Joint Routing and Sleep Scheduling for Lifetime Maximization of

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 7, JULY 2010

Joint Routing and Sleep Scheduling for Lifetime Maximization of Wireless Sensor Networks Feng Liu, Chi-Ying Tsui, Member, IEEE, and Ying Jun (Angela) Zhang, Member, IEEE

Abstract—The rapid proliferation of wireless sensor networks has stimulated enormous research efforts that aim to maximize the lifetime of battery-powered sensor nodes and, by extension, the overall network lifetime. Most work in this field can be divided into two equally important threads, namely (i) energyefficient routing that balances traffic load across the network according to energy-related metrics and (ii) sleep scheduling that reduces energy cost due to idle listening by providing periodic sleep cycles for sensor nodes. To date, these two threads are pursued separately in the literature, leading to designs that optimize one component assuming the other is pre-determined. Such designs give rise to practical difficulty in determining the appropriate routing and sleep scheduling schemes in the real deployment of sensor networks, as neither component can be optimized without pre-fixing the other one. This paper endeavors to address the lack of a joint routing-and-sleep-scheduling scheme in the literature by incorporating the design of the two components into one optimization framework. Notably, joint routing-andsleep-scheduling by itself is a non-convex optimization problem, which is difficult to solve. We tackle the problem by transforming it into an equivalent Signomial Program (SP) through relaxing the flow conservation constraints. The SP problem is then solved by an iterative Geometric Programming (IGP) method, yielding an near optimal routing-and-sleep-scheduling scheme that maximizes network lifetime. To the best of our knowledge, this is the first attempt to obtain the optimal joint routing-andsleep-scheduling strategy for wireless sensor networks. The near optimal solution provided by this work opens up new possibilities for designing practical and heuristic schemes targeting the same problem, for now the performance of any new heuristics can be easily evaluated by using the proposed near optimal scheme as a benchmark. Index Terms—Lifetime maximization, routing, sleep scheduling, wireless sensor networks, non-convex optimization.

I. I NTRODUCTION

R

ECENT advances in wireless communication technologies as well as VLSI and MEMs (Micro-ElectroMechanical-Systems) have opened up potentials for largescale deployment of wireless sensor networks (WSNs) in fields Manuscript received May 1, 2009; revised December 7, 2009 and March 8, 2010; accepted April 11, 2010. The associate editor coordinating the review of this paper and approving it for publication was T. Hou. F. Liu is with The Hong Kong Applied Science and Technology Research Institute (e-mail: [email protected]). C. Y. Tsui is with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology (e-mail: [email protected]). Y. J. Zhang is with the Department of Information Engineering, The Chinese University of Hong Kong (e-mail: [email protected]). This work was supported in part by the Competitive Earmarked Research Grant (Project Number 418707), established under the University Grant Committee of Hong Kong, and the Direct Grant for Research (Project Number 2050370), established by The Chinese University of Hong Kong. Digital Object Identifier 10.1109/TWC.2010.07.090629

sensor nodes

gateway

Fig. 1.

Wireless sensor networks.

such as habitat monitoring, battlefield surveillance, environment sensing, etc. Wireless sensor networks consist of a group of sensor nodes randomly distributed in a given area as shown in Fig. 1. A fundamental challenge of such networks lies in the energy constraint of battery-powered sensor nodes, which poses a performance limit on achievable network lifetime. Prolonging battery life in sensor nodes and, by extension, the overall network lifetime is therefore a foremost task in the design of practical WSNs [3], [4], [7]. A typical definition of network lifetime is the time till the first sensor node runs out of its battery energy [1]. This is a reasonable definition for WSNs, where the nodes near sinks (e.g., gateways or data collection centers) have to relay much more information and consume much more energy than those that are far away. The survival of these bottleneck nodes is critical to the operation of the whole network. Network lifetime maximization involves all levels of sensor network hierarchy, from hardware/software design to communication protocols. Recent efforts dealing with communication-related energy costs mainly focus on two separate but equally important fronts: energy-efficient routing and sleep scheduling. In particular, energy-efficient routing aims to balance traffic loads, and hence energy consumption,

c 2010 IEEE 1536-1276/10$25.00 ⃝

LIU et al.: JOINT ROUTING AND SLEEP SCHEDULING FOR LIFETIME MAXIMIZATION OF WIRELESS SENSOR NETWORKS

among sensor nodes across the network. Existing literature has formulated this problem into various linear programming (LP) problems depending on power consumption models, ranging from simple models that only consider payload transmission power [5], [8] or both transmission and reception power [1] to more realistic ones that include power consumption on control message passing [16] and idle listening [11]. The problem formulation is also subject to different medium access control (MAC) constraints such as half-duplex constraint, link capacity constraint [5], [8], and interference constraint [6]. On the other hand, network traffic load is light most of the time in many sensor network applications, and thus idle listening is a major source of power depletion in WSNs [13]. Thus, allowing sensor nodes to have periodic sleep cycles greatly reduces their energy consumption. While a large part of existing work on sleep scheduling focuses on striking a good tradeoff between energy efficiency and latency/reliability, the design of duty cycles and active/sleep patterns has a big impact on the balance of energy consumption across the network as well. This is because an upstream node may keep resending a packet to its downstream node and waste a lot of transmission power, if the downstream node sleeps too much during the transmission phase of the upstream node. This problem has been directly or indirectly addressed by synchronizing wake-up slots (e.g., S-MAC [2] and T-MAC), lengthening the preamble in a data packet (e.g., B-MAC [14]), or sending multiple short preambles till one is heard by the receiver (e.g., TICER [15]). By and large, all existing sleep scheduling schemes assume a pre-determined routing table. In other words, upstream-downstream (i.e., transmitter-receiver) node pairs as well as the amount of traffic going through the pairs are fixed in advance. Most prior work has treated energy-efficient routing [1], [5] and sleep scheduling [17]–[19] as two separate tasks, oftentimes assuming one component is pre-given when optimizing the other one. This includes the recent work in [12] which, although claimed to consider joint routing and sleep scheduling, in fact focuses on the optimization of routing only, except that the pre-given sleep schedules are explicitly formulated as MAC constraints. The separate design of routing and sleep scheduling poses a serious problem in the real implementation of WSNs, for neither component can be optimally adjusted before the other one is fixed. In addition, it would not be surprising that the network lifetime thus achieved is suboptimal compared with that when routing and sleep scheduling are jointly optimized. To address the above-mentioned issues, this paper endeavors to jointly optimize energy-efficient routing and sleep scheduling to maximize overall network lifetime. In particular, we formulate the integrated design of route selection, traffic load allocation, and sleep scheduling into a constrained optimization problem. As opposed to [1], our formulation considers a more realistic power consumption model which includes energy costs due to payload transmission and reception, preamble transmission, as well as idle listening. One reason for the lack of joint routing-and-sleepscheduling schemes in current literature lies in the mathematical difficulty in solving joint optimization problems. Without exception, the constrained optimization problem formulated in

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Sleep period (fixed length) node i Tslp

Active period (variable length)

Fig. 2.

Asynchronous sleep scheduling mechanism.

this work is a mathematical challenge by itself due to its nonconvex nature. We tackle the problem by transforming it into an equivalent Signomial Programming (SP) problem through the relaxation of an equality constraint. It can be proved that the relaxation does not affect the optimality of the solution. The SP problem can then be efficiently solved via an iterative convex approximation method, where a convex Geometric Programming (GP) problem is solved in each step. To the best of our knowledge, this is the first attempt to obtain the optimal joint routing-and-sleep-scheduling scheme for WSNs. An important application of the proposed scheme is to address the noticeable lack of a benchmark for the achievable network lifetime by jointly designing energy-efficient routing and sleep scheduling. The solution provided by this work opens up new possibilities for evaluating the performance of existing or newly proposed heuristics targeting the same problem. Moreover, the engineering insights gained from the near optimal solution may serve as important guidelines for the design of practical heuristics in future WSNs. The rest of the paper is organized as follows. In Sections II and III, we introduce the system model and formulate the joint routing-and-sleep-scheduling problem into a nonconvex optimization problem. In Section IV, we show how the problem can be transformed to an equivalent form that is amenable to efficient solution algorithms through iterative convex optimization. The performance of the proposed algorithm is evaluated through numerical simulations in Section V. Through simulation, we compare the performance of the proposed algorithm with schemes that optimize the routing and sleep scheduling separately. Finally, the paper is concluded in Section VI. II. S YSTEM M ODEL This paper considers sensor networks where static sensor nodes are randomly located in a given region. Similar to most previous work, this paper assumes that the traffic in the network is light, and hence transmissions are collision free. Let the routing matrix be described by R = {𝑟𝑖𝑗 }, the average rate at which packets are flowed over the link from node 𝑖 to 𝑗, where 𝑟𝑖𝑗 is fixed to be zero if nodes 𝑖 and 𝑗 are not within the RF range of each other. We define 𝑁𝑖 as the neighbor set of node 𝑖 with { ∈ 𝑁𝑖 node 𝑗 is in the RF range of node 𝑖 𝑗 (1) ∈ / 𝑁𝑖 otherwise A sensor node is either in an active mode or a sleep mode according to its sleep schedule, as depicted in Fig. 2. In the sleep mode, a node is turned off and no power is consumed. Every time a node 𝑖 goes into the sleep mode, it sleeps for 𝑖 of time before waking up. Note that the sleep a period 𝑇𝑠𝑙𝑝

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Data Ready

TX RF initialization

(T ) rf

Channel listening

(Tsav )

power saving

Tlis

RX

Tdet

Channel listening

ACK detection Data packet transmission

…...

( )

Tdata

CTS detection Tpre

( )

Sleep time Tslp

Tlis Tdet

Active Period: Idle listening

Fig. 3.

Data Done

Active period: RX Ready Data Transmission RTS (Tpre )

Channel listening

Data packet receiving

( )

CTS Tpre

ACK

(T ) pre

Active Period: Data receiving

Timing diagram in different active periods.

time of a node is fixed, but different nodes have different sleep times due to different traffic and battery distributions. A node wakes up and enters an active mode from time to time to see if it has any packets to transmit or if there are any other nodes attempting to transmit to it. An active period can be further categorized as an idle listening slot, a data transmission slot, or a data reception slot. More specifically, it is said to be an idle listening slot if the node neither transmits nor receives data packets within this active period. In this case, the active period consists of two parts, i.e., RF initialization and channel detection. An active period is said to be a data transmission slot if the node transmits a packet in this slot. Likewise, an active period is a data reception slot if the node receives a packet in this slot. In the following, we will elaborate in detail how nodes transmit and receive in the proposed system, and how much energy is consumed in each mode. Readers are referred to Fig. 2 and Fig. 3 for illustrations. A node 𝑖 initializes its RF circuits immediately after it wakes up. Assume that it takes a node 𝑇𝑟𝑓 time and 𝐸𝑟𝑓 amount of energy to initialize its RF circuits. If the node has packets to transmit, it first listens to the channel for a period 𝑖 time to see if any of its neighbors are transmitting. of 𝑇𝑙𝑖𝑠 𝑖 We define 𝑇𝑑𝑒𝑡 = 𝑇𝑟𝑓 + 𝑇𝑙𝑖𝑠 as the total time from when a node wakes up to when it goes to sleep again. The length of 𝑖 𝑖 will be discussed later. If the channel is idle for 𝑇𝑑𝑒𝑡 𝑇𝑑𝑒𝑡 time, the node sends a Request To Send (RTS) preamble. If the target receiver is also in an active mode and receives the RTS preamble, it replies with a Clear To Send (CTS) packet. The transmitter will then send a data packet, which is acknowledged by an ACK packet from the receiver. Otherwise, if the receiver happens to be in a sleep mode, the transmitter will resend the RTS preamble after going to a power saving status for a short time 𝑇𝑠𝑎𝑣 . This process is repeated until the receiver wakes up and captures the RTS preamble. Throughout the paper, it is assumed that all packets, including preambles, are transmitted at a constant power level 𝑝𝑡𝑥 . Meanwhile, it takes a receiver a power of 𝑝𝑟𝑥 to receive any packet. Likewise, we denote by 𝑇𝑑𝑎𝑡𝑎 the duration of a data packet by assuming that all the packet lengths are the same. Without loss of generality, we assume that RTS, CTS, and ACK packets are of the same length 𝑇𝑝𝑟𝑒 .

To ensure that all transmitter-receiver pairs are “sufficiently" connected and communication of other links in the same RF 𝑖 should be long enough to range is correctly detected, 𝑇𝑑𝑒𝑡 cover at least two consecutive RTS preambles (including the 𝑇𝑠𝑎𝑣 between them) of all sensor nodes within its RF range. 𝑖 is too short, a node may not be able to Otherwise, if 𝑇𝑑𝑒𝑡 capture an RTS preamble transmitted in its RF range. Hence, it requires 𝑖 𝑗 ≥ max (2𝑇𝑟𝑓 + 3𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ). 𝑇𝑑𝑒𝑡 𝑗∈𝑁𝑖

(2)

𝑖 Note that 𝑇𝑑𝑒𝑡 is dominated by the longest 𝑇𝑠𝑎𝑣 in node 𝑖’s neighbor set. This means if different nodes adopt different 𝑇𝑠𝑎𝑣 ’s, the ones with smaller 𝑇𝑠𝑎𝑣 ’s can always increase their 𝑖 𝑇𝑠𝑎𝑣 without increasing the idle listening time 𝑇𝑑𝑒𝑡 of node 𝑖. Hence, it is the most energy efficient way to let all sensor nodes have the same 𝑇𝑠𝑎𝑣 , for otherwise we can always let the ones with smaller 𝑇𝑠𝑎𝑣 ’s increase their 𝑇𝑠𝑎𝑣 ’s to save energy. In the rest of the paper, we assume that 𝑇𝑠𝑎𝑣 is the same across all nodes. Moreover, we assume that 𝑇𝑑𝑒𝑡 is equal to the shortest allowable duration, i.e., 𝑇𝑑𝑒𝑡 = 2𝑇𝑟𝑓 + 3𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 , unless otherwise stated. Likewise, the channel detection power is 𝑝𝑟𝑥 . We are now ready to compute the energy consumptions of a node during an active period, which could be an idle listening slot, a data transmission slot, or a data receiving slot. In a data transmission slot, the average energy consumption for node 𝑖 to transmit one packet to node 𝑗 is given by 𝑖𝑗 ¯𝑡𝑥 =(𝐸𝑟𝑓 + 𝑝𝑟𝑥 𝑇𝑑𝑒𝑡 ) 𝐸 − 𝐸𝑟𝑓 + (𝐸𝑟𝑓 + (𝑝𝑡𝑥 + 𝑝𝑟𝑥 )𝑇𝑝𝑟𝑒 ) ( 𝑗 ) 𝑇𝑠𝑙𝑝 /2 − (2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 + 𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 ) × +2 𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 + 𝑝𝑡𝑥 𝑇𝑑𝑎𝑡𝑎 + 𝑝𝑟𝑥 𝑇𝑝𝑟𝑒 ,

(3)

where the first line on the right hand side is the energy cost due to RF initialization and channel detection, the second and third lines are the energy cost to transmit the RTS packets until an RTS packet is acknowledged, while the forth line represents the energy cost to transmit the data packet and receive the 𝑗 /2 is the ACK packet. In the third line, in particular, 𝑇𝑠𝑙𝑝 average residual sleep time of node 𝑗 seen by node 𝑖 when


node 𝑖 initiates a transmission to node 𝑗. 𝐸𝑟𝑓 +(𝑝𝑡𝑥 +𝑝𝑟𝑥)𝑇𝑝𝑟𝑒 represents the energy consumption to transmit an RTS packet and to detect possible CTS packet. Note that there is no RF initialization in the first RTS packet transmission, as the sensor node is already on. Likewise, there is no 𝑇𝑠𝑎𝑣 following the last RTS/CTS exchange, as can be seen in Fig. 3. Hence, the 𝑇 𝑗 /2−(2𝑇𝑝𝑟𝑒 +𝑇𝑠𝑎𝑣 +𝑇𝑟𝑓 +2𝑇𝑝𝑟𝑒 )

term 𝑠𝑙𝑝 + 2 denotes the average 𝑇𝑟𝑓 +2𝑇𝑝𝑟𝑒 +𝑇𝑠𝑎𝑣 number of RTS preambles the transmitter has to transmit until one is captured by node 𝑗, and the term −𝐸𝑟𝑓 at the begin of the second line corresponds to the fact that there is no RF initialization in the first RTS packet transmission. The energy consumption for a node to receive a packet is calculated as 𝑇𝑑𝑒𝑡 𝐸¯𝑟𝑥 = (𝐸𝑟𝑓 + 𝑝𝑟𝑥 ) + 𝑝𝑟𝑥 𝑇𝑑𝑎𝑡𝑎 + 2𝑝𝑡𝑥 𝑇𝑝𝑟𝑒 , (4) 2 where the first term on the right hand side is the energy cost to initialize RF circuit and the average energy cost to detect the RF channel before desired RTS is received, the second term is the energy cost to receive the data packet, while the third term is the energy cost to transmit the CTS and ACK packets. Furthermore, the energy cost due to idle listening is given by 𝐸𝑑𝑒𝑡 = 𝐸𝑟𝑓 + 𝑝𝑟𝑥 (𝑇𝑑𝑒𝑡 − 𝑇𝑟𝑓 ).

(5)

To calculate the average power consumption of node 𝑖, consider a very long period of time 𝑇 , within which node 𝑖 𝑖𝑗 𝑖 packets to node 𝑗, received 𝑁𝑟𝑥 packets, has transmitted 𝑁𝑡𝑥 𝑖 and experienced 𝑁𝑖𝑑𝑙𝑒 active/sleep cycles without data trans𝑖 idle listening slots). The average mission/reception (i.e., 𝑁𝑖𝑑𝑙𝑒 power consumption is readily calculated as ∑ 𝑖 𝑖 ¯ 𝑖𝑗 𝑖𝑗 ¯ 𝑗∈𝑁𝑖 𝐸𝑡𝑥 𝑁𝑡𝑥 + 𝐸𝑟𝑥 𝑁𝑟𝑥 + 𝐸𝑑𝑒𝑡 𝑁𝑖𝑑𝑙𝑒 , (6) 𝑝𝑖 = 𝑇 To be specific, 𝑖𝑗 𝑁𝑡𝑥 = 𝑟𝑖𝑗 (7) 𝑇 is the average rate at which node 𝑖 transmits packets to node 𝑗, and 𝑖 ∑ 𝑁𝑟𝑥 = 𝑟𝑗𝑖 (8) 𝑇 𝑗∈𝑁 𝑖

is the average rate at which node 𝑖 receives packets from other nodes. Substituting (7) and (8) to (6), we have ∑ 𝑖𝑗 ∑ 𝐸𝑑𝑒𝑡 ¯ 𝑟𝑖𝑗 + 𝐸 ¯𝑟𝑥 𝐸 𝑝𝑖 = 𝑟𝑗𝑖 + , (9) 𝑡𝑥 𝑖 𝑇𝑑𝑒𝑡 + 𝑇𝑠𝑙𝑝 𝑗∈𝑁 𝑗∈𝑁 𝑖

𝑖

where we have assumed light traffic load so that a node is in idle modes most of the time, i.e., 𝑇 ≈ (𝑇𝑑𝑒𝑡 +

𝑖 𝑖 𝑇𝑠𝑙𝑝 )𝑁𝑖𝑑𝑙𝑒 .

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III. P ROBLEM F ORMULATION With the above derivations, joint routing and sleep scheduling can be formulated into the following optimization problem. To be specific, we endeavor to find the optimal routing matrix 𝑖 } so that the network R = {𝑟𝑖𝑗 } and sleep time Tslp = {𝑇𝑠𝑙𝑝 lifetime, defined by the time till the first sensor node runs out of its battery power, is maximized. We have 𝑖 max min 𝑇𝑙𝑖𝑓 𝑒

𝑠.𝑡.

(12a)

𝑖

R,Tslp

𝑖 𝑝𝑖 𝑇𝑙𝑖𝑓 ∀𝑖 𝑒 ≤ 𝐵𝑖 ∑ ∑ 𝑟𝑗𝑖 − 𝑟𝑖𝑗 = 𝐷𝑖 − 𝑆𝑖 𝑗∈𝑁𝑖

(12b) ∀𝑖

(12c)

𝑗∈𝑁𝑖

𝑟𝑖𝑗 ≥ 0, ∀𝑖, 𝑗

𝑖 𝑇𝑠𝑙𝑝 > 0, ∀𝑖

(12d)

𝑖 where 𝑝𝑖 is expressed as a function of 𝑟𝑖𝑗 and 𝑇𝑠𝑙𝑝 using (3)(5), and (9). In particular, the constraint (12c) is a flow conservation constraint, where 𝑆𝑖 is the average rate at which packets are generated by node 𝑖 as a source and 𝐷𝑖 is the average rate packets are absorbed by node 𝑖 as a destination. 𝑆𝑖 and 𝐷𝑖 can be set to zero if node 𝑖 is not a source or destination. The max-min problem in (12) can be straightforwardly rewritten into the following maximization problem, where minimizing 1 𝑡 is equivalent to maximizing 𝑡.

min

R,Tslp ,𝑡

𝑠.𝑡.

1 𝑡

(13a)

𝑝𝑖 𝑡 ≤1 ∀𝑖 𝐵𝑖 ∑ ∑ 𝑗∈𝑁𝑖 𝑟𝑗𝑖 − 𝑗∈𝑁𝑖 𝑟𝑖𝑗

=1 𝐷 𝑖 − 𝑆𝑖 𝑖 𝑟𝑖𝑗 ≥ 0, ∀𝑖, 𝑗 𝑇𝑠𝑙𝑝 > 0, ∀𝑖

(13b) ∀𝑖

(13c)

𝑡 > 0 (13d)

In the following theorem, we will prove that the equality constraints in (13c) can be relaxed to inequality constraints without affecting the optimal solution. Such a relaxation will be useful later on when we develop an efficient algorithm to solve the problem. Theorem 1. Problem (13) is equivalent to the following problem where the equality constraints (13c) are replaced by inequality constraints. min

R,Tslp ,𝑡

s.t.

1 𝑡

(14a)


≤1 𝐷 𝑖 − 𝑆𝑖 𝑖 𝑟𝑖𝑗 ≥ 0, ∀𝑖, 𝑗 𝑇𝑠𝑙𝑝 > 0, ∀𝑖

(14b) ∀𝑖

(14c)

𝑡 > 0 (14d)

(10)

To be more precise, there exists an optimal solution to (13) that satisfies (14c) with equality.

It is obvious from (9) that the lifetime of node 𝑖, denoted 𝑖 by 𝑇𝑙𝑖𝑓 𝑒 , is limited by its battery capacity 𝐵𝑖 through the following inequality:

Proof: For a given set of R and T𝑠𝑙𝑝 , the 𝑡 that solves Problem (14a) to (14d) is given by,

𝑖 𝑇𝑙𝑖𝑓 𝑒 ≤

𝐵𝑖 . 𝑝𝑖

(11)

1 𝑝𝑖 (R, T𝑠𝑙𝑝 ) = max . 𝑖 𝑡∗ (R, T𝑠𝑙𝑝 ) 𝐵𝑖 Let 𝑓 ∗ (R, T𝑠𝑙𝑝 ) =

1 𝑡∗ (R,T𝑠𝑙𝑝 )

(15)

denote the value of the

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𝑝𝑖 (R, T𝑠𝑙𝑝 ) 𝑖 𝐵𝑖 denote the bottleneck node that has the minimum lifetime. Assume that with R and T𝑠𝑙𝑝 , (14c) is inactive for the bottleneck node. That is objective function and 𝑖∗ (R, T𝑠𝑙𝑝 ) = arg max

∑

𝑟𝑗𝑖∗ (R,T𝑠𝑙𝑝 ) −

𝑗∈𝑁𝑖∗ (R,T𝑠𝑙𝑝 )

∑

𝑟𝑖∗ (R,T𝑠𝑙𝑝 )𝑗

𝑗∈𝑁𝑖∗ (R,T𝑠𝑙𝑝 )

(16)

< 𝐷𝑖∗ (R,T𝑠𝑙𝑝 ) − 𝑆𝑖∗ (R,T𝑠𝑙𝑝 ) . ˆ ⪯ R to satisfy In this case, we can always find another R constraint (14c) with equality for 𝑖 = 𝑖∗ (R, T𝑠𝑙𝑝 ). Specifiˆ is obtained by letting 𝑟ˆ𝑖𝑗 ≤ 𝑟𝑖𝑗 for 𝑖 = 𝑖∗ (R, ˆ T ˆ 𝑠𝑙𝑝 ) cally, R and 𝑟ˆ𝑖𝑗 = 𝑟𝑖𝑗 otherwise. In other words, we can always reduce the outgoing packet rate of node 𝑖∗ (R, T𝑠𝑙𝑝 ) to make the equality hold. We now analyze the situation in two cases. Case 1: Node 𝑖∗ (R, T𝑠𝑙𝑝 ) continues to be the bottleneck ˆ i.e., node with the new routing matrix 𝑅, ˆ T𝑠𝑙𝑝 ) = 𝑖∗ (R, T𝑠𝑙𝑝 ). 𝑖∗ (R,

(17)

From (9), it is obvious that 𝑝𝑖 is an increasing function of R for any 𝑖. Therefore, 𝑝𝑖∗ (R,T ˆ 𝑠𝑙𝑝 ) < 𝑝𝑖∗ (R,T𝑠𝑙𝑝 ) ,

(18)

which implies that the lifetime of the bottleneck node is increased. Hence, we obtain a better objective function value 𝑓 , i.e., ˆ T𝑠𝑙𝑝 ) < 𝑓 ∗ (R, T𝑠𝑙𝑝 ). (19) 𝑓 ∗ (R, Case 2: Node 𝑖∗ (R, T𝑠𝑙𝑝 ) is no longer the bottleneck node with the new routing matrix R, i.e., ˆ T𝑠𝑙𝑝 ) ∕= 𝑖∗ (R, T𝑠𝑙𝑝 ). 𝑖∗ (R,

(20)

In this case, the following inequality holds, ˆ 𝑝𝑖∗ (R,T ˆ 𝑠𝑙𝑝 ) (R, T𝑠𝑙𝑝 ) 𝐵𝑖∗ (R,T ˆ 𝑠𝑙𝑝 )

≤ ≤

𝑝𝑖∗ (R,T𝑠𝑙𝑝 ) (R, T𝑠𝑙𝑝 ) 𝐵𝑖∗ (R,T𝑠𝑙𝑝 )

IV. S OLVING THE J OINT ROUTING AND S LEEP S CHEDULING P ROBLEM This section presents an efficient algorithm to solve the joint routing and sleep scheduling algorithm. Before presenting the algorithm, we briefly introduce some mathematical preliminaries in subsection III.A. Interested readers are referred to [22] for more details on the definitions and proofs. A. Geometric Programming and Signomial Programming 𝑛 1 Definition 1. [Monomial]: A function 𝑓 (x) : 𝑅++ → 𝑅+ is a monomial if (1)

𝑓 (x) = 𝑑𝑥𝑎1 𝑥𝑎2

(2)

(𝑛)

⋅ ⋅ ⋅ 𝑥𝑎𝑛

,

(22)

where the multiplicative constant 𝑑 ≥ 0 and the exponential constants 𝑎(𝑗) are real numbers. 𝑛 1 Definition 2. [Posynomial]: A function 𝑓 (x) : 𝑅++ → 𝑅+ is a posynomial if it is a sum of monomials. That is

𝑓 (x) =

𝐾 ∑

𝑎

(1)

𝑎

(2)

𝑎

(𝑛)

𝑑𝑘 𝑥1 𝑘 𝑥2 𝑘 ⋅ ⋅ ⋅ 𝑥𝑛𝑘 ,

(23)

𝑘=1 (𝑗)

where 𝑑𝑘 ≥ 0 and 𝑎𝑘 are real numbers. Definition 3. [Signomial]: A signomial is a function with the same form as a posynomial except that the multiplicative coefficients 𝑑𝑘 are allowed to be negative. Definition 4. [Geometric Program]: A geometric program (GP) is an optimization problem of the form

𝑝𝑖∗ (R,T ˆ 𝑠𝑙𝑝 ) (R, T𝑠𝑙𝑝 ) 𝐵𝑖∗ (R,T ˆ 𝑠𝑙𝑝 )

implies that out of the many optimal solutions, there exists one solution that satisfies (14c) with equality for all nodes. This proves the theorem. Thanks to Theorem 1, Problem (13) and (14) are equivalent to each other. In the following, we will focus on solving (14) efficiently.

(21) ,

ˆ ⪯ R and 𝑝𝑖 where the first inequality is due to the fact that R is an increasing function of R. The second inequality is due to the fact that 𝑖∗ (R, T𝑠𝑙𝑝 ) is the bottleneck node under the original routing matrix R. Similar to Case 1, (21) implies a ˆ better objective function value can be achieved with R. Note that the above statements hold for any T𝑠𝑙𝑝 , including the optimal one. This implies that (14c) must be satisfied with equality for the bottleneck node at the optimal point. Or else, there is always another routing strategy that yields a better objective function value. On the other hand, changing 𝑝𝑘 of a non-bottleneck node 𝑘 through scaling 𝑟𝑘𝑗 does not have an effect on the objective function value. In other words, there could be a number of equally good optimal solutions which satisfy (14c) with equality for the bottleneck node, but either with equality or inequality for non-bottleneck nodes. In other words, we can scale 𝑟𝑘𝑗 such that constraints in (14c) are active for all nodes without affecting the optimality of the solution. This

min 𝑓0 (x) 𝑠.𝑡.

𝑓𝑖 (x) ≤ 1

∀𝑖 = 1 ⋅ ⋅ ⋅ 𝑚

ℎ𝑖 (x) = 1

∀𝑖 = 1 ⋅ ⋅ ⋅ ,

(24)

where 𝑓𝑖 are posynomial functions and ℎ𝑖 are monomial functions. Remark 1. A GP problem can be transformed into an equivalent convex optimization problem and solved efficiently. Readers are referred to [22] for detailed methods for solving GP. Definition 5. [Signomial Program]: A signomial program (SP) is an optimization problem of the same form as (24), except that 𝑓𝑖 and ℎ𝑖 can be signomial functions. Remark 2. A SP problem is an intractable NP-hard problem in general. [22] B. Posynomial approximation We first show that Problem (14) can be represented as a 𝑖 𝑖 𝑖 = 𝑇𝑠𝑙𝑝 + 𝑇𝑑𝑒𝑡 . Optimizing 𝑇𝑠𝑙𝑝 is SP problem. Define 𝑇𝑐𝑦𝑐𝑙𝑒


𝑖 equivalent to optimizing 𝑇𝑐𝑦𝑐𝑙𝑒 , as 𝑇𝑑𝑒𝑡 is a constant. Thus, Problem (14) can be rewritten as

min

R,Tcycle ,𝑡

s.t.

𝑡−1

(25a)


(25b)

≤1 𝐷 𝑖 − 𝑆𝑖 𝑖 𝑟𝑖𝑗 ≥ 0, ∀𝑖, 𝑗, 𝑇𝑐𝑦𝑐𝑙𝑒 > 0,

𝑖 𝑇𝑑𝑒𝑡 (𝑇𝑐𝑦𝑐𝑙𝑒 )−1

≤ 1, ∀𝑖

∀𝑖

𝑡>0

where the left hand side (LHS) of (25b) is, ∑ 𝑐2 ∑ 𝐸 ∑ 𝑐1 ¯𝑟𝑥 𝑗 𝑟𝑖𝑗 𝑡 + 𝑟𝑖𝑗 𝑡𝑇𝑐𝑦𝑐𝑙𝑒 + 𝑟𝑗𝑖 𝑡 𝐵𝑖 𝐵𝑖 𝐵𝑖 𝑗∈𝑁𝑖

𝑗∈𝑁𝑖

𝑗∈𝑁𝑖

+

𝐸𝑑𝑒𝑡 𝑖 (𝑇𝑐𝑦𝑐𝑙𝑒 )−1 𝑡. 𝐵𝑖

(25c) (25d) (25e)

𝑐2 =

𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ) . 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 )

(28)

According to the definitions in the previous subsection, (25) is a SP problem, as (25a), (25b), and (25e) consist of posynomials, while (25c) contains signomials. In what follows, we approximate the signomial of (25c) with a posynomial near a point R′ . In particular, we re-write (25c) as, ∑ 𝑗∈𝑁𝑖 𝑟𝑗𝑖 + 𝑆𝑖 ∑ 𝑓𝑖 (R) = ≤ 1 ∀𝑖, (30) 𝑗∈𝑁𝑖 𝑟𝑖𝑗 + 𝐷𝑖 where ∑ 𝑓𝑖 (R) is a ratio of linear functions. Denote ℎ𝑖 (R) = 𝑟𝑖𝑗 + 𝐷𝑖 . We would like to approximate ℎ𝑖 (R) by a

𝑗∈𝑁𝑖

𝑎

∂𝑔𝑖 (X′ ) 𝑟𝑖𝑗 ∂ℎ𝑖 (R) = , ∂𝑥𝑖𝑗 ℎ𝑖 (R) ∂𝑟𝑖𝑗 𝑟𝑖𝑗 =𝑟′

(33)

and

𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ) (2𝑇𝑟𝑓 + 𝑇𝑝𝑟𝑒 + 3𝑇𝑠𝑎𝑣 ) 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) > 0. (29)

∏

𝑎𝑖𝑗 =

𝑖𝑗

=

ˆ 𝑖 (R)= 𝑐𝑖 monomial ℎ

Define logarithm∑ transformations 𝑥𝑖𝑗 = log 𝑟𝑖𝑗 , 𝑔𝑖 (X) = 𝑒𝑥𝑖𝑗 + 𝐷𝑖 ) and 𝑔ˆ𝑖 (X) = log ˆ ℎ𝑖 (X) = log ℎ𝑖 (X) = log ( 𝑗∈𝑁 𝑖 ∑ log 𝑐𝑖 + 𝑗 𝑎𝑖𝑗 𝑥𝑖𝑗 . Equating first-order Taylor expansion of 𝑔𝑖 (X) at X′ with 𝑔ˆ𝑖 (X), we get ∑ ∂𝑔𝑖 (X′ ) ∑ (𝑥𝑖𝑗 − 𝑥′𝑖𝑗 ) = log 𝑐𝑖 + 𝑎𝑖𝑗 𝑥𝑖𝑗 , (32) 𝑔𝑖 (X′ )+ ∂𝑥𝑖𝑗 𝑗

(26)

In general, the energy needed to transmit a data packet, 𝑝𝑡𝑥 𝑇𝑑𝑎𝑡𝑎 , is much larger than that is needed to initialize RF circuit and transmit an RTS packet, 𝐸𝑟𝑓 + 𝑝𝑡𝑥 𝑇𝑝𝑟𝑒 . Thus, ( 𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ) 𝑐1 > 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) ) +𝑇𝑠𝑎𝑣 − 3𝑇𝑝𝑟𝑒

𝑗∈𝑁𝑖

(31)

𝑗∈𝑁𝑖

= 𝑝𝑟𝑥 𝑇𝑑𝑒𝑡 + 𝑝𝑡𝑥 𝑇𝑑𝑎𝑡𝑎 + 𝑝𝑟𝑥 𝑇𝑝𝑟𝑒 (𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ))(𝑇𝑠𝑎𝑣 − 3𝑇𝑝𝑟𝑒 ) + 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) ( 𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ) = 𝑇𝑠𝑎𝑣 − 3𝑇𝑝𝑟𝑒 + 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) 𝑝𝑟𝑥 𝑇𝑑𝑒𝑡 + 𝑝𝑡𝑥 𝑇𝑑𝑎𝑡𝑎 + 𝑝𝑟𝑥 𝑇𝑝𝑟𝑒 × 𝐸𝑟𝑓 + 𝑇𝑝𝑟𝑒 (𝑝𝑡𝑥 + 𝑝𝑟𝑥 ) ) 2(𝑇𝑟𝑓 + 2𝑇𝑝𝑟𝑒 + 𝑇𝑠𝑎𝑣 ) , (27)

and

(30) can be approximated by a posynomial ∑ 𝑗∈𝑁𝑖 𝑟𝑗𝑖 + 𝑆𝑖 ˆ ∀𝑖. 𝑓𝑖 (R) = ˆ 𝑖 (R) ℎ

which implies that

𝑐1 and 𝑐2 in (26) are constants as follows: 𝑐1

2263

𝑟𝑖𝑗𝑖𝑗 , so that the left hand side of

( ∑ ∂𝑔𝑖 (X′ ) ) ℎ𝑖 (R) 𝑥′𝑖𝑗 = ∏ . 𝑐𝑖 = exp 𝑔𝑖 (X′ )− 𝑎𝑖𝑗 ∂𝑥𝑖𝑗 ′ 𝑗∈𝑁𝑖 𝑥𝑖𝑗 𝑟𝑖𝑗 =𝑟𝑖𝑗 𝑗∈𝑁𝑖 (34) Once 𝑎𝑖𝑗 and 𝑐𝑖 are computed, the posynomial approximation to the original signomial in (30) is obtained. Detailed discussions on monomial approximation can be found in [20]. C. An Iterative GP algorithm In this subsection, an Iterative GP (IGP) algorithm is presented to solve Problem (25) efficiently through a series of approximations. At each step, we have a current guess (𝑘) (R(k) , T𝑐𝑦𝑐𝑙𝑒 ), near which a posynomial approximation of (25c) is constructed using formulas (31)-(34). Replacing (25c) by the posynomial approximation, we convert the SP problem into a GP that can be solved efficiently using convex optimization techniques [20]. The solution of the GP is taken as (𝑘+1) the next iterate (R(k+1) , T𝑐𝑦𝑐𝑙𝑒 ). The flowchart of the IGP algorithm is shown in Fig. 4. In theory, any feasible solution can be the initial solution. In our simulations, we obtain the initial solution by randomly choosing a sleep time for all nodes from a range of values. With fixed sleep time T𝑐𝑦𝑐𝑙𝑒 , (25) can be reformulated as a LP problem. We then solve the LP problem to obtain the routing matrix. This routing decision together with the initial node sleep time is used as the initial solution of the IGP algorithm. On the other hand, the algorithm stops when the improvement of network lifetime becomes marginal for several consecutive iterations. In our simulations, the iterative algorithm stops when the improvements of network life-time achieved in the latest 3 iterations are less than 1%. In the following theorem, we will prove that the IGP algorithm yields a series of solutions (0)

(𝑘)

(R(0) , T𝑐𝑦𝑐𝑙𝑒 ), ⋅ ⋅ ⋅ , (R(𝑘) , T𝑐𝑦𝑐𝑙𝑒 ), ⋅ ⋅ ⋅ that converges to a Karush-Kuhn-Tucker (KKT) solution of (25). We first introduce the following lemma which will be useful in later proofs. Lemma 1. If the posynomial approximation satisfies the following three properties, then solutions of the series of approx-

2264


TABLE I SYSTEM POWER AND TIME PARAMETERS

Initial feasible solution (k ) (R ( k ) , Tcycle ) , k=0

𝑝𝑡𝑥 𝑝𝑟𝑥 𝐸𝑟𝑓 𝑇𝑟𝑓 𝑇𝑑𝑎𝑡𝑎 𝑇𝑝𝑟𝑒

Posynomial approximation of (25c) near R (k ) k=k+1

𝑇𝑠𝑎𝑣

Transmission power Receiving power Energy of initializing RF circuits Time to initialize RF circuits Time to transmit one data packet Time to transmit one RTS/CTS/ACK preamble Power saving interval between two consecutive RTS preambles

60 mW 45 mW 20.05 𝜇J 2.1 ms 14.98 ms 0.832 ms 4.16 ms

( k +1) Solve for ( R ( k + 1) , Tcycle )

iteratively and alternatively fixing T𝑐𝑦𝑐𝑙𝑒 or R in (25) and optimizing the other. Note that the optimization problem becomes a linear program (LP) when T𝑐𝑦𝑐𝑙𝑒 is fixed, and a geometric programming (GP) when R is fixed. Both of the two problems are convex and can be solved efficiently. This S-RS method performs better than the other separate design methods since in both routing and sleep scheduling phase in each iteration, the lifetime maximization task is formulated and solved optimally. Instead of exhausting all potential separate design methods that use different heuristic routing and sleep scheduling algorithms in each phase, it is sensible to compare the network lifetime of the proposed joint optimization scheme with this S-RS method.

N

Stopping criteria satisfied? Y

(

* Optimal solution R* , Tcycle

Fig. 4.

)

Iterative geometric programming algorithm.

imation converge to a point that satisfy the KKT conditions of the original problem [21]: Property 1: 𝑓𝑖 (R) ≤ 𝑓ˆ𝑖 (R) for all R; Property 2: 𝑓𝑖 (R) = 𝑓ˆ𝑖 (R′ ) where R is the optimal solution of the approximation problem in the previous iteration; Property 3: ∇𝑓𝑖 (R′ ) = ∇𝑓ˆ𝑖 (R′ ). Theorem 2. The series (0) (𝑘) (R(0) , T𝑐𝑦𝑐𝑙𝑒 ), ⋅ ⋅ ⋅ , (R(k) , T𝑐𝑦𝑐𝑙𝑒 ), ⋅ ⋅ ⋅ converges to a KKT solution of (25).

of obtained

solutions by IGP

Proof: To prove Theorem 2, we show that the posynomial approximation (31) satisfies the three properties listed in Lemma 1. ∑ To see Property 1, note that 𝑔𝑖 (X) = log 𝑗∈𝑁𝑖 𝑒𝑥𝑖𝑗 is convex on X [22], and hence its first order Taylor approximation is a global underestimator of the function. That is, ˆ 𝑖 (X) ≤ ℎ𝑖 (X) for 𝑔ˆ𝑖 (X) ≤ 𝑔𝑖 (X) for all X. Consequently, ℎ all R, due to the monotonicity of logarithm and exponential functions. Therefore, Property 1 holds. Property 2 and 3 are trivial by straightforward calculations.

D. Performance of Separate Routing and Sleep Scheduling Design Most previous work has either only considered energyefficient routing or sleep scheduling, or treated the joint problem as two separate tasks and solved them independently. One of the best method for the separate routing and sleep scheduling (S-RS) design is to iteratively update the optimal routing decisions and sleep scheduling separately through mathematic programming, with one component fixed while adjusting the other one. Mathematically, this is done by

V. N UMERICAL R ESULTS In this section, we illustrate the performance of the proposed IGP algorithm for the joint routing and sleep scheduling problem in different scenarios through simulation. We use a similar simulation network setup as that in [1]. In the setup, there are 20 nodes randomly located in a 50m-by50m area. The gateway node lies in the centre of the square area. We use the power consumption model of Mica2 mote and CC1000 transceiver [14], which is summarized in Table I. The nodes are working at constant RF transmission power and the RF range of each node is 20 meters. The initial node battery capacity follows an independent normal distribution (𝜇, 𝛿 2 ) with 𝜇 = 2500𝑚𝐴ℎ and 𝛿 = 25𝑚𝐴ℎ. This corresponds to the case when all nodes have the same initial battery capacity with 1% process variation. We compare the network lifetime obtained by three different schemes: a routing scheme with fixed node sleep time, the iterative two-phase separate routing and sleep scheduling (SRS) method, and the proposed IGP algorithm. Each node has the same packet generation rate of 4.19 × 10−3 packet/second. We randomly generate 20 different network topologies according to the above setup. For each network topology, the initial sleep time of nodes is randomly picked from an interval [10 × 𝑇𝑑𝑒𝑡 , 1000 × 𝑇𝑑𝑒𝑡 ]. Both the IGP algorithm and S-RS method terminate when the network lifetime obtained at all of the latest 3 iterations does not outperform that of the previous iteration by 1%. The average network lifetime obtained using the three different schemes is listed in Table II. From Table II we can see that the average network lifetime obtained by IGP algorithm outperforms that of S-RS method by 26.4% and outperforms that of fixed-sleep-time routing by 278.7%.


TABLE II

AVERAGE NETWORK LIFETIME (N ODE PACKET R ATE =4.19 × 10−3 )

S-RS Method

(second) 1.03 × 109

(second) 8.15 × 108

8

12

x 10

TIGP

TSRS

network

Fixed-Sleep-Time Routing (second) 2.72 × 108

Tmax

network

network

10

8 Network Lifetime (second)

IGP Algorithm

G ENERATION

2265

6

4

IGP Algorithm S−RS Method Fixed Sleep Time, LP Routing 2

9

10 Network Lifetime (second)

0

Fig. 6.

5

10

15 20 Different Initial Conditions (Node sleep time)

25

30

Performance over different initial conditions.

8

10

9

−3

10

Fig. 5.

−2

10 Node Packet Generation Rate (packet/second)

Network lifetime over varying traffic density.

We then compare the performance of the three schemes under different network traffic density by changing the node packet generation rate. The results are shown in Fig. 5, in which network lifetime is plotted versus the packet generation rate at each node. Each point in the figure is an average of 20 different network topologies. It is not surprising to see that the higher the packet generation rate, the more energy that is needed to transmit the generated packets to the gateway, and hence the shorter the network lifetime. With joint routing and sleep scheduling, IGP outperforms the S-RS method by around 29% on average for different traffic densities, and outperforms the fixed-sleep-scheduling routing by around 284% on average. The network lifetime obtained by multiple runs of IGP algorithm using different initializations can be taken as the upper-bound performance in future evaluation of any practical heuristics. Since the network lifetime maximization problem is a nonconvex problem by nature, no efficient algorithms is guaranteed to find the global optimal solution. Theorem 2 proves that the proposed IGP algorithm converges to a solution that satisfies the KKT condition of Problem (17). This implies that the IGP algorithm may converge to a local optimal solution depending on the initial condition, due to its non-convex nature. However, it would be interesting to find out how often the algorithm can converge to the global optimal solution. Since global optimal solution cannot be guaranteed due to the non-convex nature of the optimization problem, here we use the global optimal solution to denote the solution that has the maximum lifetime obtained from simulations with several different initializations. To see this, we pick a network topology as an example and run the IGP algorithm with 30 different initializations. Each initialization randomly picks an

Maximized Lifetime In Each Iteration (second)

10

8

10

7

10

6

10

Fig. 7.

1

2

3

4

5

6

7

8 9 10 IGP Iteration

11

12

13

14

15

16

Convergence speed of IGP algorithm.

initial sleep time within an interval [10 × 𝑇𝑑𝑒𝑡 , 1000 × 𝑇𝑑𝑒𝑡 ]. 𝑚𝑎𝑥 The result is plotted in Fig. 6. We use 𝑇𝑛𝑒𝑡𝑤𝑜𝑟𝑘 to present the best solution obtained among the different initial conditions. It is very likely to be the global optimal solution (with probability one asymptotically) [9]. For comparison purpose, 𝑆−𝑅𝑆 achieved by the S-RS method the network lifetime 𝑇𝑛𝑒𝑡𝑤𝑜𝑟𝑘 is also plotted. From the figure, we can see that IGP approaches the best 𝑚𝑎𝑥 solution, 𝑇𝑛𝑒𝑡𝑤𝑜𝑟𝑘 , most of the time. In 18 of the 30 (60%) different initial conditions, IGP achieves the best solution. By contrast, the solutions obtained by S-RS are much worse. It approaches the best solution in only 1 of the 30 (3%) initial conditions. It also worths to point out that in the rest 6 of the 30 (20%) initial conditions, IGP approaches more than 99% of the best network lifetime. We then demonstrate the convergency speed of the proposed IGP algorithm. Using the same example as in Fig. 6, the network lifetime is plotted versus the number of iterations in Fig. 7. The average number of iterations until the stopping criterion is satisfied is around 11. Finally we show an example of joint routing and sleep scheduling solution in Fig. 8. There exists a set of bottleneck

2266


50

n5

research topic is to design good distributed algorithms with limited complexity and control overhead based on the existing work of distributed algorithms for routing [5][8] and GP problems [9]. Moreover, we have assumed a collision-free system in this paper due to the light traffic in sensor networks. In our future work, we will extend this model to take into account the effect of collisions on routing and sleep scheduling decisions.

n17

45 n4

40

n2 n18

35 30

n1

n10

n9 n8

n7

25 n13 20

Tslp(n3)=4.395 s Tslp(n6)=20.000 s

15

n3

T

=8.242x10 s

network

10 T

(n16)=11.300 s n12

n19

Fig. 8.

R EFERENCES

n15

50% n16 50%

slp

5 0

(n14)=4.364 s

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An example of joint routing and sleep scheduling solution.

nodes (𝑛3 , 𝑛6 , 𝑛14 , 𝑛16 ) which run out of their battery energy earlier than the other nodes and limit the network lifetime. The routing paths across the bottleneck nodes are marked by solid lines and the other routing paths are marked by dotted lines. The nodes 𝑛3 and 𝑛14 serves as one-hop neighbors of the always-on gateway and no RTS transmission power is consumed when trying to transmission data packets to the gateway. Therefore 𝑛3 and 𝑛16 have much shorter sleep time than 𝑛16 in order to reduce 𝑛16 ’s power consumption on RTS transmission at the cost of spending more idle listening power of 𝑛3 and 𝑛14 . On the other hand, since 𝑛6 does not serve as relay node and its sleep time does not affect the power consumption of any other node, we set its sleep time to the maximum allowed value which is 20 second in this example. The sleep time of 𝑛3 , 𝑛14 and 𝑛16 is decided to make the four bottleneck nodes have the same lifetime as marked in the figure. VI. C ONCLUSION AND F UTURE W ORK In this paper, we have studied network life time maximization of wireless sensor networks through joint routing and sleep scheduling. Optimal joint routing and sleep scheduling is known to be a difficult problem due to its non-convexity. We tackle the problem by transforming it into a special form of SP that only has inequality constraints. The problem is then solved through an IGP algorithm, where the non-convex problem is approximated by a standard GP in each iteration. It is proved that the IGP algorithm converges to a solution that satisfies the KKT conditions. Our work in this paper has demonstrated the importance of joint routing and sleep scheduling. The proposed IGP algorithm drastically outperforms the performance of optimal iterative separate routing and sleep scheduling method by an average of 29% over a large range of traffic rates. Compared with the traditional designs with optimal routing but fixed sleep scheduling, the proposed IGP algorithm prolongs the lifetime by an average of 284%. The proposed algorithm serves as a useful benchmark to evaluate practical heuristics that endeavor to maximize the network lifetime. Based on this work, one interesting future

[1] J.-H. Chang and L. Tassiulas, “Maximum lifetime routing in wireless sensor networks," IEEE/ACM Trans. Networking, vol. 12, no. 4, pp. 609-619, Aug. 2004. [2] W. Ye, J. Heidemann, and D. Estrin, “Medium access control with coordinated adaptive sleeping for wireless sensor networks," IEEE/ACM Trans. Networking, vol. 12, no. 6, pp. 493-506, June 2004. [3] V. Raghunathan and S. Ganeriwal, “Emerging techniques for long lived wireless sensor networks," IEEE Commun. Mag., vol. 44, no. 4, pp. 108-114, Apr. 2006. [4] J. Li and G. Alregib, “Network lifetime maximization for estimation in multihop wireless networks," IEEE Trans. Signal Process., vol. 57, no. 7, pp. 2456-2466, June 2009 [5] R. Madan and S. Lall, “Distributed algorithms for maximum lifetime routing in wireless sensor networks," IEEE Trans. Wireless Commun., vol. 5, no. 8, pp. 2185-2193, Aug. 2006. [6] R. Madan, S. Cui, S. Lall, and A. Goldsmith, “Cross-layer design for lifetime maximization in interference-limited wireless sensor networks," IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3142-3152, Nov. 2006. [7] J. C. Dagher, M. W. Marcellin, and M. A. Neifield, “A theory for maximizing the lifetime of sensor networks," IEEE Trans. Commun., vol. 55, no. 2, pp. 323-332, Feb. 2007. [8] S.-J. Kim, X. Wang, and M. Madihian, “Distributed joint routing and medium access control for lifetime maximization of wireless sensor networks," IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2669-2677, July 2007. [9] M. Chiang, C. V. Tan, D. P. Palomar, D. O’Neill, and D. Julian, “Power control by geometric programming," IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2640-2651, July 2007. [10] Y. T. Hou, Y. Shi, and H. D. Sherali, “Rate allocation and network lifetime problems for wireless sensor networks," IEEE/ACM Trans. Networking, vol. 16, no. 2, pp. 321-334, Apr. 2008. [11] C. Hua and T.-S. Yum, “Optimal routing and data aggregation for maximizing lifetime of wireless sensor networks," IEEE/ACM Trans. Networking, vol. 16, no. 4, pp. 892-903, Aug. 2008. [12] H. Wang, Y. Yang, M. Ma, J. He, and X. Wang, “Network lifetime maximization with cross-layer design in wireless sensor networks," IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 3759-3768, Oct. 2008. [13] W. Ye, J. heidemann, and D. Estrin, “An energy-efficient MAC protocol for wireless sensor networks," in Proc. IEEE INFOCOM, vol. 3, pp. 1567-1576, June 2002. [14] J. Polastre and D. Culler, “Versatile low power media access for wireless sensor networks," in Proc. ACM Conf. Embedded Netw. Sensor Syst., pp. 95 C107, Nov. 2004. [15] E.-T. A. Lin, J. M. Rabaey, and A. Wolisz, “Power-efficient rendezvous schemes for dense wireless sensor networks," in Proc. 2004 IEEE International Conf. on Commun., vol. 7, pp. 3769-3776, June 2004. [16] Q. Dong, “Maximizing system lifetime in wireless sensor networks," in Proc. International Conf. Inf. Process. Sensor Netw., pp. 13-19, Apr. 2005. [17] S. Chachra and M. Marefat, “Distributed algorithm for sleep scheduling in wireless sensor networks," in Proc. IEEE International Conf. Robotics Automation, pp. 3101-3107, May 2006. [18] R. Subramanian and F. Fekri, “Sleep scheduling and lifetime maximization in sensor networks—fundamental limits and optimal solutions," in Proc. International Conf. Inf. Process. Sensor Netw., pp. 218-225, Apr. 2006. [19] E. Bulut and I. Korpeoglu, “DSSP: a dynamic sleep scheduling protocol for prolonging the lifetime of wireless sensor networks," in Proc. 21st International Conf. Advanced Inf. Netw. Applications Workshop, pp. 725-730, May 2007. [20] M. Avriel, Advances in Geometric Programming. Plenum Press, 1980.


[21] B. R. Marks and G. P. Wright, “A general inner approximation algorithm for nonconvex mathematical programs," Operations Research, vol. 26, no. 4, pp. 681-683, 1978. [22] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. Feng Liu received his Ph.D. degree in Electrical and Electronic Engineering from the Hong Kong University of Science and Technology, Hong Kong in 2009. Since Jul. 2009, he has been with the Hong Kong Applied Science and Technology Research Institute (ASTRI), where he is currently an senior engineer. His research interests cover energy efficient circuit, algorithm and system design for wireless networks.

Chi-Ying Tsui received his B.S. degree in Electrical Engineering from the University of Hong Kong and Ph.D. degree in Computer Engineering from the University of Southern California in 1994. He joined the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology in 1994 and is currently an Associate Professor in the department. His research interests cover both VLSI design and system optimization. His current research includes designing low power multimedia architecture and wireless applications, developing power management circuits and techniques for embedded portable devices and ultra-low power systems. Dr. Tsui has published more than 140 referred technical journal and conference papers. He received the Best Paper Awards from the IEEE T RANSACTIONS ON VLSI S YSTEMS in 1995, IEEE/ACM ISLPED in 2007, and supervised the Best Student Paper Award of the 1999 IEEE ISCAS.

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Ying Jun (Angela) Zhang (S’00, M’05) received her PhD degree in Electrical and Electronic Engineering from the Hong Kong University of Science and Technology, Hong Kong in 2004. Since Jan. 2005, she has been with the Department of Information Engineering in The Chinese University of Hong Kong, where she is currently an assistant professor. Dr. Zhang is on the Editorial Boards of IEEE T RANSACTIONS OF W IRELESS C OMMUNICATIONS and Wiley Security and Communications Networks Journal. She has served as a TPC Co-Chair of Communication Theory Symposium of IEEE ICC 2009, Track Chair of ICCCN 2007, and Publicity Chair of IEEE MASS 2007. She has been serving as a Technical Program Committee Member for leading conferences including IEEE ICC, IEEE GLOBECOM, IEEE WCNC, IEEE ICCCAS, IWCMC, IEEE CCNC, IEEE ITW, IEEE MASS, MSN, ChinaCom, etc. Dr. Zhang is an IEEE Technical Activity Board GOLD Representative, 2008 IEEE GOLD Technical Conference Program Leader, IEEE Communication Society GOLD Coordinator, and a Member of IEEE Communication Society Member Relations Council (MRC). Her research interests include wireless communications and mobile networks, adaptive resource allocation, optimization in wireless networks, wireless LAN/MAN, broadband OFDM and multicarrier techniques, MIMO signal processing. As the only winner from Engineering Science, Dr. Zhang has won the Hong Kong Young Scientist Award 2006, conferred by the Hong Kong Institution of Science.