Modeling and Optimization of Transmission Schemes in ... - CiteSeerX

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Modeling and Optimization of Transmission Schemes in Energy Constrained Wireless Sensor Networks Ritesh Madan, Shuguang Cui, Sanjay Lall, and Andrea Goldsmith

Abstract— We consider a wireless sensor network with energy constraints. We model the energy consumption in the transmitter circuit along with that for transmission. We model the bottom three layers of the traditional networking stack - the link layer, medium access control (MAC) layer, and the routing layer. Using these models, we consider the optimization of the transmission schemes to maximize the network lifetime. We first consider the optimization of a single layer at a time, while keeping the other layers fixed. We make certain simplifying assumptions to decouple the layers and formulate optimization problems to compute a strategy that maximizes the network lifetime. We then extend this approach to cross-layer optimization of time division multiple access (TDMA) wireless sensor networks. In this case, we construct optimization problems to compute the optimal transmission schemes exactly and efficiently. We then consider networks with interference, and propose methods to compute approximate solutions to the resulting optimization problems. We give numerical examples that illustrate the computational approaches as well as the benefits of cross-layer design in wireless sensor networks. Index Terms— optimization, cross-layer design, wireless sensor networks, energy efficiency, network lifetime

I. I NTRODUCTION There are a number of fundamental optimization problems that arise when designing or controlling a wireless sensor network. In this paper, the objective is to construct these problems and the associated constraints from the essential characteristics of the wireless network model. Specifically, we focus on those problems that arise due to energy constraints. We consider a wireless sensor network of nodes distributed in a certain region (see, for example, [1], [2]). We assume that each node has a limited energy supply and generates information that needs to be communicated to a sink node. Also, each node can vary its transmission power, duty cycle, and modulation scheme. We will use transmission scheme/strategy to refer to the data rates, transmission powers, and link schedule for This work is supported by funds from National Semiconductor, Toyota Corporation, Robert Bosch Corporation, the Sequoia Capital Stanford Graduate Fellowship, the Stanford URI Architecture for Secure and Robust Distributed Infrastructures, and AFOSR DoD award number 49620-01-1-0365. R. Madan is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305. Email: [email protected]. S. Cui is with the Wireless System Lab, Department of Electrical Engineering, Stanford University, Stanford, CA 94305. Email: [email protected]. S. Lall is with the Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305. Email: [email protected]. A.J. Goldsmith is with the Wireless System Lab, Department of Electrical Engineering, Stanford University, Stanford, CA 94305. Email: [email protected].

a network. For energy-constrained wireless networks, we can increase the network lifetime by using transmission schemes that have the following characteristics. 1) Multihop routing: In wireless environments the received power typically falls off as the mth power of distance, with 2 ≤ m ≤ 6. Hence we can conserve transmission energy by using multihop routing in long-range applications [3], [4]. 2) Load Balancing: If a node is on the routes of many source destination pairs, it will run out of energy very quickly. Hence load balancing is necessary to avoid the creation of hot spots where some nodes die out quickly and cause the network to fail [5]. 3) Interference mitigation: Links that strongly interfere with each other should be scheduled at different times to decrease the average power consumption on these links [6]. 4) Scheduling: Given a fixed number of bits to transmit over a link, we can reduce the average transmission power over the link by scheduling it for as long as possible. Hence, weakly interfering links should be scheduled together so that each link has a longer duration of time to transmit the same amount of data. We use the standard approach of viewing the wireless system as consisting of layers. The lowest layer is the physical or the link layer, where the objective is to choose the transmission power and rate to minimize the energy consumption to transmit a given number of bits. The next layer is the MAC layer, where the objective is to select a link schedule. The uppermost layer is the routing layer. We first consider the optimization of one layer at a time, keeping the other layers fixed. We then extend our models to compute cross-layer schemes that are energy/lifetime optimal. Most of the problems considered in this paper are either convex optimization problems, or can be approximated by convex optimization problems. In addition, we will also consider a mixed integer-linear programming formulation for energy-efficient link scheduling towards the end of this paper. Some of the optimization problems formulated in this paper can be solved using partially or fully distributed algorithms. Hence, we can implement these algorithms as protocols to solve the relevant optimization problems in wireless networks in a distributed manner.

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A. Prior Work

B. Outline

Over the past few years, optimization techniques have been used to solve many problems arising in wireless as well as wireline networks. The problem of flow control in the Internet was formulated as a convex optimization problem in [7], [8]. In these papers, dual decomposition methods were used to derive distributed congestion control algorithms. In wireless networks, achievable rate combinations were computed in [9]. Cross-layer optimizations to maximize throughput have been considered in [10], [11], [12], [13], [14], [15], [16].

The rest of the paper is organized as follows. The system model for the wireless sensor network is described in Section 2. In Section 3, we model one layer at a time and formulate optimization problems to optimize each layer to maximize the network lifetime. In Section 4, we extend the models to cross-layer optimizations in TDMA-based wireless sensor networks. Section 5 describes the methodology for crosslayer optimization in interference-limited sensor networks. In Section 6, we consider a special case of interference-limited networks, where the routing flow is fixed, and the transmission powers and link scheduling need to be optimized for energy consumption.

In this paper, we consider the optimization of various layers for energy constrained wireless networks. A good overview of the design challenges and the importance of cross-layer design in energy constrained wireless networks was given in [17]. Recent results shows that the processing/circuit energy consumption can have a great impact on the capacity as well as the optimum transmission schemes for MAC channels [18], [19]. For wireless sensor networks, we may not always need to operate on the boundary of the achievable rates region. Hence, we have a choice among various transmission strategies (routing, power control, scheduling) that we can exploit to increase the lifetimes of such networks. An overview of the synergy between the various layers in a wireless network was given in [20]. If we assume that all the links in a network transmit at any given time, we can use power control to minimize the power consumption in the network while satisfying the rate constraints. An overview of algorithms for power control was given in [21]. Routing algorithms in energy constrained wireless networks have traditionally focused on minimizing the total energy consumption. Algorithms to form sparse topologies containing energy efficient routes were considered in [3], [4], [22], [23]. As pointed out in [5], minimum energy routing lead to some nodes in the network being drained of energy very quickly. Hence, instead of trying to minimize the total energy consumption, routing to maximize the network lifetime was considered in [5]. Distributed algorithms to compute a routing scheme to maximize the network lifetime were proposed in [5], [24], [25], [26]. Joint scheduling and power control to reduce energy consumption and increase single hop throughput was considered in [15]. Cross-layer design based on computation of optimal power control, link schedule, and routing flow was described in [6]. The aim of that paper was to minimize average transmission power over an infinite horizon. Also, the routing flow was computed in an incremental manner: it used the Lagrange multipliers obtained at each step by solving an optimization problem of possibly exponential complexity in the number of links. Energy efficient power control and scheduling, without rate adaptation on links, for QoS provisioning were considered in [20]. In [27], the authors describe methods for joint routing, power control, and scheduling for a TDMA-CDMA network. We note that parts of the work in this paper appear in [18], [28], [29], [30], [31], [32].

II. S YSTEM M ODEL A. Network Graph S2

S3

S1

sink

S4

S5

Fig. 1. Network Graph for a Wireless Sensor Network

We consider a static wireless sensor network, modeled by a directed graph G = (V, L), where V is the set of wireless nodes and L is the set of directed links. Each link l is from a transmitter (initial) node i to a receiver (destination) node j. A link exists from node i to node j, if the received power at node j, when node i transmits at maximum transmission power, is greater than a predefined threshold. We define the incidence matrix of the graph G ∈ R|V |×|L| as follows.  if v is the transmitter of link l  1 −1 if v is the receiver of link l Av,l =  0 otherwise

− Let us write A = A+ − A− , such that A+ v,l , Av,l = 0 if Av,l = + − 0, and A , A have only 0 and 1 entries. We assume that each node i ∈ V generates information at rate Si and has total battery energy Ei . The goal of the network is to communicate the information generated by each node P to a sink node. For the sink node, we take Ssink = − i∈V,i6=sink Si . Thus we consider a single commodity flow to simplify notation. However, the methods in this paper apply to the multicommodity flow scenario as well. This scenario is illustrated in Figure II-A. The network in the figure has five source nodes generating data that needs to be communicated to the sink node. At any given time, we consider the network to be alive, as long as all nodes in the network have some battery energy. Let

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Ti denote the time at which node i runs out of energy. Then we define the network lifetime as Tnet = mini∈V Ti . We will generalize this definition when we consider the routing layer in Section III-C.

B. Slotted Time We consider a model where time is slotted into discrete time slots numbered 0, 1, 2, . . ., where each slot is of equal length h. Let pl (m) denote the transmission power used by the transmitter of link l to send data over link l and time slot m. Similarly, let rl (m) denote the corresponding rate. We will restrict pl (m) and rl (m) to be periodic functions of the slot index m, with period M . Thus the network time shares between M transmission modes in a periodic fashion. We will use the term schedule frame to refer to a period of M time slots. Let xl denote the average flow rate over link l. Thus we have M 1 X xl = rl (m) (1) M m=1 C. Physical Layer Associated with the graph G is the link gain matrix G ∈ R|L|×|L| . Glk , l 6= k denotes the power gain from the transmitter of link k to the receiver of link l. Note that Glk depends upon the path loss from the transmitter of link k to the receiver of link l. It is also a function of the links l and k - it depends on the antenna gains, and the correlation between the code sequences if code division multiple access (CDMA) is used. The channel over each link is assumed to be an additive white Gaussian noise (AWGN) channel, with noise power spectral density N0 , over the bandwidth of operation B. The total interference and noise power at the receiver of link l during slot m is given by X Il (m) = Glk pk (m) + N0 B (2) k6=l

The signal to interference and noise ratio (SINR) at the receiver of link l during slot m is defined to be γl (m) = P

Gll pl (m) k6=l Glk pk (m) + N0 B

(3)

We assume that the bit error rate (BER) is constant and same for all links. Then the rate that can be supported over link l with SINR γl (m) satisfies rl (m) ≤ B log(1 + Kγl (m))

(4)

where K = −1.5(ln(5BER)). This is a standard model for modulation schemes such as M-quadrature amplitude modulation (MQAM) schemes with constellation size greater than or equal to 4 [33]. Note that we have relaxed the constraint that the constellation size is an integer; thus rl (m) can take all non-negative real values.

D. Power Consumption Model For each link, we will model the transmission and circuit energy consumption at the transmitter. We will not consider the energy consumption for reception; the methods described in this paper can be easily extended to model the energy consumption at the receiver as well. The nature of the problems or the computational approaches to solve them do not change when we include the receiver circuit energy as well; only the optimal operating point of the network changes. Consider a slot m and link l with transmission rate over link l given by rl (m). Assume that the transmission powers over all other links are fixed. From equations (3),(2),(4), we can see that the minimum transmission power over the link to transmit at rate rl (m) is given by pl (m) =

Il (m) rl (m) (2 B − 1) KGll

Hence, the transmission energy needed to transmit at rate rl (m) for one slot is hpl (m) Then the total energy consumption in slot m at the transmitter of link l can be approximated as (see, Section 5 in [18]) h((1 + α)pl (m) + β) (1 + α)Il (m)h rl (m) (2 B − 1) + hβ = KGll

(5)

where α and β are system constants. The physical interpretation for these constants is given below. α = ratio of overhead power consumption in the power amplifier to the transmissionpower β = energy consumption in the transmitter circuit, excluding that in the power amplifier We will denote the vector of transmission powers over the links in slot m by p(m). Let us define the function 1 : R|V | 7→ R|V | as follows. ½ 1 xi > 0 ∀i = 1, . . . , |V | (1(x))i = 0 otherwise Then 1(A+ p(m)) gives the set of nodes active in slot m. The average power consumption at node i during a schedule frame of M slots is given by   M 1 X X (1 + α)pl (m) + (1(A+ p(m)))i β  (6) M m=1 l∈O(i)

where O(i) denotes the set of outgoing links at node i. Here, we have neglected the overhead energy consumed each time that a node turns on to transmit in a schedule frame [18]. Usually, this value is small and hence can be neglected if a node transmits for a time much longer than the time it takes to turn on.

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Parameter α N0 B K β κ

Value 3 10−10 mW 0.15 100 mW 10−3 m4

TABLE I S YSTEM PARAMETERS

E. Flow and Energy Conservation The total outgoing flow at a node should be equal to the sum of the total incoming flow and the flow generated at the node itself. Thus, we have the flow conservation constraint at node i given by X X xl − xl = S i (7) l∈O(i)

l∈I(i)

where O(i) and I(i) denote the set of outgoing and incoming links, respectively, at node i. Note that the flow conservation equations are satisfied at the frame timescale. If a node is alive for time Ti , the energy dissipated by it in time Ti should be less than or equal to the initial energy Ei . Hence, we have the following energy conservation constraint at each node i.   M Ti X  X (1 + α)pl (m) + (1(A+ p(m)))i β  ≤ Ei M m=1 l∈O(i)

(8)

F. System Parameters The various system parameters that we will use for the computations in this paper are listed in Table 2.1. For the rest of the paper, to simplify notation, we will take the bandwidth B = 1. The values of transmission rates can then be interpreted as being normalized with respect to the bandwidth. Also, we take N0 B = N0 = 0.5mW. In addition, we will take Glk = dκ4 , where κ is a constant given in Table 2.1 and dlk lk is the distance between the receiver of link l and transmitter of link k. This corresponds to a deterministic path loss model where the receiver power falls off as the fourth power of the distance from the transmitter. III. L AYER - WISE O PTIMIZATION We first consider optimizing each layer separately to increase the lifetime of the network. The three layers that we consider are the link, MAC, and routing layers. Also, in this section we consider transmission schemes without interference - we assume TDMA for the MAC layer. Thus we allow only one link in the network to transmit at any given time. This is a realistic scenario in small networks in which links interfere strongly, i.e. the terms Glk , l 6= k, are comparable to Gll . For TDMA transmission schemes, we have Il (m) = N0 if link l is active in slot m. From equation (5), we see that the power consumption to transmit at rate r over a link is of the form r

f (r) = k1 (e k2 − 1) + k3

where k1 , k2 > 0. The function f (r) is convex for r ≥ 0. Hence, time sharing between two transmission modes with different rates consumes more power than transmission at a uniform rate equal to the average rate. Thus if a link is active over ml slots, the energy efficient scheme is to transmit at a uniform rate during the ml slots. In this section, we will denote by rl and pl the transmission rate and power, respectively, over link l when it is active. Thus we have suppressed the dependence of the transmission rate and power on the slot index m, since the optimal scheme for a link is to transmit at the same rate and hence same power over all active time slots. The average power consumption at node i, given by equation (6), can be written as   M 1 X X (1 + α)pl (m) + (1(A+ p(m)))i β  M m=1 l∈O(i) (9) 1 X = ml ((1 + α)pl + β) M l∈O(i)

where ml it the number of slots per frame during which link l transmits at rate rl using transmission power pl =

N0 (2rl − 1) KGll

(10)

A. Link Layer We fix the MAC and the routing layer, i.e. the average flow rate xl and the number of time slots ml allocated to each link l in every frame are fixed. We assume that the MAC layer mitigates interference between active links. Hence, we neglect the cross-link interference terms and approximate Il (m) = N0 if link l is active in slot m. The goal is to select the transmission rate for each link to minimize the energy consumption on the links and hence maximize the network lifetime. The only constraint is that each link l has to transmit at an average rate xl . This can be written as rl ≥

M xl ml

Also, in Section II-B, we assumed that the transmission power pl (m) and rate rl (m) remain constant over a single slot of length h. Hence, the amount of time that each link is active in a given frame should be a multiple of h, i.e. xl ∈ {0, . . . , M } (11) rl h The objective is to minimize the total power consumption in the network. This can be written as the following optimization problem with variables rl , ∀l ∈ L. Ã ! 1 X (1 + α)N0 xl rl xl minimize (2 − 1) + β M KGll rl rl l∈L

subject to

rl ≥

M xl ml

(12)

xl ∈ {0, . . . , M } rl h Note that the objective function and the constraints are separable in the variables rl ’s. Hence, we can solve the optimization

5

150

150 d=5m

125

g(r) = average transmission power (mW)

f(r) = total average power (mW)

d=5m d=10m d=20m 100

75

50

25

0 2

4 6 8 10 transmission rate r (bits/s/Hz)

12

(a) total average power

d=10m

125

d=20m 100

75

50

25

0 2

4 6 8 10 transmission rate r (bits/s/Hz)

12

(b) average transmission power

Fig. 2. Power consumption on a single link

problem for each link separately to minimize the transmission power over each link and hence maximize the network lifetime. Moreover, for each link we have an optimization problem in one variable rl , with rl being allowed to take only finitely many values. This problem can be solved easily. 1) Example: Here, we illustrate the trade-off between reducing transmission power by transmitting at a low rate for a longer time, and reducing circuit power by transmitting in bursts. Consider two nodes separated by a distance d, with one node transmitting data to the other node at an average rate of x =2 bits/s/Hz. If r is the rate of transmission, the transmitter turns on for a fraction xr of time. The total average power consumption is given by (see equation (5)) µ ¶ x (1 + α)N0 r f (r) = (2 − 1) + β r KG where G = dκ4 is the link gain. Also, the average transmission power used by the link is given by

x (1 + α)N0 r (2 − 1) r KG For r ≥ 2, g(r) is minimized at r = 2. Thus to minimize the average transmission power, we should choose the lowest possible data transmission rate. Figure 2 shows the total average power consumption and the average transmission power as a function of the transmission rate, for different distances d. We can see that as the distance between the two nodes increases, the optimal transmission rate decreases. This is because as the distance increases the transmission power increases but the circuit power remains constant. Hence, lowering of transmission power by reducing the transmission rate becomes increasingly important to decrease the average power consumption. Thus the optimal transmission scheme for a small distance is more bursty than that for a large distance. g(r) =

B. MAC Layer Here we consider link scheduling for time division multiple access (TDMA) schemes, where only one link is active in

a given time slot. The general case of link scheduling with interference is a much harder problem. We will give a partial solution to this problem in Sections V-C and VI-B. We first fix the routing flow and the transmission rate on each link. If a link l is active for ml slots, we have the following constraint for TDMA schemes: X ml ≤ M l∈L

Also, if the routing layer assigns a flow rate of xl to link l, we have M xl ml ≥ rl Thus we would like to find a solution to the following feasibility problem. M xl rl X ml ≤ M ml ≥

(13)

l∈L

ml ∈ {0, 1, . . . , M }, ∀l ∈ L

This is a mixed-integer linear program in the variables ml . The standard techniques for solving such problems include branch and bound methods [34] and semi-definite programming (SDP) relaxations [35]. If we replace the integer constraints on ml by ml ≥ 0, the solution will be a variablelength TDMA scheme [32], where the length of the time slots is allowed to vary. Then the problem is a linear program and can be solved efficiently. C. Routing Layer We now fix the link and the MAC layers. Thus the transmission rate rl and the maximum number of slots ml for which each link can transmit during each frame, is fixed. We have the following constraint on the maximum flow over a link l. r l ml = xl ≤ xmax l M

6

0.1 0.1 2

0.015 0.1

0.1

0.1

0.1

50 m 3

1

5 sink

0.14

0.03

0.11

0.07 0.03

0.14

0.015 0.1

0.1

4

(a) topology

(b) optimal flow

Fig. 3. Load balancing to maximize the network lifetime

We also have the constraint given in (11). However, in this section, we will assume that each frame is divided into a large enough number of slots such that we can drop this constraint while computing the optimal routing flow. We can thus approximate the number of slots for which link l transmits data by Mrlxl . This is a good approximation if »

¼ M xl M xl − rl rl

is small compared to M , which would be the case if each frame is divided into a large number of slots. The average power consumption at node i is given by equation (9). X 1 X ml ((1 + α)pl + β) = cl xl M l∈O(i)

(14)

Since, rl and hence pl are fixed, cl is a constant for a given link l. Also, since the total initial energy at node i is Ei , the lifetime of node i under flow {xl } is given by Ti (x) = P

Ei l∈O(i) cl xl

(15)

Then Tnet (x) = mini∈V Ti (x), ∀i ∈ V . The goal is to select a flow {xl } such that it satisfies the maximum rate constraint and the flow conservation constraints in (7), and maximizes the network lifetime Tnet (x). This can be written as the following optimization problem in variables xl , ∀l ∈ L. maximize subject to

Tnet (x) Ax = S 0 ¹ x ¹ xmax

(16)

We note that this problem formulation was first considered in [5]. Using the definition of Tnet and equation (15), the above

T Ax = S 0 ¹ x ¹ xmax

T A+ diag(c)x ¹ E

where diag(c) denotes the matrix C such that Cii = ci and Cij = 0 for i 6= j. The variables are T, xl , ∀l ∈ L. The last constraint implies that no node runs out of energy as long as the network is alive. The value of T obtained by solving this problem gives the optimal value of the network lifetime. Using a change of variables q = 1/T , we obtain the following linear programming problem.

subject to

where (1 + α)pl + β rl

maximize subject to

minimize

l∈O(i)

cl =

problem can be rewritten as follows.

q Ax = S 0 ¹ x ¹ xmax

(17)

A+ diag(c)x ¹ qE

This problem can be solved efficiently in a centralized manner to compute a maximum lifetime routing flow. 1) Subgradient-based Distributed Algorithms: If we assume that there is a flow x such that 0 ≺ x ≺ xmax , and it satisfies the flow conservation equations, we can choose a value of q large enough to satisfy the energy conservation constraints with strict inequality. Then the Slater’s condition for constraint qualification is satisfied (see, for example, [36]) and hence, strong duality holds. Thus we can solve the primal problem via the dual. It can be seen that the dual problem is separable in the variables xl . Hence, we can exploit dual decomposition methods to obtain distributed subgradientbased algorithms to solve problem (17). For details, see [29]. 2) Example: We now illustrate the importance of load balancing in maximizing the network lifetime. We consider a simple topology shown in Figure 3 (a). Nodes 1, 2, 3, and 4 are source nodes generating data at an average rate of 0.1 bits/s/Hz. There are 16 links in the network, each pair of links in opposite directions is shown by a single edge in the figure. We assume that the transmission rate over each link is

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8 bits/s/Hz when active. Thus we model a scenario where the network supports low data rates compared to the achievable rates over the network. From equations (10) and (14), we have β (1 + α)d4ll N0 8 (2 − 1) + 8Kκ 8

cl =

where dll is the distance between the transmitter and the receiver of link l. We assume that each source node has the same amount of initial battery energy. The optimal flow that maximizes the network lifetime is shown in Figure 3 (b). The flow direction and value is indicated for each link. We see that node 1 distributes its flow over 3 different paths instead of sending all its flow along the minimum cost path 1 → 3, 3 → 5. This helps to conserve energy at node 3 and hence prolong the network lifetime, which is the time at which the first node in the network runs out of energy. 3) Generalized Network Lifetime: Until now we considered the network lifetime to be the time at which the first node runs out of energy. Thus we assumed that all nodes are of equal importance and critical to the operation of the sensor network. However, for a heterogeneous wireless sensor network, some nodes may be more important than others. Also, if there are two nodes collecting highly correlated data, the network can remain functional even if one node runs out of energy. Moreover, for the case of nodes with highly correlated data, we may want only one node to forward the data at a given time. Thus we can activate the two nodes in succession, and still be able to send the required data to the sink. In this section, we will model the lifetime of a network to be a function of the times for which the nodes in the network can forward their data to the sink node. In order to state this precisely, we redefine the node lifetime and the network lifetime for the analysis in this section. In addition, we will drop the constraint on the maximum rate over a link. We will assume that the network operates in a low-data rate regime, and hence we can easily re-allocate ml such that xl ≤ mMl rl . Each node k ∈ V generates data at rate Sk . Let xkl be the rate at which link l transmits data, originally generated by node k. The flow conservation equations at node i for the flow originating at node k can be written as follows. X X xkl = (δk )i Sk , ∀i, k ∈ V (18) xkl − l∈O(i)

l∈I(i)

where δk ∈ R|V | is such that (δk )i = 1 if i = k, and 0 otherwise. We define Tk , the lifetime of node k, to be the total time for which node k generates data that is transmitted over the network to thePsink node. Then the total number of bits sent over link l is k∈V xkl Tk . Thus we have the energy constraint at each node given by X X xkl Tk ≤ Ei , ∀i ∈ V cl l∈O(i)

k∈V

where cl is as defined in equation (14). We consider a generic definition of network lifetime given by a concave function of the node lifetimes. In particular, we define Tnet = f (T1 , . . . , T|V | )

where f : R|V | → R is a concave function in the vector of node lifetimes. In the previous sections, we considered the special case of Tnet = min(T1 , . . . , T|V | ). We can write the problem of maximizing the network lifetime as follows. maximize

f (T1 , . . . , T|V | )

subject to

Axk = Sk δk ,

∀k ∈ V

k

X

k∈V

x º 0,

+

∀k ∈ V

k

Tk A diag(c)x ¹ E

We apply a change of variables ylk = xkl Tk . We can interpret ylk as the total number of bits generated by node k, transmitted over link l. We can rewrite the above problem as the following convex optimization problem. maximize

f (T1 , . . . , T|V | )

Ay k = Tk Sk δk ,

subject to

k

X

k∈V

+ k

y º 0,

A y diag(c) ¹ E

∀k ∈ V (19)

∀k ∈ V

Then, we can interpret the first set of constraints as bit conservation equations. Since the data from all the nodes is routed to a single sink, solving the above problem is equivalent to solving maximize subject to

f (T1 , . . . , T|V | ) Az = diag(S)T,

(20)

z º 0,

A+ diag(c)z ¹ E

P where zl = k∈V ylk is the total number of bits transmitted over link l. From the solution zl of this problem, we can easily obtain an optimal flow ylk , k ∈ V , which is a solution to problem (19). We remark that for certain realistic scenarios, the function f may not be differentiable. For such functions we can use the analytic center cutting plane method (see, for example, [37], [10]) or the subgradient method [38] to solve the optimization problem (20). 4) Example: Consider a sensor network of randomly distributed nodes in a certain region, as shown in Figure 4. The deployment region of the network has been divided into a grid of 16 regions. Two nodes are connected by an edge if they can directly communicate with each other using a transmission power less than the maximum allowable transmission power at each node. Let Gi denote the set of nodes in region i. We assume that the network is functional as long as at least one node in each region can sense data. This would correspond to a network lifetime given by the following concave function of node lifetimes.   X Tj  Tnet = min  i=1,...,16

j∈Gi

Note that the time P for which at least one node is active in region i is j∈Gi Tj . This is because all nodes in a

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Substituting the above relation in equations (9) and (10), the average power consumption at node i is 1 X ml ((1 + α)pl + β) M l∈O(i) µ ¶ N0 1 X (2rl − 1) + β = ml (1 + α) M KGll l∈O(i) µ ¶ M xl 1 X N0 = ml (1 + α) (2 ml − 1) + β M KGll l∈O(i) X gl (xl , ml ) =

(22)

l∈O(i)

where Fig. 4. Network lifetime for a gridded network

particular region can take turns to transmit data. Now lets further consider a network to be partially functional as long as some regions have at least one alive node. Moreover, sensing in some regions can be more important than that in others. In such a scenario, we can consider the network lifetime to be a weighted linear sum of the lifetimes of the individual regions, i.e., Tnet =

16 X

ki

i=1

X

Tj

j∈Gi

³ M xl ´ gl (xl , ml ) = al ml 2 ml + bl

gives the average power consumption over link l as a function of xl and ml . Also, al and bl are suitably defined constants for link l. We note that the function g(x, m) is convex over x, m ≥ 0. As in Section III-C, we can write the lifetime of node i as Ti = P

In the previous section, we considered optimization of the individual layers, while keeping the other layers fixed. Also, we made certain simplifying assumptions about the various layers. For example, the MAC layer was assumed to be TDMA for interference mitigation. In this section, we start with similar assumptions, and further develop the analysis to optimize two or more layers at a time. We consider pairwise optimization of layers for TDMA networks. Then we consider optimization of all three layers, again with the restriction of the MAC layer to TDMA. These problems can be solved exactly by standard techniques. In fact, if we relax the TDMA schemes to variablelength TDMA, these problems can be solved very efficiently.

A. Pairwise Layer Optimizations We consider two joint optimizations - between the link and MAC layers, and between the MAC and routing layers. 1) Link and MAC: We fix the routing layer. Thus the average flow rate xl over link l is fixed. Again, we consider the simple case where the MAC is restricted to TDMA. Then if a link l is allocated ml slots, the rate at which it transmits data is M xl rl = (21) ml

Ei l∈O(i) g(xl , ml )

By analysis similar to that in Sections III-C and III-B, the problem of maximizing the network lifetime (with fixed routing layer) can be written as the following optimization problem with the inverse network lifetime q and the ml ’s as variables. minimize

IV. TDMA BASED W IRELESS S ENSOR N ETWORKS

(23)

subject to

q X

l∈O(i)

g(xl , ml ) ≤ qEi , X l∈L

∀i ∈ V

ml ≤ M ml ∈ {0, . . . , M },

∀l ∈ L (24)

The solution q, ml , ∀l ∈ L to the above optimization problem automatically determines the link layer transmission rate rl (see equation (21)). The function g is convex in ml . Hence, the problem is a mixed integer-convex optimization problem. It can be solved using branch and bound methods. Since the function g is smooth, in practice the branch and bound methods converge quickly [30]. If we replace the integer constraints on the ml ’s by ml ≥ 0, ∀l ∈ L, the solution gives the optimal variable-length TDMA schedule. The resulting problem is convex and the solution can be computed efficiently using interior point methods (see for example, [36]). 2) MAC and Routing: For joint optimization of the MAC and routing layers, we fix the link layer parameters rl ’s. The average power consumption at each node is then given by equation (14). The analysis from Section III-C almost completely applies here. The only difference is that xmax is a variable now. If link l is allocated ml slots, then we have = xmax l

1 r l ml ⇒ xmax = diag(r)m M M

9

Hence, the problem of maximizing the network lifetime is an optimization problem similar to the one in (17). minimize subject to

q Ax = S 1 diag(r)m 0¹x¹ M A+ diag(c)x ¹ qE X ml ≤ M

(25)

l∈L

ml ∈ {0, . . . , M },

∀l ∈ L

a large wireless network, consider two links such that the transmitter of one link is separated by a large distance from the receiver of another link, and vice versa. Then if the two links are scheduled simultaneously, they do not interfere much. Thus we can take advantage of frequency reuse to schedule each link for a longer time to reduce the average transmission power on the links. If each of these links supports a high enough rate, the power consumption in the power amplifier dominates the power consumption in the rest of the transmitter circuit (see equation (5)). Then the simultaneous scheduling of weakly interfering links leads to a reduction in the net power consumption.

The variables are q, ml ’s. The problem is a mixed integerlinear program. We can relax the integer constraints on ml to obtain a linear program; the solution will then give a variablelength TDMA scheme. A. Cross-Layer Optimization

B. Cross-Layer Optimization We now consider the joint optimization of the link, MAC and routing layers. Thus the variables are rl , xl and ml for each link l ∈ L. Note that the values of two of these variables automatically determine the third one. For example, if the average flow rate and the number of slots is fixed, we have rl given by equation (21). Also, the flow variables xl have to satisfy the flow conservation equations in (7). Thus in addition to constraints in problem (24), we have additional constraints given by the flow conservation equations. Hence, the problem of maximizing the network lifetime can be written as minimize subject to

l∈O(i)

Ax = S g(xl , ml ) ≤ qEi , X l∈L

rl (m) ≤ log(1 + Kγlm ) Ã

Gll pl (m) = log 1 + K P G k6=l lk pk (m) + N0

∀i ∈ V

l∈O(i)

ml ∈ {0, . . . , M },

∀l ∈ L

(27)

  M M X X 1  X X rl (m) − rl (m) = Si , ∀i ∈ V M m=1 m=1

ml ≤ M xº0

!

PM 1 The average rate over link l is xl = M m=1 rl (m). Thus the flow conservation equations in (7) can be written as

q X

The transmission rate over link l, during slot m, is bounded by (see equations (3), and (4)).

(26)

where the variables are q, xl ’s, and ml ’s. This is a mixed integer-convex program, similar to the one in problem (24). Hence, it can be solved using branch and bound methods. Also, relaxing ml ∈ {0, . . . , M } to ml ≥ 0 again gives a convex optimization problem which can be solved efficiently. The resulting solution gives the optimal transmission scheme where the MAC is variable-length TDMA. V. I NTERFERENCE - LIMITED W IRELESS S ENSOR N ETWORKS We now develop models for wireless networks where interfering links are allowed to transmit simultaneously. For such networks, we state the optimization problem exactly, and then suggest an iterative method to obtain approximate solutions to the problem. Here, we consider a MAC layer that is more general than TDMA. TDMA divides the spectral resources in an orthogonal manner by scheduling interfering links at different times. In

l∈I(i)

⇒

A

M X

r(m) = M S

m=1

(28)

Also, we have the energy conservation constraint in (8).   M Ti X  X (1 + α)pl (m) + (1(A+ p(m)))i β  ≤ Ei M m=1 l∈O(i)

M ´ Tnet X ³ + A (1 + α)p(m) + β1(A+ p(m)) ≤ E M m=1 (29) where Tnet = min(T1 , . . . , T|V | ).

⇒

The goal is to compute a transmission scheme, i.e. pl (m), rl (m) (and hence xl ), to maximize the network lifetime. The constraints are flow conservation and energy conservation constraints given by equations (28) and (29), respectively. Moreover, any physically possible transmission scheme satisfies the relation (27). This can be written as the following optimization problem in the variables Tnet , pl (m), rl (m), for

10

all l ∈ L, m = 1, . . . , M . Tnet

maximize

A

subject to

M X

r(m) = M S

m=1

Ã

log 1 + K P

r(m) º 0 !

Gll pl (m) G k6=l lk pk (m) + N0

≥rl (m)

M ´ Tnet X ³ (1 + α)A+ P m + β1(A+ p(m)) ¹ E M m=1 (30)

P The function log( i ai exi ) is convex if ai ≥ 0, xi ∈ R (see, for example, [36]). Composition with an affine function preserves convexity. Hence, the function µ ¶ X N0 rl (m)−Ql (m) Glk rl (m)+Qk (m)−Ql (m) e + e log Gll Gll k∈L(m),k6=l

is convex over r(m), Q(m). Using the above change of variables and the approximate rate constraint (32) in problem (31), we obtain the following optimization problem.

A

for all m = 1, . . . , M and l ∈ L. As in Section (III-C), we can use the change of variables q = 1/T to rewrite the above problem as the following optimization problem.

A

subject to

M X

log 1 + K P

r(m) = M S

M ³ X

r(m) º 0 !

Gll pl (m) k6=l Glk pk (m) + N0

+

(1 + α)A P

m

m=1

+

≥ rl (m)

´

r(m) = M S

m=1

r(m) º 0,

µ

m=1

Ã

M X

m = 1, . . . , M

rl (m) = 0, ∀l ∈ / L(m), m = 1, . . . , M ¶ X Glk rl (m)+Qk (m)−Ql (m) N0 rl (m)−Ql (m) e + e ≤ 0, log Gll Gll

q

minimize

q

minimize subject to

+ β1(A p(m)) ¹ qM E (31)

for all m = 1, . . . , M and l ∈ L. This problem is not a convex optimization problem, or even a mixed integerconvex problem. Hence, it is hard to solve. We propose an iterative approach below where, in each iteration, we solve an approximate problem that is convex. B. Routing and Power Control

M ³ X

k∈L(m),k6=l

X

m=1 l∈O(v)∩L(m)

∀l ∈ L(m), m = 1, . . . , M ¡ ¢´ (1 + α)eQl (m) + β ≤ qM Ei ,

∀i ∈ V (33)

The variables are q, rl (m), Ql (m), for l ∈ L(m), m = 1, . . . , M . Thus we can efficiently solve the approximate problem for optimal transmission powers and rates over each link, for a given link schedule. Geometrically, for a fixed link schedule and the convex approximation to the rate constraint, the feasible set in problem (33) is a convex subset of the feasible set in problem (31). C. Link Scheduling

The convex optimization problem (33) is feasible only if the constraints rl (m) ≥ 0, l ∈ L(m), m = 1, . . . , M , are feasible. For the approximate rate constraint, it implies that each link has an SINR≥ 1 during the scheduled slots. If we schedule all links during all slots, the problem may be infeasible. There is no simple characterization of the set of link schedules for which the constraints rl (m) ≥ 0, l ∈ L(m), m = 1, . . . , M are feasible. Hence, in order to use problem formulation (33) to This is a good approximation if the SINR over link l and time compute an optimal transmission scheme, we need to solve slot m is high. For any SINR γ, log(γ) is a lower bound on the this problem for all possible link schedules. However, for achievable rate. Hence, the feasible set corresponding to the a network of L links, and for a schedule frame that is M optimization problem with the above approximation is a subset time slots long, there are 2M L different link schedules. Thus of the feasible set of the original optimization problem (31). the complexity of this approach is doubly exponential in the Thus the network lifetime computed under this approximation number of slots and the number of links. is a lower bound on the optimum network lifetime. Using a We propose a heuristic for adapting a link schedule to a change of variables Ql (m) = log(pl (m)), we can rewrite the given routing flow and a set of transmission powers over each approximate rate constraint for link l ∈ L(m) as follows (see, link, during the M slots. Combining this heuristic with the for example, [39]). problem formulation in (33), we propose an iterative approach ¶ that alternates between link scheduling and computing a µ X Glk rl (m)+Qk (m)−Ql (m) N0 rl (m)−Ql (m) e + e log solution to problem (33) for a fixed link schedule. Thus the Gll Gll k∈L(m),k6=l algorithm solves a series of convex optimization problems with feasible regions given by different convex subsets of ≤0

The rate constraint in the problem formulation (31) is not convex. For a fixed link schedule, let L(m), m = 1, . . . , M denote the sets of links allowed to be active in each slot m. We approximate the rate constraint by ! Ã Gll pl (m) (32) rl (m) ≤ log P k∈L(m),k6=l Glk pk (m) + N0

11

the feasible region of the original optimization problem (31). Each convex subset corresponds to a link schedule and an approximation of the rate constraints by convex constraints (see equation (32)). D. Iterative Cross-Layer Optimization Initialize with feasible suboptimal schedule

Compute optimal rates, powers

YES

Feasible? NO YES

The algorithm is modular in the sense that different link scheduling approaches can be combined with the solution of problem (33). For small networks, we can use a TDMA schedule for step (1). The algorithm uses a greedy heuristic to adaptively schedule links at each iteration, and then resolves the convex optimization problem (33) to determine an optimal routing flow and transmission powers in each slot. The heuristic is motivated by a scenario in which bandwidth is scarce and so each link is forced to transmit at a rate above the power optimal rate shown in Figure 2. It then becomes essential to reduce the transmission power by scheduling links for as long as possible without causing too much interference to other links. It is found that even such a simple greedy heuristic can give strategies with a higher network lifetime than that given by static approaches to scheduling (e.g. TDMA and time sharing between modes in which links separated by a minimum distance are scheduled together). The gains in network lifetime are due to energy-efficient multihop routing, frequency reuse, and load balancing.

SINR>1 feasible?

E. Distributed Implementation Turn off links with SINR appr. 1 NO NO

Allocate an additional slot to a link with the max. avg. power

Repeated schedule?

YES Quit

Fig. 5. Iterative approach to compute powers, rates and link schedule.

The iterative approach used to compute an approximate optimal strategy is summarized in the flowchart in Fig. 5. The various steps are as follows. 1) Find an initial suboptimal, feasible schedule to begin with. A good candidate would be a schedule in which most of the links are activated at least once in each frame of M slots, and also links that are activated in the same slot only interfere weakly. 2) Solve problem (33) to find an optimal routing flow and transmission powers during each slot under the high SINR approximation. If the problem is infeasible, quit. 3) Turn off links during slots in which they have an SINR close to 1. Since we approximated the rate as r = log(SINR), links carry very little traffic over the slots in which they have an SINR of about 1. 4) Find a link that consumes the maximum average power over the entire frame. Schedule this link to be on during an additional time slot. The selected slot should be the one in which there is minimum interference to this link. If the resulting schedule is one that was used in a previous iteration, quit. 5) Check if SINR≥ 1 is feasible over all slots. If yes, go to (2), else quit.

Like in Section III-C, we can exploit the structure of the problem (33) to derive partially distributed algorithms using dual decomposition. The details can be found in [30]. F. Example Consider the topology shown in Figure 6(a). We take the source rates to be S1 , . . . , S9 = 0.15 bits/s/Hz. Also, we take the distance between neighboring nodes as d = 56 m. Thus the distance is large enough such that the transmission power dominates over the power consumption in the circuit. Note that the links closer to the sink support a higher average data rate than the links further away from the sink. We take the number of slots in each frame M = 18. We apply the algorithm proposed in the previous section to compute an efficient transmission scheme for this network. The results after 25 iterations are shown in Figure 6. As we can see, the links carrying a higher data rate are scheduled for a higher number of slots to reduce the average transmission power. The total power consumption is shown in Figure 2(d). Also, note that during each slot, more than one link is active. VI. J OINT S CHEDULING AND P OWER C ONTROL IN I NTERFERENCE - LIMITED N ETWORKS In this section, we consider the special case where the routing flow is fixed in interference-limited wireless networks. This would be the case, for example, in a cluster with a star topology, in a sensor network. For this scenario, we can still use the approach in Section V-D. Here, we propose an alternative heuristic approach that involves the solution of a mixed integer-linear program and computation of a power control strategy that involves a matrix inversion. The approach is motivated by the special case in which the topology is regular, i.e. Gll is the same for all links l ∈ L. We will make the assumption that the interference in the network is high enough such that the energy for transmission dominates. Thus

12

1.4

s1

s2

s9

L1

L2

1

L8

L9

sink

9

2

avg. rate (bits/s/Hz)

1.2 1 0.8 0.6 0.4 0.2

10

0

d

1

2

(a) topology

5 link

6

7

8

9

8

9

200 avg. power consumption (mW)

6 no. of slots allocated

4

(b) average rates

7

5 4 3 2 1 0

3

1

2

3

4

5 link

6

7

8

150

100

50

0

9

(c) number of slots

1

2

3

4

5 link

6

7

(d) average power Fig. 6. Linear Topology

in this section, we only model the energy consumption in the power amplifier and for transmission; we neglect the energy consumption in the rest of the transmitter circuit components. Hence, we set β = 0 in equation (5). We consider energy efficient link scheduling for fixed average link rates. We first study variable-length TDMA schedules to minimize a linear combination of the power consumption on all the links. We then consider generic link schedules with the property that each link is activated for a time proportional to the optimal slot length in a variable-length TDMA schedule.

We consider the following optimization problem.

A. Optimal TDMA Scheduling

The objective function is a strictly increasing linear function of the average power consumption on each of the links. If the Gll ’s are equal for all links l ∈ L (for example, in regular networks), then the problem formulation minimizes the total power consumption in the network. Also, the objective function is such that it favors the links with high link gains Gll ’s. The variables are the ml ’s. Note that we consider variable-length TDMA schemes, and hence there are no integer constraints. Then we have the following proposition.

We will motivate our heuristic method by first considering optimal variable-length TDMA schedules to minimize a weighted sum of the power consumption on all the links in the network. The choice of the objective function is one which allows for an analytical solution to the optimization problem. We will use the analytical solution to motivate our proposed heuristic in the next section. Neglecting the energy consumption in the circuit components other than the power amplifier, the average power consumption over link l is given by equation (23) with bl = −1, i.e. g(xl , ml ) = (1 + α)

´ N0 ml ³ Mmxl 2 l −1 KM Gll

(34)

minimize

X

Gll g(xl , ml )

l∈L

subject to

X

ml ≥ 0,

∀l ∈ L

(35)

ml = M

l∈L

Proposition 6.1: Assume that for all links l ∈ L, xl > 0. The optimal variable-length TDMA schedule that solves l problem (35), is given by ml = kxl , where m M is the fraction of time for which link lPis active, and k is a constant of proportionality such that l∈L ml = M .

13

Proof: Using equation (34), the objective function can be written as µ Mx ¶ X X l (1 + α)N0 ml ml Gll g(xl , ml ) = 2 Gll −1 Gll KM l∈L l∈L X M xl aml (2 ml − 1) =

written as the following mixed-integer linear program. minimize

X

subject to

k∈L,k6=l

l∈L

a2

M xl ml

µ

λl ≥ 0, λl ml = 0, ∀l ∈ L ¶ M xl +(ν − a) + λl = 0, ∀l ∈ L 1− ml

Any m and (λ, ν) satisfying these conditions are primal and dual optimal, with zero duality gap. Using the fact that for an optimal solution, ml > 0 (and hence λl = 0) for all l, the KKT conditions are satisfied if X ml = M, m º 0 a2

M xl ml

l∈L ¶ µ M xl = a − ν, ∀l ∈ L 1− ml

We can P satisfy these conditions by choosing ml = kxl , such that l∈L ml = M .

Intuitively, a link that supports a higher average rate is scheduled for a larger fraction of time. For a regular topology where all the diagonal entries of the matrix G are equal, i.e. Gll is the same for all l ∈ L, this gives a TDMA schedule with minimum total average power consumption in the network.

Glk vk (m) ≤ λ + D(1 − vl (m))

M X

l∈L

for a suitably defined constant a. The Karush-Kuhn-Tucker (KKT) conditions (see, for example, [36]) for problem (35) can be written as follows. X ml = M, m º 0

λ

vl (m) =ml

m=1

vl (m) ∈{0, 1}

(36) for all l ∈ L, m = 1, . . . , M where D is an arbitrary large constant. The variables are λ and the vl (m)’s. The first set of constraints ensure that the maximum total cross-link gain to an active link is less than or equal to λ. The second set of constraints activate each link for a number of slots proportional to the TDMA slot lengths. For a path loss model, where the received power decays monotonically with the distance from the transmitter, the above problem formulation can be thought of as spacing the active links in all time slots “as far apart as possible”, subject to the scheduling constraints. We can interpret the above optimization problem as follows. Proposition 6.2: Consider a regular topology where Gll = G for all links l ∈ L. If all the links are constrained to use the same transmission power p, when active, the schedule computed by the mixed integer-linear program (36) minimizes the total power consumption among all schedules with M slots and each link l active for a number of slots proportional to xl . Proof: Consider a schedule {vl (m)} that satisfies X Glk vk (m) ≤ λ + D(1 − vl (m)) k∈L,k6=l

Also, let λ be such that this relation is satisfied with equality for at least one m and l such that vl (m) = 1. For every link l and slot m such that vl (m) = 1, let λl (m) be given by X λl (m) = Glk vk (m) k∈L,k6=l

B. Minimum Cross-Link Interference Gain Schedules Suppose given xl ’s, we want to design a periodic slotted schedule to support the rates. Let ρ be such that xl = ml ρ, ∀l ∈ L, where the ml ’s are integers. We can always find such a ρ to an arbitrary degree of accuracy. We consider link schedules where each link l is activated for ml number of slots. Thus the time allocated to each link is in proportion to the optimal slot length in a variable-length TDMA schedule. In addition, we now consider frequency reuse which allows more than one link to be active in each slot. Also, each active link transmits at the same data rate ρ. Let vl (m) be a scheduling variable such that vl (m) = 1 if link l is active in slot m, and 0 otherwise. For a fixed number of time slots M per frame and a schedule v = {vl (m), l ∈ L, m = 1, . . . , M }, the maximum sum of the cross-link gains from the interfering links to an active link is X Glk vk (m) λ(v) = max {l,m:vl (m)=1}

k∈L,k6=l

The problem of computing a feasible schedule that minimizes the maximum total cross-link gain to an active link can be

Then for every link l and slot m such that vl (m) = 1, the common transmission power satisfies (see equation (4)) ! Ã KGp ≥ρ log 1 + P k∈L,k6=l Glk vk (m)p + N0 ⇒

⇒

P

KGp ≥ 2ρ − 1 k∈L,k6=l λl (m)p + N0 N0 (2ρ − 1) p≥ KG − (2ρ − 1)λl (m)

Since λ = max{l,m:vl (m)=1} (λl (m)), the minimum possible common transmission power is p=

N0 (2ρ − 1) KG − (2ρ − 1)λ

Thus p is a increasing function of λ for KG − (2ρ − 1)λ > 0. Since the solution to the optimization problem in (36) minimizes λ, it minimizes the value of p. This also minimizes the total average power consumption in the network, which is given by 1 X (1 + α)pml M l∈L

14

However, if the links are allowed to use different transmission powers, then we can use the problem formulation as a heuristic to compute a schedule with low interference. C. Power Control For a schedule computed as above, we can use optimal power control (see, for example, [21], and the references therein) to obtain the minimum transmission power vector for each slot such that each active link transmits at the common rate x. We summarize the relevant results from [21] below. Let L(m) be the set of links that transmit data in slot m, at rate x. For notational convenience, let us renumber these links as 1, 2, . . . , |L(m)|. Then each link needs a minimum SINR 1 (2ρ − 1). Let the vector u ∈ R|L(m)| be such that of γ0 = K ui =

γ 0 N0 Gii

Also, consider the matrix F ∈ R|L(m)|×|L(m)| given by Fij =

γ0 Gij 1i6=j Gii

where 1i6=j = 1, if i 6= j, and 0 otherwise. Then the rate ρ over each link l ∈ L(m) is feasible if ρF < 1, where ρF is the Perron-Frobenious eigenvalue of F . Then the Pareto-optimal transmission power vector is given by (I − F )−1 u D. Example Consider the grid topology shown in Fig. 7(a). Each link in the grid supports the same average rate x = 0.15 bits/s/Hz. This may correspond to, for example, an aggregation pattern with data fusion and redundant transmission for robust multipath routing in sensor networks. We assume a path loss model, where the received power decays as the fourth power of distance from the transmitter. We used a branch and bound method (see, for example, [34]) to compute a schedule that minimizes the maximum total cross-link gain for a fixed M (see problem (36)). The computation was repeated for different number of slots M per frame, for a constant frame-length. The resulting schedule for M = 5 is shown in Fig. 7(a). The number next to each link gives the slot index over which it is scheduled. For the schedule thus obtained for each value of M , we computed the optimal transmission power on each link to support an average data rate of x = 0.15 bits/s/Hz. The variation of the total power consumption and the maximum total cross-link gain with M is shown in Fig. 7(b). The values are normalized so that we can plot both the cross-link gains and the power consumption on the same graph. The rate vector is infeasible for the schedule computed for M = 2, 3 slots. For M ≥ 4, we see that the power consumption first increases with M because the increase in power due to less transmission time dominates the power savings due to interference mitigation. However, for M = 8, the schedule computed using the mixed integer-linear program in (36) mitigates the interference to such a large extent that the power savings due to interference

mitigation dominates. For M ≥ 8, the power savings due to less interference are not significant, and hence power consumption increases with M . The values of M at which the interference decreases sharply depend on the topology under consideration. Also, note that M = 12 is same as a TDMA schedule, where each link is active for one-twelfth of the time. VII. C ONCLUSIONS We described a framework for modeling energy constrained wireless networks. The constraints imposed by the underlying system were studied and optimization problems were constructed to design such networks. In particular, we modeled the circuit energy consumption and the traditional physical, MAC, and the routing layers. We considered the optimization of individual layers as well as cross-layer optimization. The optimization problems can be solved exactly for TDMA networks. For networks with interference, we proposed approximation approaches to solve the optimization problems. The computational approaches were illustrated by several numerical examples. R EFERENCES [1] D. Estrin, L. Girod, G. Pottie, and M. Srivastava, “Instrumenting the world with wireless sensor networks,” ICASSP, 2001. [2] G. Pottie and W. Kaiser, “Wireless sensor networks,” Communications of the ACM, vol. 43, no. 5, pp. 51–58, 2000. [3] V. Rodoplu and T. H. Meng, “Minimum energy mobile wireless networks,” IEEE J. Selected Areas in Communications, vol. 17, no. 8, pp. 1333–1344, 1999. [4] L. Li and J. Y. Halpern, “Minimum energy mobile wireless networks revisited,” IEEE International Conference on Communications (ICC), 2001. [5] J.-H. Chang and L. Tassiulas, “Energy conserving routing in wireless ad-hoc networks,” Proc. IEEE INFOCOM, pp. 22–31, 2000. [6] R. L. Cruz and A. V. Santhanam, “Optimal routing, link scheduling and power control in multi-hop wireless networks,” IEEE INFOCOM, 2003. [7] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control for communication networks: shadow prices, proportional fairness and stability,” Journal of the Operational Research Society, vol. 49, pp. 237–252, 1998. [8] S. H. Low and D. E. Lapsley, “Optimization flow control - I: Basic algorithm and convergence,” IEEE/ACM Trans. Networking, vol. 7, pp. 861–874, 1999. [9] S. Toumpis and A. Goldsmith, “Capacity regions for wireless ad hoc networks,” IEEE Transactions on Wireless Communications, vol. 4, pp. 736–748, 2003. [10] L. Xiao, M. Johansson, and S. Boyd, “Simultaneous routing and resource allocation via dual decomposition,” Accepted for publication in IEEE Transactions on Communications. [11] D. O’Neill, D. Julian, and S. Boyd, “Seeking Foschini’s genie: optimal rates and powers in wireless networks,” Accepted for publication in IEEE Transactions on Vehicular Technology, 2004. [12] M. Johansson and L. Xiao, “Scheduling, routing and power allocation for fairness in wireless networks,” IEEE Vehicular Technology Conference, 2004. [13] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, “Impact of interference on multi-hop wireless network performance,” MobiCom, 2003. [14] M. Kodialam and T. Nandagopal, “Characterizing achievable rates in multi-hop wireless networks: The joint routing and scheduling problem,” MobiCom, 2003. [15] T. ElBatt and A. Ephremides, “Joint scheduling and power control for wireless ad hoc networks,” IEEE Transactions on Wireless Communications, vol. 1, pp. 74–85, 2004. [16] B. Radunovic and J. Y. L. Boudec, “Joint scheduling, power control and routing in symmetric, one-dimensional, multi-hop wireless networks,” Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), 2003.

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10

11

12

no of slots (M)

(b) power consumption vs. M

Fig. 7. Numerical Results - Power consumption in normalized units.

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