Optimal replicator factor control in wireless sensor

J Control Theory Appl 2011 9 (1) 115–120 DOI 10.1007/s11768-011-0230-0

Optimal replicator factor control in wireless sensor networks Meng ZHENG 1,2 , Haibin YU 1 , Wei LIANG 1 , Xiaoling ZHANG 1,2 1.Key Laboratory of Industrial Informatics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang Liaoning 110016, China; 2.Graduate School of the Chinese Academy of Sciences, Beijing 100039, China

Abstract: For TDMA MAC protocols in wireless sensor networks (WSNs), redundancy and retransmission are two important methods to provide high end-to-end transmission reliability. Since reliable transmissions will lead to more energy consumption, there exists an intrinsic tradeoff between transmission reliability and energy efficiency. For each link, we name the number of its reserved time slots in each MAC superframe as a replicator factor. In the following paper, we propose a reliability-lifetime tradeoff framework (RLTF) for WSNs to study replicator factor control problem. First, for the redundancy TDMA MAC, we formulate replicator factor control problem as convex programming. By the gradient projection method, we develop a fully distributed algorithm to solve the convex programming. Second, for the retransmission TDMA MAC, we set the retransmission upper bound for each link according to the optimal replicator factors under the redundancy MAC and compute the total communication overhead for the retransmission MAC. Finally, we compare the communication overhead of these two MAC protocols under different channel conditions. Keywords: Wireless sensor networks; Transmission reliability; Network lifetime; Replicator factors

1

Introduction

WSNs always consist of low-cost, low-power, and energy-constrained sensors responsible for monitoring a physical phenomenon and reporting to access points (APs) where the end-user can access the data [1]. In many applications, wireless sensor nodes usually carry limited and irreplaceable power supply, and thus, we have to design energy-aware protocols in order to prolong the operational lifetime of WSNs [2∼6]. Besides energy constraints, transmission reliability is another concern in data-centric WSNs, especially for industry wireless networks. Transmission reliability research in WSNs has been addressed in many works from different perspectives [7, 8]. Various network techniques, such as congestion control in transport layer [9], multipath routing in network layer [10], CSMA [11] and TDMA scheduling [12] in MAC layer, power control, routing and ARQ control in the cross-layer method [13], are used to enhance the reliability of wireless transmission. In this paper, we study TDMA MAC protocol to provide high end-to-end reliable transmission while prolonging the network lifetime. For reliable data delivery, redundancy and retransmission are two methods that are widely adopted in TDMA MAC protocols [12]. The difference between these two methods lies in whether acknowledgement (ACK) is used when a data packet is correctly received. Both of these two methods send multiple copies of source data. We name the number of copies on links as replicator factors. Usually, replicator factors are all set as a constant in a heuristic fashion. However, the heuristic way of choosing replicator factors is not efficient, since larger factors may lead to a huge waste of energy, and smaller factors may not guarantee the desired transmission reliability. Thus, there is an in-

trinsic tradeoff between transmission reliability and energy efficiency. Does there exist an optimal setting for replicator factors? This question is with high value in academic and engineering field. However, as far as we know, so far, there has been little work devoted to devising optimal replicator factors for TDMA MAC in WSNs. Motivated by the fact above, the RLTF is proposed to rigorously study the impact of replicator factors on system performance (transmission reliability and energy efficiency) in a mathematical manner. First, we concentrate on the redundancy TDMA MAC, i.e., hop-by-hop redundancy transmission without ACK. We model the utility of network reliability as the weighted sum of the utility of each route’s reliability, which can be expressed by the logarithm function of the end-to-end transmission reliability. Due to the fact that both the reliability and network lifetime are correlated with replicator factors, we combine these two conflicting objectives into a single weighted objective by introducing a weighted parameter; consequently, the RLTF is formulated as convex programming with respect to replicator factors. Based on the gradient projection method, a fully distributed way is proposed to achieve optimal replicator factors. Then, we extend the above research to the retransmission TDMA MAC, i.e., hop-by-hop retransmission with ACK. By setting the retransmission upper bound for each link according to the optimal replicator factors, we may save energy under the retransmission TDMA MAC while guaranteeing the same reliability level as that of the redundancy TDMA MAC. The overhead comparison between these two MAC protocols helps us choose the more energy-efficient MAC under different channel states. Our contribution is therefore threefold. First, for two

Received 13 October 2010. This work was supported by the National Science Foundation of China (No. 60704046, 60725312, 60804067), the National Science Foundation of Liaoning Province (No. 20092083), the National 863 high technology research and development Plan (No. 2007AA04Z173, 2007AA041201). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011

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M. ZHENG et al. / J Control Theory Appl 2011 9 (1) 115–120

kinds of TDMA MAC protocols, we propose RLTF to rigorously study the replicator factor control problem. Then, we formulate the RLTF model as a convex optimization problem, and a distributed algorithm is derived to optimize replicator factors. Finally, theoretical analysis on the communication overhead between these two MAC protocols is carried out. The rest of the paper is organized as follows. In Section 2, related works are reviewed. In Section 3, we study the RLTF model under redundancy TDMA MAC. In Section 4, we study the RLTF model under retransmission TDMA MAC. In Section 5, simulation results illustrate the effectiveness of our algorithm. In Section 6, we conclude this paper and point out the future work.

work in WSN [17∼20]. Most of them concentrated on the development of fully distributed implementation or the development of extended models in [16]. This work is similar to the rate-lifetime tradeoff in WSNs [16∼20]. However, the objective (i.e., reliability-lifetime tradeoff) and optimizing variables (i.e., replicator factors) in this paper are different from those (i.e., rate-lifetime tradeoff and power, scheduling, and other parameters) of references [16∼20]. Though the RLTF model in this paper is also convex and solvable by distributed algorithm, the modeling method and the problem context are totally different. For data-centric WSNs, transmission reliability and energy consumption are usually the two main concerns. Thus, this work is with more practical meaning than references [16∼20].

2

3 RLTF under redundancy TDMA MAC

Related work

There has been much research on energy-aware protocols and transmission reliability for WSNs. On the one hand, energy-aware algorithms could be roughly classified into two cases: minimizing the total energy consumption [2] and maximizing the network lifetime [3∼6]. Reference [3] pointed out that the work in [2] may lead to some nodes in the network running out of energy quickly. In [3∼6], network lifetime was defined as the period from the time instant when the network starts functioning to the time instant when the first node runs out of energy. On the other hand, transmission reliability could be enhanced by multiple network techniques. In transport protocols, such as RMST [9], the usage of NACK packets and of caching data within the network is chosen for data block transfer and is beneficial in terms of energy. In the multipath routing scheme ReInForM [10], multiple copies of the same packet are transmitted over randomly chosen routes. Packet duplication can occur at every intermediate node. The number of duplicates is determined from the locally estimated error rate, the hop distance to the sink and the target delivery probability. In [11, 12], CSMA and TDMA scheduling technologies are considered to guarantee prescribed end-to-end transmission reliability, respectively. In cross-layer paper [13], Kwon et al. maximized the network lifetime in WSNs under the constraint of the target end-to-end transmission success probability by adopting a cross-layer strategy that considers physical layer (i.e., power control), MAC layer (i.e., ARQ control) and network layer (i.e., routing protocol) jointly. To sum up, the above literature either reduces energy consumption or enhances transmission reliability. However, none of them has considered these two conflicting objectives together and controlled the balance between them by the optimization of replicator factors. There is also research on the network utility maximization (NUM) for both wired and wireless networks. In the seminal paper by Kelly [14], NUM framework that successfully addressed the rate control problem in wired networks was developed. As for wireless networks, M. Chiang et al. have proposed the layering as optimization decomposition (LAOD) framework for wireless networks in [15]. Nama et al. extended the traditional energy-aware research and jointly studied the rate control problem and the energy conservation problem for WSNs [16]. However, the subgradient-based approach in [16] did not allow a fully distributed solution. After that, many scholars have devoted themselves to the study on the rate-lifetime tradeoff frame-

We present the WSN as a directed graph G(V, L), where V denotes the set of sensor nodes, and L denotes the set of logical links. Let Lout (v) denote the set of outgoing links from node v and Lin (v) the set of incoming links to node v. Let R denote the set of routes. Each route r is assumed to carry data from a unique sensor to the sink. Let L(r) denote the set of links that route r passes through, and let R(l) denote the set of routes whose data flow through link l. We assume a TDMA-based MAC is employed. Time is divided into periodic MAC superframes, and each MAC superframe comprises multiple small time slots. Each sensor is allowed to transmit in its allocated time slots in each MAC superframe so that no collision could occur. Since there exists no interference from the other nodes’ transmission, the transmission failure could only be due to channel errors whose negative effect on transmission reliability could be alleviated by redundancy transmissions. 3.1 Network transmission reliability In this paper, we will adopt the NUM framework to study the reliability allocation among sensors. The objective funcωr log(Pr ), tion of the NUM can be formulated as r∈R

where Pr denotes the end-to-end transmission reliability of route r, and ωr is the weight associated with route r. Suppose the packet loss or error rate on all links is a constant p. Then, the end-to-end reliability of transmission P l (αl ), P l (αl ) = 1 − pαl , route r is defined as Pr = l∈L(r)

where αl denotes the replicator factor of link l. In the following sections, we will mainly concentrate on how to obtain optimal replicator factors. Comment 1 The reasons why we use the logarithm function of each route’s reliability to model its utility are two fold: 1) The logarithm function can guarantee the proportional fairness of reliability allocation among all routes [21]. 2) The logarithm function can change the original expression of each route’s end-to-end reliability from the product of every unit reliability on its route to the sum of each unit’s log reliability. This transformation plays a key role in proving the convexity of the replicator factor control model. 3.2 Network lifetime In a WSN, sensor nodes usually have much tighter energy constraints than the sink node; thus, we only focus on the energy dissipated in the sensor nodes. Let Tv denote the

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lifetime of node v, v = 1, 2, · · · , N , i.e., the time at which it runs out of energy. Definition 1 [2] We consider a general definition of network lifetime (1) Tnet = min(Tv ). v∈V

The lifetime maximization problem [2∼6] maximizes the time at which the first node dies, i.e., it minimizes the maximum ratio of average power consumption to initial energy among all nodes. Definition (1) balances the data flow in the network such that no node incurs high power consumption. 3.3 Tradeoff between network transmission reliability and network lifetime By introducing a system parameter θ ∈ [0, +∞), we can combine these two objectives as a single weighted objective. The weighted objective function can be obtained as follows: ωr log(1 − pαl ) + min(Tv ). (2) θ r∈R

l∈L(r)

v∈V

3.3.1 Energy dissipation constraint Let εs and εr denote the energy consumed per bit in hardware in sensing and receiving data, respectively. We assume that all nodes have identical power dissipation characteristics in sensing and receiving. Let εtl denote the energy consumed per bit in transmitting on link l. εtl is given by εtl = μ + ηdnl , where μ is the energy cost of transmit electronics of node v, η is a coefficient term corresponding to the energy cost of transmit amplifier, and dl is the distance between two terminal nodes of link l. n is the path loss factor, 2 n 4. According to references [18, 19], we assume the total average power for each node consists of three parts, i.e., the energy consumption for transmitting, receiving and sensing. Then the total average power dissipated in node v is given by εtl fl αl + εr fl αl + εs sv , (3) Pvavg (α) = l∈Lout (v)

l∈Lin (v)

where fl denotes the flow rate of link l, and sv denotes the 1 and qv is defined as the source rates of node v. Let qv = Tv normalized power dissipation of sensor node v. As a result, we get energy constraints (4) Ev qv = Pvavg (α). 3.3.2 RLTF optimization model As indicated in [21], the max-min rate allocation problem can be approximated in a distributed way within the NUM framework. Motivated by reference [21], we adopt the same approach to approximate the network lifetime problem (max-min energy allocation problem). Thus, we intro1 q β+1 , and max Tv duce a utility function Wvβ (qv ) = v∈V β+1 v and min Tv can be well approximated by − Wvβ (qv ) as v∈V

v∈V

β approaches ∞. Next, we will combine the weighted approximation objective function and constraints above as follows: ωr log(1 − pαl ) − Wvβ (qv ) (5) max θ r∈R

l∈L(r)

v∈V

s.t. Ev qv = Pvavg (α), v ∈ V, αl 1, l ∈ L. Obviously, the feasible domain is convex, since constraints (4) are linear functions of {αl , qv }. We can get the Hessian

H(α, q) of objective function: −Cpα1 −CpαL diag α1 , · · ·, , −βq1β−1 , · · ·, −βqVβ−1 , 2 α 2 L (p −1) (p −1) ωr . H(α, q) is negative definite where C = θ(ln p)2 r∈R

in the feasible domain, and thus, the objective function is strictly convex. Further, we can conclude that the RLTF model is strictly convex. Though the nonlinear convex optimization problem (5) may be solved by centralized computation by using the interior-point method for convex optimization [22], as the network scale increases, the centralized computation cannot scale well. Thus, we will design a fully distributed algorithm to solve problem (5). 3.4 Distributed algorithm Since {qv } are dummy variables that can be expressed by α, we define ωr log(1 − pαl ) − Wvβ (qv ). Q(α) = θ r∈R

l∈L(r)

v∈V

Then, the RTFL model (5) can be transformed into a new equivalent model: (6) max Q(α), α1

where 1 denotes the column vector whose components are all one. Obviously, Q(α) is differentiable. To describe the gradient G(α) of Q(α), we introduce the symbol matrix A = [Avl ]V ×L , where ⎧ ⎨ 1, if v is the transmitter of link l, Avl = −1, if v is the receiver of link l, ⎩ 0, otherwise. Then, the component of G(α) is given by β ∂qv (ln p)pαl ωr α qv , − Gl (α) = θ l p −1 ∂αl v∈Vl r∈R(l)

(7)

where Vl = {v|Avl = 0, v ∈ V }. The formula for updating αl can be stated as , (8) αl (k + 1) = [αl (k) + λGl (α(k))]+∞ 1 where λ is the positive constant stepsize and [f ]ba = max{a, min{f, b}}. We now formally state the update procedure for the algorithm: Step 1 Initialize α(0), ωr , λ, β and θ. Step 2 At the kth iteration, each sensor node v com∂qv putes qvβ and sends back this information to their up∂αl stream neighbors. Step 3 Each sensor v computes Gl (α(k)), l ∈ Lout (v) as (7) and updates flow rates of its outgoing links as (8). Step 4 Go to Step 2 until {αl (k)} converges. Proposition 1 The convergence of the algorithm is guaranteed provided that the step size λ is sufficiently small. Proof The core of the algorithm is formula (8) that is based on the traditional gradient projection method. As pointed in [22], the gradient projection method is guaranteed convergent when the constant stepsize in formula (8) is small enough. Then we complete Proposition 1. Comment 2 1) From the above update procedure, we can see that each node updates its replicators of outgoing links according to the energy information of its downstream neighbors. Thus, the algorithm is fully distributed. 2) Variables {αl } in the RLTF model are assumed to be real num-

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bers in [1, +∞). However, optimal {αl∗ } should be integers in practice. Thus, we adopt a random way to tackle with this problem: ∗ αl + 1, if rand(0, 1) αl∗ − αl∗ , (9) αl = otherwise. αl∗ , In this way, we can simply verify that E(αl ) = (αl∗ − αl∗ ) × (αl∗ + 1) +(1 − αl∗ + αl∗ ) × αl∗ = αl∗ , where E(αl ) denotes the mathematical expectation of αl .

4

simplicity.

RLTF under retransmission TDMA MAC

In Section 3, we have studied the RLTF under redundancy TDMA MAC. Sometimes the redundancy TDMA MAC may cause unnecessary waste of energy, since sensor nodes keep transmitting data to their next-hop neighbors according to optimal replicator factors regardless of whether the last transmission succeeds. Thus, we will study the RLTF under the retransmission TDMA MAC in this section. The duration of each slot is set long enough to transmit a data packet and receive an ACK. In the retransmission TDMA MAC, a source node receives an ACK if its data packet is correctly received by its next-hop neighbor. Retransmission stops as soon as the source receives an ACK. We assume that each link is symmetric, i.e., the transmitter can communicate with the receiver and receive an ACK from the receiver. Usually, ACK packets are forwarded with very high probability, since ACK packets are small in size, and forward error correction (FEC) codes can be introduced in the packet. Without loss of generality, we assume ACK packets are not lost. Instead of remodeling RLTF under the retransmission TDMA MAC, we set the retransmission upper bound of each link as the optimal replicator factor computed in Section 3. In this way, we may save energy while retaining the same transmission reliability as redundancy TDMA MAC. When using redundancy TDMA MAC, the expected number of transmissions on link l, l ∈ L, is αl∗ ; When using retransmission TDMA MAC, the expected number of transmissions on link l including an ACK packet, l ∈ L, is given by ∗

αl

∗

1 − pαl . (10) 1−p i=1 To choose an energy-efficient MAC, have toevaluate we αl∗ (p) and Nl (p). the quantity relationship between Nl (p) = 1 +

pi−1 = 1 +

l∈L

Fig. 1 The small-sized topology of WSN.

First, we validate the effectiveness of the algorithm by MATLAB. Fig. 2 presents the convergence property of the algorithm. In Fig. 2, replicator factors converge fast (after 30 iterations). The parameters in the algorithm are listed as follows: p = 0.05, λ = 0.05, θ = 1.0 × 10−60 , β = 10, α(0) = 1. The optimal replicator factor vec T tor is 1.7209 1.7893 1.7209 3.1974 3.3827 1.9513 . Due to the symmetry in positions, link 1 and link 3 are with the same optimal replicator factor.

Fig. 2 The evolution of replicator factors.

Then, according to formula (10), we vary p and study the impact of p on communication overhead of these two MAC protocols. From Fig. 3, we can see that network communication overhead, which is computed as the sum of each link’s overhead, increases as rate p increases, which is consonant with our intuition.

l∈L

In the next section, we will plot the overhead equations to evaluate the impact of packet loss or error rate on communication overhead.

5

Simulation

In this section, the algorithm is first simulated over a simple network. In Fig. 1, the distances between any two adjacent nodes are equal, i.e., dl = d = 50 m, l ∈ L. Here, we use the energy dissipation parameters in [19], where εs = 50 nJ/bit, εr = 50 nJ/bit, μ = 50 nJ/bit, η = 0.0013 pJ/bit/m4 , and path loss factor n = 4. Each node is assumed to have equal initial energy of 1 kJ. There are six routes in Fig. 1 and sr = 4 kbps, r ∈ R. The weight factors ωr , r ∈ R are all equal. Here, we set ωr = 1 for the

Fig. 3 Effect of increasing p.

In Fig. 3, initially, redundancy outperforms retransmission permanently (p 0.016) due to the extra ACK packet


in retransmission. However, retransmission performs better than redundancy quickly (p > 0.016) as wireless channel deteriorates. Further, the overhead gap between redundancy and retransmission becomes larger as p increases. Consequently, it is wise to use retransmission MAC in WSNs when wireless channel is in a mild or bad condition. Finally, the scalability of the algorithm will be tested by a medium-sized network as shown in Fig. 4. In Fig. 4, there are fourteen sensor nodes and one sink node randomly located in the square 120×120 m2 . Route ri , i = 1, · · · , 8 are generated by the source node i, i = 1, · · · , 8, respectively, and the other sensors are relay nodes. Fig. 5 validates the convergence of the algorithm again.

Fig. 4 The medium-sized topology of WSN.

Fig. 5 The evolution of replicator factors.

6

Conclusions

There exists an intrinsic tradeoff between transmission reliability and energy efficiency in WSNs. In this paper, we have proposed the RLTF to study the replicator factor control problem under two kinds of TDMA MAC protocols. First, for the redundancy TDMA MAC, we have formulated the replicator factor control problem as convex programming. By the gradient projection method, we have developed a fully distributed algorithm to solve the convex programming. Further, we have extended the above research to the replicator factor control under the retransmission MAC. The communication overhead comparison between these two MAC protocols helps us to choose the more energyefficient MAC according to the variable channel state. Future work may incorporate routing into the RLTF and jointly consider routing and replicator factor control in a cross-layer optimization method. At this time, routing variables and replicator factors are coupled in energy constraints. Some new methods are needed to deal with the RLTF due to its non-separable and non-convex character. References [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, et al. A survey on sensor networks[J]. IEEE Communications Magazine, 2002, 40(8): 104 – 112.

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Meng ZHENG was born in Liaoning Province, China, in 1983. He received his B.S. degree in Applied Mathematics, and M.S. degree in Operational Research and Cybernetics at Northeastern University, Shenyang, China, in 2005 and 2008, respectively. He is working on his Ph.D. degree at the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current interests include wireless sensor networks, industry wireless networks and networked control systems. E-mail: zhengmeng [email protected]. Haibin YU was born in Heilongjiang Province, China, in 1964. He received his Ph.D. degree in Automatic Control at Northeastern University, Shenyang, China. He is currently a professor of Key Laboratory of Industrial Informatics at Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current interests include wireless sensor networks and networked manufacturing. E-mail: [email protected].

Wei LIANG received her Ph.D. degree in Mechatronic Engineering from Shenyang Institute of Automation, Chinese Academy of Sciences, in 2002. She is currently serving as an associate professor of Shenyang Institute of Automation. Her research interests are in the areas of wireless sensor network, industry communication and system simulation. Email: [email protected].

Xiaoling ZHANG received her B.S. degree in Taiyuan University of Technology, Taiyuan, Shanxi, China, in 2005. She is currently working towards her M.S. and Ph.D. degrees in the area of wireless industrial sensor networks in Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, Liaoning, China, where her current research focus is on the industrial wireless standards and optimal scheduling algorithms for increasing the reliability and timeliness in wireless networks. E-mail: [email protected].