# Reliability Improvement of Circular $k$-out-of-$n$: G Balanced Systems
through Center of Gravity

Yongkyu Cho  Department of Industrial Engineering, Kangnam University, South
Korea
E-mail<EMAIL_ADDRESS>Seung Min Baik  Department of Industrial
and Management Engineering, Pohang University of Science and Technology, South
Korea
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>Young Myoung Ko
Correspondence to Young Myoung Ko<EMAIL_ADDRESS>Department of
Industrial and Management Engineering, Pohang University of Science and
Technology, South Korea
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper considers a circular $k$-out-of-$n$: G balanced system equipped
with homogeneous and stationary units. Building on previous research by
Endharta et al. [5], we propose a new balance definition in circular $k$-out-
of-$n$: G balanced systems based on the concept of center of gravity.
According to this condition, a circular $k$-out-of-$n$: G balanced system is
considered balanced if its center of gravity is located at the origin. This
new balance condition is not only simple but also advantageous as it covers
the previous two balance conditions of symmetry and proportionality. To
evaluate the system’s reliability, we consider the minimum tie-sets, and
extensive numerical studies verify the enhancement of system reliability
resulting from the proposed balance definition.

_Keywords: Reliability evaluation, Circular $k$-out-of-$n$: G balanced
systems, Balance condition, Center of gravity, Minimum-tie set, Minimal path
set_

## Notation

  * •

$n$: number of units in a system

  * •

$k$: minimum number of non-failed units for a functioning system

  * •

$r$: probability that a unit is functioning properly

  * •

$\mathcal{U}$: index set of all units in the system;
$\mathcal{U}=\\{1,2,...,i,...,n\\}$

  * •

$U$ or $U_{j}$: ($j^{\textrm{th}}$) subset of $\mathcal{U}$ denoting the set
of non-failed units; $U_{j}=\\{u_{1},u_{2},...,u_{l},...,u_{|U_{j}|}\\}$ where
each $u_{l}$ corresponds to a unit index $i\in\mathcal{U}$ such that
$u_{1}<u_{2}<\cdots<u_{|U_{j}|}$.

  * •

$\mathbf{U}_{k}$: set of $k$-combinations of set $\mathcal{U}$;
$\mathbf{U}_{k}=\\{U_{1},U_{2},...,U_{|\mathbf{U}_{k}|}\\}$ where
$|\mathbf{U}_{k}|=\binom{n}{k}.$

  * •

$d_{l}$: distance between units $u_{l+1}$ and $u_{l}$ in a subset $U$ for
$l=1,2,...,|U|$; $d_{l}=u_{l+1}-u_{l}$ if ${l}\neq|U|$ and $n+u_{1}-u_{|U|}$
if $l=|U|$.

  * •

$D_{U}$: distance tuple for a subset $U$; $D_{U}=(d_{1},d_{2},...,d_{|U|})$

  * •

$E^{(l)}$: reverse tuple of $D_{U}$ starting from the $l^{\textrm{th}}$
distance; $E^{(l)}=(d_{l},d_{l-1},...,d_{2},d_{1},d_{|U|},$
$d_{|U|-1},...,d_{l+1})$

  * •

$\mathbf{E}_{U}^{B}$: set of $E^{(l)}$’s such that $E^{(l)}=D_{U}$ for a
subset $U$; the system is balanced if $|\mathbf{E}_{U}^{B}|$ is an even number
or if $|\mathbf{E}_{U}^{B}|>1$.

  * •

$T$ or $T_{t}$: ($t^{\textrm{th}}$) tie-set of a system;
$T_{t}=\\{u_{1},u_{2},...,u_{l},...,u_{|T_{t}|}\\}$

  * •

$\mathbf{T}$: set of all tie-sets of a system;
$\mathbf{T}=\\{T_{1},T_{2},...,T_{t},...,T_{|\mathbf{T}|}\\}$

  * •

$T^{M}$ or $T_{m}^{M}$: ($m^{\textrm{th}}$) minimum tie-set of a system;
$T_{m}^{M}=\\{u_{1},u_{2},...,u_{l},...,u_{|T_{m}^{M}|}\\}$

  * •

$\mathbf{T}^{M}$: set of minimum tie-sets;
$\mathbf{T}^{M}=\\{T_{1}^{M},T_{2}^{M},...,T_{m}^{M},...,T_{|\mathbf{T}^{M}|}\\}$

  * •

$\mathbf{T}^{N}$: set of non-minimum tie-sets;
$\mathbf{T}^{M}=\mathbf{T}\backslash\mathbf{T}^{N}.$

  * •

$X_{i}$: binary state variable of unit $i$; $X_{i}=1$ if unit $i$ is
functioning, $X_{i}=0$ otherwise.

  * •

$\mathbf{X}$: system state vector; $\mathbf{X}\equiv[X_{1},...,X_{n}]$

  * •

$\phi(\cdot)$: system structure function; $\phi(\mathbf{X})=1$ if the system
is functioning, $\phi(\mathbf{X})=0$ otherwise.

  * •

$R_{S}$: system reliability;
$R_{S}\equiv\mathbb{P}\left[\phi\left(\mathbf{X}\right)=1\right]=\mathbb{E}\left[\phi\left(\mathbf{X}\right)\right].$

##  1 Introduction

This paper considers systems with spatially distributed units such as balanced
engine systems in planetary descent vehicles and Unmanned Aerial Vehicles
(UAVs), commonly called drones equipped with multiple rotary-wings as shown in
Fig. 1. In the balanced engine system, if an engine in a pair fails during
landing, system dynamics mandate that the other engine in the pair must be
turned off in order to uphold the balance of the descent vehicle [9]. The
$k$-out-of-$n$ balanced system can be used to analyze the reliability in such
a situation because there may be a minimum number of pairs of engines required
to prevent rapid descent during landing. Similarly to the descent system,
there may be a minimum number of operating rotary-wings to operate drones
reliably. In fact, many systems with spatially distributed units have a
circular configuration with evenly distributed units. The definition of
balance varies depending on the context of the analysis, so reliability
evaluation does not have a clear-cut solution. Additionally, considering the
units’ positions is the most significant factor that makes the problem
difficult when defining the system’s balance.

Inspired by drones and descent systems, Hua and Elsayed [10, 11, 12], along
with a series of papers, defined this situation as a $k$-out-of-$n$ pairs: G
balanced system and conducted extensive reliability studies for this system.
Unlike descent systems where each engine provides propulsion in a vertical
direction to the ground, however, the rotors of a drone generate lift force
while rotating, making the definition of balance more complicated. In the
series of research studies regarding drone reliability, the balance condition
was mainly defined based on system symmetry.

Meanwhile, Endharta et al. [5] proposed a new balance condition based on the
proportionality of the system for a similar situation. That is, as long as the
operating units are evenly distributed throughout the system, balance can be
maintained. Furthermore, considering such balance conditions, the units of the
system no longer need to necessarily form pairs, which can lead to a
relaxation of the target system from a $k$-out-of-$n$ pairs system to a
$k$-out-of-$n$ system. As a result, Endharta et al. [5] investigated a
reliability analysis problem for circular $k$-out-of-$n$: G balanced systems.
Note that a $k$-out-of-$n$ pairs system can be seen as a special case of a
$k$-out-of-$n$ system. For example, if Fig. 1(b) is considered a $k$-out-
of-$4$ pairs system, it can be regarded as a $2k$-out-of-$8$ system with some
additional conditions. Therefore, a balance condition based on symmetry can
also be applied to examine the balance of $k$-out-of-$n$ systems, which can
lead to a direct comparison between two different balance conditions.

Building on the previous research study by Endharta et al. [5], this paper
presents another new balance definition for circular $k$-out-of-$n$: G
balanced systems. This new balance definition is advantageous due to its
simplicity and generality, as it covers the two balance conditions previously
investigated. In essence, the new balance definition directly takes into
account the concept of center of gravity. The main contributions of our work
are summarized as follows.

  * •

We propose a new balance definition for circular $k$-out-of-$n$: G balanced
systems that can enhance the system reliability.

  * •

We investigate the inclusion relationships among the three different balance
conditions discovered so far through mathematical proofs and numerical
examples.

  * •

We demonstrate the reliability improvement resulting from the new balance
definition using extensive numerical analysis. We also discuss the effect of
system parameters on the overall system reliability.

The remainder of this paper is organized as follows. Section 2 conducts a
literature review of existing reliability studies on a variety of $k$-out-
of-$n$: G balanced systems. Section 3 describes the target system and outline
the key modeling assumptions. Section 4 presents three definitions of balance
conditions, including the newly proposed one. This section also provides an
in-depth investigation of the relationships among the three balance
conditions. Section 5 explains the reliability evaluation method with a
descriptive numerical example. Section 6 provides extensive numerical results
showing the enhancement of reliability resulting from the proposed balance
definition. Finally, Section 7 concludes and suggests possible extensions of
the research study.

##  2 Literature Review

The $k$-out-of-$n$ models can be largely divided into two types: $k$-out-
of-$n$: F and $k$-out-of-$n$: G. A $k$-out-of-$n$: F system fails when at
least $k$ units are failed whereas a $k$-out-of-$n$: G system functions when
at least $k$ units are non-failed. In a $k$-out-of-$n$: G system with
homogeneous units of binary-states (failed and non-failed), the number of non-
failed units follows the binomial distribution; the system reliability
coincides with the probability that $k$ or more units are non-failed [3].

The circular $k$-out-of-$n$: G balanced system, which is the target system of
this paper, is a variant of $k$-out-of-$n$: G model in which the units are
arranged circularly, and the reliable system requires to satisfy a certain
balance condition. Among the variants of circular $k$-out-of-$n$: G balanced
models, the most relevant special case is the $k$-out-of-$n$ pairs: G balanced
system which has $n$ pairs of units distributed evenly on a circular
configuration. For example, the graphical model in Fig. 1(b) can be considered
to have four pairs of units: (1,5), (2,6), (3,7), and (4,8). Sarper and Sauer
[17] presented the first reliability study for such a system: balanced engine
systems in planetary descent vehicles, e.g., four-engine (i.e., two pairs) and
six-engine (i.e., three pairs) configurations. The authors provided a basic
reliability evaluation framework based on simple stochastic models such as
Bernoulli distributed unit state and exponentially distributed failure time.
Applying the $k$-out-of-$n$ pairs: G balanced model to drone systems, Hua and
Elsayed [10, 11] conducted the extensive reliability studies regarding
degradation analysis and reliability estimation. Notably, Hua and Elsayed [11]
were the first to define the balanced state of a drone system by symmetry,
considering the concept of Moment Difference to measure the degree of
symmetry. The authors considered two different scenarios (rebalancing is
allowed or not) and develop a systematic approach for estimating the
reliability of $k$-out-of-$n$ pairs: G balanced systems. Although the proposed
approach was effective for reliability evaluation, the computational
intractability when systems are large remained a limitation. In this regard,
Hua and Elsayed [12] proposed a Monte Carlo simulation-based reliability
approximation method which is fast as well as accurate hence applicable even
for large systems. Considering more complex configurations of drone systems,
Guo and Elsayed [6] presented a $(k_{1},k_{2})$-out-of-$(n,m)$ pairs: G
balanced system. The system modeled a rotary UAV with multi-level of rotors
where $m$ rotors are arranged vertically in parallel in the same position.
Reliability estimations for two scenarios (forced-down rotors are considered
as failed and forced down rotors are considered as standbys) have been
obtained by enumerating operational states and calculating the probability of
their occurrences.

Unlike the above literature that only considered system symmetry as a measure
of balance in drone systems and conducted extensive reliability studies,
Endharta et al. [5] proposed a new balance definition based on system
proportionality that can be applied to drone systems. By considering the new
balance definition, the target system can be relaxed from a $k$-out-of-$n$
pairs to a $k$-out-of-$n$ system. Endharta et al. [5] considered a system with
homogeneous and stationary units, and the minimum tie-sets were enumerated to
evaluate the system reliability. The results showed the reliability
improvement compared to when considering only the symmetry-based balance
definition. In the subsequent study by Endharta and Ko [4], a load-sharing
system was considered where the amount of load is equally distributed among
the working units. The failure time of each unit was assumed to follow an
exponential distribution, and the load-sharing relationship between the amount
of load and the failure rate is assumed to follow a power rule. System failure
paths were arranged to evaluate reliability, and for larger systems,
reliability was approximated using Monte Carlo simulation.

Regarding the technique of reliability evaluation, the Inclusion-Exclusion
(IE) method is a well-known approach that can be used for the systems
consisting of units with constant failure probability. Using the IE method,
reliability expression of such systems can be derived based on the concepts of
minimum tie-sets or cut-sets [3]. To improve the computational efficiency of
the IE method, Heidtmann [8] proposed an improved version that eliminates
certain terms in the expression. Another popular method called the sum-of-
disjoint-products (SDP) method, which is also related to minimum tie-sets or
cut-sets, has been explored by McGrady [15]. For $k$-out-of-$n$: G systems
with independent but non-identical units, Rushdi [16] developed a pivotal
decomposition approach for evaluating reliability. Recently, Hao et al. [7]
proposed a new IE method called Quick Inclusion-Exclusion (QIE) to increase
the efficiency of the IE method and reduce the amount of required memory.

There have been a body of recent reliability studies on a variant of balanced
systems. Wang et al. [23] studied a multi-state $k$-out-of-$n$: F balanced
system with a rebalancing mechanism, which is important in engineering fields
like new energy storage and aeronautics. The components and system are assumed
to have multiple states, and an age maintenance strategy and optimization
model were investigated to obtain the optimal results. The proposed model was
demonstrated through numerical examples based on a product line balancing
problem. Dui et al. [2] proposed a model for studying the mission reliability
and structure optimization of an UAV swarm based on importance measures. The
mission reliability model was based on polygonal linear consecutive-$k$-out-
of-$n$: F systems, and the structure optimization has been analyzed using
conditional reliability, conditional failure rate, and remaining useful life.
The proposed method was demonstrated through numerical examples of triangular
and quadrilateral UAV swarms. Wang et al. [21] presented a condition-based
preventive maintenance policy for balanced systems with identical components.
The system state was evaluated based on the deterioration levels of all
components, and PM activities are employed to avoid competing failures. The
optimal preventive maintenance thresholds are determined by minimizing the
system maintenance cost within the framework of semi-Markov decision process.
Zhao and Wang [29] suggested a new maintenance policy optimization method for
systems with two balanced components, assuming that the components degrade
over time according to a bivariate Wiener process. The objective of the
maintenance actions was to eliminate the differences of degradation levels of
system components at the cost of aggravating the degradation, and the model
was optimized using Markov decision process with both finite and infinite
planning horizons. Wang et al. [22] proposed a maintenance policy optimization
model for balanced systems composed of multiple functionally-exchangeable
units, where a unit may fail due to self-failure or external stress. The
objective was to find the optimal number of operating units to minimize the
maintenance cost per unit time, and an illustrative example was used to
demonstrate the effectiveness of the proposed policy. Wu et al. [26] studied a
load-sharing consecutive-$k$-out-of-$r$-from-$n$ subsystems: F balanced system
with linear and circular structures that maintains balance by forcing working
components to standby or resuming standby components to operate. The system
fails if there are $r$ consecutive subsystems in which at least $k$ subsystems
fail, and the system reliability was analyzed using the Markov processes and
finite Markov chain imbedding approach. Tian et al. [19] presented a new
reliability evaluation method for Performance-based balanced systems with
common bus performance sharing (PBSs-CBPS), which considers the balance degree
threshold, transmission loss, and transmission capacity limit. A continuous-
time discrete-state Markov model was built to address the transition behaviors
of components, and the universal generation function method combined with
nonlinear programming was proposed to calculate system reliability. Wang et
al. [25] investigated two reliability models for balanced systems with multi-
state protective devices, considering rebalancing mechanisms and failure
criteria of both dynamic and static balanced concepts. The models involved new
triggering and variable protection mechanisms to reduce the impact of shocks
and internal degradation on units, and the reliability was derived using the
Markov process imbedding method with Monte-Carlo simulation.

In the next section, we describe the target system and introduce the key
assumptions considered in this paper.

##  3 System Description

Consider a system with circularly arranged units such as descent systems
equipped by $n$ engines or a drone product equipped by $n$ rotary-wings.
Although the system starts up in the perfect state with $n$ operational units,
some of them can go wrong as time goes by. The system with some failed units
(say a subsystem), however, still has possibility to be reliable if the number
of non-failed units are enough to generate sufficient forces to remain the
system in the air, for example, at least $k$ units are non-failed then the
drone can stay afloat in the air. In such situation, what we must consider
together is the balance of the overall system. The set of non-failed units
that can keep the balance will generate the total thrust in the same direction
to the lifting forces so that the system stays in the air. The circular
$k$-out-of-$n$: G balanced system is a reliability model that abstracts out
such a situation. Fig. 1 shows a pair of such system (a drone product) and its
graphical model.

((a))

((b))

Figure 1: An illustrative example of a multi-rotor drone and its corresponding
system model

Here, we explain some of the main notations to mathematically describe the
target system. First of all, we consider a set of indexes of units
$\mathcal{U}$ (${U}=\\{1,2,...,n\\}$) that comprise a circular $k$-out-of-$n$:
G balanced system. Then, $U$ (or $U_{j}$) denotes the ($j^{\textrm{th}}$)
subset of $\mathcal{U}$ containing the set of indexes non-failed units (i.e.,
$U=\\{u_{1},u_{2},...,u_{|U|}\\}$ where each $u_{l}$ corresponds to each non-
failed unit index). For example, if we have $\mathcal{U}=\\{1,2,3,4\\}$ and
unit $2$ has been failed, then we have a subset $U=\\{1,3,4\\}$ where
$u_{1}=1,u_{2}=3,u_{3}=4$. Note that $u_{l}$’s are in an ascending order:
$u_{1}<u_{2}<\cdots<u_{|U|}$. We also introduce a collection of sets denoted
by $\mathbf{U}_{k}$ that consists of all the $k$-combinations of a unit set
$\mathcal{U}$, that is,
$\mathbf{U}_{k}=\\{U_{1},U_{2},...,U_{|\mathbf{U}_{k}|}\\}$ where
$|\mathbf{U}_{k}|=\binom{n}{k}$.

A circular $k$-out-of-$n$: G balanced system is said to be operational if at
least $k$ among $n$ units are non-failed as well as the system maintains a
balance with the non-failed units. Since On-Off control is not a highly
sophisticated control mechanism in modern electronic devices, it is reasonable
to assume each non-failed unit can be forced up and down. This On-Off
mechanism can be utilized to rebalance the system by forcing down some
operational units that are harming the system balance. To evaluate the system-
wise reliability of such systems, it is necessary to take the unit-wise
reliability into account. In this regard, we assume that the system consists
of independent and homogeneous units each of which has a stationary survival
probability. In short, we assume a constant survival and failure probabilities
$r$ and $1-r$ for each unit and for all the planning horizon.

Below, we summarize the key assumptions considered for the last of this paper.

  * •

The system consists of independent and homogeneous units each of which has a
constant survival and failure probabilities $r$ and $1-r$.

  * •

Each non-failed unit is subject to On-Off control. For example, any non-failed
operating unit can be forced down for rebalancing the system.

In the next section, we formally define the balance conditions that can be
considered regarding the reliability analysis of circular $k$-out-of-$n$: G
balanced systems.

##  4 Balance Conditions

We consider three balance conditions for circular $k$-out-of-$n$: G balanced
systems: Balance Condition I (symmetry, BC1 [11]), Balance Condition II
(proportionality, BC2 [5]), and the newly proposed Balance Condition III
(center of gravity at origin, BC3). Note that just satisfying one of the
balance conditions does not necessarily mean that the system is functioning;
for a $k$-out-of-$n$: G system to be operational, there should be at least $k$
non-failed units as well. Taken together, a circular $k$-out-of-$n$: G
balanced system is operational when it satisfies at least one of the balance
conditions with at least $k$ non-failed units. Although this section
introduces all the three balance conditions for explanatory purpose, we want
to emphasize that the proposition of BC3 is the contributing part of this
paper. Hence, we refer readers to the previous literature [11, 5, 4] for in-
depth explanation on BC1 and BC2.

###  4.1 BC1: System is symmetric

The first balance condition BC1 is related to the symmetry of the target
system. That is, if the system is symmetric in terms of a certain definition,
it is considered as balanced. BC1 has been originally suggested to evaluate
the circular $k$-out-of-$n$ pairs: G balanced systems. Since the system can be
regarded as a special case of circular $k$-out-of-$n$: G balanced system in
which an additional pair constraint is considered, BC1 can also be applied to
any of the circular $k$-out-of-$n$: G balanced systems for evaluating its
balance. The following definition states BC1 that is suggested by Hua and
Elsayed [11].

######  Definition 1 (BC1).

A circular $k$-out-of-$n$: G balanced system is said to be balanced if it is
symmetric in the sense that all its operating units are symmetric w.r.t at
least a pair of perpendicular axes as well as the number of pairs is an even
number.

Fig. 2 depicts illustrative examples that explain BC1. Regarding all the
graphical models that will be shown in this paper, white circles express the
operating units whereas black-colored circles correspond to the failed or
turned-off units. First, the graphical model in Fig. 2(a) describes a
subsystem with $U=\\{4,5,6,10,11,12\\}$. As shown in the figure, it forms a
pair of perpendicular axes of symmetry so that is identified to satisfy BC1
hence balanced. In comparison, Fig. 2(b) provides a subsystem with
$U=\\{4,8,12\\}$ that forms three distinct axes of symmetry in which no pair
of perpendicular axes is observed. According to definition 1, it is identified
not to satisfy BC1 hence unbalanced. Later in Section 4.2, we will show that
the subsystem in Fig. 2(b) also can be identified to be balanced by
considering more generalized balance condition (see Fig. 3(a)).

((a))

((b))

Figure 2: Illustrative examples for BC1

To explain the quantitative way of judging whether or not the system satisfies
BC1, we introduce some mathematical notations. First, let $d_{l}$ denote the
distance between units $u_{l+1}$ and $u_{l}$ defined by $d_{l}=u_{l+1}-u_{l}$
if $l\neq|U|$ and $d_{l}=n+u_{1}-u_{|U|}$ if $l=|U|$, for a subsystem
$U=\\{u_{1},u_{2},...,u_{|U|}\\}$ where $l=1,2,...,|U|$. We also define
$D_{U}$ be a tuple that collects all the distance enumeration:
$D_{U}=(d_{1},d_{2},...,d_{|U|})$. Then, let $E^{(l)}$ be the reverse tuple of
$D_{U}$ which starts from $d_{l}$;
$E^{(l)}=(d_{l},d_{l-1},...,d_{2},d_{1},d_{|U|},d_{|U|-1},...,d_{l+1})$. Note
that an $E^{(l)}$ that satisfies $E^{(l)}=D_{U}$ corresponds to an axis of
symmetry. By collecting all such $E^{(l)}$’s, we generate a set of reverse
tuples denoted by $\mathbf{E}_{U}^{B}$. Combined with definition 1, the
following proposition states a quantitative way of figuring out if a system
satisfies BC1.

######  Proposition 1.1 (Endharta et al. [5]).

For a circular $k$-out-of-$n$: G balanced system with an index set of non-
failed units $U$, the axis of symmetry is located between units $u_{1}$ and
$u_{l+1}$ for $l$ such that $E^{(l)}=D_{U}$. Thus, $|\mathbf{E}_{U}^{B}|$ can
represent the number of axes of symmetry in the system and the system balance
is examined as follows.

  1. (a)

If $|\mathbf{E}_{U}^{B}|$ is an even number, the system is symmetric w.r.t. at
least a pair of perpendicular axes and balanced.

  2. (b)

If $|\mathbf{E}_{U}^{B}|$ is an odd number, the system is symmetric w.r.t.
certain axes, but not balanced.

  3. (c)

If $|\mathbf{E}_{U}^{B}|$ is 0, the system is not symmetric, then it is not
balanced.

Simply put, the result of proposition 1.1 leads towards the following remark.

######  Remark 1.1.

A circular $k$-out-of-$n$: G balanced system with an index set of non-failed
units $U$ satisfies BC1 if $|\mathbf{E}_{U}^{B}|$ is a nonzero even number.

In the next subsection, we will explain an expanded balance definition (BC2),
which is the main result of the previous study by Endharta et al. [5].

###  4.2 BC2: System is spread proportionally

The second balance condition BC2 focuses on the proportionality of the
operating units in a system. According to BC2, a circular $k$-out-of-$n$: G
balanced system is considered as balanced if the system is spread
proportionally by the operating units.

######  Definition 2 (BC2).

A circular $k$-out-of-$n$: G balanced system is said to be balanced if the
operating units are spread proportionally within the system.

To quantify the definition 2, we investigate the angles between two non-
consecutively neighboring operating units (say sector angle), and then examine
the patterns observed from the sector angles. Fig. 3 shows some representative
examples of the systems satisfying BC2 and the following remark states the
quantitative way of evaluating BC2 proposed by Endharta et al. [4].

######  Remark 2.1.

Let $a_{s}$ be the $s^{\textrm{th}}$ sector angle between the two non-
consecutively neighboring operating units and $N$ be the total number of such
angles. Then, a circular $k$-out-of-$n$: G balanced system showing at least
one of the following patterns is said to be balanced.

  1. (a)

All sector angles $a_{1},a_{2},...,a_{N}$ are congruent angles;
$a_{1}=a_{2}=\cdots=a_{N}$.

  2. (b)

When $N$ is an even number, all sector angles $a_{1},a_{2},...,a_{N}$ are the
opposite angles of one another; $a_{s}=a_{\frac{N}{2}+s}$ for
$s=1,...\frac{N}{2}$.

((a))

((b))

((c))

((d))

Figure 3: Representative examples of the systems satisfying BC2

Regarding the method of examining BC2 given a subsystem $U$, Endharta and Ko
[4] previously found that the existence of more than one reverse distance
tuple $E^{(l)}$ such that $E^{(l)}=D_{U}$ implies the spread proportionality
of the system. Hence, the following remark can be used to check if a subsystem
satisfies BC2.

######  Remark 2.2 (Endharta and Ko [4]).

A circular $k$-out-of-$n$: G balanced system with an index set of non-failed
units $U$ satisfies BC2 if $|\mathbf{E}_{U}^{B}|>1$.

Combining remark 2.2 with remark 1.1, we draw a conclusion that a subsystem
satisfying BC1 also satisfies BC2 whereas the opposite is not always true. For
example, among the graphical models that depicted in Fig. 3, only three of
them also satisfies BC1 with at least a pair of perpendicular axes of symmetry
(see Figs. 3(a), 3(c) and 3(d))). As such, BC2 is regarded as a more
generalized balance condition than BC1. For another descriptive example, Fig.
4(a) shows a subsystem which is the same graphical model as previously shown
in Fig. 2(b), which has been identified as unbalanced according to BC1. Now,
we notice that it can be regarded as a balanced system by considering BC2.

((a))

((b))

Figure 4: Illustrative examples for BC2

In the next subsection, we will introduce the most generalized balance
condition (to the best of our knowledge) BC3 that covers both BC1 and BC2. For
example, although the subsystem in Fig. 4(b) cannot be identified as balanced
even if we consider both BC1 and BC2, it will be regarded as balanced by
taking BC3 into account (see Fig. 5(a)).

###  4.3 BC3: System has a center of gravity at origin

The motivation of BC3 stems from the following research question: What is the
essence that BC1 and BC2 are trying to evaluate? In consequence, both BC1 and
BC2 can be considered as the conditions representing the states in which a
subsystem comprised of only a part of units can maintain the balance. Such
condition may correspond to the states in which the center of gravity formed
by the operating units is located at the geometric center of the system. To be
specific, the center of gravity stands for the mean point of an object’s or
system’s weight distribution. By the assumption that we have homogeneous
units, the system’s direction of total thrust should be in the same direction
as the lift force given the center of gravity located at the origin. Based on
this simple and intuitive concept, we state the following definition for the
new as well as more generalized balance condition.

######  Definition 3 (BC3).

A circular $k$-out-of-$n$: G balanced system is said to be balanced if the
system with operating units has a center of gravity at the origin.

For example, plots in Fig. 5 depict two graphical models describing the
meaning of BC3. First, Fig. 5(a) corresponds the case where BC3 is satisfied
because the operating units in a subsystem $U=\\{1,4,5,9,10\\}$ forms the
center of gravity at the origin; the blue-colored ‘x’-marker indicates the
location of the center of gravity. Note also that the subsystem is the same as
the previous figure Fig. 4(b) which was examined as unbalanced by considering
both BC1 and BC2. On the other hand, Fig. 5(b) shows another subsystem
$U=\\{3,6,8,12\\}$ in which the center of gravity (the red-colored ‘x’-marker)
is not located at origin, which is identified as unbalanced although the most
generalized condition BC3 is considered.

((a))

((b))

Figure 5: Illustrative examples for BC3 (the colored ‘x’-markers represent the
center of gravity for each subsystem)

To quantitatively examine BC3 for a circular $k$-out-of-$n$: G balanced
system, we consider a conventional two-dimensional coordinate system with
x-axis and y-axis. Without loss of generality, we assume that each unit of the
system is located on a unit circle with radius 1. That is, the distance
between the origin $(0,0)$ and each unit is $1$ for all units. Again, without
loss of generality, we adjust the coordinate of unit $1$ to $(1,0)$ and
arrange all the other units counterclockwise in ascending order of unit index
such that the x-y coordinate of unit $i$, $(x_{i},y_{i})$, becomes
$\left(\cos\left(\left(i-1\right)\theta\right),\
\sin\left(\left(i-1\right)\theta\right)\right)$ for $i=1,...,n$ where
$\theta=2\pi/n$. Then, the coordinate of the center of gravity given a
subsystem $U=\\{u_{1},u_{2},...,u_{|U|}\\}$ can be simply calculated by
$\left(\left(\sum_{u\in U}x_{u}\right)/|U|,\left(\sum_{u\in
U}y_{u}\right)/|U|\right)$. In summary, the following remark states a
quantitative way of examining whether or not a subsystem $U$ satisfies BC3.

######  Remark 3.1.

A circular $k$-out-of-$n$: G balanced system with an index set of non-failed
units $U$ satisfies BC3 if the following equality holds:

$\displaystyle\begin{bmatrix}0\\\
0\end{bmatrix}=\begin{bmatrix}\displaystyle\frac{\sum_{u\in
U}\cos\left(\left(u-1\right)\theta\right)}{|U|}\\\
\displaystyle\frac{\sum_{u\in
U}\sin\left(\left(u-1\right)\theta\right)}{|U|}\end{bmatrix}.$ (1)

For example, consider the graphical model in Fig. 5(a) showing a subsystem
$U=\\{1,4,5,9,10\\}$ of a circular $k$-out-of-$12$: G balanced system. Noting
that $\theta=2\pi/12=\pi/6$ and $|U|=5$, the center of gravity, say
$(\bar{x}_{U},\bar{y}_{U})$, can be obtained as follows

$\displaystyle\begin{bmatrix}\bar{x}_{U}\\\ \bar{y}_{U}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}\displaystyle\frac{\sum_{u\in
U}\cos\left(\left(u-1\right)\theta\right)}{|U|}\\\
\displaystyle\frac{\sum_{u\in
U}\sin\left(\left(u-1\right)\theta\right)}{|U|}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{\cos\left(0\cdot\frac{\pi}{6}\right)+\cos\left(3\cdot\frac{\pi}{6}\right)+\cos\left(4\cdot\frac{\pi}{6}\right)+\cos\left(8\cdot\frac{\pi}{6}\right)+\cos\left(9\cdot\frac{\pi}{6}\right)}{5}\\\
\displaystyle\frac{\sin\left(0\cdot\frac{\pi}{6}\right)+\sin\left(3\cdot\frac{\pi}{6}\right)+\sin\left(4\cdot\frac{\pi}{6}\right)+\sin\left(8\cdot\frac{\pi}{6}\right)+\sin\left(9\cdot\frac{\pi}{6}\right)}{5}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0\\\ 0\end{bmatrix},$

which verifies that a subsystem $U=\\{1,4,5,9,10\\}$ satisfies BC3 hence is
identified as balanced.

For another example, consider the graphical model in Fig. 5(b) showing a
subsystem $U=\\{3,6,8,12\\}$ with $\theta=\pi/6$ and $|U|=4$. Similarly, the
center of gravity can be obtained as follows

$\displaystyle\begin{bmatrix}\bar{x}_{U}\\\ \bar{y}_{U}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{\cos\left(2\cdot\frac{\pi}{6}\right)+\cos\left(5\cdot\frac{\pi}{6}\right)+\cos\left(7\cdot\frac{\pi}{6}\right)+\cos\left(11\cdot\frac{\pi}{6}\right)}{4}\\\
\displaystyle\frac{\sin\left(2\cdot\frac{\pi}{6}\right)+\sin\left(5\cdot\frac{\pi}{6}\right)+\sin\left(7\cdot\frac{\pi}{6}\right)+\sin\left(11\cdot\frac{\pi}{6}\right)}{4}\end{bmatrix}$
$\displaystyle\approx\begin{bmatrix}-0.0915\\\
0.0915\end{bmatrix}\neq\begin{bmatrix}0\\\ 0\end{bmatrix},$

which concludes that a subsystem $U=\\{3,6,8,12\\}$ does not satisfy BC3 hence
is unbalanced.

###  4.4 Relationship between the balance conditions

In this subsection, we discuss the relationship among the three balance
conditions BC1, BC2, and BC3. The main purpose of this investigation is to
show that BC3 is the most generalized balance condition. First, we start with
the following proposition which shows that BC3 is a more generalized balance
condition than BC1.

######  Proposition 3.1.

BC1 implies BC3.

###### Proof.

See Appendix A. ∎

Next, the following proposition states that BC3 is a more generalized balance
condition that BC2. Recall that we have observed an example of this
relationship graphically by comparing Fig. 4(b) with Fig. 5(a).

######  Proposition 3.2.

BC2 implies BC3.

###### Proof.

See Appendix B. ∎

Simply combining the propositions 3.1 and 3.2, we have the following
conclusion.

######  Corollary 1.

BC3 is the most generalized balance condition among BC1, BC2, and BC3.

###### Proof.

By propositions 3.1 and 3.2. ∎

In the next section, we explain how to evaluate the system reliability based
on the well-known minimum tie-set method considering the balance conditions
discussed so far.

##  5 Reliability Evaluation

The reliability of circular $k$-out-of-$n$: G balanced systems can be
evaluated considering the balance conditions. We apply the minimum tie-set
method (also known as minimal path set method) [3]. That is, we interpret the
system as a parallel system consists of minimum tie-sets and find the
probability that at least one minimum tie-set is operational. In the following
two subsections, We will introduce the notion of minimum tie-sets in the
context of circular $k$-out-of-$n$: G balanced systems and then explain how to
evaluate the system reliability.

###  5.1 Minimum tie-sets

A tie-set (also known as path set) is a subset of units in the system which by
operating ensures the system is functioning. Hence, a tie-set for a circular
$k$-out-of-$n$: G balanced system should include at least $k$ elements and
satisfy one of the balance conditions. Among the ordinary tie-sets, the
minimum tie-sets (also known as minimal path sets) are the tie-sets that
cannot be reduced without losing their property as a tie-set. The minimum tie-
set plays an important role when we evaluate the system reliability since the
system can be interpreted as a parallel structure of minimum tie-sets [3].

To examine the minimality of a tie-set, we need to check its inclusion
relationship with other tie-sets; a tie-set which is a superset of another
tie-set cannot be minimal. The flowchart in Fig. 6 summarizes the procedure to
find all the minimum tie-sets in a circular $k$-out-of-$n$: G balanced system.
Note that the procedure applies for all the balance definitions by changing
the balance condition checking step.

Figure 6: Procedure for enumerating all the minimum tie-sets

###  5.2 System reliability

In this subsection, we explain how to evaluate the system reliability using
the set of minimum tie-sets. Let $X_{i}$ be the binary state variable of unit
$i$; $X_{i}=1$ if unit $i$ is functioning, $X_{i}=0$ otherwise. Collecting the
states of all units, we let $\mathbf{X}$ be the system state vector;
$\mathbf{X}=[X_{1},X_{2},...,X_{n}]$. Then, we define the system structure
function $\phi(\cdot)$ of $\mathbf{X}$ as follows:

$\displaystyle\phi(\mathbf{X})\equiv
1-\prod_{T^{M}\in\mathbf{T}^{M}}\left(1-\prod_{u\in T^{M}}X_{u}\right).$

By the above definition, $\phi(\mathbf{X})$ becomes an indicator random
variable that is 1 if $\mathbf{X}$ corresponds to an operational system and 0
otherwise.

Using the system structure function, the system reliability $R_{S}$ can be
directly derived by calculating $\mathbb{P}\left[\phi(\mathbf{X})=1\right]$
which is equal to $\mathbb{E}\left[\phi(\mathbf{X})\right]$ due to the
property of indicator random variable. Since the probability that a unit is
functioning properly is a constant $r$ and all units are assumed to be
homogeneous, we have $\mathbb{E}[X_{i}]=r$ for all $i$. Together with the
assumption of independent units, $R_{S}$ is obtained by

$\displaystyle R_{S}=\mathbb{E}\left[\phi(\mathbf{X})\right]$
$\displaystyle=\mathbb{E}\left[1-\prod_{T_{M}\in\mathbf{T}^{M}}\left(1-\prod_{u\in
T^{M}}X_{u}\right)\right]\quad\textrm{(by the definition of reliability)}$
$\displaystyle=1-\prod_{T^{M}\in\mathbf{T}^{M}}\left(1-\prod_{u\in
T^{M}}\mathbb{E}\left[X_{u}\right]\right)\quad\textrm{(by independence)}$
$\displaystyle=1-\prod_{T^{M}\in\mathbf{T}^{M}}\left(1-\prod_{u\in
T^{M}}r\right)\quad\textrm{(by homogeneity)}$
$\displaystyle=1-\prod_{T^{M}\in\mathbf{T}^{M}}\left(1-r^{|T^{M}|}\right).$
(1)

From the last Eq. (1) for the system reliability, calculating the value of
$R_{S}$ is straightforward once the set of minimum tie-sets $\mathbf{T}^{M}$
is acquired by enumeration. Next, we provide a specific example of system
reliability evaluation for descriptive purpose.

###  5.3 A descriptive numerical example

Consider a circular $4$-out-of-$12$: G balanced system. Following the
procedures in Fig. 6, we enumerate all the minimum-tie sets according to each
balance definition and visualize them in Fig. 7. As shown in the figure, we
have 15 minimum tie-sets considering only BC1, 19 minimum tie-sets considering
BC2, and 31 minimum tie-sets considering BC3, for a circular $4$-out-of-$12$:
G balanced system. Since the Eq. (1) implies that the system reliability
$R_{S}$ is proportional to the number of minimum tie-sets $|\mathbf{T}^{M}|$,
the increment of the number of minimum tie-sets will result in the enhancement
of the system reliability. To better understand the systems with different
parameters, Table 1 compares the number of minimum tie-sets for various
combinations of $(k,n)$ for each balance condition. Although not all the
systems show the difference in the number of minimum tie-sets between BC3 and
BC2, we observe significant differences (larger than 10) for some systems such
as $4$-out-of-$12$ ($\textrm{diff}=12$), $6$-out-of-$12$ ($\textrm{diff}=21$),
$6$-out-of-$14$ ($\textrm{diff}=14$), and $8$-out-of-$14$
($\textrm{diff}=14$).

Figure 7: Enumeration of all the graphical models of minimum tie-sets in a
circular $4$-out-of-$12$: G balanced system (each legend exhibits the balance
conditions that the corresponding minimum tie-set satisfies)

In the next section, we provide extensive numerical studies on the reliability
enhancement effect resulted from introducing the new balance condition BC3.

Table 1: Comparison of the number of minimum tie-sets considering BC1, BC2,
and BC3

System Parameters |  |  Number of Minimum Tie-sets |  |  | Difference |
---|---|---|---|---|---|---
$k$ | $n$ |  BC1 | BC2 | BC3 | BC2$-$BC1 | BC3$-$BC2
2 | 6 |  3 | 5 | 5 | 2 | 0
4 |  |  3 | 3 | 3 | 0 | 0
2 | 8 |  4 | 4 | 4 | 0 | 0
4 |  |  6 | 6 | 6 | 0 | 0
6 |  |  4 | 4 | 4 | 0 | 0
2 | 10 |  5 | 7 | 7 | 2 | 0
4 |  |  10 | 12 | 12 | 2 | 0
6 |  |  10 | 10 | 10 | 0 | 0
8 |  |  5 | 5 | 5 | 0 | 0
2 | 12 |  6 | 10 | 10 | 4 | 0
4 |  |  15 | 19 | 31 | 4 | 12
6 |  |  11 | 15 | 36 | 4 | 21
8 |  |  15 | 15 | 19 | 0 | 4
10 |  |  6 | 6 | 6 | 0 | 0
2 | 14 |  7 | 9 | 9 | 2 | 0
4 |  |  21 | 23 | 23 | 2 | 0
6 |  |  21 | 23 | 37 | 2 | 14
8 |  |  21 | 21 | 35 | 0 | 14
10 |  |  21 | 21 | 21 | 0 | 0
12 |  |  7 | 7 | 7 | 0 | 0

##  6 Numerical Analysis

We investigate the reliability of circular $k$-out-of-$n$: G balanced systems
with various system parameters $k$ (minimum number of non-failed units for
working system), $n$ (number of units in the system), and $r$ (unit
reliability), for different balance condition consideration. Though the main
purpose of this section is to show the reliability enhancement effect thanks
to the new balance definition BC3 compared to BC2, we include the BC1’s
results together in order to provide richer experimental data that helps
understanding the gradual improvement from BC1 to BC2 to BC3.

###  6.1 Reliability comparison for the systems with various parameters

Fig. 8 depicts the effect of the unit reliability on the system reliability
for each balance condition (say $R_{S}^{\textrm{BC1}}$,
$R_{S}^{\textrm{BC2}}$, and $R_{S}^{\textrm{BC3}}$) for $k$-out-of-12 (three
plots in upper row) and $k$-out-of-14 (another three plots in lower row)
systems where $k=4,6,8$. The first important but anticipatory observation is
that considering more generalized balance condition consistently improves the
system reliability for all the systems: $R_{S}^{\textrm{BC3}}\geq
R_{S}^{\textrm{BC2}}\geq R_{S}^{\textrm{BC1}}$ for any $r$ in $[0,1]$. For
some systems such as $6$-out-of-$12$, the level of improvement is quite
significant so that introducing the new balance condition seems to be
promising in terms of operating circular $k$-out-of-$n$: G balanced systems.
Second, the system reliability shows definite “S”-shaped curve regardless of
the system parameters. Together with the first observation, it is implied that
the effect of unit-wise reliability enhancement on the system reliability is
stronger in the middle of $r$’s range $[0,1]$ and goes weaker as it approaches
extreme values.

((a))

((b))

((c))

((d))

((e))

((f))

Figure 8: System reliability plots with varying unit reliability $r$ for the
three balance conditions (upper row: $k$-out-of-12, lower row: $k$-out-of-14)

Fig. 9 shows the system reliability with varying $k$’s and $r$’s for the
systems with $n=12$ (upper row) and $n=14$ (lower row) under each balance
condition. Similar to the observations from the previous plots, the system
reliability shows “S”-shaped curve in terms of unit reliability $r$ with the
increasing trends from BC1 to BC2 to BC3, for all the cases. In addition, we
find the decreasing tendency of $R_{S}$ as $k$ increases because the systems
with larger $k$ require more non-failed units for the system to be
operational. Note that the degree of decrement looks more significant when $k$
increases from an even number to an odd number (say $k_{o}$), for example, see
the changing slopes of $R_{S}$ for $k=4,5,6$ for each plot in Figs. 9(d)–9(f).
This phenomenon aligns with our intuition because maintaining the physical
balance with the odd number of units must be more difficult than with the even
number of units. Hence, there must be only few additional minimum tie-sets for
the system with $k=k_{o}$ compared to the system with $k=k_{o}+1$ (the
smallest even number greater than $k_{o}$).

((a))

((b))

((c))

((d))

((e))

((f))

Figure 9: System reliability plots with varying $k$ and $r$ for the systems
with $n=12$ (upper row) and $n=14$ (lower row) under each balance condition

Fig. 10 provides the system reliability plots with varying $k$ and $n$ for
$r=0.5,0.7,0.9$ and different balance conditions. It is worth mentioning that
the reliability is only evaluated for the systems with $k$ and $n$ such that
$k\leq n-1$ hence the plots have empty space for some pairs of $(k,n)$. From
the plots in Fig. 10, we understand the overall changing shape of the system
reliability $R_{S}$ affected by changing $k$, $n$, and $r$. For example, we
observe the decreasing trend of $R_{S}$ as $k$ increases for a fixed $n$ for
all the cases. Similar to the observed phenomena in Fig. 9, the level of
$R_{S}$’s decrement looks significant when $k$ is increased from an even
number to an odd number. On the other hand, $R_{S}$ shows increasing trend as
$n$ increases for a fixed $k$ for all the cases. The sensitivity of $R_{S}$ to
$(k,n)$ looks greater for smaller $r$’s (see Figs. 10(a)–10(f)) whereas we
mostly observe few changes except for the marginal area of the plots when $r$
is larger (see Figs. 10(g)–10(i)).

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

Figure 10: System reliability plots with varying $k$ and $n$ for the systems
with $r=0.5,0.7,0.9$ under each balance condition

###  6.2 Summary with interpretation

In this section, we summarize the experimental results followed by the
corresponding interpretations. Firstly, we consistently observed
$R_{S}^{\textrm{BC3}}\geq R_{S}^{\textrm{BC2}}\geq R_{S}^{\textrm{BC1}}$ and
it demonstrates that the introduction of BC3 can enhance the system
reliability of circular $k$-out-of-$n$: G balanced systems. The analysis in
Section 4.4 provides the supporting theory, which shows that BC3 covers BC2
and BC1. Although BC3 is considered as the most generalized balance condition
for circular $k$-out-of-$n$: G balanced systems to the best of our knowledge,
the system reliability can be further improved if one can find another
effective balance condition.

Secondly, $R_{S}$ shows definite “S”-shaped curve to the unit reliability $r$
when all the other parameters fixed. That is, the unit-wise reliability
improvement is more effective in terms of enhancing the system reliability
when $r$ is not around the extreme values $0$ or $1$. This tendency can be
useful for the impact-effort analysis of unit-wise reliability improvement.

Finally, it is trivial that the increment of $k$ results in the decrement of
$R_{S}$. However, we consistently observed more significant $R_{S}$’s
reduction when $k$ is increased from an even number to an odd number from
Figs. 9-10. From this observation, we realize that determining $k$ in a
circular $k$-out-of-$n$: G balanced system as even as possible must be
advantageous for most of the systems.

##  7 Conclusion

This article examines the reliability of circular $k$-out-of-$n$: G balanced
systems based on several definitions of the balance condition. Motivated by
two previously investigated conditions, BC1 (symmetry) and BC2
(proportionality), we propose a new balance condition BC3 based on the center
of gravity concept. Among the three conditions, the newly proposed BC3 is
advantageous in terms of its simplicity and generality because both BC1 and
BC2 can be regarded as special cases of BC3. We investigate the inclusion
relationships among the balance conditions through mathematical proofs and
several numerical examples. For system reliability, we apply the minimum tie-
set method in which the system is interpreted as a parallel system consists of
minimum tie-sets. Since the system reliability is proportional to the number
of minimum tie-sets, considering BC3 can discover more minimum tie-sets,
thereby enhancing system reliability. Extensive numerical studies demonstrate
the effect of considering BC3 and the changing trends of system reliability
for varying system parameters.

There are several future research suggestions that extend this paper. First,
the reliability evaluation method used in this paper requires complete
enumeration of all working states, which can be computationally challenging
when $k$ and/or $n$ are large. In this regard, a more efficient method that
reasonably approximates system reliability in a shorter time can be explored.
Second, assumptions considered throughout this paper can be relaxed. For
example, we have assumed that survival and failure probabilities for each unit
are constants at any time and that the unit is only of two states (failed and
non-failed). The constant probability assumption can be relaxed by introducing
a failure time distribution such as exponential and Weibull, etc. Also,
considering multi-state units would relax the binary-state assumption, thereby
opening up another interesting research direction regarding circular $k$-out-
of-$n$: G balanced systems.

## Acknowledgment

This work was supported in part by the National Research Foundation of Korea
(NRF) grants (No. 2021R1G1A1094924, 2021R1A2C1094699 and 2021R1A4A1031019)
funded by the Korea government (Ministry of Science and ICT, MSIT) and in part
by Korea Institute for Advancement of Technology (KIAT) grant funded by the
Korea Government (MOTIE) (P0008691, HRD Program for Industrial Innovation).

## Appendix

##  Appendix A Proof of Proposition 3.1

Consider a conventional two-dimensional coordinate system with x-axis and
y-axis on which the units are circularly and evenly located. Assume that we
have a subsystem $U=\\{u_{1},u_{2},...,u_{|U|}\\}$ that satisfies BC1. Then,
the subsystem $U$ is symmetric with respect to at least a pair of
perpendicular axes by the definition 1. That is, for any operating unit $u\in
U$, we have another operating unit $u^{\prime}\in U$ such that the angle
between $u$ and $u^{\prime}$ is $\pi$. Letting $(x_{u},y_{u})$ be the x-y
coordinate of $u$, we have

$\displaystyle\begin{bmatrix}x_{u}\\\
y_{u}\end{bmatrix}=\begin{bmatrix}r\cos{\theta}\\\
r\sin{\theta}\end{bmatrix},$

where $r$ is the Euclidean distance from the origin $(0,0)$ to the unit $u$
and $\theta$ is the angle between x-axis and unit $u$. Since the angle between
$u$ and $u^{\prime}$ is $\pi$, we have

$\displaystyle\begin{bmatrix}x_{u^{\prime}}\\\
y_{u^{\prime}}\end{bmatrix}=\begin{bmatrix}r\cos{(\theta+\pi)}\\\
r\sin{(\theta+\pi)}\end{bmatrix}.$

Letting $(\bar{x}_{\\{u,u^{\prime}\\}},\bar{y}_{\\{u,u^{\prime}\\}})$ be the
coordinate of a center of gravity formed by the units $u$ and $u^{\prime}$, we
have

$\displaystyle\begin{bmatrix}\bar{x}_{\\{u,u^{\prime}\\}}\\\
\bar{y}_{\\{u,u^{\prime}\\}}\end{bmatrix}=\begin{bmatrix}\frac{1}{2}\\{r\cos{(\theta)}+r\cos{(\theta+\pi)}\\}\\\
\frac{1}{2}\\{r\sin{(\theta)}+r\sin{(\theta+\pi)}\\}\end{bmatrix}=\begin{bmatrix}-r\left\\{\cos{\left(\theta+\frac{\pi}{2}\right)}\cos{\left(\frac{\pi}{2}\right)}\right\\}\\\
r\left\\{\sin{\left(\theta+\frac{\pi}{2}\right)}\cos{\left(\frac{\pi}{2}\right)}\right\\}\end{bmatrix}=\begin{bmatrix}0\\\
0\end{bmatrix},$

applying the simple trigonometric angle addition formula and because
$\cos(\pi/2)=0$.

Since the above result applies for any unit in a subsystem satisfying BC1, the
center of gravity of such a subsystem is always $(0,0)$. Therefore, BC1
implies BC3.

##  Appendix B Proof of Proposition 3.2

Assume that we have a subsystem $U=\\{u_{1},u_{2},...,u_{|U|}\\}$ that
satisfies BC2. Then, the sector angles are either identical for all
$s=1,2,\dots,N$, as described in remark 2.1(a), or the opposite angles of one
another as described in remark 2.1(b). In this proof, we will only show that
remark 2.1(a) implies BC3 because remark 2.1(b) is trivially considered as
satisfying BC1 (symmetry) hence implies BC3 by proposition 3.1.

Given the identical sector angles, we can partition the subsystem $U$ into
$N_{\textrm{SG}}$ mutually exclusive and exhaustive subgroups $U_{j}$’s each
of which have the same number of units such that

$\displaystyle
U=\\{u_{1},u_{2},\dots,u_{|U|}\\}=\bigcup_{j=1}^{N_{\textrm{SG}}}U_{j}\quad\textrm{and}\quad|U_{1}|=|U_{2}|=\cdots=|U_{N_{\textrm{SG}}}|.$

Because all the subgroups are homogeneous and all the sector angles are
congruent, we can interpret the subsystem $U$ as an equivalent system $V$
comprised of $N_{\textrm{SG}}$ virtually aggregated units, say units
$v_{j}$’s, such that $V=\\{v_{1},...,v_{N_{\textrm{SG}}}\\}$ where the angle
between any two such units is $\theta=2\pi/N_{\textrm{SG}}$. Considering the
physical properties, we define the x-y coordinate of $v_{j}$ to be the center
of gravity formed by the units in the associated subgroup $U_{j}$:
$(x_{v_{j}},y_{v_{j}})\equiv(\bar{x}_{U_{j}},\bar{y}_{U_{j}})$. Then, the
center of gravity formed by the subsystem $U$, $(\bar{x}_{U},\bar{y}_{U})$ can
be expressed as follows:

$\displaystyle\begin{bmatrix}\bar{x}_{U}\\\
\bar{y}_{U}\end{bmatrix}=\begin{bmatrix}\bar{x}_{V}\\\
\bar{y}_{V}\end{bmatrix}=\begin{bmatrix}\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}x_{v_{j}}\\\
\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}y_{v_{j}}\end{bmatrix}=\begin{bmatrix}\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}\bar{x}_{U_{j}}\\\
\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}\bar{y}_{U_{j}}\end{bmatrix}.$

Let $r$ be the Euclidean distance between the origin $(0,0)$ and a virtual
unit $v$. Without loss of generality, we adjust the coordinate of unit $v_{1}$
to $(r,0)$ and arrange all the other units $v_{j}$’s counterclockwise in
ascending order such that the x-y coordinate of unit $v_{j}$,
$(x_{v_{j}},y_{v_{j}})$, becomes
$\left(\cos\left(\left(j-1\right)\theta\right),\sin\left(\left(j-1\right)\theta\right)\right)$
for $j=1,...,N_{\textrm{SG}}$ where $\theta=2\pi/N_{\textrm{SG}}$. Then,
$(\bar{x}_{U},\bar{y}_{U})$ is

$\displaystyle\begin{bmatrix}\bar{x}_{U}\\\
\bar{y}_{U}\end{bmatrix}=\begin{bmatrix}\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}r\cos{\left((j-1)\frac{2\pi}{N_{\textrm{SG}}}\right)}\\\
\frac{1}{N_{\textrm{SG}}}\sum_{j=1}^{N_{\textrm{SG}}}r\sin{\left((j-1)\frac{2\pi}{N_{\textrm{SG}}}\right)}\end{bmatrix}.$
(B.1)

To show that the RHS of Eq. (B.1) is indeed the origin $(0,0)$, we will use
the following lemma.

######  Lemma 1 (Knapp [14]).

If $a,d\in\mathbb{R}$, $d\neq 0$, and $n$ is a positive integer, then

$\displaystyle\sum_{k=0}^{n-1}\cos{(a+kd)}=\frac{\sin{(nd/2)}}{\sin{(d/2)}}\cos{\left(a+\frac{(n-1)d}{2}\right)}$
(B.2)

and

$\displaystyle\sum_{k=0}^{n-1}\sin{(a+kd)}=\frac{\sin{(nd/2)}}{\sin{(d/2)}}\sin{\left(a+\frac{(n-1)d}{2}\right)}.$
(B.3)

First, substituting $k=j-1$, $a=0$, $n=N_{\textrm{SG}}$, and
$d=2\pi/N_{\textrm{SG}}$ into the both sides of Eq. (B.2), we have the
following equation:

$\displaystyle\sum_{j=1}^{N_{\textrm{SG}}}\cos{\left((j-1)\frac{2\pi}{N_{\textrm{SG}}}\right)}=\frac{\sin{\pi}}{\sin{(\pi/N_{\textrm{SG}})}}\cos{\left(\frac{(N_{\textrm{SG}}-1)\pi}{N_{\textrm{SG}}}\right)},$
(B.4)

which reduces to $0$ because we have $\sin{\pi}=0$ in the RHS.

Similarly, substituting $k=j-1$, $a=0$, $n=N_{\textrm{SG}}$, and
$d=2\pi/N_{\textrm{SG}}$ into the both sides of Eq. (B.3), we have the
following equation:

$\displaystyle\sum_{j=1}^{N_{\textrm{SG}}}\sin{\left((j-1)\frac{2\pi}{N_{\textrm{SG}}}\right)}=\frac{\sin{\pi}}{\sin{(\pi/N_{\textrm{SG}})}}\sin{\left(\frac{(N_{\textrm{SG}}-1)\pi}{N_{\textrm{SG}}}\right)},$
(B.5)

which reduces to $0$ because we have $\sin{\pi}=0$ in the RHS. Combining the
results from Eqs. (B.4) and (B.5), we notice that the RHS of Eq. (B.1) is
$[0,0]^{\top}$. Therefore, BC2 implies BC3.

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