Cross-layer Optimization of Correlated Data Gathering in Wireless ...

Cross-layer Optimization of Correlated Data Gathering in Wireless Sensor Networks ∗ State

Shibo He

∗

Jiming Chen

∗

David K. Y. Yau

†

Youxian Sun

∗

Key Lab. of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China † Department of Computer Sciences, Purdue University, West Lafayette, IN, USA

Abstract—We consider the problem of gathering correlated sensor data by a single sink node in a wireless sensor network. We assume that the sensor nodes are energy-constrained and design efficient distributed protocols to maximize the network lifetime. Many existing approaches focus on optimizing the routing layer only, but in fact the routing strategy is often coupled with power control in the physical layer and link access in the MAC layer. This paper represents a first effort on network lifetime maximization that jointly considers the three layers. We first assume that link access probabilities are known and consider the joint optimal design of power control and routing. We show that the formulated optimization problem is convex and propose a distributed algorithm, JRPA, for the solution. We also discuss the convergence of JRPA. When the optimal link access probabilities are unknown, as in many practical networks, we generalize the problem formulation to encompass all the three layers of routing, power control, and link-layer random access. In this case, the problem cannot be converted into a convex optimization problem, but there exists a duality gap when the Lagrangian dual method is employed. We propose an efficient heuristic algorithm, JRPRA, to solve the general problem, and show through numerical experiments that it can significantly narrow the gap between the computed and optimal solutions. Moreover, even without a priori knowledge of the best link access probabilities predetermined for JRPA, JRPRA achieves extremely competitive performance with JRPA. Other numerical results are provided to show the convergence of the algorithms and their advantages over existing solutions.

I. I NTRODUCTION In monitoring and surveillance applications in wireless sensor networks (WSNs), sensor nodes are used to monitor and collect data about their local areas, and report the data to a sink node for analysis. The sensors are normally powered by energy-constrained batteries and deployed at a high density. Because of the high density, information sensed by different sensors is usually spatially correlated. Hence, for energy efficiency, the sensors need to decide their source data rates to the sink in order to communicate the total network information while minimizing redundancies in their data reports. Besides, the communication strategy itself must be designed to maximize the network lifetime. Cristescu et al. [1] propose Slepian-Wolf coding to decide the amount of data reported for each sensor node and try to find an optimal routing tree to minimize the total transmission cost. Based on the Slepian-Wolf coding, they give a closedform expression for the source rate allocation and devise a shortest-path tree algorithm for the optimal routing. Their approach is amenable to a fully distributed implementation by employing a localized form of the Slepian-Wolf coding, which we will also use in our work. However, an important drawback of their work is that they do not fully consider

the impact of the source rate allocation on the underlying communication layers. Specifically, they ignore the capacity of the network in trying to route data over a minimum-energy network path. Hence, links on the selected paths will have a high probability of congestion, which severely degrades the network performance. This problem is partially addressed by Yuen et al. [2], who incorporate link capacity constraints in their problem formulation. They derive a cluster of links that cannot transmit data at the same time to account for channel contention constraints. Via congestion control, they try to avoid the overloading of each link. For example, when a link is congested, data may be routed to bypass the link although doing so may incur added network overhead. However, they assume that the capacity of each link is fixed and cannot be adapted to optimize the transmission. In practice, the capacity of each link depends on the power level of the transmission and the MAC protocol, so that physical-layer power control and MAC-layer random access strategies must also be incorporated into the overall solution. Another important issue is that, in routing sensor data to the sink node, we must truly optimize for the network lifetime. Indeed, each sensor node may behave both as a data source and a data relay. Nodes that are frequently used by other nodes as relays will have their energy drained quickly. When these nodes die, the network may become partitioned. A good routing strategy is thus needed to balance the traffic load and prolong the network lifetime. This illustrates again the intricate interactions between the communication layers. For example, nodes that are chosen to be on busy data paths by the routing will need high data rate allocations supported by the MAC and physical layers, and an optimal solution must consider their joint effects. In addition, solutions that focus on minimizing the total energy cost of the network [1], [2] will not work well in practice, because if lots of data are routed through the same path to minimize the global energy use, the nodes on this path will run out of energy quickly, thus partitioning the network. The above examples illustrate the importance of a comprehensive cross-layer design in truly maximizing the network lifetime, which is the focus of this paper. To the best of our knowledge, ours is a first attempt to solve the problem by fully recognizing the interactions between all the routing, data link, and physical layers. For example, unlike the approach in [2], we account for dynamic link capacities obtainable by optimal power control in the physical layer. We also account for the energy consumption of the individual network nodes, rather than that of the total network only, in order to maximize the network lifetime. Our proposed solutions can be implemented

2

in a fully distributed manner. We first consider a special case of our problem in which the link-layer random access probabilities are given a priori. We formulate this special case as a joint power control and routing optimization problem, and prove that it is convex. Through the Lagrangian dual method, the problem is decomposed into two independent convex optimization problems, for power control in the physical layer and routing in the network layer, respectively, coordinated by suitable Lagrangian multipliers. The proposed solution, which we call the joint routing and power control algorithm (JRPA), is thus realized by two independent distributed protocols, which we call the power control protocol (PCP) and routing strategy protocol (RSP), for the physical and network layers, respectively. We prove theoretically that the proposed protocols are convergent, and that the optimal solution of the dual problem equals that of the primal problem. In practice, the best link access probabilities are frequently unknown a priori. Hence, after solving the special case, we discuss the general case, in which all the issues of routing, power control, and random link access are considered. The general problem cannot be converted into a convex problem, and there exists a duality gap between optimal solutions to the dual and primal problems. We give a heuristic distributed algorithm, which we call the joint routing, power control, and random access algorithm (JRPRA), to solve the general problem. We show through numerical experiments that JRPRA can significantly narrow the gap between the computed and optimal solutions. Moreover, even without a priori knowledge of the best link access probabilities made available to JRPA, JRPRA has extremely competitive performance with JRPA in the experiments. The experimental results also show the convergence of the proposed algorithms and their advantages over existing approaches. II.

RELATED WORK

Cristescu et al. use the Slepian-Wolf model to determine the rate allocation and routing strategies for gathering spatially correlated data in WSNs. Their goal is to minimize the total cost of data transmission, and they propose approximation algorithms as solutions [1]. Rickenbach et al. employ singleinput coding to aggregate correlated data along communication paths [3]. Given n nodes, they propose a MEGA algorithm for a foreign coding model, which aims to find a minimumenergy data gathering topology in O(n3 ) time. Yuen et al. point out that link capacities could be an important constraint for data gathering. They propose a distributed framework for gathering correlated data in WSNs through the Lagrangian dual method [2]. All the above efforts focus on minimizing the total transmission cost. They may over-stress nodes on the minimum energy paths. An alternative approach is to maximize the network lifetime, which is defined as the time until the first node in the WSN runs out of energy [4]. Chang et al. formulate maximum lifetime routing in WSNs as a linear programming problem. They propose distributed algorithms for constant information generation rates and arbitrary generation processes [5]. Madan et al. adopt a Lagrangian dual approach for the maximum

lifetime routing, and propose partially and fully distributed algorithms as solutions [4]. Hua et al. use data aggregation in an optimal routing strategy to maximize the network lifetime [6]. They present an approximation function for the integrated data aggregation and routing, based on which a distributed gradient search algorithm is designed. Kalpakis gives novel algorithms to solve the optimal routing problem using in-network data aggregation [7]. The above efforts all focus on the network layer only, and do not consider the impact of other layers. Cross-layer design has been applied in network protocol design. Madan et al. use the approach in optimizing routing and link scheduling (but not the physical layer) for the lifetime of WSNs [8]. Their problem deals with multiple independent traffic flows and does not exploit the data correlations between these flows. Long et al. use joint congestion control, random access, and power control to maximize the utility (e.g., throughput) of a wireless mesh network [9]. Chiang et al. provide an extensive survey on cross-layer network utility maximization (NUM) [10]. Addressing the network layer only, Cui et al. [11] approximate the lifetime of a wireless network with an objective function that network nodes can compute using local information only. We leverage their technique in our distributed implementation. However, none of the existing efforts optimize across all the routing, data link, and physical layers for our network lifetime maximization problem. III. P ROBLEM S TATEMENT Assume that there are N sensing nodes and one sink node in the region of interest. We can model the WSN as a directed graph G = (V, L), where V includes N sensing nodes and one sink node, and L denotes the directed link set. (i, j) ∈ L means that sensor node i can transmit data to sensor node j and dij is the distance between i and j. A sensor node monitors a local region (of arbitrary shape) determined by its sensing range for information about the region. For sensor node i, denote by yi the random variable of the information monitored by i. Given a subset X of N , let yX = (yi )i∈X denote the vector formed by random variables for the rates of data flows generated by the sensor nodes in X. A summary of the notation used in this paper is given in Table I. Note that we denote sets by capital letters, variables by lowercase letters, vectors by bold lowercase letters, and matrices by bold capital letters. Where there is no confusion, we may abuse notations and use capital letters to denote both sets and their cardinalities. A. Slepian-Wolf coding for correlated data A physical quantity (e.g., temperature) in the sensing region monitored by sensor node i is given as a random variable denoted by yi . If we want to gain full information about the physical quantity monitored by i, we have to transmit at least H(yi ) amount of data if i is the only node communicating its data to the sink node. However, if the physical quantities associated with different sensing regions are correlated and all the sensor nodes send their data, then it is possible for each node i to transmit less than H(yi ) amount of data without causing any information loss. Slepian and Wolf [12] show that, for two correlated sensor nodes i and j, their full information can be encoded with a total rate equal to their joint entropy

3

TABLE I N OTATION DEFINITIONS Symbol ri pij p¯ij xij qij hij Sij

Definition rate of data flow generated by sensor node i; transmission power allocated to link (i, j) maximum transmission power of link (i, j) total rate of data flows over link (i, j) random access probability associated with link (i, j) gain of link from transmitter i to receiver j set of sensor nodes whose transmissions may interfere with the receiver of link (i, j), excluding i set of links whose transmissions are interfered by transLi missions from node i, excluding outgoing link from i γij signal-to-interference-and-noise ratio H(y) entropy of random variable y H(y1 |y2 ) entropy of y1 conditioned on y2 background noise associated with link (i, j) σij

H(yi , yj ), if the individual data rates of the sensors are at least equal to the conditional entropies H(yi |yj ) and H(yj |yi ) for the two nodes, respectively. In our problem, allocating a data rate ri , i ∈ N , for each sensor node i can communicate full information about the region of interest if and only if these rates satisfy X ri ≥ H(yX |yX c ), (1) i∈X

where X is an arbitrary subset of N , X c = N − X, and yX c = (yi )i∈X c . B. Power control and random access In this section, we present an interference-based model for random access at the link layer. A link (i1 , j1 ) is under the interference from another link (i2 , j2 ) if the distance between the nodes j1 and i2 is less than some threshold. For communicating data (both its own data and data relayed for other nodes) to the sink, each sensor node i has a transmission probability of qi . When i determines to transmit data, it chooses one of its outgoingPlinks, say (i, j) ∈ L, with a probability qij , such that qij = qi . For link (i, j) to j:(i,j)∈L

be used for successful data transmission, it is necessary that (i) i chooses link (i, j) to transmit the data, (ii) none of i’s neighbors choose i as the receiver of data, and (iii) j does not choose to send data on any of its outgoing links. The probability that (i)–(iii)Pare satisfied is given P by ρij , and we have ρij = qij (1 − qj1 i )(1 − qji1 ). j1 :(j1 ,i)∈L

i1 :(j,i1 )∈L

Further to (i)–(iii) above, for the transmission on link (i, j) to be successful, it is necessary that the received signal at j is not garbled by another concurrent transmission not involving i and j. We use the signal to interference and noise ratio (SINR) model in [9] to characterize the condition of non-interference, in which the average SINR of link (i, j) is given by γij =

2 +θ σij

P

hij pij P

i1 ∈Sij (i1 ,j1 )∈L

hi1 j pi1 j1 qi1 j1

,

where θ is the orthogonality factor. Then, the transmission over link (i, j) is successful if the SINR, γij , is above a threshold (0) γij . The capacity of link (i, j) can in turn be determined by

the interference-based communication model as ½ (0) ρij log γij , γij ≥ γij cij = . 0, otherwise C. Flow conservation constraints We adopt multi-path routing to forward the sensed data to the sink node. For each sensor node i, associate a routing variable xij with each link (i, j) ∈ L. xij > 0 means that the link (i, j) is selected by sensor node i to forward messages to sensor node j, and xij = 0 means that the link (i, j) is not selected. Hence x is the routing decision variables as well as variables of flow rates. We assume lossless transmissions in this paper. For each sensor node i, i ∈ N , it has to satisfy the flow conversation constraint that the total data transmission rate by i is equal to the received data rate by i plus the rate of data generated by i itself. Formally, we have X X (2) xij − xji = ri , i ∈ N, j:(i,j)∈L

j:(j,i)∈L

where ri is the rate of data generated by i after Slepian-Wolf coding of the data. D. Network lifetime model We now give the energy model used in this paper. We mainly consider the energy consumed for communicating the sensor data to the sink node. Let etij and erij denote the power consumption for transmitting and receiving one unit of data over link (i, j), respectively. The total power consumption at sensor node i, denoted by wi , is equal to X X (3) wi = xij etij + xji erji . j:(i,j)∈L

j:(j,i)∈L

We assume that the sensor node i has a given limited amount of initial energy ei . Hence, its energy lifetime ti is given by ei ti = (4) . wi The lifetime of the network is defined as the time from the initial deployment of the network to the time that the first sensor runs out of energy in the network. The maximum network lifetime problem is thus: max min ti . i∈N

(5)

E. Cross-Layer optimization model We need to maximize the network lifetime while guaranteeing that full information about the network can be communicated to the sink node. The goal can be expressed as the following cross-layer design problem: max min ti  (0)  γij ≥ γij , (i, j) ∈ L     xij ≤ cij , (i, j) ∈ L s.t. 0 ≤ pij ≤ p¯ij , (i, j) ∈ L    1 − qi ≥ 0, i ∈ N   constraints (1), (2), (3), (4)

(6)

There are two challenges in solving the above problem in a distributed manner. First, we will need to disseminate

4

information about the remaining energies of the sensor nodes throughout the network, which results in high communication overhead. Existing work has used approximation algorithms for solving the maximum network lifetime problem. Following [11], we can approximate the objective function (6) as α → ∞ P ziα−1 by min α−1 , where zi = 1/ti . As discussed in [11],

Let peij = log pij . Taking log of (8), we change the inequality to (0)

log γ eij ≥ γ eij , (0)

(0)

where γ eij = log γij and γ eij is given by

i∈N

the computation of zi only needs local information exchange between the sensor nodes. We will use this approximation approach to design effective distributed algorithms with much less communication overhead than a brute-force approach. Second, global network information is needed when using Slepian-Wolf coding to allocate data rates to the sensor nodes. Cristescu et al. show that a localized form of the Slepian-Wolf coding can be employed to give a distributed approximation algorithm for the rate allocation with a known approximation ratio [1]. Specifically, let Ni denote the subset of sensor node i’s neighbors that are closer to the sink node than i, where node j is said to be i’s neighbor if the distance between them is less than a certain threshold. The localized Slepian-Wolf coding specifies that each sensor node i should encode its data at a rate ri equal to the entropy of its associated RV conditioned on the RVs of Ni , i.e., ri = H(yi |yNi ). Then the optimization of (6) can be approximated as the problem: PP :

min

(7)

(0) γij ,

γij ≥ (i, j) ∈ L 0 ≤ xij ≤ cij , (i, j) ∈ L 0 ≤ pij ≤ p¯ij , (i, j) ∈ L X X xij − xji = ri , i ∈ N j:(i,j)∈L

2 σij

hij epeij P

P

+θ

i1 ∈Sij (i1 ,j1 )∈L

hi1 j qi1 j1 epei1 j1

.

(13)

We have the following theorem. Theorem 1: Assume that q is fixed. After a log change of p in inequality (8), (PP) is a convex optimization problem. Proof: When α ≥ 2, the objective function (7) is convex in x, and z are dummy variables given by (3) and (4). The constraints (10) and (11) are clearly convex. The constraints (9) and (12) are convex provided that log γ eij is a concave function. In fact, log γ eij

= log hij + peij 2 − log(σij +θ

X

X

hi1 j epei1 j1 qi1 j1 ).

i1 ∈Sij (i1 ,j1 )∈L

The function log(

P

ai exi ) is convex if ai ≥ 0 [8]. In that

case, log γ eij is a concave function. Hence (PP) is convex. A. Lagrangian dual decomposition

i∈N

s.t.

γ eij =

i

X z α−1 i α−1

(12)

(8) (9) (10) (11)

j:(j,i)∈L

The power allocation variables p and routing variables x are coupled by the inequality (9). We use the Lagrangian dual method to decouple them in the dual problem. Since the original problem is convex (Theorem 1), there is no duality gap between the optimal primal and dual solutions. Define the Lagrangian as [13, Sec.6]

1 − qi ≥ 0, i ∈ N constraints (3), (4) .

L(x, p, µ)

= L1 (x, µ) − L2 (p, µ),

where To solve (PP), we have to decide a rate allocation for each sensor node, the power allocation and random access probability for each link, and the routing of data from the sensors to the sink. Using Slepian-Wolf coding, we can use a shortest path algorithm such as Bellamn-Ford to compute the distance between each sensor node and the sink, and then assign a rate to each sensor node accordingly. The remaining problem is then to jointly optimize the power control, link random access, and routing to maximize the network lifetime. IV. J OINT POWER CONTROL AND ROUTING The cross-layer optimization problem above is non-convex and non-separable. We now consider a restricted special case of the problem in which the link random access probabilities are known a priori. With the random access fixed, we only focus on the joint power control and routing problem. We will show that after suitable transformations of the variables and inequality constraints, the problem (PP) can be expressed as an equivalent convex problem. We will consider the general cross-layer optimization problem in Section V.

X X z α−1 i + µij xij , α−1 i∈N (i,j)∈L X = µij ρij log γ eij .

L1 (x, µ)

=

L2 (p, µ)

(i,j)∈L

Let D1 (µ) be the solution of the following minimization: DP1 : L (x, µ) P P 1  xij − xji = H(yi |yNi ), i ∈ N    j:(i,j)∈L j:(j,i)∈L P P xij etij + xji erji = ei zi , i ∈ N s.t. ,  j:(i,j)∈L j:(j,i)∈L   xij ≥ 0 and D2 (µ) be the maximum of the problem DP2 :

( s.t.

where e p¯ij = log p¯ij .

L2 (p, µ) (0)

log γ eij ≥ γ eij peij ≤ e p¯ij

,

5

Algorithm 1 Power Control Protocol (PCP)

The dual problem (DP) to the primal problem (PP) is max D(µ), µº0

where the Lagrangian dual objective function D(µ) = D1 (µ) − D2 (µ). Because of the convexity of (PP), the dual problem (DP) can be solved by a gradient projection method, i.e., µij can be updated according to µij (t + 1) = [µij (t) + δ(xij − ρij log γ eij )]+ ,

(14)

where δ is the step size and [a]+ = max(0, a). The dual objective function D(µ) is decomposed into two independent subproblems: DP1 , for power control in the physical layer, and DP2 , for routing in the network layer. The decomposition corresponds to a vertical decomposition across the protocol stack. The two subproblems are coordinated by the Lagrangian multipliers µ. In the following, we will design two distributed algorithms for the power control and routing, respectively, which corresponds to a horizontal decomposition across the protocol stack. B. Power control

e µ, β) L2 (p, X X (0) = µij ρij log γ eij − βij (e γij − log γ eij ). (i,j)∈L

C. Routing design We proceed to solve DP1 , i.e., determine routing of the data from the sensor nodes to the sink. The network flows have to follow the flow conservation constraint and be routed such that the energy consumption can be balanced among the sensors. Associating Lagrangian multiplier λi with each flow conservation constraint, we get the relaxed function: L1 (x, λ, µ) P ziα−1 P P (µij + λi − λj )xij − λ i ri . = α−1 +

∂L1 ∂xij ∂L1 ∂λi

etij erij + zjα−2 + µij + λi − λj , ei ej X X xij − xji − ri .

= ziα−2 =

j:(i,j)∈L

j:(j,i)∈L

As DP1 is a convex problem, it can be solved by

(15)

λi (t + 1)

=

xij (t + 1)

=

∂L1 (t)]+ , ∂λi ∂L1 [xij (t) − δ (t)]+ . ∂xij

[λi (t) + δ

(19) (20)

Algorithm 2 gives the Routing Strategy Protocol (RSP). Algorithm 2 Routing Strategy Protocol (RSP)

(0)

log γ eij − γ eij ,

= µij ρij + βij −

(18)

i∈N

(i,j)∈L

Taking derivative of L1 (x, λ, µ), we obtain

(i,j)∈L

It can be easily obtained that =

each iteration of protocol, for each link (i, j): Collect pi1 j1 and hi1 j from the interference set Sij ; Update βij according to (16); Communicate the updated βij to the sensors in Li . each iteration of protocol, for each sensor node i: Collect βi1 j1 , hij1 , hi1 j1 , and pi1 j1 from link (i1 , j1 ), (i1 , j1 ) ∈ Li ; b) Update peij according to (17); c) Communicate the updated power allocation to Sij .

i∈N

In this section, we give a distributed protocol for optimal power allocation in the physical layer. The first objective for power control is to make sure that the SINR of link (i, j), γij , (0) is larger than the threshold γij . The second objective is to determine suitable link capacities that can attain the required data rates while maximizing the network lifetime. The two objectives can be achieved by solving DP2 . Note that DP2 is also a convex optimization problem, and we can use further Lagrangian multipliers β to relax the problem. Let

∂L2 ∂βij ∂L2 ∂ peij

1) At a) b) c) 2) At a)

X

(µi1 j1 ρi1 j1 + βi1 j1 ) ×

(i1 ,j1 )∈Li

hij1 epeij . γ ei1 j1 qij hi1 j1 epei1 j1 The primal-dual method [14] can be used to solve the problem (15), i.e., the power allocation variables and Lagrangian multipliers can be updated according to ∂L2 βij (t + 1) = [βij (t) − δ (t)]+ , ∂βij ∂L2 peij (t + 1) = [e pij (t) + δ (t)]ep¯ij , ∂ peij where [a]ep¯ij = min(a, e p¯ij ). We integrate the above design choices in Algorithm 1:

(16) (17)

1) At a) b) c) 2) At a) b) c)

each iteration of protocol, for each link (i, j): Collect zi , zj , λi , λj , ei , ej from nodes i and j; Update the routing variable xij according to (20); Communicate the new xij to the nodes i and j. each iteration of protocol, for each sensor node i: Get xij , (i, j) ∈ L, xji , (j, i) ∈ L, from network; Update λi according to (19); Communicate the new λi to the links for which i is either the sender or receiver.

D. Joint routing and power control algorithm (JRPA) We are now ready to establish the Joint Routing and Power control Algorithm (JRPA) for the correlated data gathering problem. In the physical and network layers, respectively, PCP and RSP operate independently to update their power allocation and routing strategies. The interface variables µ are used to coordinate the two strategies. That is, JRPA utilizes µ to control the performance of the two sub-algorithms and

6

Algorithm 3 Joint Routing and Power Control Algorithm (JRPA) 1) Initialization: a) For each link (i, j), choose initial routing variable xij and Lagrange multiplier µij . b) For each sensor node i, choose initial power level pij for each link (i, j); use shortest path method to decide the distance from the sink node and communicate with its neighbors to decide its rate allocation ri . 2) At each iteration, use PCP to solve the power allocation; 3) Perform RSP to solve the routing; 4) Update each µij according to (14).

regulate the power allocation and routing strategies towards the optimal solution. The algorithm is given in Algorithm 3. Theorem 2: Choosing a small enough δ and starting from an arbitrary initial point, 0 ¹ p ¹ p¯, x º 0, the algorithm JRPA converges statistically to the optimal solution of (PP). Moreover, the solution obtained is primal-dual optimal. Proof: Because of limited space, we only sketch the proof. Define the Lyapunov function as: 1 X (µij (t) − µ∗ij )2 . V(µ(t)) = 2δ (i,j)∈L

Following the steps in [9], we can derive V(µ(t+1))+

t X

(D(µ(τ ))−D(µ∗ )) ≤ V(µ(1))+

τ =1

βB , (21) 2

where B is a bounded constant. From (21), by utilizing the concavity of the dual function D(µ), we get ¯ lim sup (D(µ(t)) − D(µ∗ )) ≤ t→∞

¯ = where µ(t)

1 t

t P

Proof: JRPA solves the primal-dual problem by the subgradient method, whose convergence rate is shown to be linear in [16]. V. J OINT ROUTING , P OWER CONTROL , R ANDOM ACCESS JRPA works well when the best link access probabilities are known a priori. Such knowledge is frequently unavailable in real networks. In this section, we consider the general case of problem (PP), i.e., we do not assume that the link access probabilities are given, but will jointly optimize the routing, power control, and random access. With the change, the link capacity constraint (9) cannot be transformed into a convex function f (x, p, q) ≤ 0, and problem (PP) is no longer convex. Thus there exists a duality gap between the primal and dual problems. In this section, we will develop a heuristic algorithm for solving the general optimization problem. In the next section, we will show by numerical results that the algorithm is effective in approximating the optimal solutions. P Let qeij = log qij , qei = eqeij and ρeij = eqeij (1 − P P j:(i,j)∈L eqej1 i )(1 − eqeji1 ). We now give the Joint j1 :(j1 ,i)∈L

i1 :(j,i1 )∈L

Routing, Power control, and Random access Algorithm (JRPRA). The main procedure is similar to that of JRPA (see Algorithm 3) in the previous section, and we will just state the differences between the algorithms, in terms of updates for the variables βij , peij , qeij , xij and µij , κi . In JRPRA, the PCP component of JRPA is replaced by a corresponding PCRAP protocol, where the link power βij , peij , κi and qeij are updated according to

δB , 2

βij (t + 1)

=

κi (t + 1)

=

(0)

[βij (t) − δ(log γ eij − γ eij )]+ , X [κi (t) − δ(1 − eqeij )]+ ,

µ(τ ).

Then from [15], given a small enough δ, µ(t) converges statistically to the optimal value µ∗ . From Theorem 1, the primal problem (PP) is convex, the solution obtained by JRPA is primal-dual optimal. A key issue about JRPA is its communication complexity in terms of the number of message exchanges required by the algorithm. The analysis is as follows. For PCP, at each iteration, information about each link (i, j) only has to be communicated to sensor nodes in the interference set Sij to obtain pi1 j1 and hi1 j , and each sensor node i only has to communicate with links in the set Li to obtain βij , hij , hij , and pij . This information is local-dependent only. For RSP, although the function (18) is coupled with x, and updating xij and λi needs information from the network, the required information is all local-dependent since it concerns the sensor nodes i and j and the link (i, j) only. Hence, at each iteration of the protocols, each node only has to exchange a (small) constant amount of information with each of its neighbors. The following theorem states the convergence rate of JRPA. Theorem 3: The JRPA algorithm converges to the optimal solutions of (PP) at a linear rate.

(23)

j:(i,j)∈L

τ =1

E. Complexity discussion

(22)

peij (t + 1)

=

qeij (t + 1)

=

∂L2 (t)]ep¯ij , ∂ peij ∂L2 [e qij (t) + δ (t)]− . ∂ qeij

[e pij (t) + δ

Here, [a]− = min(0, a) and ∂L2 ∂ peij

= βij +

µij − log γ eij

∂L2 ∂L2 ∂p eij , ∂ qeij

X

(24) (25)

are given by

(βi1 j1 +

(i1 ,j1 )∈Li

µi1 j1 ) log γ ei1 j1

hij1 epeij , hi1 j1 epei1 j1 X µji1 eqeij = µij − P 1− eqej1 j i :(j,i )∈L ×θe γi1 j1 eqeij

∂L2 ∂ qeij

1

−

1

j1 :(j1 ,j)∈L

X j1 :(j1

1− ,i)∈L

X (i1 ,j1 )∈Li

µj1 i eqeij P

eqeii1

+ κi eqeij −

i1 :(i,i1 )∈L

(βi1 j1 +

hij1 epeij µi1 j1 )θe γi1 j1 eqeij . log γ ei1 j1 hi1 j1 epei1 j1

We summarize the PCRAP protocol in Algorithm 4

7

Algorithm 4 Power Control & Random Access Protocol (PCRAP) 1) At each iteration of protocol, for each link (i, j): a) Collect pi1 j1 , qi1 j1 and hi1 j from the interference set Sij and κi from sensor node i; b) Update βij and qeij according to (22) and (25), respectively; c) Send the new βij , qij to the sensor nodes in Li . 2) At each iteration of protocol, for each sensor node i: a) Collect βi1 j1 , hij1 , hi1 j1 , and pi1 j1 from link (i1 , j1 ), (i1 , j1 ) ∈ Li ; b) Update κi and peij by (23) and (24), respectively; c) Communicate the new power allocation to Sij .

In the network layer, RSP remains the same as in JRPA, except that the routing variables xij are updated according to xij (t + 1) = [xij (t) + δ where

∂L1 ∂xij

∂L1 (t)]+ , ∂xij

is given by

etij erij ∂L1 µij = ziα−2 + zjα−2 + + λi − λj , ∂xij ei ej xij and µij is updated according to µij (t + 1) = [µij (t) + δ(log xij − log(log γ eij ) − log ρeij )]+ .

(26)

We summarize the JRPRA algorithm in Algorithm 5. Algorithm 5 Joint Routing, Power Control, and Random Access Algorithm (JRPRA) 1) Initialization: Similar to JRPA. 2) At each iteration, perform PCRAP to decide the power allocation and link access probabilities; 3) Use modified RSP to solve the routing; 4) Update each µij according to (26).

VI. I MPLEMENTATION I SSUES We discuss how the JRPA and JRPRA algorithms can be implemented in a practical WSN. A. Modeling the data correlations We assume that correlations in the components of y can be modeled as conditional entropies. In practice, joint distributions of information about a monitored region may not be exactly known. However, a widely accepted model found to provide excellent approximations in many real application scenarios is the Gaussian process [17]. For example, successful applications of the Gaussian process to model temperature distributions in real life can be found in [18]. In our problem context, since there is a finite number of sensor nodes in the RoI, we can adopt the multivariate normal distribution, a special case of the Gaussian process, to estimate the distribution of the correlated data. Specifically, the joint N dimensional multivariate normal distribution, GN (ν, K), for the spatial data YN measured at the N nodes, is given by T −1 1 f (YN ) = √ e−(0.5(YN −ν) K (YN −ν)) , (27) 1/2 2π det (K)

where K is the covariance matrix of YN , and ν is the mean vector. Obviously, the multivariate normal distribution is decided by the parameters K and ν. One advantage of this distribution is that, for yi , the conditional distribution f (yi |yic ) is also a normal distribution, whose variance Kyi |yic and mean vector νyi |yic are given by νyi |yic ky2i |yic

= νyi + Kyi ,yic Ky−1 (yic − νic ), ic ,yic =

ky2i ,yi

−

Kyi ,yic Ky−1 Kyic ,yi . ic ,yic

(28) (29)

In a field deployment, sensors can measure their distances from each other using a specialized protocol [1] or a more general localization protocol [2]. Correlations of data between pairs of (close-by) sensors can then be estimated based on the measured distances and domain knowledge, e.g., gradual variations of temperature over space [18]. Alternatively, the correlations can be calibrated. During a startup phase, all the sensor nodes transmit what they sense to the sink node. After collecting enough data samples, the sink node computes the conditional probability distributions of the sensed data (for pairs of reasonably close-by nodes) and sends the results to the sensors. The sensor nodes then use the conditional distributions to implement the proposed protocols and algorithms. To account for dynamic changes in the external environment, if and when some of the sensors observe significant changes in their measurements, re-calibrations of the correlations may be performed for these sensors. B. Implementation of Lagrangian multipliers We now address the implementation of Lagrangian multipliers for JRPA and JRPRA. We will focus on the case of JRPA only, as JRPRA can be handled in a similar way. The main issue is how to implement the Lagrangian multipliers µ, β, and λ. Lagrangian multipliers, in general, can be interpreted as the prices for the supply/demand of a goods [19]. µij , (i, j) ∈ L, corresponds to the congestion of link (i, j) and can be interpreted as the congestion price. From (14), we can see that when the rate of link (i, j) exceeds the capacity of (i, j), µij will increase, and vice versa. Random Exponential Marking (REM) [20] is known to be a highly effective algorithm for evaluating the path congestion price. In our problem, each link (i, j) adopts a similar REM process to measure its congestion level and update the price µij accordingly. For β, it can be interpreted as the price of not reaching a specified signal-to-interference-and-noise ratio (SINR). Hence, each sensor node can measure its SINR at each iteration, and update the SINR price according to (16). For λ, it can be interpreted as the price of inconsistent coordination of the incoming and outgoing flows [21]. It can be implemented similarly as µ and β. .

. .



.

Fig. 1.

N

.

.

Topology of the simulated WSN.

8

0.3

p12

p23

0.2 p46

0.18

1.5 x24 1

p56

0.16

x12 x23

0.5

0.14

0

1000

2000 3000 Iteration number (n)

4000

0

5000

(a) The convergence of PCP in JRPA.

x35

0

1000

2000 3000 Iteration number (n)

4000

0.5 0.45 q

24

0.4

q35

0.35 0.3 q

0.25

23

q12

0.2

5000

(b) The convergence of RSP in JRPA. Fig. 2.

Acess probability of each link

2

0.24 0.22

q56

0.55 Rate of each link (kbps)

Power of each link (w)

q46 0.6

p35

0.26

0.12

0.65

x46 x 56

2.5

p24

0.28

0

1000

2000 3000 Iteration number (n)

4000

5000

(c) The convergence of PCRAP in JRPRA.

The convergence of JRPA and JRPRA.

VII. N UMERICAL R ESULTS In this section, we report numerical experiments to evaluate the performance of JRPA and JRPRA proposed in Sections IV and V, respectively. We also discuss the advantages of these algorithms over the proposed solution in [2]. We consider the WSN topology shown in Fig. 1, which consists of k + 4 sensor nodes and one sink node. There are hence n = k + 5 network nodes and the sink node has index n. The distances between (1, 2), (2, 3), · · · , (k +1, k +2), (k + 1, k+3) and (k+2, k+4) are all equal to 15, and the distances between (k + 3, k + 5) and (k + 4, k + 5) are equal to 10. A transmission on one link interferes with a transmission on another if the distance between the receiver of the first link and the sender of the second link is at least 15 m. For the SINR 2 interference model, we set σij = 5 × 10−13 , hij = 0.097/dij , (0) γij = 50 and θ = 1/256. The correlation model f (YN ) can be decided in applications as discussed in section VI-A. For simplicity, we set kij = 2 e−0.0006dij . According to [1], the entropy of yi conditioned on yNi is given by µ ¶ det K[yi , yNi ] 1 (30) , H(yi |yNi ) = log 2πe 2 det K[yNi ] where K[y] denotes the covariance matrix formed by the random variables y. When the network starts, each node uses Bellamn-Ford to get its shortest path from the sink node. Then a node, say i, forms the set Ni by exchanging local messages with its neighbors. By using (30), the data rate originating from node i, ri , can be obtained. Note that by using SlepianWolf coding, the sensor nodes which are closer to the sink node will have higher original data rates, thus reducing the communication cost. Following the communication model in [21], we set erij = 50n J/bit and etij = % + ςdm , where % = 50nJ/bit, ς = 0.0013p J/b/m4 , and m = 4. The initial energies of the sensors 1–5 are set to be 2500 J. The sink node 6 is assumed to have enough energy, and we do not account for its energy use. We will first set k = 1, i.e., there are five sensor nodes and one sink node in the network. We use the baseline topology to show the convergence of our proposed algorithms. Then we vary k to be 2, 3, 4, 5, and show the advantages of our proposed algorithms over existing approaches, as the size of the network increases.

A. Algorithm performance evaluation a) JRPA algorithm: In this special case problem, we assume that the best random access probabilities (separately determined) are known, and use them as fixed input to the JRPA algorithm. (More generally, any consistent set of the probabilities could be used, although the performance would vary.) Setting the step size α = 0.12, we collect the values of p and x at each iteration, and the results are shown in Fig.2. From Fig. 2(a) and Fig. 2(b), we can see that p and x all converge to the optimal solutions. As discussed, the optimal dual solution is also primarily optimal. b) JRPRA algorithm: We evaluate the JRPRA algorithm in Section V. In this case, the random access probabilities q are no longer fixed. Instead, they are decided in the iterative process of the algorithm. Fig. 2(c) gives the results. The updates of p and x are very similar to those in JRPA. So we plot the updates for q only and list the final achieved solution in Table II. From Fig. 2(c), note the convergence of JRPRA. Table II shows the solutions obtained by JRPA and JRPRA. The power allocation p and routing strategy x are very similar in the two algorithms. These results indicate that JRPA is a special case of JRPRA (with q fixed). More importantly, JRPRA performs extremely competitively with JRPA, but does so without a priori knowledge of the best link access probabilities. TABLE II S OLUTION COMPARISON BETWEEN JRPA AND JRPRA. Routing variables (kb/s) JRPA JRPRA Power variables (W) JRPA JRPRA Access probabilities JRPA JRPRA

x12 x23 x24 x35 x46 x56 0.2610 0.0980 0.8611 0.7961 2.2799 2.2149 0.2610 0.0980 0.8611 0.7961 2.2799 2.2149 p12 p23 p24 p35 p46 p56 0.2012 0.2011 0.2663 0.2528 0.1624 0.1426 0.2009 0.2006 0.2666 0.2531 0.1625 0.1428 q12 q23 q24 q35 q46 q56 0.1746 0.1871 0.3602 0.3431 0.5704 0.5641 0.1746 0.1871 0.3602 0.3431 0.5704 0.5641

B. Algorithm comparison We proceed to discuss the advantages of the proposed algorithms over existing solutions. Similar to [2], much existing work focuses on the minimum energy consumption (MinE) problem for correlated data gathering in WSNs. Also, many

9

TABLE III S OLUTION COMPARISON BETWEEN OUR AND Routing variables MinE MinE-NC JRPA JRPRA Bandwidth (JRPA)

EXISTING APPROACHES .

x12 x23 x24 x35 x46 x56 Lifetime 0.2610 0 0.9590 0.698 2.378 2.117 4161 h 1.42 0 2.84 1.42 4.26 2.84 1957 h 0.2610 0.0980 0.8611 0.7961 2.2799 2.2147 4421 h 0.2610 0.0980 0.8611 0.7961 2.2799 2.2147 4421 h 0.4474 0.6655 0.8611 0.7962 2.280 2.2147

existing efforts on the optimal routing problem do not account for the spatial correlation of information between the sensors. We denote the use of MinE in this case by MinE-NC. These metrics do not optimize the network lifetime as we seek, since minimizing the total energy consumption may unduly stress certain sensor nodes and cause them to die quickly, and not exploiting the spatial correlation of information may lead to the communication of unnecessary data and waste energy. We can see these effects in Table III. The solutions obtained by MinE [2], MinE-NC, JRPA, and JRPRA are listed in the table. Like before, the performance of JRPRA is very similar to that of JRPA (also like before, JRPRA does not require a priori knowledge of the best link access probabilities). This is shown in Table II, and the network lifetimes achieved by JRPA and JRPRA are both about 4421 h. The network lifetime obtained by MinE is 4161 h, which is significantly shorter than that obtained by JRPA. MinE-NC obtains a still shorter network lifetime of 1957 h. We conclude that the limitations of MinE and MinE-NC in prolonging the network lifetime are apparent relative to JRPA and JRPRA. Lastly, Yuen et al. [2] assume that the capacity of each link is fixed. If the transmission data rate over a shortest-path link exceeds the link’s capacity, the extra traffic has to be routed to bypass the link, causing extra energy consumption. For example, in Fig. 1, if node k + 3 has to transmit 4 kb of data to node k + 5 and the capacity of link (k + 3, k + 5) is 3 kb, then node k + 3 is forced to transmit 1 kb of data to node k +4. In contrast to their approach, we use power control to adaptively provision sufficient bandwidth for each link. By comparing the actual link rates on the 5th row of Table III and the provisioned link bandwidth on the 7th row, we can see that the power control is effective. C. Network performance under varying number of nodes In this section, we vary k to be 2, 3, 4, 5, and report results for MinE, MinE-NC, JRPA, and JRPRA. The network lifetimes achieved by the four algorithms are listed in Table IV. We can see that as the number of sensor nodes increases, the network lifetime decreases sharply for MinE and MinENC, whereas the decreases are much slower for JRPRA and JRPA. The performance gain is achieved by optimizing for the network lifetime globally, rather than for the total energy consumption. We conclude that the proposed algorithms are significantly more energy efficient than MinE and MinE-NC. Importantly, the energy savings become more pronounced as the network size increases. VIII. C ONCLUSIONS We solved the problem of optimal gathering of correlated data in WSNs for maximum network lifetime. We adopted

TABLE IV L IFETIME OF THE NETWORKS WITH INCREASING NUMBER OF SENSORS . Sensor node MinE MinE-NC JRPRA (JRPA)

6 7 4161 h 3598 h 1957 h 1398 h 4421 h 4045 h

8 9 3169 h 2831 h 1087 h 900 h 3737 h 3461 h

10 2559 h 753 h 3303 h

a comprehensive cross-layer approach, which has advantages over existing solutions. We first considered a special case of the problem and showed that it is convex. The JRPA solution decomposed the problem into two independent convex subproblems of power control and routing. We presented two distributed protocols, PCP and RSP, for their solutions. We then solved the general case of the optimization involving all three layers of routing, power control, and link access, using the heuristic distributed algorithm JRPRA. Numerical results validated our analysis and confirmed the effectiveness of the solutions. They demonstrated the advantages of the proposed algorithms over existing approaches. R EFERENCES [1] R. Cristescu, B. Beferull-Lozano, and M. Vetterli. On network correlated data gatherings. In Proc. INFOCOM, 2004. [2] K. Yuen, B. Liang, and B. Li. A distributed framework for correlated data gathering in sensor networks. IEEE Trans. Vehicular Tech., 2008. [3] P. Rickenbach and R. Wattenhofer. Gathering correlated data in sensor networks. In Proc. of ACM DIALM-POMC, 2004. [4] R. Madan and S. Lall. Distributed algorithms for maximum lifetime routing in wireless sensor networks. IEEE Trans. Wireless Comm., 2006. [5] J. Chang and L. Tassiulas. Maximum lifetime routing in wireless sensor networks. IEEE Trans. Networking, 2004. [6] C. Hua C. and T. Yum. Optimal routing and data aggregation for maximizing lifetime of wireless sensor networks. IEEE/ACM Trans. Networking, 2008. [7] K. Kalpakis, K. Dasgupta, and P. Namjoshi. Efficient algorithms for maximum lifetime data gathering and aggregation in wireless sensor networks. Computer Networks, 2003. [8] R. Madan, S. Cui, S. Lall, and A. Goldsmith. Cross-layer design for lifetime maximization in interference-limited wireless sensor networks. IEEE Trans. Wireless Comm., 2006. [9] C. Long, B. Li, and Q. Zhang. The end-to-end rate control in multiplehop wireless networks: Cross-layer formulation and optimal allocation. IEEE JSAC, 2008. [10] M. Chiang, S. Low, A. Calderbank, and J. Doyle. Layering as optimization decomposition: A mathematical theory of network architectures. Proceedings of IEEE, 95(1):255–312, 2007. [11] Y. Cui, Y. Xue, and K. Nahrstedt. A utility-based distributed maximum lifetime routing algorithm for wireless networks. IEEE Trans. Vehicular Technology, 2006. [12] D. Slepian and J. Wolf. Noiseless coding of correlated information sources. IEEE Trans. Information Theory, 1973. [13] S. Mokhtar and C. Shetty. Nonlinear Programming : Theory and Algorithm. John Wiley & Sons, 1979. [14] D. Palomar and M. Chiang. Alternative distributed algorithm for network utility maximization: Framework and applications. IEEE Trans. Automatic Control, 2007. [15] L. Chen, S. Low, and J. Doyle. Joint congestion control and media access control design for ad hoc wireless networks. In Proc. INFOCOM, 2005. [16] N. Shor. Minimization Methods for Non-Differentiable Functions. Springer-Verlag, 1979. [17] N. A. Cressie. Statistics for Spatial Data. Wiley, 1991. [18] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg. Near-optimal sensor placements: Maximizing information while minimizing communication cost. In Proc. of IPSN, 2006. [19] F.P. Kelly. Charging and rate control for elastic traffic. European Trans. Telecommunications, 1997. [20] S. Athuraliya, D. Lapsley, and S. Low. An enhanced random early marking algorithm for Internet flow control. In IEEE Infocom, 2000. [21] J. Chen, S. He, Y. Sun, P. Thulasiraman, and X. Shen. Optimal flow control for utility-lifetime tradeoff in wireless sensor networks. Computer networks, 2009.