Battery Recovery Aware Sensor Networks - Temple University

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Battery Recovery Aware Sensor Networks ∗ EE

Chi-Kin Chau∗† , Muhammad Husni Wahab∗ , Fei Qin∗ , Yunsheng Wang∗ , Yang Yang∗ Department, University College London. † Computer Laboratory, University of Cambridge. Email: [email protected], {m.wahab, f.qin, uceeywa, y.yang}@ee.ucl.ac.uk

Abstract—Many applications of sensor networks require batteries as the energy source, and hence critically rely on energy optimisation of sensor batteries. But as often neglected by the networking community, most batteries are non-ideal energy reservoirs and can exhibit battery recovery effect — the deliverable energy in batteries can be replenished per se, if left idling for sufficient duration. We made several contributions towards harnessing battery recovery effect in sensor networks. First, we empirically examine the gain of battery runtime due to battery recovery effect, and found this effect significant and durationdependent. Second, based on our findings, we model the battery recovery effect in the presence of random sensing activities by a Markov chain model, and study the effect of duty cycling and buffering to harness battery recovery effect. Third, we propose a more energy-efficient duty cycling scheme that is aware of battery recovery effect, and analyse its performance with respect to the latency of data delivery. Index Terms—Sensor Networks, Energy Efficiency, Battery Recovery, Duty Cycling

I. I NTRODUCTION Wireless sensor networks are created by networks of small sensors integrated with tiny embedded processors, wireless interfaces, and MEMS micro-sensors. Many applications of wireless sensor networks require batteries as energy source for the sensors. However, small form factors of devices often prohibit the uses of large and long lasting batteries. Moreover, ad-hoc deployment of sensor networks and the inconvenience of battery recollection usually constrain frequent replacement of batteries. Hence, the design of energy-efficient control algorithms and protocols is a crucial topic in sensor networking. There are a variety of energy optimisation studies in the literature that mostly consider batteries as ideal energy reservoirs, from which energy can be drained at constant discharging voltage, and can be halted and resumed at anytime to regain the same voltage. However, most commercial batteries are governed by complex non-linear internal chemical reactions to provide energy. Such chemical reactions are known by chemical engineers to be dependent on a variety of environmental factors and operational parameters (e.g. temperature, discharging duration, discharging current, memory of past discharging profiles) [1]. 1 Chi-Kin Chau is grateful to The Croucher Foundation for financial support, and to Basu Prithwish for helpful discussion. Yang Yang’s research was partially supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under the project EP/F004532/1. The authors also would like to thank the reviewers for detailed comments. This research is continuing through participation in the International Technology Alliance sponsored by the U.S. Army Research Laboratory and the U.K. Ministry of Defence.

Particularly, there is a subtle phenomenon called battery recovery effect, which refers to the process that the chemical substances in a battery will replenish themselves if left idling for sufficient duration, and hence, the deliverable energy of a battery can be recharged per se. Thus, we are motivated to design and engineer control algorithms and protocols that can harness battery recovery effect. In this paper, we first empirically examine the gain of battery runtime due to battery recovery effect, through extensive test-bed experiments on commercial sensors. We found that such gain can be significant under appropriate duty cycling control. Our experiments also show that there exists a saturation threshold, by which more consecutive idling periods will not contribute to more recovery. The ramification is that if we carefully adjust the idle periods of batteries before reaching the saturation threshold, we can maximise battery recovery effect without wasting too much idling time that may hamper the quality of service. Our experiments consider periodic constant battery consumptions. To investigate the behaviour of battery recovery effect in the presence of random sensing activities, we formulate a Markov chain model and provide analytical insights on the gain of expected battery runtime. Then, we study the effect of duty cycling and buffering to harness battery recovery effect. It is common that RF transceiver operations in a sensor will consume most of the energy (even in listening mode), as compared to the processing and sensing activities. Thus, duty cycling is frequently employed to regulate the on/off periods of the RF transceiver, while keeping the rest of sensor module on (e.g. the sensing and processing units are on to detect and buffer the sensing data before the RF transceiver is awake for transmission.2) It is important to design proper duty cycling and buffering strategies that can maximise the battery recovery effect. Here, we propose a more energy-efficient duty cycling scheme by setting the sleep duration of the RF transceiver as the saturation threshold of the battery, which can take the maximal advantage of the duration-dependent battery recovery effect. We verify the usefulness of our scheme by simulation studies. Furthermore, we consider the setting of multi-hop sensor network, where each sensor can act as a relay to forward the sensing data for other sensors to the sink. In this setting, there requires a coordination scheme among the duty cycling 2 Here we do not want to jeopardise the accuracy of sensing data, and hence do not consider to turn off the sensing and processing units for the sake of energy conservation. Note that such duty cycling control on the RF transceiver only affects the timeliness of sensing data.

This paper appears in the proceedings of the 7th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), 2009.

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sensors, such that each sensor can discover the appropriate timeslot to transmit data without wasting energy to probe the availability of their relays. We adapt a distributed randomised coordination scheme from [2], and extend it to be aware of battery recovery effect. In this scheme, each sensor infers the random duty-cycling schedule of the relay based on pseudorandom sequence, and the random duty-cycling schedule is set to take advantage of battery recovery effect by forcing a sensor to sleep when it has been awake for certain duration. We then analyse the performance of our scheme, by extending the latency analysis of pseudo-random duty cycling scheme in previous work [3]. We obtain analytical results of the latency of data delivery. In this paper, our contributions are threefold, as summarised as follows: 1) We provide experimental evidence on the significant gain of battery runtime by battery recovery effect, and found that the gain is duration-dependent, characterised by a saturation threshold. 2) We study the behaviour of duration-dependent battery recovery effect in the presence of random sensing activities, and the effect of duty cycling and buffering, through analysis and simulations. 3) We propose a more energy-efficient duty cycling scheme and a distributed coordination scheme among duty cycling sensors, extending the pseudo-random duty cycling scheme to be aware of battery recovery effect. We also analyse the performance on the latency of data delivery. Because of space constraint, the proofs of analytical results are omitted in this paper, but can be found in the full paper [4]. II. R ELATED W ORK Sensor networks commonly use rechargable batteries, such as Nickel-cadmium (NiCd), Nickel-metal hydride (NiMH), Sealed lead-acid (SLA), Lithium-ion (Li-ion), and Lithiumpolymer (similar to Li-ion). Different batteries have different properties. NiCd and NiMH are more commonly used in sensor networks. NiCd has longer cycle life, while NiMH has higher energy density. There have been numerous studies about the performance of batteries in chemical engineering [1]. In networking, [5] carried out an empirical study to measure the performance of battery-powered sensors, but did not examine the saturation threshold. Although battery consumption has been modeled extensively in networking, many extant models ignore or over-simplify the realistic battery characteristics (e.g. allowing unlimited battery recovery). There are two main models considering realistic battery characteristics. First, the kinetic battery models [6]–[8] attempt to model the detailed chemical reactions and diffusion process between the electrode and electrolyte in a battery through a set of partial differential equations. These models aim to fully capture the non-linear dynamics in a battery. However, these models are less tractable, and different form factors of batteries can significantly affect the accuracy of the models.

Second, there are stochastic battery models [9]–[13] that capture the battery dynamics using probabilistic Markovian models. But most of them do not consider the effect of idle time. [14] considers idle time in embedded systems, but does not address the performance in sensor networking. We remark that while all these stochastic battery models attempt to imitate the kinetic battery model with smaller complexity, the uses of probabilistic battery recovery is different from the deterministic kinetic battery models. Moreover, these models are also less tractable, with few analytical insights provided. In this paper, we present a more analysable Markov chain model that simplifies the stochastic battery models [10], [11], [14]. Particularly, our model uses deterministic battery recovery, yet is able to capture realistic battery characteristics, such as limited recovery and the effect of idle time. More importantly, useful analytical insights of realistic battery behaviour can be derived from our model. There are a number of approaches of energy managament in sensor networks, including topology management and network layer optimisation. But relevant to our work are the ones based on MAC layer, which aim to reduce redundant radio operations in MAC protocols: 1) idle listening, 2) overhearing, 3) collisions, and 4) protocol overhead (headers or signalling messages). Since both listening and reception consume significant energy in common sensors. In this paper, we focus on reducing idle listening and overhearing by duty-cycling and exploiting battery recovery effect. III. E XPERIMENTAL R ESULTS We present the experimental results on the significance of battery recovery effect from our sensor network test-bed. The experiments have been carried out on two types of commercial sensors from Crossbow: TelosB and Imote2. Both are popular models for wireless sensor networking. TelosB is consisted of MSP430 (1.8mA active and 5.1µA standby) as MCU and CC2420 (23mA active and 21µA standby) as the RF transceiver. Imote2 is consisted of PXA271 as CPU and CC2420 as the RF transceiver. TelosB and Imote2 have different system architectures — TelosB involves more energysaving designs with less energy overhead, whereas Imote2 is equipped with high computation ability (from 13MHz to 400MHz) that requires higher energy overhead. The two models reflect different applications with two extreme energy consumption requirements. In the experiments, we use an analogue-digital conversion (ADC) interface card and LabVIEW to measure and record the discharging profiles of a pair of communicating sensors (see Fig. 1). Each sensor is powered by standard AAA NiMH 600 mAh batteries (TelosB has two batteries, whereas Imote2 has three). When the supply voltage of the battery is lower than a certain threshold (called stop voltage), the device can no longer operate, which is considered as to be completely discharged. We set different duty cycling rate on the sensors by putting the sensor in wake-up and sleep modes periodically, and measure the induced battery runtime. The duty cycling rate is defined as the fraction of wake-up periods. Figs. 3 and 5 show


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the discharging profiles for TelosB3 and Imote2 respectively, which are rescaled by multiplying the duty-cycling rate. Also, Figs. 4 and 6 show the gain of battery runtime under different duty-cycling rates, as compared to continual discharging. There are a number of key observations from the experiments: • There are clear signs of battery recovery effect. Keeping the same wake-up duration, longer sleep duration can induce longer (rescaled) battery runtime, and hence larger deliverable energy of battery. • The effect of sleep duration is non-linear. It appears that sleep periods more than a certain threshold will contribute much less to battery recovery, we call a saturation threshold. • TelosB has a lower stop voltage, whereas Imote2 has a higher one. This is due to TelosB involves lower energy overhead. Since Imote2 has high energy overhead and long boot-up time, battery recovery effect appears less prominent when shorter wake-up duration is used. • Both models of sensors exhibit non-uniform discharging profiles. For TelosB, the initial stage of discharging appears to be slightly convex, followed by a concave drop. For Imote2, the whole discharging profile is convex. This is due to the higher stop voltage of Imote2, which can only show partial discharging process, compared to TelosB’s one (or the well known turn ‘S’-shaped discharging curve of Ni-based battery). • Even the sensor is in sleep mode, there is still energy consumption due to the timer and other background processes. TelosB consumes 6.1µA in sleep mode, whereas Imote2 is 0.38mA. Since battery recovery effect can still happen under low battery consumption, we found the impact of background consumption is not substantial. • We observe that most AAA NiMH batteries can be overrecharged. Although the standard initial voltage of an 3 Because TelosB has a much longer battery runtime, we use a lower sampling rate, which gives a smoother profile curve.

Fig. 2.

Measurement from LabVIEW on Imote2.

AAA battery is stated as 1.2V, we found that a battery can be recharged up to 1.3V by standard battery charger. The exact value is dependent on battery memory. • Although our measurement of gain of battery recovery effect differs in other environmental settings (e.g. temperature), the insight revealed by our experiments will still be useful to the modelling and optimisation of battery recovery effect in sensor networks. Finally, we discuss a major difference in the system architectures of TelosB and Imote2. In general, Imote2 cannot be switch into and back from sleep mode fast. TelosB is of MCU+RF architecture, where it is easy to turn the MCU into sleep mode by issuing internal commands. But Imote2 is of CPU+RFIC and PowerControl IC. When Imote2 switches into sleep mode, it first needs to store it memory into flash, sending command to PowerControl IC, then hibernates itself. When Imote2 switches into awake mode, it simply restarts itself. It is found that in the experiment that Imote2 need several seconds to boot up, and in this period Imote2 consumes much more than the normal operations (see the current plot for Imote2 in Fig. 2). We remark that the difference in the system architectures reflects the different roles of both models: TelosB usually acts as a leaf node, whereas Imote2 is supposed to be the data processing node. IV. M ODEL OF BATTERY C ONSUMPTIONS Our experimental findings confirm the usefulness of battery recovery effect, and show the characteristics of durationdependence. We are motivated to model and study durationdependent battery recovery effect in more realistic situations considering random sensing activities as follows. For simplicity, we assume a discrete setting. The state of a battery can be characterised by a tuple hn, c, ti, where n, c, t are nonnegative integers. c is the theoretical capacity determined by the amount of chemicals in the electrode and electrolyte, n is the nominal capacity determined by the amount of available active chemicals for chemical reactions in the battery, and t is the idle duration since the battery has stopped discharging.


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Fig. 4. The maximum gain of battery runtime is around 25% for TelosB. The saturation threshold is around 5 sec.

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Fig. 5. The discharging profiles of Imote2 with respect to different duty cycling rates.

In the discharging process, both n and c are decreasing. The amount of available active chemicals constrains the energy of a battery can deliver, despite the presence of unused chemicals in the battery. Hence, n ≤ c. But when the battery stops discharging, there is a recovery process, as a diffusion process between electrode and electrolyte to replenish available active chemicals, and effectively increases n (though cannot increase the theoretical capacity c)4 . There is a saturation threshold for t, such that more consecutive idle periods will not contribute more recovery. A. Normal Operations To model random sensing activities under normal operations without duty cycling, we assume the consumption of a battery is a Poisson random variable, which is reasonable if the sensing data follow Poisson distribution. We model the battery behaviour as a Markov chain Mbat with state set defined as: n o hn, c, ti : n, c, t are non-negative integers, and n ≤ c And the transition probabilities from states of n ≥ 1 is defined as:

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Fig. 6. The maximum gain of battery runtime is up to 37% for Imote2. The saturation threshold is around 0.5 min.

a) Discharging: The transition probability of a

hn, c, ti −→ hn-k, c-k, 0i is pλpo (k) =

λk e-λ k!

for k ≥ 1 and n-k ≥ 1. The consumption is a Poisson random variable, where λ captures the rate of battery consumption in a timeslot. b) Completely Discharged: The transition probability of b

hn, c, ti −→ h0, c-n, 0i is

∞ X

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k!

That is, whenever the consumption is higher than or equal to the nominal capacity n, the battery will be completely discharged. c) Idling with Recovering: The transition probability of hn, c, ti −→ hn+1, c, t+1i is pλpo (0) = e-λ c

for c ≥ n+1 and t < tsat . That is, there is recovery when no battery consumption, and the consecutive idle duration is lesser than the saturation threshold tsat . d) Idling without Recovering: The transition probability of hn, c, ti −→ hn, c, t+1i is pλpo (0) = e-λ d

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studies [1] have shown that the recovery process is a nonlinear dynamics, which critically depends on the state of battery hn, c, ti. In general, the higher values of n and c are, the more significant the recovery effect is observed. The recovery effect is more prominent when the battery has consecutive idle periods than sporadic idle periods. There is also a saturation threshold for t, such that more consecutive idle periods will not contribute more recovery.

for c < n+1 or t ≥ tsat . That is, there is no recovery when the nominal capacity reaches the theoretical capacity, or the consecutive idle duration reaches the saturation threshold.


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e) Otherwise, the transition probabilities for all other state transitions equal 0. Note that h0, c, ti is an absorption state that will not transit to other states, corresponding to a complete discharged state. In Figs. 7-8 we illustrate the state transitions, assuming tsat = 1. We remark that there are several simplifications in this model. First, we assume the saturation threshold tsat is independent of n and c. Second, the recovery process is the same for any state below the saturation threshold. While these assumptions cannot capture the complete non-linear dynamics in real batteries, they are sufficient to represent a generic model of battery behaviour. With the minimum number of parameter (only tsat ), our model does not rely on extensive experiments to determine further parameters as in a more complete model that otherwise critically depend on the types of batteries and other environmental factors (e.g. temperature)5. Moreover, this model enables simple analytical results to yield useful insights for harnessing battery recovery effect. Let Wtm (n) be the expected battery runtime at state hn, n+m, ti in Markov chain Mbat (i.e. the expected number of timeslots a battery can last until reaching an absorption state as being completely discharged). By the definition of state transitions in Mbat and the linearity of expected values, we obtain: W00 (n)

=1+

n-1 X

pλpo (k)W00 (n-k),

Wt0 (n) = W00 (n)

proceed to immediate discharging. We let the buffered battery behaviour be a Markov chain Mbuf with state set as: n o hn, c, bi : n, c, b are non-negative integers, and n ≤ c Define the transition probabilities from states of n ≥ 1 as: a’) Discharging: The transition probability of λk e-λ a0 hn, c, 0i −→ hn-k, c-k, 1i is pλpo (k) = k! for k ≥ 1 and n-k ≥ 1. b’) Completely Discharged: The transition probability of b0

hn, c, bi −→ h0, c-n, 0i is

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c’) Idling: The transition probability of hn+1, c, 0i if c ≥ n+1 c0 hn, c, 0i −→ is pλpo (0) = e-λ hn, c, 0i if c < n+1 d’) Buffering: Define b+ , b+1(mod bmax +1) (i.e. the number of buffered timeslots increases by 1, until reaching the limit of buffered timeslots as bmax ). The transition probability of λk e-λ hn-k+1, c-k, b+ i if c ≥ n+1 d0 λ hn, c, bi −→ is p (k) = po hn-k, c-k, b+ i if c < n+1 k!

(1)

for 1 ≤ b ≤ bmax and n-k ≥ 1. e’) Otherwise, the transition probabilities for all other state That is, if the consumption is of k charges and is less than transitions equal 0. the nominal capacity n, then the expected battery runtime is Note that the above Markov chain will record the battery W00 (n-k) + 1. Otherwise, it will be completely discharged (the consumption during buffering periods, and hence automatexpected battery runtime is 1). ically deduct the battery consumption when releasing the For m ≥ 1, buffer. In Figs. 9-10 we illustrate the state transitions when  bmax = 1. n 1 P λ  t+1  ppo (k)W0m (n-k) if t < tsat Similarly, let Bbm (n) be the expected battery runtime with 1 + pλpo (0)Wm -1 (n+1) + k=1 buffering at state hn, n+m, bi in Mbuf . Like Wtm (n), we Wtm (n) = n -1 P  λ 0 λ tsat  ppo (k)Wm (n-k) if t = tsat obtain: 1 + ppo (0)Wm (n) + k=1 Bb0 (n) = W00 (n) (3) (2) k=0

B. Duty Cycling and Buffering Next, we define Markov chain Mbuf with duty cycling on RF transceiver and buffering. We assume that the sensor will sleep for bmax timeslots, immediately after the battery is consumed for a burst of RF transmission. During the sleep periods, the sensing unit is still on, and the data is buffered for at most bmax timeslots. To take the maximal advantage of battery recovery effect, we set bmax ≤ tsat . Now the state is defined as hn, c, bi, where b ≥ 1 means the battery has been in buffered state for b timeslots. If b + 1 > bmax , then the buffer will not hold any consumption and 5 Other models (e.g. [11], [14]) attempt to incorporate more parameters, such as probabilistic recovery effect to capture non-linear battery behaviour. These models are often inconvenient for analysis, and yield little analytical insights.

For m ≥ 1,  n -1 P +   pλpo (k)Bbm-1 (n-k+1) if b > 0 1 + k=0 Bbm (n) = n -1 P   pλpo (k)B1m (n-k) if b = 0 1 + pλpo (0)B0m-1 (n+1) + k=1

(4) where b+ , b+1(mod bmax +1) (i.e. the number of buffered timeslots increases by 1, until reaching the limit of buffered timeslots as bmax ).

C. Analytical Results In general, Wtm (n) and Bbm (n) appear with no simple closed-form expression. However, we can obtain several analytical results as follows, which can offer useful insights of the behaviour.


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Theorem 1: The expected battery runtime is bounded by: 1 n+m-1 + W0m (n) ≤ λ 1 − e-λ

The bounds are intuitive, because the battery runtime should not exceed Θ(n+m) = Θ(c). Indeed, these upper bounds can be shown as tight.

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the effectiveness decreases as bmax and tsat increase, because the larger saturation threshold means more likely battery recovery effect take place, so duty-cycling and buffering will not improve too much. Max Gain H%L tsat =bmax =1

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Theorem 2: When λ is small or n is large, for some constants α, n+m W0m (n) ≈ +α λ Theorem 3: Considering bmax = tsat , buffering is always better: Bbm (n) ≥ Wtm (n). We carried out simulation studies for specific n and m. Figs. 11-12 depict the plots for m = 40. We can see that duty-cycling and buffering facilitates battery recovery, and can reach the upper bound closer. We remark that more drastic gap between W0m (n) and B0m (n) can be observed for larger m. Figs. 11-12, we have seen that by duty-cycling and buffering, there is a significant improvement of the battery runtime. In Fig. 13, we examine the gain of using duty-cycling and buffering. We found the n’s in Figs. 11-12 that maximise the difference between W040 (n) and B040 (n) with respect to different values of λ and bmax , tsat , and plot the corresponding gain. We observe that the maximum gain can be up to 200%. However,

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Fig. 13. The maximum gain of using duty-cycling and buffering against normal operations over all n.

V. M ULTI - HOP S ENSOR N ETWORKS In this section, we consider multi-hop sensor networks, where each sensor can act as a relay to forward the sensing data for other sensors to the sink. If we employ the duty cycling and buffering scheme as in Sec. IV on all the sensors, then there requires a coordination scheme among the duty


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Fig. 11. When bmax = tsat = 1, we plot of the upper bound battery runtime with duty-cycling and buffering B040 (n).

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cycling sensors, such that each sensor can discover the appropriate timeslot to transmit data. A simple scheme is to employ a global coordinator that assigns a periodic/deterministic duty cycling schedule to each sensor in a-priori manner, and all sensors are provided the knowledge of the duty cycling schedules of their neighbours. However, this scheme suffers serious scalability issue, and cannot cope with the rapid changing network topology due to ad hoc sensor deployments. Another simple duty cycling scheme is to turn RF transceivers off and on independently randomly. Then, packet forwarding can only occur when the transmitting and receiving nodes are both awake. This does not require a global coordinator among the devices, and is more resilient and self-configuring for ad hoc deployments. On the other hand, in periodic/deterministic duty cycling schedules, the on/off schedules of network nodes have to be carefully coordinated such that efficient routing and forwarding can occur, which are less robust than random duty cycling schemes. However, random duty cycling also suffers performance issue, such that a sensor needs to probe the availability of its relays. Here, we adapt a distributed randomised coordination scheme from [2], by which each sensor infers the random duty-cycling schedule of the relays based on pseudo-random sequence. We then extend this pseudo-random duty-cycling scheme to be aware of battery recovery effect, is set to take advantage of battery recovery effect by forcing a sensor to

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sleep when it has been awake for certain duration. We also obtain analytical results of the latency of data delivery, based on the latency analysis of pseudo-random duty cycling scheme in previous work [3]. A. Pseudo-random Duty-cycling Scheme First, we describe the pseudo-random duty-cycling scheme. As in Sec. IV, we assume that when a sensor switches to sleep mode, only the RF transceiver is off, keeping the processing and sensing units for sensing activities on. Hence, it will not undermine the sensing functionality of the sensor networks. Pseudo-random duty cycling [2], [3] is proposed as a more energy-efficient approach than purely random duty cycling. Suppose that transmissions occur in slotted time. Usually, random duty cycling is determined by a pseudo-random sequence generator at a node. In pseudo-random duty cycling scheme, if a node knows the seed and the cycle position of the neighbours’ pseudo-random sequence generators, then it can deterministically predict its neighbours’ wake-up and sleeping timeslots. This prevents a node from sending packets in the timeslots that all its neighbours sleep, and can effectively reduce energy for unnecessary RF operations. We suppose that all transmitters will buffer all their outgoing packets until the timeslot when their respective receivers are awake. The pseudo-random duty-cycling scheme is described as follows:


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1) At bootstrapping, neighbouring sensors exchange the seed, cycle position and duty-cycling rate (the probability threshold of a slot that it is awake or asleep) of pseudo-random sequence generators. 2) At each slot, a sensor determines its state (being asleep or awake), and all the neighbours’ states in the next slot. 3) To forward packets, a sensor will wait until there is an awake neighbour on its shortest path to the sink, and then transmit the packets with corresponding receiver ID in the header. The receiver of the corresponding ID will carry out the forwarding of the packets until reaching the sink. Note that we assume the arrival packet rate is not high, and ignore the interference of simultaneous transmitters6 . We also assume that the sensing events of all sensors follow the same independent Poisson distribution pλpo (k). We defer the study of correlated sensing events to the future work. In the following, we study the battery runtime of pseudo-random duty-cycling scheme. Give a network topology, we pick one node to be the sink, and all other nodes must send the random sensing data (as distributed as Poisson distribution pλpo (k)) to the sink using multi-hop forwarding. We define the network life as the expected time that there is a relaying node running out of its battery. By simulation studies, we compare the network life of pseudo-random duty-cycling scheme and the normal case where all the sensor is always on. We consider that only transmission will consume energy (not listening), which is proportional to the amount of data transmitted. First, we consider linear network topology, where there are n nodes. The rightmost node is a sink. Fig. 14 shows that the gain of pseudo-random duty-cycling scheme is around 20% − 35%. Next, we consider 2D lattice network topology, and index each node as (i, j) by integers i, j. There is a link between nodes (i, j) and (i0 , j 0 ), if (|i-i0 | = 1 and j = j 0 ) or (|j-j 0 | = 1 and i = i0 ). Without loss of generality, we denote node (0, 0) as the sink, and all other nodes send packets destined to it. (i,j)

(k,l) (0,0)

Fig. 16. A lattice where the orange node is the sink (0, 0), we consider the number of shortest paths from (i, j) via (k, l).

Suppose that each node only uses greedy forwarding (i.e. 6 Even the packet rate is more moderate, duty cycling on sensors already cut down the number of potential collisions.

forwarding packets to only the neighbours on the shortest path to the sink). 1) For i ≥ 1, node (i, 0) will forward to (i-1, 0). 2) For i, j ≥ 1, node (i, j) for i, j ≥ 1 will randomly forward to (i-1, j) or (i, j-1) with equal probability. Fig. 15 shows that the gain of pseudo-random duty-cycling scheme can be up to 50%. B. Battery Recovery Awareness To extend the pseudo-random duty cycling scheme to take advantage of battery recovery effect, we propose a simple scheme by forced sleep. Suppose that a sensor has been awake for more than wmax consecutive slots at the current slot, then it must go to sleep for the next bmax slots for some bmax ≤ tsat . This allows sufficient battery recovery process to maximise the deliverable energy in the sensor network. A typical setting will be wmax = 1 and bmax = tsat . VI. L ATENCY A NALYSIS Increasing the sleep periods of sensor to maximise battery recovery effect will inevitably increase the latency of delivering a packet to the sink. In this section, we provide analytical results for the latency of data delivery in sensor networks with pseudo-random duty cycling. Suppose node i is waiting to forward data, which has a set of neighbours Ni and degree as di . Each of these neighbours is performing pseudo-random duty cycling with probability ρdc , such that in one time slot, each node is awake with i.i.d. probability ρdc , and is sleep of probability 1-ρdc . Let L(i) be the random number of slots at i before one of the neighbours of i wakes up. Therefore, L(i) = min{L1 , L2 , . . . , Ldi }, where Lj is the waiting time random variable for neighbour j ∈ Ni . Theorem 4: (See [3]) E[L(i) ] =

1 1 − (1 − ρdc )di

Note that the expected per-hop latency decreases quickly with decreasing node degree di . For extremely low duty cycling rate (i.e. small value ρdc ), we obtain: 1 1 E[L(i) ] ≈ = 1 − (1 − ρdc di ) ρdc di Theorem 5: For 2D lattice, let `(i, j) be the end-to-end latency from (i, j) to (0, 0). j-1 i-1 and E[`(0, j)] = 1 + (1) E[`(i, 0)] = 1 + ρdc ρdc i+j-1 (2) For i, j ≥ 1, E[`(i, j)] ≤ 1 + ρdc The above theorems enable sensor network designers a useful tool to balance and optimise the trade-off between increasing battery runtime of sensor networks and the incurred latency of data delivery. Here we discuss a useful application of our theorem. Suppose that we are designing a sensor network with a latency constraint. We let the maximum tolerable latency be E[L(i) ], and obtain the corresponding ρdc from Theorems 4-5.


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Network Life HslotsL 400

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Fig. 14. Considering linear network topology, and bmax = tsat = 3, we compare the network life of pseudo-random duty-cycling scheme and the normal case where all the sensor is always on. Network Life HslotsL 100

Duty-cycling Λ=1

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Fig. 15. Considering 2D lattice network topology, and bmax = tsat = 3, we compare the network life of pseudo-random duty-cycling scheme and the normal case where all the sensor is always on.

VII. C ONCLUSION This paper examines the gain of battery recovery effect and provides analytical results to shed light on harnessing the battery recovery effect in sensor networks. In particular, we analyse battery recovery effect in the presence of random sensing activities, based on our experiments. We derive upper bounds of battery runtime and study the benefit of dutycycling and buffering. We then propose a more energy-efficient duty cycling scheme that is aware of battery recovery effect, by extending the pseudo-random duty cycling scheme. We provide analytical results that predict the latency of data delivery in sensor networks when considering battery recovery optimisation. In future work, we aim to study a broader scope of optimising battery recovery effect in conjunction with a variety of qualities of service observed in sensor networks, such as coverage, connectivity, reliability. R EFERENCES [1] D. Linden, Handbook of Batteries and Fuel Cells. McGraw-Hill, 1994. [2] J. Redi, S. Kolek, K. Manning, C. Partridge, R. Rosales-Hain, R. Ramanathan, and I. Castineyra, “Javelen: An ultra-low energy ad hoc wireless network,” Ad Hoc Networks Journal, vol. 5, no. 8, 2008. [3] P. Basu and C.-K. Chau, “Opportunistic forwarding in wireless networks with duty cycling,” in Proc. of ACM Workshop on Challenged Networks (CHANTS), September 2008.

[4] C.-K. Chau, M. H. Wahab, F. Qin, and Y. Yang, “Harnessing battery recovery effect in sensor networks,” Tech. Rep. [5] C. Park, K. Lahiri, and A. Raghunathan, “Battery discharge characteristics of wireless sensor nodes: An experimental analysis,” in Proc. IEEE Secon, 2005, pp. 430–440. [6] J.F.Manwell and J.G.McGowan, “Extension of the kinetic battery model for wind/hybrid power systems,” in Proc. EWEC, 1994, pp. 284–289. [7] M. Doyle and J. S. Newman, “Analysis of capacity-rate data for lithium batteries using simplified models of the discharge process,” J. Applied Electrochem, vol. 27, pp. 848–856, JUL 1997. [8] T. F. Fuller, M. Doyle, and J. S. Newman, “Newman. modeling of galvanostatic charge and discharge of the lithium polymer insertion cell,” J. Applied Electrochem, vol. 140, pp. 1526–1533, 1993. [9] C. F. Chiasserini and R. R. Rao, “A model for battery pulsed discharge with recovery effect,” in Proc. IEEE WCNC, 1999, pp. 636–639. [10] C.-F. Chiasserini and R. R. Rao, “Improving battery performance by using traffic shaping techniques,” IEEE J. Selected Areas in Communications, vol. 19, pp. 1385–1394, JUL 2001. [11] C. F. Chiasserini and R. R. Rao, “A traffic control scheme to optimize the battery pulsed discharge,” in Proc. MILCOM, vol. 2. IEEE, 1999, pp. 1419–1423. [12] S. Sarkar and M. Adamou, “A framework for optimal battery management for wireless nodes,” IEEE J. Selected Areas in Communications, vol. 21, no. 2, 2002. [13] T. L. Martin and D. P.Siewiorek, “Nonideal battery and main memory effects on CPU speed-setting for low power,” Very Large Scale Integration (VLSI) Systems, IEEE Trans., vol. 9, pp. 29–34, FEB 2001. [14] V. Rao, G. Singhal, A. Kumar, and N. Navet, “Battery model for embedded systems,” in Proc. Intl. Conf. on VLSI Design, 2002.
