desired schedule and power allocation may be derived. This algorithm, detailed in [15], is briefly reviewed for the sake of completeness. In Section 3, an approximation to the single link capacity function (based on the UWB model) is applied to the dual optimization sub-problem, simplifying it. A PSD limit is shown to result in a simple algorithmic variation. Thus, the framework is useful to solve the UWB scheduling problem. Subsequently, in Section 4 the topology constraints presented by a cellular network are applied to the dual optimization sub-problem. An analysis of the dual problem, for the single-cell case, demonstrates that the optimal max-min uplink schedule is a FDMA scheme (a single transmitter per sub-band). The multi-cell case is also analyzed and the result is a per-cell FDMA scheme assigning at most one transmitter per cell per sub-band. Thus, the framework is instrumental in demonstrating the optimality of

RAJESWARAN, KIM AND NEGI

this FDMA solution. Simulation results shown in Section 5, include results for medium-sized UWB networks with a PSD limit, single-cell and multi-cell scenarios. The simulation results match the expected theoretical results demonstrating the effectiveness of this general scheduling framework. 2. Scheduling in ad-hoc networks In the subsequent exposition, bold-font x refers to vectors and bold-font capital X refers to matrices. For vector x, (x)l refers to its lth element, while xT refers to its transpose. Inequality ≥ for vectors and matrices refers to element-wise inequality, while A  0 refers to a positive definite matrix. The letters p, q will denote powers of various kinds, while b, c, r will denote data rates of various kinds. In this section, we provide a short overview of the scheduling framework derived in [15]. Consider a static wireless adhoc network with several nodes. A transmission from a given node (transmitter Si ) to another node (receiver Di ) constitutes a single wireless link i. n links are assumed to be active. The ambient Gaussian noise PSD is N0 . Let gij ≥ 0 be an arbitrary gain from transmitter Si to receiver Dj . The gains gij are determined by the environment and could account for path loss and shadowing, among other effects. Channel fading, which represents short-term gain fluctuations, is not considered. A gain model [16] could be a signal power of 1/dα with distance d (α ≥ 1) resulting in gij = |Si −Dj |−α . However, we make no such special assumptions on gij in this paper. The links could be determined by a routing algorithm [2], which is not considered in this paper. The links could have the same receivers (Dj = Di ), such as in the uplink of a wireless LAN with a single receiver (access point) . Also, the receiver Dj of link j could be the transmitter of link i (Dj = Si ) resulting in gij = ∞. Intuitively, this will imply that the links i and j should not be scheduled in the same sub-band. Thus, the model chosen here is general enough to model a broad class of network designs. We assume that a central scheduler can schedule each link as desired, by allotting appropriate sub-band(s) and powers to the link. The total transmit power pi of Si may be upper bounded by its respective power bound pimax . The general scheduler is described with this total transmit power bound of pimax . However other bounds, such as PSD bounds, may also be introduced as described in Section 3. pmax = [p1max , . . . , pnmax ]T is denoted as the ‘power-bound vector’. The scheduler partitions the total bandwidth W into M sub-bands of size f W, where the vector of fractional bandwidth f = [f1 , f2 , . . ., fM ]T ≥ 0 satisfies eT f = 1. The bandwidth W can be written as W = Wx0 , where W0 can be considered as a constant (unit) bandwidth (corresponding to a wideband, as opposed to UWB system), while x is the UWB parameter of interest. This normalization of W by W0 is used to demonstrate the UWB effect (Section 3) as well as a variation of operating SNR in the cellular case (Section 4). The introduction of this parameter in the general scheduling problem is for ease of presentation and does not affect the generality of the scheduler.

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The scheduler also partitions the total power pi of . each transmitter Si among the M sub-bands as pi = . [pi1 , pi2 , . . . , piM ]. Also, pm = [p1m , p2m , . . . , pnm ]T is the ‘power vector’ denoting the power allocation to the various nodes in sub-band m. Thus, the n × M matrix P defined as T .
P = pT1 , . . . , pTn = [p1 , . . . , pM ], is the power allocation to the n nodes in the m sub-bands. The signal power received by receiver Di in sub-band m on link i is pim gii , while the interference power it receives from node j in sub-band m is pjm gji . Capacity-achieving Gaussian channel codes are assumed for each link. Thus, each link is assumed to support a data rate corresponding to the Shannon capacity [4] of that link. The rate bim achieved by link i in sub-band m (of size fm W = fm Wx0 ) depends on the power vector pm as below,   pim gii W0 m log 1 + bi = fm . (1)  x fm N0 Wx0 + j =i pjm gj i Throughout this paper, log (·) refers to the natural logarithm, and thus, all capacities are in units of ‘nats’. Note that bim is bounded as bim ≤ limx→0 bim (x) = (pim gii )/N0 . Let . B = [b1 , . . . , bM ] denote the rate matrix corresponding to the power matrix P (bm is determined by pm ). Denote pmax g b¯ imax = Ni 0 W0ii as the maximum received ‘Signal to Noise Ratio’ achievable by link i, of a wideband system, which uses bandwidth W0 . It is achieved when it transmits its maximum power pi max over its entire bandwidth W0 . Link i is assumed to have a minimum rate requirement (perhaps req specified by a higher layer protocol) of ri = βi b¯ imax W0 . Here 0 ≤ β i ≤ 1 can be thought of as a normalized rate requirement. The various normalizations are required to provide a compact description of the scheduling problem, as well as for a simple parameterization of simulation results. These do not, in any manner, result in loss of generality. . req req Denote the ‘rate-requirement vector’ rreq = [r1 , . . . , rn ]T . A max-min fairness criterion is assumed, to ensure fairness amongst the various links. The scheduling problem may now be formalized as below, Power-Schedule: subject to

max

{M,γ ,f,P} T

e f = 1,

(2)

γ f ≥ 0,

(3)

P e ≤ pmax , P ≥ 0,

(4)

Be ≥ γ r .

(5)

req

The vector e = (1, 1, . . . 1)T is a column vector of the appropriate dimension. Here γ is the index of max-min fairness [7,17], which ensures that all rates achieve a minimum fraction γ of their required rates. The constraint (3) represents the partition of the allotted W into M sub-bands, while (4) represents the total power limitation and positivity of powers respectively. This can be replaced by other constraints, such as a PSD limit in Section 3. Equation (5) requires that the capacity achieved by the schedule f meet the lower bound of

γ rreq . Thus Power Schedule is a general representation of the wireless static ad-hoc max-min scheduling problem. The optimization problem Power Schedule (2) is nonconvex [12] in its variables {M, P, f, γ }, since the constraint (5) is non-convex in P. Thus, the general scheduling problem for wireless ad-hoc networks is a hard non-convex problem. To simplify the representation, we first apply a change of variables as follows. Define m pim . p / (N0 W0 fm ) (6) = qim = imax pi / (N0 W0 ) fm pimax m bim . b / (fm W0 ) cim = i req = req ri / W0 fm ri

(7)

Here, qim may be considered the true ‘relative-SNR’ in subband m (i.e., relative to other sub-bands) of the Ultra wide band receiver. Similarly ci m can be interpreted as the true ‘relative-spectral efficiency’ in sub-band m of the Ultra wide band receiver. Note that the bandwidth W0 cancels out in the ratios, justifying the generality of this representation. In subsequent sections the spectral efficiency function bi m will be substituted with different rate functions depending on the specific scenario under consideration. This will result in a redefinition of ci m , which we use (rather than introducing a new variable) to ensure a succinct notation. Next we transform this problem to a Linear Program (LP) in the variables {f, γ } by enumerating all the feasible relative SNR vectors q ≥ 0 and collect them as columns of matrix Q. Similarly, we collect the corresponding relative spectral efficiency vectors c into the matrix C. This requires that Q, C have an infinite number of columns. These infinite matrices are a conceptual tool to translate the hardness of the non-convex optimization problem (2), to a form amenable to some analysis. With these change of variables, the problem Power Schedule can be written as the Linear Program (LP), Primal LP: subject to

max {f, γ } T

(8)

γ

e f = 1,

f ≥ 0,

(9)

Qf ≤ e,

(10)

Cf ≥ γ e.

(11)

Note that Primal LP is fully specified by the parameters {x, β i , gij , b¯ i max , ∀ i, j }. Thus, it provides a compact representation of the scheduling problem (justifying the change of variables). In particular, note that parameters such as pi max , N0 , W0 are irrelevant to this representation. Since the problem (8)–(11) is an LP, if the matrices Q and C are explicitly specified, it may be easily solved. However, in this case, the matrices Q and C have an infinite number of columns, and so, a direct solution of Primal LP is impossible. Therefore, we need an algorithm which intelligently produces the columns of Q and C that must appear in an optimum solution. Note that by Carath´eodory theorem, there exists an optimal solution with at most (2n+1) pairs of vectors qm , cm [12,15]. To obtain such an algorithm, we need to look at the dual optimization problem to Primal LP.

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The dual Dual LP, of Primal LP may be derived using Karush-Kuhn-Tucker (KKT) conditions [12],    T T

 e λq + max λc C − λqT Q l  min    λc ,λq l  

(12)

I ∗ = max Il

subject to

eT λc = 1,

where the variables λc ≥ 0 and λq ≥ 0 are the dual variables corresponding to the constraints (11) and (10) respectively. Although the dual problem (12) is convex in λc , λq , it requires evaluation of I ∗ . The dual problem thus consists of a series of sub-problems, each of which involves computing I ∗ for a particular λc , λq . If I ∗ could be obtained, then the dual problem (12), could be solved using any standard interior-point algorithm used to solve convex (LP) optimization problems. Unfortunately, in general, obtaining I ∗ will involve evaluating an infinite number of Il (one for each feasible relative-SNR vector q ≥ 0). Equivalently, we can write I ∗ = maxq≥0 (λc T c − λq T q), where q ≥ 0 is any relativeSNR vector, and c is its corresponding relative-spectral efficiency vector. Thus the hardness of wireless scheduling problem has been converted to the complexity of obtaining an I ∗ . Intuitively, the scheduling problem Power Schedule is roughly similar to a continuous version of a ‘weighted graph coloring problem’ [4]. The dual problem Dual LP involves solving a series of sub-problems. Each sub-problem (I ∗ ) is roughly similar to a ‘maximum weighted independent set problem’. The dual sub-problem (I ∗ ) form renders itself to various simplifications. In Section 3 we will look at I ∗ in the UWB regime and present a methodology to simplify the problem through a lower bound approximation to the capacity function. Then, in Section 4, we will analyze I ∗ with the specific constraints imposed by a cellular topology and demonstrate the optimal solution analytically. These results will demonstrate the utility of this general representation of the scheduling problem. Prior to the results in Sections 3 and 4 we present our generic algorithm that may be used to solve Primal LP, assuming that I ∗ can be found. An iterative algorithm that obtains the primal solution, by generating the optimal relativeSNR vectors qm , the relative-spectral efficiency vectors cm corresponding to qm using (7), and the sub-band assignments fm , can be devised by using any standard interior-point algorithm to solve a convex (LP) optimization problem. We used the logarithmic-potential method [9,12] to obtain the iterative algorithm SCHEDULER. The logarithmic-potential method is a method to introduce a barrier function for the various constraints, thus converting the problem to an unconstrained optimization problem [12]. The logarithmic-potential function t chosen here is n t  . [log((ρ)j − γ ) t (γ , ρ, π ) = log γ + n j =1 + log(1 − (π)j )]

Here ρ = Cf is the current feasible relative-spectral efficiency, π = Qf is the current feasible relative-SNR, while γ e is the barrier to ρ, and t is a weighting parameter. Let γ (ρ) be the solution to maximizing (over γ ), the logarithmic potential function. SCHEDULER computes a solution to the Primal LP, whose objective function is at least (1−)γ ∗ , for any specified  > 0, where γ ∗ is the optimal solution to the primal. The algorithm, similar to [9] is, Algorithm: SCHEDULER 1. Initialize π , ρ with an appropriate initial solution, e.g., set f = (1, 0, 0, . . .)T , Q = (π, 0, 0, . . .), C = (ρ, 0, 0, . . .).. 2. While ‘duality-gap’ >  do: (a) Compute γ (ρ). γ (ρ)·t/n (ρ)·t/n (b) Set (λc )j = (ρ) and (λq )j = γ1−(π . )j j −γ (ρ) ∗ (c) Find qˆ as the solution to I and the corresponding cˆ . Set fˆ = (. . .,0,1,0,. . .)T , where the single ‘1’ is placed in the next non-zero position (indicating a new subband). ˆ ρ ← (1 − τ )ρ + τ cˆ , π ← (d) Update f ← (1 − τ )f + τ f, ˆ Here, τ is a chosen step size. Also, (1 − τ )π + τ q. ˆ , C← update the optimal matrices, i.e., Q ← (Q, q) (C, cˆ ). 3. Apply a Carath´eodory-theorem based algorithm to reduce C, Q, f to a C∗ , Q∗ , f∗ with not more than (2n + 1) columns, while maintaining Qf = Q∗ f ∗ , Cf = C∗ f ∗ and eT f = eT f ∗ = 1. The above interior-point solver is guaranteed to converge within polynomial number of iterations of the algorithm, since the dual problem is an LP. To summarize, SCHEDULER solves the general scheduling problem (8) by iteratively solving the dual optimization sub-problem, I ∗ , and generating the optimal Q and sub-band sizes f. Next, we consider the dual sub-problem I ∗ and analyze it in particular scenarios. 3. UWB ad hoc scheduling UWB wireless networks have recently become significant from a research as well as a practical perspective [8]. In this section, we consider the scheduling problem in UWB networks. A model of the UWB physical layer is applied towards simplifying the dual sub-problem I ∗ , and hence solving the general ad-hoc scheduling problem. First, the UWB physical layer is analyzed, with particular attention to the UWB effect on the spectral efficiency b¯ im . 3.1. UWB communication model The communication model assumed is: 1. Power: Transmitting node Si is constrained (4) to a maximum transmit power of pimax . 2. Bandwidth: The underlying communication system has a very large but finite bandwidth. As noted in Section 2, this

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effect is modelled as W = Wx0 , whereW0 , is unit bandwidth, while 0 ≤ x ≤ 1 is a ‘UWB parameter’ representing the extent of the ultra wide band nature of the system, with small x denoting a UWB system. The key characteristic of such a UWB model [8,18] is the low pmax g spectral efficiency (i.e., x · b¯ imax = Ni 0 Wii 1), due to large W. FCC guidelines in the United States mandate that such a system have low spectral efficiency, to prevent an increase in the ambient noise power level, and this requirement is modelled. In this regime, it is likely that the Shannon capacity is ‘almost’ a linear function of power, with a small correction term. This motivates the following approximation b˜ im to the rate function bim , for large W = Wx0 (i.e., x 1). m . p gii b˜ im = i N0   pjm gj i  pm gii 1 pim gii (13) −x· i + N0 2 fm N0 W0 j =i fm N0 W0     L 1

L2

Here, b˜ im is the first-order Taylor series approximation to bim . Further, b˜ im ≤ bim , and so b˜ im is a lower bound approximation to bim as shown in [15]. The UWB parameter x determines the accuracy of the approximation b˜ im . As x decreases (i.e., more UWB-like), the lower bound b˜ im increases to the upper bound pim gii , and so, the approximation becomes accurate. There are N0 two quadratic (in the SNR variables pim ) 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 (unlike the case W → ∞), which is represented by L2 . The effect of both terms disappears when x → 0 (i.e., W → ∞). As mentioned in Section 2, we will redefine ci m corresponding to the approximation b˜ im of b¯ im . Thus for UWB ad-hoc scheduling redefine m qim . b˜ i = −x cim = req βi fm ri   qim 1 m ¯ max  m gj i ¯ max q b + b qj · . (14) βi 2 i i gjj j j =i

roughly similar to a weighted graph coloring problem. The sub-problem I ∗ is a quadratic optimization problem, and although usually non-convex, is simpler to solve than the general non-convex problem (2). Note that the quadratic term (in q) is weighted by x, the UWB parameter. Intuitively, the ‘hardness’ of solving the quadratic problem I ∗ reduces for smallx. Since A
0 in general, I ∗ may require a brute force search over the (infinitely many) q ≥ 0 vectors. 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 As −1 hs . ∗ > 0, evaluate Val(s) = 14 hTs A−1 s hs . (3) I = 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.
Proof. The optimal solution is characterized by As  0. This has been proved in [15].
Since there are 2n possible principal sub-matrices As , the algorithm requires 2n checks for positive definiteness and inverse computations (each of complexity O(n3 )). This is certainly tractable for n ≤ 15, a sufficient size to solve mediumsized wireless networking applications, such as home networks. Since I ∗ can be evaluated exactly, it can be used to solve the Dual LP (12), and therefore, also the Primal LP (8). The PSD limit case is considered next and shown to result in a simple algorithmic change. 3.3. UWB scheduling with power spectral density limit The UWB scheduling problem has been presented with a per node power constraint (10). However, the US FCC regulation specifies a transmit PSD mask to ensure that UWB systems do not interfere with existing licensed systems. To incorporate this practical constraint, we apply a PSD. limit to our scheduling problem, Primal LP and drop the power constraint (10). The PSD limit, dmax is imposed as, pim gnom ≤ d max fm W0 /x

Next, this definition is applied to the scheduling framework. 3.2. UWB ad-hoc scheduling The scheduling framework has been derived with a per node power constraint and we first apply the new definition of cim , (14) to the dual sub-problem I ∗ resulting in, I ∗ = max hT q − qT Aq, q ≥0

. where A(i, j ) =

(λc )i b¯ max gj i 1 x( βi gj jj 2 ∗

+

(λc )j b¯ imax gij βj gii

(15) . )and hi =

− (λq )i . As mentioned, I is roughly similar to the maximum weighted independent set problem, while the primal is (λc )i βi

(16)

where the constant gnom is the gain measured at some fixed nominal distance. It is assumed that this PSD limit imposes a more stringent power limitation than practical constraints such as battery limitations. Thus the maximum transmission max power pimax , of node i is pimax = dgnomWx0 . Hence, qim is upper bounded as qim =

a d max W0 /x pim ≤ =1 fm pimax pimax gnom

(17)

where (a) is due to the the power limit pimax imposed by the PSD constraint. In effect, the power constraint (10) in Primal LP is replaced by (17) and so, the Primal LP becomes

RAJESWARAN, KIM AND NEGI

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PSDPrimal LP under the PSD constraint (16) as below. PSDPrimal LP :

max{f, γ } γ

subject to eT f = 1 , f ≥ 0 ,

Cf ≥ γ e.

(18)

where C consists of all possible relative spectral efficiency vectors c corresponding to the feasible (0 ≤ q ≤ 1) vectors q. PSDPrimal LP differs from Primal LP in the lack of a power constraint (no constraint on Qf and an upper bound on q). Note that PSDPrimal LP is fully specified by the parameters {βi , gij , d max , ∀ i, j }. Also, the accuracy of the capacity approximation b˜ im now depends on d max . This may be seen by applying (16) to the definition in (13) and parallels the effect of the UWB parameterx in the power limit case. PSD Primal LP results in the altered dual, T

min max λc c λc 0≤q≤ 1   I∗

subject to eT λc = 1.

(19)

The change in the dual problem maybe easily accounted for in the search for I ∗ . Result 2 Similar to Result 1, I ∗ may be obtained as follows. (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) Choose a subset su , of the remaining links to operate at the PSD limit (qi = 1) and the rest to remain silent (qi = 0). (3) If . T T ∗ 0 < A−1 s hs < 1, evaluate V al(s, su ) = h q−q Aq (4) I = ∗ ∗ max V al(s, su ) is achieved for index set s , su , at the optimal ∗ ∗ q∗ . Here q∗s∗ = 12 A−1 s∗ hs∗ , qsu ∗ = 1 and q = 0 otherwise.
The proof parallels the proof of Result 1 in [15]. The search is over all possible As  0 and choice of su . In the power limited case the lower bound of 0 was the only extreme value, whereas in the PSD limit case we have 0 ≤ q ≤ 1. Thus the algorithm requires 3n (3 choices for each qi ) evaluations for the possible q vectors, as compared to the 2n evaluations in the power limited case. The rest of the algorithm is similar to Algorithm SCHEDULER, except that λq = 0 since the power constraint is no more required. Thus, the framework is easily modified to handle the PSD limit case for UWB ad-hoc scheduling. UWB ad-hoc scheduling may thus be solved for both the power limit and PSD limit cases by utilizing our scheduling framework. In the next section, we look at a different scenario for our scheduling framework - the cellular topology. 4. Cellular topology

points on an infrastructure backbone. We model the uplink transmission in these kind of networks as a set of nodes (wireless devices) transmitting to a single receiver (access point or base station). This model is referred to as ‘Cellular topology’. In this section we apply the constraints imposed by the Cellular topology to our scheduling framework and demonstrate that as expected, the solution is much simpler than in the general ad-hoc case. We begin by analyzing the case of a single cell before presenting the multi-cell network analysis. 4.1. Single-cell Consider the single-cell cellular uplink case with n nodes transmitting to the base station (forming n links). We have, gj i = gjj ∀ (i, j ).

(20)

The scheduling framework in Section 2 makes no specific assumptions on the gains gij . Thus the cellular constraint (20) may be directly applied to the dual sub-problem I ∗ , [21]. De. . n fine S = x i=1 qi b¯ imax = xqT b¯ max , a weighted sum of all qi ’s (relative-SNR) in the cell, to simplify analysis. Applying the condition (20) to the definition of ci in (7) and using (1), (6), yields   1 1+S ci = . log xβi b¯ imax 1 + S − qi b¯ imax x I ∗ can be rewritten as,



I ∗ = max max λc T c − λq T q . S ≥0 q≥0

(21)

For a fixed value of S, say S = s, ci is convex in q and so is the inner objective function of I ∗ , (λc T c − λq T q). 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. The solution (for fixed s) consists of a single transmitting node (qi > 0) while the remaining nodes are silent (qj = 0 ∀j = i). Thus, the solution to I ∗ is FDMA. i.e., only one link transmitting per sub-band. Recognizing the FDMA nature of the solution simplifies the cellular dual sub-problem (21) to finding the sole transmitting link (amongst the n links) which achieves the maximum value of (21) (FDMA solution). The objective function with )i log(1+qi b¯ imax x)−(λq )i qi , is only one transmitting link, xβ(λi b¯c max i a differentiable concave function of qi . The maximum of this (λc )i −(λq )i . function is achieved at (qi )max = max(0, (λβiq )i b¯ max x ). Hence I ∗ i is  

(λc )i max ¯ max max , b log 1 + (q ) x − (λ ) (q ) maxi i q i i i xβi b¯ max i

Cellular networks are a well established form of wireless communication for both the legacy application of voice as well as emerging data services. The centralized control with base stations on an infrastructure backbone, makes network design simpler than in ad-hoc networks. Wireless Local Area Networks such as 802.11 based networks also rely on access

the maximum objective function among the n links. This search presents a complexity of O(n). The specific Cellular network topology reduced the complexity of the hard scheduling problem to polynomial order. The scheduling framework was instrumental in demonstrating this effect. Note that this formulation utilizes the exact

SCHEDULING FRAMEWORK FOR UWB

& CELLULAR NETWORKS

15

Shannon capacity and the solution is applicable to all SNR’s of operation (unlike the UWB scheduling case, where an approximation is utilized). Next, we generalize this solution to a multi-cell architecture. 4.2. Multi-cell An uplink multi-cell scenario consists of a number of cells, each with its set of nodes transmitting to its base station. Towards deriving a methodology to obtain I ∗ for the multi-cell case, we first analyze the 2-cell case for simplicity of exposition. Assume two cells with n and n − n nodes respectively. Similar to the single-cell case, we define a weighted .  relative SNR sum for each cell as S1 = x ni=1 qi b¯ imax +  n 

. ¯ max gi1 , S2 = x ni=1 qi b¯ imax gginii + ni=n +1 qi b¯ imax i=n +1 qi bi gii respectively. Note that S1 and S2 consist of intra-cell and intercell interference terms. Here link 1 and n operate in the cell 1 and 2 respectively. Applying S1 and S2 , ci can be rewritten as below,   

−1 1  , ∀ 1 ≤ i ≤ n log 1+S11+S  xβi b¯ imax −qi b¯ imax x ci =  

 1+S2  xβi b¯ max −1 log , ∀ n < i ≤ n. i 1+S2 −qi b¯ max x i

The 2-cell dual sub-problem is n  ((λ ) c −(λ ) q ) ∗ c i i q i i i=1  I = max max S1 ,S2 ≥0 q≥0 + ni=n +1 ((λc )i ci −(λq )i qi )

(22)

As in the single-cell case, for fixed S1 , S2 , the inner objective function is convex in q, and so, one of the vertices of the feasible region is the optimal point. Note that the vertices are defined by S1 = s1 , S2 = s2 and at least n−2 active non-negativity constraints (qi = 0). Thus the optimal point has one or two positive qi ’s. Also, at most one link per cell has a positive qi , since the single-cell case solution is FDMA. Therefore, the solution is FDMA for each cell and (n + 1) × (n − n + 1) such combinations need be considered to find the optimal value of the objective function. This ‘per cell FDMA’ nature of the solution is applied to simplify the 2-cell dual sub-problem, (22). Unfortunately, the objective function (for a particular choice of links (i, j), is not a concave function of q due to the inter-cell interference. However, the ranges of the relative-SNR’s of the two links (i, j), are     (λc )i (λc )j − (λq )j − (λq )i βj βi  and 0, 0, (λq )i b¯ max x (λq )j b¯ max x i

j

. This is because interference can only reduce the capacity of each link. Using a brute force search over this range, the maximum value may be obtained for a particular choice of links (i, j). Finally, comparing the maximum values of all the possible (n + 1) × (n − n + 1) pairs, we obtain the solution of the 2-cell dual sub-problem I ∗ . The 2-cell case analysis may be extended to solve the scheduling problem of a general cellular network, consisting

of k cells. The optimal solution is ‘per cell FDMA’ again, where at most one link is active in each cell. A comparison of all the k-cell FDMA combinations, results in an O(( nk )k ) complexity of the k-cell dual sub-problem I ∗ . The scheduling framework was effectively applied to solving the single-cell scheduling problem analytically. It also resulted in a reduced complexity (as compared to the general non-convex scheduling problem) search algorithm to solve the multi-cell scheduling problem. Next, simulation results are presented to confirm the results. 5. Simulations The algorithm SCHEDULER was implemented for both the UWB scheduling problem as well as the cellular scheduling problem. The solution to the dual sub-problem I ∗ varies depending on the scenario, while the overall iterative algorithm SCHEDULER maintains the same structure. First, simulation results for the UWB scheduling problem are presented. 5.1. UWB network simulations The performance of SCHEDULER for the power limited UWB scheduling problem was presented in [15]. The results demonstrated that SCHEDULER resulted in solutions that matched the theoretically expected results. CDMA-like solutions at small x and FDMA-like solutions at large x were obtained, demonstrating the UWB physical layer effect. Here, we present results for the PSD limit UWB scheduling problem as described in Section 3.3. As in [15] we simulate SCHEDULER for certain typical as well as random networks. The assumption is that the UWB network scenario involves up to 12 links, such as in a home network. A transmitter and receiver were placed for each of the n links, on a unit area. Assuming a distance-loss model with α = 3, the gains gij = |Si − Dj |−α were calculated. The constant gnom was set as the gain corresponding to the minimum possible length of a link so as to normalize the value of γ . The PSD limit d max was set by choosing the value of d max /N0 , i.e the PSD limit relative to the noise PSD. The range for this normalized PSD limit d max /N0 was [0.01,0.3]. In these simulations, βi = min (gii /gnom ) ∀ i. Note that these parameter choices completely specify the PSDPrimal LP, (18). An approximation error of  = 0.1 was allowed in the iterative algorithm, thus guaranteeing γ within 90 % of the optimal. The step size τ was chosen adaptively, and the simulations converged within 1000 iterations in all cases. Figures 1, 2 and 3 show the typical network topologies investigated. These are respectively, case 1: Common transmitter (all links share the same transmitter), case 2: Common receiver (all links share the same receiver), case 3: Two clusters. The squares denote the transmitters and the circles denote the receivers in all figures. The typical optimal schedules given by SCHEDULER for these networks is shown in Table 1 for d max /N0 = 0.1 (Wide-Band) and in Table 2 for d max /N0 = 0.01 (Ultra Wide Band). At d max /N0 = 0.01, it is

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Table 2 Schedules for d max /N0 = 0.01.

1 0.9

Topology

0.8

3

2

0.7 Transmitter

0.6 0.5

4

0.3

5 6

0.2 0.1 0.2

0.4

0.6

0.8

1

Figure 1. CDMA topology. 1 0.9 0.8

3

2

0.7 0.6 0.5

4 1

0.4 0.3

5 6

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Figure 2. FDMA topology.

clear that the scheduler chooses CDMA-like solutions (several parallel transmissions in close proximity), for all the networks, demonstrating the effect of the UWB physical layer. At d max /N0 = 0.1 (higher PSD), the scheduler begins to schedule some links in separate bands, for example choosing to schedule (1, 2, 3, 4, 5) but not 6 in both Cases 1 and 2. However, the solutions are still CDMA-like in nature and most of the nodes transmit at the PSD limit (i.e., qim = 1). In

Common transmitter Common receiver Two cluster

log approx.

Common transmitter

(1,2,3,4,5,6)

0.98

1

Common receiver Two cluster

(1,2,3,4,5,6)

0.99

1

(1,2,3,4,5,6)

0.99

1

Case 3, the solution is a pure CDMA-like solution even for d max /N0 = 0.1. This strong bias toward CDMA-like solutions even at larger PSD limits is in contrast to the power limited case [15] (where higher received signal to noise ratios caused by x = 0.1 resulted in more FDMA-like solutions). Intuitively, the power constrained case imposed an average power  ( m fm Pim ) constraint across the entire bandwidth, but no constraint on the power in a particular sub-band (of any size). This allowed the choice of arbitrarily small sub-bands with large power to increase the achieved data rate of a node, while satisfying the average power constraint. The PSD limit however, limits the maximum power in a sub-band (depending on the size of the band) and this results in CDMA-like solutions. With the PSD limit, it is favorable for nodes to transmit simultaneously (in the presence of interference), rather than in their own sub-band of a smaller size, i.e., in PSD limit case, bandwidth is more precious than power. In all cases, γ ∗ is indeed achieved within an approximation error of 10%. The last column in both tables shows the minimum ratio of capacities computed using b˜ im and bim . From there, it can be seen that the accuracy of the quadratic approximation to the true ‘log-capacity’ function is good even for d max /N0 as high as 0.1. These results parallel the performance in the power limited case [15]. Finally, a randomly chosen network was investigated (figure 4), with the same parameters as earlier and n = 12. At a low PSD limit (d max /N0 = 0.01), SCHEDULER 0.7 0.6 0.5 0.4

2

5

0.3 1

0.2

4

0.1

Table 1 Schedules for d max /N0 = 0.1. Topology

γ approx.

1

0.4

0 0

Typical schedules

0

Typical schedules

γ approx.

log approx.

(1,2,3,4,5) (1,2,4,5,6)

0.90

0.82

3

6

–0.1 –0.2

(1,2,3,4,5) (1,3,4,5,6)

0.90

0.82

(1,2,3,4,5,6)

0.98

0.93

0

0.2

0.4

0.6

0.8

Figure 3. The two cluster topology.

1

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17

resulted in a pure CDMA-like schedule of all links operating simultaneously in a single sub-band at their peak PSD. At a larger PSD limit of d max /N0 = 0.1 the schedules were again CDMA-like with most links transmitting simultaneously. The number of selected sub-bands increased to 3. This is in contrast to the power limited case, where at higher x, the solutions were FDMA-like [15]. Only link 9 (figure 4) is not scheduled in the third sub-band and this is because it is a short link (compared to the others) and hence is able to achieve its requirement in the first 2 sub-bands. The silence of link 9 in the third sub-band also aids other proximate constrained links (such as 10), by reducing interference. To demonstrate the effect of the PSD limit d max /N0 on the optimal schedule, we choose the entropy H(f) [4] of the optimal sub-band sizes f (for the random network) as a metric. This is possible since eT f = 1. Intuitively, a CDMA-like solution will have a low entropy (few large sized sub-bands), while an FDMA solution will have a high entropy. Figure 5 shows that H(f) increases with d max /N0 thus demonstrating the effect of the PSD limit on scheduling. However, note that the change in entropy is not as large as the change in the power limited case [15], contrasting the effect of a PSD limit and a power limit. Thus, SCHEDULER was successfully used to solve medium sized UWB ad-hoc network scheduling problems for both the PSD limit and the power limit case. It also demonstrated that with a PSD limit, CDMA-like solutions are optimal. Additional comment: The solution presented by SCHEDULER was reduced by applying a Carath´eodory based algorithm, to ensure a solution with at most 2n + 1 sub-bands. However, the reduction can be based on a preference for particular MAC characteristics. Such a preference may be guided by efficient or practical implementations. For example, if a CDMA-like solution is desired, a higher fraction of the bandwidth should be assigned to the qm ’s with several active links. To choose such a solution, a linear program Transform LP is 1.5

2

1

4

1.5

1 Entropy of f

SCHEDULING FRAMEWORK FOR UWB

0.5

0 0

0.02

0.04

Relative p.s.d d

0.06

0.08

0.1

0.12

max

/No – UWB parameter

Figure 5. Entropy of the optimal schedule f: Small d max /N0 indicates a UWB system. Small entropy indicates a CDMA-like solution.

developed. wT f

(23)

subject to eT f = 1 , Qf ≤ e , Cf ≥ Cf.

(24)

Transform LP :

max

f ≥0

where the f is the new fractional bandwidth vector and . H (q¯ m ) of normalized relative w m = H (q¯ m ). The entropy  m m . m SNR vector, q¯ = q / i qj , is chosen as a metric which reflects the degree of power distribution in qm . This choice of metric is expected to bias the result towards a CDMAlike solution. The constraint (24) ensures that the solution to Transform LP is at least as good as the original solution. On the other hand, using a metric such as −H (q¯ m ) will yield a more FDMA-like solution, by choosing qm ’s with small H (q¯ m ). Thus, the choice of metric may be used to determine the bias of the solution. Transform LP was applied to the result of the power limited UWB scheduling with x = 0.1. The original solution consisted of a large number of sub-bands allowing some flexibility in the MAC choice. With the choice of metric as in (23) (CDMA-like bias), the resulting solution contained 13 sub-bands and the entropy of f was 2.15. With the FDMA-like metric, the resulting solution contained 22 sub-bands and the entropy of f was 2.76. This demonstrates the utility of Transform LP. Next we present results for the cellular scenario.

5 0.5

3

5.2. Cellular topology simulations

10

1

9 0 0

0.2

0.4

0.6

0.8

Figure 4. The random topology.

1

1.2

First, to simulate a single-cell scenario, 20 transmitters were distributed uniformly and randomly on an annulus (radius range [0.25, 1]), with center as the base station (figure 6). In this single-cell uplink scenario, a large number of transmitters could be simulated by the virtue of low complexity of the single cell dual sub-problem I ∗ , (21). We set

RAJESWARAN, KIM AND NEGI

18 1.5

1.5

1

1

0.5

0.5

0

0 –0.5

–0.5 –1

–1 –1.5 –1.5

–1

–0.5

0

0.5

1

1.5

–1.5

–1

Figure 6. The cellular topology: single-cell.

7

7

6

5 4 3 2

3

5 4 3 2 1

1 0 –3

2

is no longer a good choice since interference from other links is dominant. With x 1, the gain of SCHEDULER over CDMA is at least a factor of 3. We study the multi-cell scheduling problem, using a simple 2-cell simulation scenario. Two cells were considered in a typical hexagonal architecture (center to center distance 2 cos( π3 )), as in figure 8. 20 nodes are distributed, with 10 nodes in each cell. The search algorithm developed in Section 4.2 was implemented. As shown in figure 9, the performance of SCHEDULER has a significant gain over the CDMA solution in the large SNR region. This parallels the result for the single-cell case. The scheduling solution largely consisted of two nodes (one per cell) transmitting simultaneously. Thus, SCHEDULER utilizes a frequency reuse factor close to 1. Recall that a frequency reuse factor of 7 is typical for FDMA cellular systems with omnidirectional base station antennas. For a quantitative comparison with FDMA, the γ of a traditional fixed bandwidth FDMA

8

6

1

Figure 8. The cellular topology: 2-cell.

Scheduling gain over CDMA

Scheduling gain over CDMA

βi = 1, b¯ imax = 1 ∀ i, and the distance path-loss exponent α = 3. To study the effect of bandwidth on the scheduling solution, simulation was performed over a wide range of ‘UWB parameter’ x. Note that x is proportional to the maximum possible signal to ambient noise ratio of a link. Thus, equivalently, the simulation is over a range of operating SNR’s. In Section 4, it was shown that FDMA is the optimal scheduling scheme for cellular networks. To provide a basis of comparison, we choose an optimal CDMA scheme (a common scheme in current cellular networks), which optimally assigns transmission power to all links in the entire frequency band. This assignment may be obtained by a linear search over γ while satisfying the inequalities (10), (11). Figure 7 shows the ratio of the γ of SCHEDULER to that of the CDMA system. At large bandwidths, i.e., low x or equivalently low operating SNR, CDMA performs as well as FDMA (the optimal scheme). This is because background noise is dominant over interference. However, with small bandwidth, or equivalently in the high SNR regime, CDMA

0

–2

–1 0 UWB parameter ( log10 ( x ) )

1

Figure 7. Cellular scheduling gain: single-cell.

2

0 –3

–2

–1 0 UWB parameter ( log10( x ) )

Figure 9. Cellular scheduling gain: 2-cell.

1

2

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19

scheme was calculated. It is trivial to obtain γ corresponding to this traditional FDMA scheme where a separate equally sized sub-band is assigned for each node. The gain of SCHEDULER in the 2-cell case over this FDMA scheme was 60% at high SNR, equivalently low bandwidth (x = 100). The gain in the 2-cell case indicates a large expected capacity gain over traditional FDMA schemes for cellular networks. Thus SCHEDULER may be successfully used to solve scheduling problems in various scenarios.

[5]

6. Conclusion

[10]

The general max-min fair scheduling problem in an ad-hoc wireless network, which consists of selecting power vectors and sub-band sizes, is non-convex. The dual of this scheduling problem consists of a series of optimization sub-problems. The paper used a recent result in optimization theory [9] to implement an iterative procedure, guaranteed to converge to the optimal schedule within any desired accuracy. This algorithm requires a computable solution to the dual-sub problem I ∗. Further, I ∗ was studied in particular cases of the general ad-hoc network. In the case of a UWB physical layer with large bandwidth, Gaussian noise, and Gaussian code books, the capacity of UWB links can be accurately approximated by the first order Taylor expansion in 1/W. The dual of the scheduling problem then involves solving a quadratic optimization problem (quadratic in power vectors), which allows an exact solution for medium sized networks (less than 15 nodes). Sample results presented, showed that the scheduler efficiently and accurately produces schedules, which maximize the max-min fair allocation, by intelligently scheduling conflicting links in different sub-bands. Results for the case with a PSD limit were CDMA-like even at higher operating SNR’s, as compared to the FDMA solution for power limited networks. Analyzing I ∗ in the case of a cellular network (a particular topology in the general ad-hoc network model) resulted in FDMA as the optimal solution for the single-cell case. Percell FDMA was shown to be the optimal schedule for the multi-cell cases. Simulation results for the single-cell and 2cell case demonstrated the gain of this optimal solution over a traditional CDMA solution. The gain was noted to increase with the operating SNR. References [1]

[2] [3] [4]

N. Bambos, S. Chen and G. Pottie, Radio link admission algorithms for wireless networks with power control and active link quality protection, in: Proc. IEEE Infocom 1996 (March 1996). D. Bertsekas and R. Gallager, Data Networks (Prentice Hall, 1992). J.A. Bondy and U. Murthy, Graph Theory with Appl. (Elsevier, 1976). T. Cover and J. Thomas, Elements of Inform. Theory (John Wiley, 1991).

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[8] [9]

[11]

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[15]

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3GPP2 C.S0002-D, Physical layer standard for cdma2000 spread spectrum systems, rel. D, (March 2004). D. Goodman and N. Mandayam, Power control for wireless data, in: IEEE Personal Communications (April 2000) pp. 48–54. X.L. Huang and B. Bensaou, On max-min fairness and scheduling in wireless ad-hoc networks: analytical framework and implementation, in: Proc. 2nd MobiHOC (2001) pp. 221–231. IEEE 802.15 WPAN High Rate Alternative PHY Task Group 3a, accessible at http://grouper.ieee.org/groups/802/15/pub/TG3a.html K. Jansen and L. Porkolab, On preemptive resource constrained scheduling: Polynomial-time approximation schemes, in: Proc.Integer Prog. Comb. Optim., 9th IPCO Conference (May 2002) pp. 329– 349. I. Katzela and M. Naghshineh, Channel assignment schemes for cellular mobile telecommunication systems: A comprehensive survey, in: IEEE Personal Communications (June 1996) pp. 10–31. H. Luo, S. Lu and V. Bharghavan, A new model for packet scheduling in multihop wireless networks, in: ACM MobiCom 2000 (Aug. 2000). S. Nash and A. Sofer, Linear and Nonlinear Prog. (McGraw, 1996). R. Negi and A. Rajeswaran, Capacity of power constrained ad-hoc networks, in: INFOCOM ’04 (March 2004). R. Negi and A. Rajeswaran, Physical layer effect on MAC performance in wireless ad hoc networks, in: Proc. IASTED Conf.CIIT ’03 (July 2003). R. Negi and A. Rajeswaran, Scheduling and power adaption for networks in the ultra wide band regime, in: GLOBECOM ’04 (Dec. 2004). J.G. Proakis, Digital Communications 3rd Edition, (McGraw-Hill, 1995). B. Radunovic and J.L. Boudec, Optimal power control,scheduling and routing in UWB networks, in: Proc. IEEE JSAC (Dec. 2004). S. Verdu, Spectral efficiency in the wideband regime, IEEE Trans. Info. Theory 48(6) (2002) 1319–1343. J. Zander, Performance of optimum transmitter power control in cellular radio systems, IEEE Trans. Veh. Technol. VT-41 (1992) 57– 62.

Arjunan Rajeswaran received his Masters 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 reserach interests lie in the area of wireless networks. His focus is in the application of information and communication theoretic tools towards wireless network design. Several IEEE publications reflect his curent research on Medium Access Control design and performance. Arjunan received the best student paper award at IEEE/ACM Broadnets 2004. E-mail: [email protected]

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 working toward the Ph.D degree in the Department of Electrical and Computer Engineering at Carnegie Mellon University. His main research interests are in wireless networks and communication theory. E-mail: [email protected]

20 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 Assistant Professor. His research interests include signal processing, coding for communications systems, information theory, networking, cross-layer optimization and sensor networks. E-mail: [email protected]

RAJESWARAN, KIM AND NEGI