JOINT POWER ADAPTATION, SCHEDULING AND ROUTING FRAMEWORK FOR WIRELESS AD-HOC NETWORKS Gyouhwan Kim, Arjunan Rajeswaran and Rohit Negi Department of Electrical and Computer Engineering, Carnegie Mellon University Emails: [email protected], [email protected] and [email protected] ABSTRACT In wireless ad-hoc networks, there exists strong interdependency between protocol layers, due to the shared wireless medium. Hence we cast the power adaptation (physical layer), scheduling (link layer) and routing (network layer) problems into a joint optimization framework. We analyze this hard non-convex optimization problem, and obtain a dual form consisting of a series of sub-problems. The sub-problem demonstrates the functionalities of the protocol layers and their interaction. We show that the routing problem may be solved by a shortest path algorithm [1]. In the case of Ultra Wide Band (UWB) networks, the power adaptation & scheduling problem is simplified and may be solved [2]. Thus, an algorithmic solution to the joint problem, in the UWB case, is developed. Comparison of results with the previous information theoretic capacity results on UWB networks [3], demonstrates the importance of this cross-layer optimization framework. 1. INTRODUCTION Recently, wireless ad-hoc networks have been the subject of significant research interest, as a network model for various potential applications including sensor networks, personal area networks (PAN’s) and military battle field communications. Thus, the design and performance of ad-hoc wireless networks, merits detailed investigation. Recently, important information theoretic bounds on the ad-hoc network capacity [4, 3] have been shown. Substantial research, in the field of protocol design for ad-hoc networks at both the medium access and routing layers, has also been achieved [5]. However, protocol design typically aims at performance improvement over legacy protocols, while methods to approach network capacity bounds are rarely addressed. Further, traditionally, power adaptation, scheduling and routing have been treated as separate problems of different protocol layers. However, in wireless ad-hoc networks, there exists strong interdependency between these three functions, due to the shared wireless medium [3]. Thus, towards efficient ad-hoc network design, it is critical to develop algorithms

that provide explicitly optimal wireless resource scheduling and routing solutions, in a joint framework. The power adaptation & scheduling problem, assuming a routing solution, is known be a hard non-convex optimization problem [2]. The hardness of this problem arises from the non-convex Shannon link capacity [6], i.e, the nonconvex relationship between the capacity of a link and the power of interfering nodes. Limited solutions, to the power adaptation & scheduling problem, have been achieved with simplified assumptions of the physical layer model. In [7, 8], link capacity was assumed to be a linear function of signal to noise ratio (SNR) and bandwidth while in [9], a discrete rate selection scheme was considered as a more practical model of the physical layer. Though these simplified models aided analysis of the power adaptation & scheduling problem, the problem remains computationally hard. At the network layer, multi-hop routing, allows for more efficient communication than direct path routing [10]. Traditionally, wireless ad hoc networking research has dealt mainly with efficient distributed protocol design [5]. The formal routing problem, of choosing paths and the associated flows, is of high dimensionality and so computationally hard. Recently, there has been interest in describing and solving this formal routing problem [9]. Thus, the joint power adaptation, scheduling and routing problem is a cross-layer combination of the above two problems, each of which are hard. In this paper, we consider this joint problem, as a tool, for evaluation of metrics such as network capacity and for providing design guidelines in the development of optimal routing and scheduling algorithms. Here, this non-convex joint routing and scheduling problem is converted to an equivalent linear program (LP), and its dual is developed. The dual form involves a series of sub-problems, which clearly consist of the routing problem and the power adaptation & scheduling problem. Utilizing this form, it is shown that the routing solution may be developed by incremental path addition, and path selection using a standard shortest path algorithm. In [2], we presented a general ad-hoc network scheduling framework, assuming a routing solution, and solved it for UWB networks using a quadratic approximation to the link capacity. The

methods from [2] are extended to this joint framework. The joint problem is thus solved, by appropriately combining solutions to the interdependent routing problem and power adaptation & scheduling problem. This framework is then applied to the case of UWB networks. Extensive simulations are performed to demonstrate the effect of bandwidth and node density, on optimal solution characteristics and network performance of UWB networks. In particular, it is shown that the UWB capacity results of [3] can be validated by our framework. The rest of the paper is organized as follows. In the next section, we introduce a general wireless ad-hoc network model. In Section 3, the joint problem is formalized as a non-convex optimization problem and dual sub-problems are investigated. Subsequently, an iterative algorithm to solve the joint problem is developed in Section 4. Simulation results and observations are discussed in Section 5.

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 5) in terms of 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 power adaptation & scheduling problem, thus consists of determining the transmit powers pm i of each 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. Thus, each link is assumed to provide a data rate equal to its Shannon capacity [6]. Thus, based on the scheduling and power adaptation choice, the rate bm i achieved by link i in sub-band m (of size fm W = fm Wx0 ) is   pm W0 i gii m log 1 + , (1) bi = fm  x fm N0 Wx0 + j=i pm j gji

2. WIRELESS AD-HOC NETWORK MODEL

where log(.) refers to the natural logarithm, and thus, 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 path gain model, i.e., 1/dα where d is the distance between transmitter and receiver 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 routing layers, the joint problem is formalized.

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 and matrices refers to element-wise inequality. 2.1. 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. 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 matrix L denotes the ‘mapping matrix between paths and links’ 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 is the ‘mapping matrix 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). Thus 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 rate of communication between every source-destination pair.

3. JOINT POWER ADAPTATION, SCHEDULING & ROUTING IN AD-HOC NETWORKS In wireless ad-hoc networks power is a scarce resource. Minimization of power and fairness are combined, in the choice of (min-max power), as the objective of our joint problem. A solution to the min-max power problem (with uniform rate requirement) may be translated to a solution of the max-min capacity problem (with equal node power limitation). 3.1. 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 :

2.2. Physical layer model The total system bandwidth W is split into M sub-bands  as fW ,f ≥ 0. Here, M 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

subject to

min

M,P0 ,f ,y,P

P0

(2)

eT f = 1 ,

f ≥ 0 , (3)

Fy = r , Be ≥ Ly ,

y ≥ 0 , (4) (5)

DPe ≤ P0 e .

(6)

The vector e = (1, 1, · · · 1)T is a column vector of the appropriate dimension. The constraint (3) represents the partition of the entire bandwidth W into M sub-bands. Equation (4) represents the the routing constraint, i.e., the rate requirement of a source being met by the path flows. The inequality (5) ensures that link capacities (Be) support the total flow rate due to all paths (Ly) utilizing them. This constraint is a combination of link layer (link capacities) and routing (path flows) functionalities. In the last inequality (6), (D)ki = 1, if node k is the source node of link i, else (D)ki = 0. Hence D imposes the power constraint per node, by taking the sum of transmission powers of all links emanating from each node. A solution to Joint Nonlinear will yield the required optimal routing, scheduling and power adaptation strategies. However, the non-convex [11] relation between the rate matrix B and the power matrix P, as in (1), renders the problem intractable [11]. Towards simplifying analysis, we introduce the normalized variables, qim

. y¯p =

pm bm m . i i N0 W0 fm , ci = fm W0 , . rs . xp 0 ¯s = W , P¯0 = NP0 W W0 , r 0 0

. =

(7) .

(8)

Here, qim is the (transmit) SNR of link i in sub-band m (at x = 1), and cm i is the corresponding spectral efficiency. y¯p , r¯s and P¯0 are normalized versions of yp , rs and P0 , respectively. Next, we transform Joint Nonlinear to a Linear ¯ } by enumerating all Program (LP) in the variables {P0 , f , y the feasible infinite number of SNR vectors, q ≥ 0, and collecting them as columns of matrix Q. The corresponding spectral efficiency vectors c are collected into the matrix C. The translation of the non-convex problem Joint Nonlinear into an infinite dimensional LP, is a tool to aid analysis. The transformed problem Primal LP is Primal LP : subject to

min P¯0

P¯0 ,f ,¯ y

eT f = 1 , f ≥0, ¯≥0, F¯ y = ¯r , y Cf ≥ L¯ y, DQf ≤ P¯0 e .

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

L and F have a column for each path. The number of paths is O(n!) resulting in large dimensional L and F, while Q and C are infinite dimensional. So, a direct solution of Primal LP is impossible. Therefore, we require an algorithm which intelligently produces the columns of L, F, Q and C that constitute an optimal solution. Note that, by Carath´eodory theorem, there exists an optimal solution with at most (L + S + 1) paths and (2L + 1) SNR vectors [2, 6]. To obtain such an algorithm, we analyze the dual optimization problem of Primal LP.

3.2. Dual problem analysis The Lagrange dual function [11] of Primal LP is, y − (λTc C − λTq DQ)f ) , min ( (λTc L)¯ 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 in equation (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,  S   ∗ ∗ Dual LP : max r¯w Rw − I , (15) λq ,λc ≥0, eT λq =1

where

R∗w

w=1

= minp∈Pw (λTc L)p

I ∗ = maxm (λTc C − λTq DQ)m This dual form demonstrates that the joint power adaptation, scheduling and routing problem may be considered as a series of sub-problems (i.e, the inner cost function in the dual problem (15) for a particular choice of λc , λq ). Further, each of these sub-problems involves solving minimum cost routing problems (R∗w ) and a weighted maximum independent set scheduling problem (I ∗ ). The power adaptation problem is implicit in the independent set problem, since a specific power vector must be chosen by I ∗ in each subproblem. Therefore, the above structure of the dual problem presents a mathematical view of the concept of ‘layering’ in networking [8, 9, 12]. The strong dependency between the layers is demonstrated by the dual variables λc , λq . Next, we construct an algorithm to solve Primal LP, based on the above analysis. 4. ALGORITHM Primal LP may be solved by applying an interior point method [13]. Towards such a solution we choose a logarithmic barrier function [11], N t  . log(P¯0 − (π)k ) Φt (P¯0 , α, ρ, π) = P¯0 − 2N k=1



t 2L

L 

log((ρ)i − (α)i ) . (16)

i=1

π = DQf , ρ = Cf and α = L¯ y are feasible power vector, the corresponding link capacity vector and the link capacity requirement (imposed by the routing solution), respectively. The solution to R∗w may be obtained by a shortest

path algorithm (such as Dijkstra’s algorithm). Hence assuming a solution to I ∗ , the dual sub-problem is solvable. Utilizing this solution, similar to [13], we devise an algorithm, JOINT SOLVER, to solve Primal LP iteratively. JOINT SOLVER,outlined below, provides a specified accuracy  ∈ (0, 1), such that the min-max power is at most P¯0∗ /, where P¯0∗ is the optimum value and is outlined below. 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. 2. While D(λq , λc )/P¯0 <  do: (a) Compute P¯0 satisfying (b) Set (λq )k =

t/2N P¯0 −(π)k

∂Φt ∂ P¯0

= 0.

and (λc )i =

t/2L (ρ)i −(α)i .

(c) Perform routing for all w sequentially. i. Find a shortest path pˆw by solving R∗w . ii. y ← ((1 − δw )y, δw r¯w ). Here δw is a chosen step size. iii. Append the corresponding columns to F, L. iv. Update λc and α. (d) Perform power adaptation & scheduling . ˆ as the solution to I ∗ and the correi. Find q sponding ˆ c. ii. f ← ((1 − τ )f , τ ). Here τ is a chosen step size. ˆ ), C ← (C, ˆ iii. Q ← (Q, q c). iv. Update π and ρ. (e) Evaluate dual cost D(λq , λc ). 3. Apply a Carath´eodory-theorem based algorithm to reduce the size of solution. This algorithm requires solving I ∗ , which is in general a hard problem due to its infinite dimensionality (or equivalently the non-convex relation between power and capacity as in (1). However, in the important case of UWB networks, I ∗ may be simplified and thus the UWB case will be used as an illustrative example in solving the joint optimization problem. In [2], we exploited the UWB characteristics, specifically the low-spectral efficiency [14], to develop a novel quadratic approximation to UWB’s Shannon capacity. The approximation (actually, a lower bound) was obtained as the Taylor expansion of Shannon capacity (1). Applying this approximation, the power adaptation & scheduling problem becomes, I ∗ = max hT q − qT Aq , q ≥0

(17)

(a) min energy path

(b) x = 0

(c) x = 0.01

(d) x = 25

Fig. 1. Routing patterns: 10 node, 5 S-D pairs where the matrix A and vector h depend on the network gains and the dual variables λc , λq . Solving (17) exactly, is tractable for medium sized networks by applying intelligent search techniques. The excellent accuracy was shown in [2]. For the analysis of narrow band systems operating in high SNR regime where the quadratic approximation cannot be applied, we use a computationally efficient heuristic to solve I ∗ . This heuristic sequentially selects links that maximize the incremental contribution to the cost function at each stage. The accuracy of this heuristic was proved in some sample cases, through the comparison with the result from the quadratic approximation. The low complexity of this heuristic allows us to analyze networks with large number of nodes (∼ 50). In the infinite bandwidth case (x = 0), each link achieves the maximum possible capacity (minimum power requirement), and hence this provides a lower bound on the network performance. Interference is no more a consideration, with infinite bandwidth, resulting in a linear link capacity power relationship ci = gii qi and a simple simultaneous schedule of all links. Applying these simplifications, the Joint NonLinear reduces to an LP, in the infinite bandwidth case, and may be solved easily. 5. SIMULATION RESULTS In this section, we present simulation results that demonstrate two important aspects of the solution to the joint problem. First, the effect of bandwidth on the optimal routing and scheduling solutions are discussed and second the the effect of node density on network performance as a function of bandwidth is demonstrated. Nodes are uniformly distributed within unit area and every node is either a source or destination, i.e., the number of S-D pairs S = n/2. The source rate r¯w = 0.05 ∀w. The path loss exponent α = 3. To achieve a target min-max power of at most 3dB larger than optimum, the approximation parameter  = 0.5. In Figure 1, we present routing solutions for the 10node network. For clarity of representation, main flows (with > 10% of total flow) for only a single S-D pair, are displayed. The thickness of a link represents the relative amount of data flow. In Figure 1-(a) the minimum energy path (MEP) routing solution is demonstrated. The MEP solution is the optimal routing solution that minimizes total power of the network (a relaxation from the per node

α−1

max power is Ω((n log n)− 2 ). As bandwidth increases (x → 0), the optimization result indeed approaches the information theoretic lower bound (solid line in Figure 2). Also, network performance at large x, indicates the opposite trend of decreasing capacity (increasing power) as expected from [4]. Therefore, the optimization results support the theoretical result from [3], i.e., UWB networks have dramatically different characteristics.

−1

10

min Po (min−max power)

Increasing bandwidth (x = 0, 0.1, 1, 3)

−2

10

Information theoretic lower bound

−3

10

6. CONCLUSION

Equal rate for all S−D pairs −4

10

0

10

1

10

2

10

N (node density)

Fig. 2. Effect of node density & bandwidth power constraint case) under an infinite bandwidth assumption [3, 7]. As shown in Figure 1-(b), even with infinite bandwidth, the optimal routing solution to the Primal LP consists of multiple paths. Though the MEP minimizes the total energy consumption of network, it is not the optimal solution to the min-max power minimizing problem, which provides fairness by satisfying per-node power constraints. In the UWB case (Figure 1-(c)), the routing solution consists of multi-hop paths including the MEP. As the bandwidth decreases to the narrow band case (Figure 1-(d)), flow shifts to the direct path. The optimal scheduling solution in the UWB case is ‘CDMA-like’, i.e, all simultaneous transmissions [3, 2], due to the noise-limited nature of the system. Also as the bandwidth decreases the solution becomes more ‘FDMA-like’, i.e., few simultaneous transmissions in each band, since the system becomes interference-limited. This effect is also visible in the optimal routing solution. In the UWB case, Figure 1-(c), a total of 7 links are active whereas in the narrow band case, Figure 1-(d), a total of only 3 links are active. The limited system bandwidth, combined with the ‘FDMAlike’ scheduling allows very few active links, resulting in routes with minimal number of links. In Figure 2, the min-max power as a function of node density is shown. At large bandwidth (small x), dense networks increase network performance (decreasing min-max power). In the narrow band regime (large x), network performance worsens with increasing node density. This performance variation with bandwidth, is justified by a comparison with known information-theoretic results below. The well known Gupta-Kumar’s capacity result shows that the uniform throughput capacity is Θ((n log n)−1/2 ) [4], a decreasing function of node density. In [3], the network capacity in the limiting case W → ∞ was derived, presenting a contrasting result to [4]. The upper bound of α−1 the throughput per node in [3], is O((n log n) 2 ) (α > 2), 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 min-

In this paper, we presented a novel cross-layer optimization framework to analyze the joint power adaptation, scheduling and routing problem. This framework was applied to UWB networks and an algorithm developed to generate the optimal solutions. The drastic variation of optimal routing and scheduling solutions with bandwidth, was shown, demonstrating the effectiveness of this framework as a protocol design and network performance evaluation tool. Future work will include investigating computationally efficient and distributed algorithms. 7. REFERENCES [1] D. Bertsekas and R. Gallager, Data Networks, 1992. [2] R. Negi and A. Rajeswaran, “Scheduling and Power Adaption for Networks in the Ultra Wide Band Regime,” GLOBECOM ’04, December 2004. [3] R. Negi and A. Rajeswaran, “Capacity of power constrained ad-hoc networks,” INFOCOM ’04, March 2004. [4] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Trans. Info. Theory, vol. 46, March 2000. [5] C. K. Toh, Ad Hoc Mobile Wireless Networks: Protocols and Systems, Prentice Hall, NJ, 2002 [6] T. Cover and J. Thomas, Elements of Inform. Theory, John Wiley, 1991. [7] B. Radunovi´c and J. L. Boudec, “Optimal power control, scheduling and routing in UWB networks,” Proc. IEEE JSAC, Dec 2004. [8] R. Cruz and A. Santhanam, “Optimal routing, link scheduling and power control in Multi-hop wireless networks,” INFOCOM ’03. [9] M. Johansson and L. Xiao, “Cross-layer optimization of wireless networks using nonlinear column generation,” Tech. Report, RTI, Stockholm, Nov. 2003. [10] S. Toumpis and A. Goldsmith, “Capacity regions for wireless ad-hoc networks ,” IEEE Trans. Wireless Comm., Vol. 2(4), pp 736-748, July, 2003. [11] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, 2004. [12] W. Yu and J. Yuan, “Joint source coding, routing and resource allocation for wireless sensor networks,” submitted to ICC 2005. [13] K. Jansen and L. Porkolab, “On Preemptive Resource Constrained Scheduling: Polynomial-Time Approximation Schemes,” Proc. 9th IPCO Conf., pp. 329-349, May 2002. [14] J. G. Proakis, Digital Communications, 3rd edition, McGraw-Hill, 1995.

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