11institutetext: Jeremiah D. Deng 22institutetext: Department of Information
Science, University of Otago, PO Box 56, Dunedin 9054, 22email:
<EMAIL_ADDRESS>
# Sizing up the Batteries: Modelling of Energy-Harvesting Sensor Nodes in a
Delay Tolerant Network

Jeremiah D. Deng

###### Abstract

For energy-harvesting sensor nodes, rechargeable batteries play a critical
role in sensing and transmissions. By coupling two simple Markovian queue
models in a delay-tolerant networking setting, we consider the problem of
battery sizing for these sensor nodes to operate effectively: given the
intended energy depletion and overflow probabilities, how to decide the
minimal battery capacity that is required to ensure opportunistic data
exchange despite the inherent intermittency of renewable energy generation.

##  1 Introduction

Recently, energy-harvesting wireless sensor networks (EH-WSN) [1] have become
a promising technology for sensing applications. The advantage of EH-WSN is
obvious - batteries on the sensor nodes can be downsized due to their energy-
harvesting capability, the network enjoys longer life time, eliminating the
need of frequent of battery replacement, which is especially challenging for
large-scale sensor deployment. However, apart from reservoirs, most renewable
energy sources are intermittent in nature, which raises new challenges in
designing EH-WSNs. For example, sensors may not get proper sunshine for
recharging for hours, and wearable devices operated by kinetic energy will not
benefit much from humans sitting for hours. This implies the necessity of
using batteries to buffer the unsteady power supply from renewable energy
sources.

We consider a generic EH-WSN scenario where mobile nodes are equipped with
capacity-limited batteries that are powered by harvested kinetic energy; data
exchange between nodes requires 1) they are within transmission range to each
other; and 2) there is sufficient energy to conduct data transmission. This is
in effect an EH-WSN operating as a delay-tolerant network (DTN) [15], where
data transmission is opportunistic. In such a scenario, it is both important
to ensure the battery size is large enough to avoid energy depletion (and
hence potential failure for transmission) and energy overflow, both
detrimental to the battery life.

In a previous work [17], we have examined battery sizing in terms of depletion
probability and overflow probability respectively, using a coupled data and
energy queue system. In this work, we intend to investigate the mathematical
properties of battery size as a function regarding the operational probability
requirement, and develop an algorithm to calculate the minimum battery size
needed to meet the given requirements.

##  2 Related Work

As a performance modelling tool, queueing theory has been employed to study
EH-WSNs. Gelenbe [5] first looked the modelling of an EH-sensor node using the
concept of discretized energy unit called “energy packets”. The arrival of
these energy packets is assumed to follow a Poisson process. A routing
approach was further developed in [6]. A more general queueing model was
introduced in [10], relaxing the assumption that exactly one energy packet is
required to transmit a data packet. A Markovian model with data buffering was
further considered in [4]. In a recent work [17] we showed that kinetic energy
harvested by fitness gears discretized as energy packets can be well modelled
by Poisson processes. These previous works, however, considered only static EH
sensors, without involving potential intermittent connections between EH-
sensor nodes due to mobility.

On the other hand, mobility has been widely investigated in ordinary wireless
sensor networks and DTNs [12, 14]. Despite some counter-arguments [3], several
mobility model studies [2, 8, 16, 12] suggested that two mobile nodes’
encounter follows a Poisson process in mobile ad hoc networks and DTNs. There
are few studies on energy harvesting networks that investigated the effects of
intermittent connections [11, 13].

##  3 System Modelling

Notations used in this article are listed as follows: alligator

energy packet arrival rate

data packet arrival rate

connection arrival rate

ratio $\lambda_{D}/\lambda_{C}$

ratio $\lambda_{E}/\lambda_{C}$

ratio $\lambda_{D}/\lambda_{E}$

proportion of time that there is no data in the system

proportion of time that system have $k$ energy packets $k=0,...,K$

utilization factor of data buffer

utilization factor of energy buffer

acceptable probability of energy depletion

acceptable probability of energy overflow

battery capacity decided based on $\alpha$

battery capacity decided based on $\beta$

ceiling, the greatest integer more than or equal to $x$

###  3.1 The queueing model

We consider a network of mobile EH-sensors. Energy harvesting leads to Poisson
arrivals of energy packets (EP) with a rate of $\lambda_{E}$. Energy
consumption occurs when there are data packets in buffer, provided that there
are nodes in proximity, which is modulated by another Poisson arrival rate
$\lambda_{C}$. Thus an Energy queue is formed at each sensor node, which can
be modelled as an $M/M/1/K$, where $K$ is the battery capacity (in terms of
number of energy packets). Data packets (DP) arrive at a Poisson rate
$\lambda_{D}$, and leave a node if there is a connection available and there
is at least an energy packet in system. As memory in a sensor node is
relatively cheap and less constrained, for simplicity we set no limit to the
data buffer, hence allowing the data queue to be modelled by an $M/M/1$.
Clearly both queues are coupled by the connection availability. Hence the
Markovian packet departures in both the Energy queue and the Data queue are
modulated by the connection arrival rate $\lambda_{C}$.

The system diagram for a sensor node is shown in Figure 1.

Figure 1: System diagram of a mobile sensor node. DPs arrive in the data
buffer, while EPs arrive in the battery. Consumption of the energy as well as
the transmission of data occur simultaneously when triggered by a connection
established with another node.

Similar to [10] and [6], we focus on modelling energy needed for data
transmission and assume that compared with data transmission the sensing
process consumes insignificant amount of energy from the battery.

It is also worth mentioning that data transmission time is much faster than
energy harvesting in a node. Given the size of sensory DP and the relatively
large bandwidth in a network, packet transmission time is negligible [5].
Finally, to simplify our analysis we assume at each encounter only one DP is
transmitted.

These assumptions allow us to have a tractable system model with the energy
and data state diagrams shown in Figure 2.

###  3.2 Queueing analysis

Queueing analysis has been carried out in the previous work [17]. Here we only
summarize some main results.

The utilization of the data queue is given by:

$\rho_{D}=\frac{\lambda_{D}}{\lambda_{C}(1-P_{E_{0}})}.$ (1)

For sake of system stability, we have $\rho_{D}<1$. According to queueing
theory, we have

$P_{D_{0}}=1-\rho_{D}.$ (2)

The utilization of the energy queue is

$\rho_{E}=\frac{\lambda_{E}}{\lambda_{C}(1-P_{D_{0}})}.$ (3)

And the probability of energy depletion is

$P_{E_{0}}=\frac{1-{\rho_{E}}}{1-{\rho_{E}}^{K+1}},$ (4)

while the probability of energy overflow is

$P_{E_{K}}={\rho_{E}}^{K}P_{E_{0}},$ (5)

0$\lambda_{E}$$\lambda_{C}(1-P_{D0})$1$\lambda_{E}$$\lambda_{C}(1-P_{D0})$$\cdots$$\lambda_{E}$$\lambda_{C}(1-P_{D0})$$K$-1$\lambda_{E}$$\lambda_{C}(1-P_{D0})$$K$
(a) The Energy queue modelled as M/M/1/K.

0$\lambda_{D}$$\lambda_{C}(1-P_{E0})$1$\lambda_{D}$$\lambda_{C}(1-P_{E0})$2$\lambda_{D}$$\lambda_{C}(1-P_{E0})$…
(b) The Data queue modelled as M/M/1.

Figure 2: Queueing models for the Energy and Data queues respectively.

By substituting Eq.(1) and Eq.(3) in Eq.(2), we have

$P_{D_{0}}=1-\frac{\lambda_{D}}{\lambda_{C}(1-P_{E_{0}})}.$ (6)

Similarly, from (4), we have

$P_{E_{0}}=\frac{\left(\lambda_{C}(1-P_{D_{0}})\right)^{K}\left(\lambda_{C}(1-P_{D_{0}})-\lambda_{E}\right)}{\left(\lambda_{C}(1-P_{D_{0}})\right)^{K+1}-{\lambda_{E}}^{K+1}},$
(7)

which, by substituting $P_{D_{0}}$ using Eq.(6), becomes

$\begin{array}[]{ll}P_{E_{0}}&=\displaystyle\frac{\lambda_{D}^{K}(\lambda_{D}-\lambda_{E}(1-P_{E_{0}}))}{\lambda_{D}^{K+1}-(\lambda_{E}(1-P_{E_{0}}))^{K+1}}\\\
&=\displaystyle\frac{\gamma^{K}(\gamma+P_{E_{0}}-1)}{\gamma^{K+1}-(1-P_{E_{0}})^{K+1}}.\end{array}$
(8)

where $\gamma=\lambda_{D}/\lambda_{E}$. Here we introduce a new variable
$\zeta=\frac{1-P_{E_{0}}}{\gamma}$, to further simplify the mathematical
formulations. From the equation above, we have:

$1-\gamma\zeta=\displaystyle\frac{1-\zeta}{1-\zeta^{K+1}}.$ (9)

Using Eq.(5), we work on the probability of energy overflow

$P_{E_{K}}=\displaystyle\frac{(1-P_{E_{0}})^{K}(\gamma+P_{E_{0}}-1)}{\gamma^{K+1}-(1-P_{E_{0}})^{K+1}},$
(10)

which can be further simplified to

$P_{E_{K}}=\begin{cases}\displaystyle\frac{1-\frac{1}{\zeta}}{1-\frac{1}{\zeta^{K+1}}}&\zeta\neq
1,\\\ \frac{1}{K+1}&\zeta=1.\end{cases}$ (11)

where the case of $\zeta=1$ is obtained by using L’Hôpital’s rule.

Figure 3: Different values of $K_{\alpha}$ with respect to $\gamma$ for
$\alpha=0.05,0.02$, and $0.1$.

##  4 Battery capacity sizing

Having obtained the formulae for $P_{E_{0}}$ and $P_{E_{K}}$, we are now set
to find the close-form solution for the battery size as required by battery
depletion and overflow probabilities. To simplify notations, let
$P_{E_{0}}=\alpha$, $P_{E_{K}}=\beta$.

###  4.1 Battery size versus depletion probability

First we look at the battery size decided by $\alpha$, denoted by
$K_{\alpha}$. $K_{\alpha}$ is in fact a function of $\alpha$ and $\gamma$.
From Eq.(9), we have

$1-\gamma\zeta=\frac{1-\zeta}{1-\zeta^{K_{\alpha}+1}},$ (12)

which leads to

$\zeta^{K_{\alpha}}=\frac{1-\gamma}{1-\gamma\zeta}.$ (13)

Note that $1-\gamma\zeta=\alpha>0$. By taking logarithm on both sides, and
substituting $\zeta$ with $\frac{1-\alpha}{\gamma}$, eventually we have

$K_{\alpha}=\displaystyle\frac{\ln\frac{1-\gamma}{\alpha}}{\ln\frac{1-\alpha}{\gamma}}.$
(14)

Note this result implies that under the required condition $\gamma<1$, we have
a positive solution of $K_{\alpha}$. One can see that if $1-\gamma>\alpha$,
then $1-\alpha>\gamma$; otherwise if $1-\gamma<\alpha$, then
$1-\alpha<\gamma$. Therefore $K_{\alpha}>0$. To further explore the properties
of $K_{\alpha}>0$, we first introduce a lemma.

###### Lemma 1

$K_{\alpha}$ is monotonously increasing in terms of $\gamma$.

The proof of Lemma 1 is given in Appendix Lemma 1.

The interpretation is rather straightforward – the larger the $\gamma$ ratio
is, the more frequent DPs arrive compared with EPs, hence causing higher
chance of battery depletion. To maintain the depletion probability under
increased $\gamma$, a larger battery capacity is therefore needed.

From Lemma 1, we arrive at Theorem 1.

###### Theorem 1

The battery size as required by the depletion probability is a monotonously
decreasing function of the latter.

###### Proof

Obviously, Eq.(14) contains some kind of symmetry between $\gamma$ and
$\alpha$. Let $\alpha^{\prime}=1-\alpha$, $\gamma^{\prime}=1-\gamma$, then the
function for calculating $K_{\alpha}$ satisfies

$\begin{array}[]{rl}f(\gamma,\alpha)&=\displaystyle\frac{\ln(1-\gamma)-\ln\alpha}{\ln(1-\alpha)-\ln\gamma}=\displaystyle\frac{\ln\gamma^{\prime}-\ln(1-\alpha^{\prime})}{\ln\alpha^{\prime}-\ln(1-\gamma^{\prime})}\\\
&=\displaystyle\frac{\ln(1-\alpha^{\prime})-\ln\gamma^{\prime}}{\ln(1-\gamma^{\prime})-\ln\alpha^{\prime}}=f(\alpha^{\prime},\gamma^{\prime})=f(1-\alpha,1-\gamma).\end{array}$
(15)

This suggests that for function $f(.)$, an increased $\alpha$ corresponds in
effect to a decreased “$\gamma$”; and an increased $\gamma$ corresponds to a
decreased “$\alpha$”. As we have already shown $K_{\alpha}$ is monotonically
increasing with $\gamma$, we can now conclude $K_{\alpha}$ is a monotonically
decreasing function of $\alpha$.

Theorem 1 is again reasonable, since a smaller energy depletion probability
would require a larger battery size. Figure 3 shows some example $K_{\alpha}$
curves with different $\gamma$ and $\alpha$ values.

Figure 4: $K_{\beta}$ values with respect to $\alpha$ and $\beta$ when
$\gamma=0.95$.

So far we have assumed $\gamma\neq 1-\alpha$. The special case of
$\gamma=1-\alpha$, however, is allowed, as again using L’Hôpital’s rule we
have

$K_{\alpha}=\frac{1}{\alpha}-1,$ (16)

which is also a positive, monotonically decreasing function of $\alpha$.

###  4.2 Battery size versus overflow probability

Let $K_{\beta}$ be the battery size decided by a given $P_{E_{K}}$ value
($\beta$). We consider the normal case given in Eq.(11). Let $z=1/\zeta$. We
have

$\beta=\displaystyle\frac{1-z}{1-z^{K_{\beta}+1}},$ (17)

which leads to

$z^{K+1}=\frac{z+\beta-1}{\beta}.$ (18)

Here we can see unless $z+\beta>1$, there is no real solution for $K_{\beta}$.
In fact, it is easy to see that when $z<1$,

$1-z<\frac{1-z}{1-z^{K_{\beta}+1}}<1,$ (19)

i.e., $\beta$ has a lower bound $1-z$.

When the condition $z+\beta>1$ is satisfied (and naturally so when $z>1$), we
have

$K_{\beta}=\displaystyle\frac{\ln\frac{z+\beta-1}{\beta}}{\ln
z}-1=\frac{\ln[\gamma-(1-\alpha)(1-\beta)]-\ln\beta\gamma}{\ln\gamma-\ln(1-\alpha)}$
(20)

For the battery size decided by the overflow probability, we have the
following theorem:

###### Theorem 2

$K_{\beta}$ is monotonically decreasing when $\beta$ increases.

The proof of Theorem 2 is given in Appendix Theorem 2.

Figure 4 shows the trend $K_{\beta}$ values display across a range of $\alpha$
and $\beta$ values when $\gamma=0.95$.

It can be proven that for the special case of $z=1$, i.e., $\gamma=1-\alpha$,
we have

$K_{\beta}=\lim_{z\rightarrow 1}\frac{\ln\frac{z+\beta-1}{\beta}}{\ln
z}-1=\frac{1}{\beta}-1.$ (21)

Clearly, Theorem 2 still holds.

###  4.3 Battery sizing algorithm

Given different requirements in terms of $\alpha$ and $\beta$ values, and the
system setup in terms of $\gamma$, we can derive the relevant $K_{\alpha}$ and
$K_{\beta}$ to size up the battery. As seen from Eq.(20), the condition

$(1-\alpha)(1-\beta)<\gamma$

has to stand for calculating $K_{\beta}$. One question remains – between
$K_{\alpha}$ and $K_{\beta}$ which one actually decides the size of the
battery? It is easy to see that when $\beta+\gamma=1$, $K_{\alpha}=K_{\beta}$.
Since we have shown that $K_{\alpha}$ is an monotonically increasing function
of $\gamma$, and $K_{\beta}$ a monotonically decreasing function of $\beta$,
we give the following corollary without needing a formal proof:

###### Corollary 1

If $\beta+\gamma<1$, $K_{\alpha}<K_{\beta}$; if $\beta+\gamma>1$,
$K_{\alpha}>K_{\beta}$.

(a) $\beta+\gamma<1$

(b) $\beta+\gamma>1$

Figure 5: Comparison of $K_{\alpha}$ and $K_{\beta}$ in two different cases.

Hence we have the following algorithm for battery sizing.

Data: $0<\alpha,\beta<1$, $(1-\alpha)(1-\beta)<\gamma<1$

Result: minimum $K$ satisfying $P_{E_{0}}<\alpha$, $P_{E_{K}}<\beta$

1 if  _$\beta+\gamma <1$_ then

2 $K\leftarrow K_{\beta}$ using Eq.(20);

3

4else

5 $K\leftarrow K_{\alpha}$ using Eq.(14);

6

7 end if

Algorithm 1 Battery sizing under constraints of $P_{E_{0}}$ and $P_{E_{k}}$

Figure 5 gives two settings when $K_{\alpha}$ and $K_{\beta}$ are compared.
These numerical examples clearly confirm the correctness of Corollary 1.

###  4.4 A numerical example

To demonstrate the feasibility of an EH node in mobile settings, let us assume
a scenario with the battery depletion probability $\alpha=0.05$, and the
overcharge probability $\beta=0.3$. Suppose the three rates in the system are
$\lambda_{D}=0.72$ data packets/sec, $\lambda_{E}=0.8$ energy packets/sec, and
$\lambda_{C}=0.9$ connection/sec. We have
$\gamma=\lambda_{D}/\lambda_{E}=0.9$, and $\beta+\gamma>1$. Hence according to
Algorithm 1, we obtain the required battery size $K=K_{\alpha}=12.82\approx
13$ using Eq.(14). If we use an energy packet size of 155$\mu W$ which can be
generated by a moderate walking activity [7], the needed battery size will be
$13\times 155\mu W=2.015mW$.

##  5 Conclusion

Despite the great potential in utilizing rechargeable nodes in wireless sensor
networks and body area networks, the wide application of energy-harvesting IoT
systems remains elusive, largely due to the uncertain nature of energy
harvesting and the lack of performance analysis results for guiding system
design. In this article, we have studied the modelling of energy charging and
consumption behaviours of sensor nodes in a DTN setting, where data
transmission is subject to both the availability of sufficient energy, and the
existence of sensor nodes in reachable proximity. A stochastic model with
coupled Poisson arrival processes on the energy and data queues of the node is
formulated and solved, based on which a closed-form solution of the optimal
battery size is derived to meet the specified probabilities of energy
depletion and overflow.

For future work, our model can be extended by considering general probability
distributions for energy or data arrivals, allowing more flexible settings of
data and energy packet sizes. This may enable more types of energy-harvesting
sources being considered, for instance solar. Beyond using discretized energy
units, continuous fluid models [9] could be employed for future investigation,
where the theoretical findings may be further validated by simulation studies
under realistic settings.

###### Acknowledgements.

I would like to dedicate this article to Emeritus Professor Martin K. Purvis,
who introduced me to the wonderful world of queueing theory and encouraged me
to brave the less-travelled roads in mobile ad hoc networks and IoT research.
This little but meticulous work of mine certainly benefitted from the pleasant
chats we had around philosophy, theology, programming languages, Donald Knuth,
etc. I would also like to acknowledge that Dr. Sophie Zareei, whose PhD thesis
Martin and I co-supervised, did some of the initial work on the same topic.

## Appendix – Proofs

### Lemma 1

$K_{\alpha}$ is monotonically increasing on $\gamma$.

###### Proof

It can be worked out that

$\frac{\partial
K_{\alpha}}{\partial\gamma}=\displaystyle\frac{\frac{1}{\gamma}\ln\frac{1-\gamma}{\alpha}-\frac{1}{1-\gamma}\ln\frac{1-\alpha}{\gamma}}{\ln^{2}\frac{1-\alpha}{\gamma}}.$

We want to show that this derivative is non-negative, i.e.

$\frac{1}{\gamma}\ln\frac{1-\gamma}{\alpha}\geq\frac{1}{1-\gamma}\ln\frac{1-\alpha}{\gamma},$

which is equivalent to

$\left(\frac{1-\gamma}{\alpha}\right)^{\frac{1}{\gamma}}\geq\left(\frac{1-\alpha}{\gamma}\right)^{\frac{1}{1-\gamma}}.$

Further transformation leads to

$\left(1-\gamma\right)^{\frac{1}{\gamma}}\gamma^{\frac{1}{1-\gamma}}\geq\alpha^{\frac{1}{\gamma}}(1-\alpha)^{\frac{1}{1-\gamma}}.$

Let $x=1-\alpha$, and

$f(z)=(1-z)^{\frac{1}{\gamma}}z^{\frac{1}{1-\gamma}},$

so to prove the inequality above we only need to show that $f(\gamma)\geq
f(x)$. To find the maximum of $f(z)$, let $f^{\prime}(z)=0$. Solving this, we
get $z=\gamma$. ∎

The derivation above also shows that the equality stands when
$\gamma=1-\alpha$.

### Theorem 2

$K_{\beta}$ is monotonically decreasing on $\beta$.

###### Proof

We consider the general case where $K_{\beta}$ can be put as

$K_{\beta}=\displaystyle\frac{\ln(\frac{z-1}{\beta}+1)}{\ln z}-1,$

where $z=\frac{\gamma}{1-\alpha}$. Consider two cases only (the special case
of $z=1$ is already handled in main text):

  1. 1.

$z>1$. Both the numerator and the denominator of the fraction term are
positive. Clearly the bigger $\beta$ is, the smaller $K_{\beta}$;

  2. 2.

$z<1$. Both the numerator and the denominator are negative. With $\beta$
increasing, $\frac{z-1}{\beta}$ will increase, albeit being negative. The
numerator will increase as an negative value, hence the value for $K_{\beta}$
will decrease as a positive value.

∎

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