The Roots of the Reliability Polynomials of Circular Consecutive-k-out-of-n:F Systems

Abstract

The zeros of the reliability polynomials of circular consecutive-k-out-of-n:F systems are studied. We prove that, for any fixed $k\ge 2$, the set of the roots of all the reliability polynomials (for all $n\ge k$) is unbounded in the complex plane. In the particular case $k=2$, we show that all the nonzero roots are real, distinct numbers and find the closure of the set of roots. For every $n\ge k$, the expressions of the minimum root and the maximum root are given, both for circular as well as for linear systems.

1. Introduction

Consecutive-k-out-of-n:F systems were introduced by Kontoleon [1] under the name of “r-successive-out-of-n:F”, being re-branded “consecutive-k-out-of-n:F” one year later [2]. They have attracted researchers’ interest due to their robustness and ability to achieve high reliability at reasonably low costs. A consecutive-k-out-of-n:F system is formed by n components placed in a row (if the system is linear), or in a cycle (for a circular system, a concept mentioned for the first time in [3]), which fails if and only if at least k consecutive components fail. The reader can find a detailed presentation of the subject in [4,5,6] and recent contributions in the field, developments, and applications in [7,8,9,10,11,12,13,14,15,16,17,18].

A linear consecutive-k-out-of-n:F system can be represented as a graph $G=(V,E)$, with $n+2$ vertices, $V=\{S=0,1,2,\dots ,n,n+1=T\}$, and having the set of edges $E=\{ij,0\le i<j\le n+1,j-i\le k\}$ (see Figure 1). The nodes $1,2,\dots ,n$ are identical and statistically independent, failing with probability $q=1-p$, while the terminals $S=0$, $T=n+1$, as well as the edges, are always operational. Note that this graph representation may be directed or undirected. The reliability of the system, denoted by $R_L(k,n;p)$, is the probability that the system is working, i.e., the probability that the subgraph induced by the operational nodes has at least one path linking the terminal nodes S and T.

A circular consecutive-k-out-of-n:F system can also be represented as a directed graph $G=(V,E)$, with n vertices, $V=\{1,2,\dots ,n\}$, and having the set of directed edges $E=\{ij,0\le i\le n,i+1\le j\le i+k(\mathrm{mod}n\left)\right\}$ (see Figure 2). As in the linear case, we assume that the nodes $1,2,\dots ,n$ are identical and statistically independent, failing with probability $q=1-p$, while the edges are always operational. The reliability of the circular system, $R_C(k,n;p)$, is the probability that the subgraph induced by the operational nodes is Hamiltonian (it has a cycle that visits each vertex exactly once).

In both cases, the reliability is a polynomial of degree n in p, for every $n\ge k$ (for $n<k$, it is equal to 1). Moreover, these reliability polynomials satisfy the same recurrence relation, for every $n\ge k$ in the linear case, and for every $n\ge 2k$ in the circular case, respectively. Philippou [19] identified the relation between the reliability polynomials of linear consecutive systems and the generalized Fibonacci polynomials (introduced by Hoggatt and Bicknell in [20]), while Charalambides [21] noted the relation between the reliability polynomials of circular consecutive systems and the Lucas polynomials of order k.

The reliability polynomial of a network formed by a number of identical components is usually considered as a function of p, the probability that a single component is functional. The value of p is normally supposed to be in the interval $[0,1]$, but studying the analytic properties of these polynomials as functions of a complex variable $z\in \mathbb{C}$ can shed new light on the landscape of reliability theory. Originally [22], the roots of reliability polynomials were investigated for the possible log-concavity of the coefficients. In fact, various properties of the roots can be linked to the shape of the coefficients [23], so finding the location and magnitude of the roots is clearly worth exploring.

For instance, using the Newton’s inequalities (see [24], 7.1), it can be easily shown that for any polynomial $P\left(z\right)={\displaystyle \sum _{j=0}^n}{a}_j{z}^j$ with positive coefficients and real roots, the sequence of coefficients is log-concave, that is,

$$a_j^2>a_{j-1}{a}_{j+1},j=1,\dots ,n-1.$$

It is well-known that a log-concave sequence is unimodal. As a matter of fact, the condition above is sufficient, but it is not necessary. A more general result [25] states that if all the (complex) roots of a polynomial with positive coefficients lie in the region $\frac{2\pi }{3}<argz<\frac{4\pi }{3}$, then the sequence of coefficients is log-concave.

Finding the set of roots of various polynomials related to different families of graphs is both a tempting and also a fruitful problem which has attracted lots of research in the past few decades (see, for instance [22,26,27,28,29,30] for reliability polynomials; [31,32,33,34,35] for chromatic polynomials; and [36,37,38,39] for independence polynomials). In many of these papers, a central role is played by a theorem on analytic functions due to Beraha, Kahane, and Weiss [40], which proves to be a highly useful instrument to study the zeros of a family of polynomials defined recursively.

Using the Beraha–Kahane–Weiss theorem, the authors have proven in a recent paper [41] that the set of the roots of the reliability polynomials of linear consecutive-k-out-of-n:F systems is unbounded, for any $k\ge 2$. In this paper, we show (in Section 2) that the result also holds for circular consecutive-k-out-of-n:F systems. In Section 3, we study the particular case when $k=2$ and prove that all the nonzero roots of the reliability polynomial of the circular consecutive-2-out-of-n:F system are real, distinct numbers (for any $n\ge 2$). Moreover, we show that any root of a circular consecutive-2-out-of-n:F system is also a root of the linear consecutive-2-out-of-$(2n-2)$:F system. Finally, in Section 4 we investigate the values of the minimum and the maximum root for different values of n, and for both kinds of systems. We show that the interval between the minimum and the maximum root is growing faster (as n increases) for circular systems than for linear systems. It is also worth mentioning that, in each case, the sequence of intervals exhibits a kind of periodicity, being formed by three monotonic subsequences.

2. The Circular Consecutive-$\mathit{k}$-out-of-$\mathit{n}$:F Systems Have Unbounded Roots

For any fixed $k\ge 2$, the reliability polynomials $R_C(k,n;p)$ of circular consecutive-k-out-of-n:F systems satisfy the following recurrence relation for any $n\ge 2k$ (see [42]):

$$R_C(k,n;p)=p{R}_C(k,n-1;p)+pq{R}_C(k,n-2;p)+\dots +p{q}^{k-1}{R}_C(k,n-k;p).$$

(1)

Since the system fails if and only if at least k consecutive components fail, the first k polynomials are equal to 1:

$$R_C(k,n;p)=1,n=0,1,\dots ,k-1,$$

(2)

and when $n=k$:

$$R_C(k,k;p)=1-q^k.$$

(3)

For $n=k+1,\dots ,2k$, the reliability polynomials have the expression [42]:

$$R_C(k,n;p)=1-q^n-np{q}^k.$$

(4)

Since $k\ge 2$ is fixed, for any $n\ge 0$, the polynomial $R_C(k,n;p)$ can be written as follows:

$$R_C(k,n;p)=p^n{\mathcal{C}}_n\left(z\right),$$

(5)

where

$$z=\frac{q}{p}=\frac{1}{p}-1,$$

(6)

and ${\mathcal{C}}_n\left(z\right)$ is a polynomial of degree n in z. From (1), we find that the polynomials ${\mathcal{C}}_n\left(z\right)$ satisfy the following recurrence relation, for any $n\ge 2k$:

$${\mathcal{C}}_n\left(z\right)={\mathcal{C}}_{n-1}\left(z\right)+z{\mathcal{C}}_{n-2}\left(z\right)+\dots +z^{k-1}{\mathcal{C}}_{n-k}\left(z\right).$$

(7)

Using Equations (2)–(6), we obtain:

$$\begin{array}{c}{\mathcal{C}}_n\left(z\right)={(1+z)}^n,\mathrm{for}n=0,1,\dots ,k-1\hfill \\ {\mathcal{C}}_k\left(z\right)={(1+z)}^k-z^k\hfill \\ {\mathcal{C}}_n\left(z\right)={(1+z)}^n-z^n-n{z}^k{(1+z)}^{n-k-1},\mathrm{for}n=k+1,\dots ,2k.\hfill \end{array}$$

(8)

In the following, we recall some basic results on recursively defined polynomials. Consider an arbitrary sequence of polynomials $\left\{P_n\right\}$, $n=0,1,\dots$, defined by a linear recurrence relation of order k:

$$P_n\left(z\right)=\sum _{j=1}^k{f}_j\left(z\right)P_{n-j}\left(z\right),n\ge k,$$

(9)

where $f_j\left(z\right)$ and $P_{j-1}\left(z\right)$, $j=1,2,\dots ,k$ are given polynomials.

We say that $x\in \mathbb{C}$ is a limit of zeros of $\left\{P_n\right\}$ if there exists a sequence of complex numbers $\left\{z_n\right\}$ such that $P_n\left(z_n\right)=0$ and ${\displaystyle \underset{n\to \infty }{lim}{z}_n=x}$.

Let ${\lambda }_j\left(z\right)$, $j=1,2,\dots ,k$ be the roots of the characteristic equation

$${\lambda }^k-\sum _{j=1}^k{f}_j\left(z\right){\lambda }^{k-j}=0.$$

(10)

Then, for any $z\in \mathbb{C}$ such that ${\lambda }_i\left(z\right)\ne {\lambda }_j\left(z\right)$ for $i\ne j$, one can write:

$$P_n\left(z\right)=\sum _{j=1}^k{\alpha }_j\left(z\right){\lambda }_j{\left(z\right)}^n,$$

(11)

where ${\alpha }_j\left(z\right)$, $j=1,\dots ,k$ are found from the linear system of equations obtained by writing (11) for $n=0,\dots ,k-1$.

Theorem 1

(Beraha–Kahane–Weiss theorem [40]). Suppose that $\left\{P_n\right\}$ is a sequence of polynomials defined by a relation of the form (9) such that $\left\{P_n\right\}$ satisfies no recursion of order less than k, and there is no constant ω, with $\left|\omega \right|=1$, such that ${\lambda }_i\left(z\right)=\omega {\lambda }_j\left(z\right)$ for some $i\ne j$.

Then, x is a limit of zeros of $\left\{P_n\right\}$ if and only if the roots of the characteristic Equation (10) can be ordered such that one of the following holds:

(i) $\left|{\lambda }_1\left(x\right)\right|>\left|{\lambda }_j\left(x\right)\right|$ for every $j=2,\dots ,k$, and ${\alpha }_1\left(x\right)=0$,

(ii) $\left|{\lambda }_1\left(x\right)\right|=\left|{\lambda }_2\left(x\right)\right|=\dots =\left|{\lambda }_l\left(x\right)\right|>\left|{\lambda }_j\left(x\right)\right|$, $j=l+1,\dots ,k$, for some $l\ge 2$.

Using this theorem, we can prove the following result.

Theorem 2.

For any fixed $k\ge 2$, the set of complex roots of all the polynomials $R_C(k,n;p)$, $n\ge k$, is unbounded.

Proof. 

First, we prove that the sequence of polynomials ${\mathcal{C}}_n\left(z\right)$ defined by the recurrence (7) of order k, with the initial conditions (8), satisfies the conditions from Theorem 1. In our case, the characteristic Equation (10) is written

$${\lambda }^k-\sum _{j=1}^k{z}^{j-1}{\lambda }^{k-j}=0.$$

(12)

Suppose that there exists a constant $\omega$ with $\left|\omega \right|=1$ with the property that for any $z\in \mathbb{C}$, there exists $\lambda \left(z\right)$ a root of (12) such that $\omega \lambda \left(z\right)$ is also a root of (12).

We can bring Equation (12) to a simpler form if we multiply it by $(\lambda -z)$. Thus, we can see that any root of (12) also satisfies the equation

$${\lambda }^{k+1}-(1+z){\lambda }^k+z^k=0,$$

(13)

while (13) has a root (namely, $\lambda =z$) which is not a root of (12).

For $z=-1$, the characteristic Equation (12) becomes

$${\lambda }^k-{\lambda }^{k-1}+\dots +{(-1)}^{k-1}\lambda +{(-1)}^k=0$$

(14)

and its roots are the roots $\ne -1$ of the Equation (13), which is written:

$${\lambda }^{k+1}={(-1)}^{k+1}.$$

If k is odd, the complex roots of this equation are

$${\lambda }_r=exp\left(\frac{2r\pi }{k+1}i\right),r\in \{0,1,\dots ,k\}\setminus \left\{\frac{k+1}{2}\right\},$$

while for k even, the roots are of the form:

$${\lambda }_r=exp\left(\frac{(2r+1)\pi }{k+1}i\right),r\in \{0,1,\dots ,k\}\setminus \left\{\frac{k}{2}\right\}.$$

In any case, the roots of the characteristic Equation (14) are all distinct complex numbers of modulus 1:

$$\left|{\lambda }_1(-1)\right|=\dots =\left|{\lambda }_k(-1)\right|=1,$$

(15)

and we can see that the ratio between any two different roots is of the form

$$\omega =exp\left(\frac{2m\pi }{k+1}i\right),m\in \{1,2,\dots ,k\}.$$

(16)

Thus, if there were a constant $\omega$ such that, for any z, there exist two roots $\lambda$ and $\omega \lambda$ of (12), then $\omega$ should be of the form (16), and hence it should satisfy ${\omega }^{k+1}=1$.

Consider $z\ne 0,-1$. Then, the roots of (13) are different from 0. If we suppose that both $\lambda$ and $\omega \lambda$ satisfy Equation (13), then we obtain that ${\omega }^k=1$, a contradiction, which implies that there is no such constant $\omega$. So we find (using (15)) that the hypotheses of Theorem 1 case (ii) are fulfilled, and hence there exists a sequence of complex numbers $\left\{z_n\right\}$ such that ${\mathcal{C}}_n\left(z_n\right)=0$ and ${\displaystyle \underset{n\to \infty }{lim}{z}_n=-1}$.

Recall that z was introduced by the change in variable (6). Therefore, we need to prove that $z_n\ne -1$. From the recurrence of order k (7), one can easily deduce the following recurrence of order $k+1$:

$${\mathcal{C}}_n\left(z\right)=(1+z){\mathcal{C}}_{n-1}\left(z\right)-z^k{\mathcal{C}}_{n-k-1}\left(z\right),\mathrm{for}\mathrm{all}n\ge 2k+1.$$

(17)

Hence,

$${\mathcal{C}}_n(-1)={(-1)}^{k+1}{\mathcal{C}}_{n-k-1}(-1),$$

for every $n\ge 2k+1$. Using (8), we deduce that ${\mathcal{C}}_n(-1)={(-1)}^{n-1}\ne 0$, for every $n=k,k+1,\dots ,2k$, so it follows that ${\mathcal{C}}_n(-1)\ne 0$, for any n. Thus, we find that $-1$ is not a root of the polynomials ${\mathcal{C}}_n$, so $z_n\ne -1$, for any $n\ge k$. Using (6), we have $z_n=\frac{1-p_n}{p_n}$, and it follows that

$$p_n=\frac{1}{z_n+1},n=k,k+1,\dots$$

is a sequence of roots of the reliability polynomial $R_C(k,n;p)$, with ${\displaystyle \underset{n\to \infty }{lim}\left|p_n\right|=\infty }$, and the theorem is proved. □

3. The Roots of Consecutive-2-out-of-$\mathit{n}$:F Systems

In this section, we study the particular case of circular consecutive-2-out-of-n systems and find the explicit formula for the reliability polynomials and their roots (proved to be all real numbers). We also aim to compare the circular and the linear consecutive-2-out-of-n systems, underlying the similarities and differences, as well as the relationships between the two families of polynomials. Therefore, we begin with a short review of the reliability polynomials of linear consecutive systems (studied in [41]).

The reliability polynomials $R_L(k,n;p)$ of linear consecutive-k-out-of n:F systems satisfy the same recurrence relation as $R_C(k,n;p)$ [4], but starting from $n=k$ (while the polynomials of circular systems satisfy the recurrence relation (1) starting from $n=2k$):

$$R_L(k,n;p)=p{R}_L(k,n-1;p)+pq{R}_L(k,n-2;p)\left(p\right)+\dots +p{q}^{k-1}{R}_L(k,n-k;p),$$

(18)

for every $n\ge k$. The initial polynomials are

$$R_L(k,n;p)=1,n=0,1,\dots ,k-1.$$

(19)

Using the same transformation, $z=\frac{q}{p}$, and $R_L(k,n;p)=p^n{\mathcal{L}}_n\left(z\right)$, we obtain the sequence of polynomials $\left\{{\mathcal{L}}_n\right\}$ satisfying the same recurrence as $\left\{{\mathcal{C}}_n\right\}$

$${\mathcal{L}}_n\left(z\right)={\mathcal{L}}_{n-1}\left(z\right)+z{\mathcal{L}}_{n-2}\left(z\right)+\dots +z^{k-1}{\mathcal{L}}_{n-k}\left(z\right),$$

(20)

for every $n\ge k$ and the initial conditions

$${\mathcal{L}}_n\left(z\right)={(1+z)}^n,\mathrm{for}n=0,1,\dots ,k-1.$$

(21)

When $k=2$, Equations (20) and (21) become

$$\begin{array}{cc}\hfill {\mathcal{L}}_n\left(z\right)& ={\mathcal{L}}_{n-1}\left(z\right)+z{\mathcal{L}}_{n-2}\left(z\right),n=2,3,\dots \hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathcal{L}}_0\left(z\right)& =1,{\mathcal{L}}_1\left(z\right)=1+z.\hfill \end{array}$$

(23)

Thus, as noted in [41], the sequence $\left\{{\mathcal{L}}_n\right\}$ is the sequence of Jacobsthal polynomials $\left\{J_n\right\}$, with ${\mathcal{L}}_n\left(z\right)=J_{n+2}\left(z\right)$. Introduced in [43], the Jacobsthal polynomials have the same coefficients as the more famous Fibonacci polynomials, being related by the formula $F_n\left(z\right)=z^{n-1}{J}_n\left(\frac{1}{z^2}\right)$ (see ([44], Chp. 44) for a detailed presentation).

Table 1. Fibonacci and Jacobsthal polynomials.

	Fibonacci Polynomials	Jacobsthal Polynomials
	$F_{n+2}\left(z\right)=z{F}_{n+1}\left(z\right)+F_n\left(z\right)$	$J_{n+2}\left(z\right)=J_{n+1}\left(z\right)+z{J}_n\left(z\right)$
$n=0$	$F_0\left(z\right)=0$	$J_0\left(z\right)=0$
$n=1$	$F_1\left(z\right)=1$	$J_1\left(z\right)=1$
$n=2$	$F_2\left(z\right)=z$	$J_2\left(z\right)=1={\mathcal{L}}_0\left(z\right)$
$n=3$	$F_3\left(z\right)=z^2+1$	$J_3\left(z\right)=1+z={\mathcal{L}}_1\left(z\right)$
$n=4$	$F_4\left(z\right)=z^3+2z$	$J_4\left(z\right)=1+2z={\mathcal{L}}_2\left(z\right)$
$n=5$	$F_5\left(z\right)=z^4+3z^2+1$	$J_5\left(z\right)=1+3z+z^2={\mathcal{L}}_3\left(z\right)$
$n=6$	$F_6\left(z\right)=z^5+4z^3+3z$	$J_6\left(z\right)=1+4z+3z^2={\mathcal{L}}_4\left(z\right)$
$n=7$	$F_7\left(z\right)=z^6+5z^4+6z^2+1$	$J_7\left(z\right)=1+5z+6z^2+z^3={\mathcal{L}}_5\left(z\right)$
$n=8$	$F_8\left(z\right)=z^7+6z^5+10z^3+4z$	$J_8\left(z\right)=1+6z+10z^2+4z^3={\mathcal{L}}_6\left(z\right)$

shows the Fibonacci polynomials $F_n$ and Jacobsthal polynomials $J_n$ for $n=0,\dots ,8$.

The characteristic equation corresponding to the recurrence relation (22) is

$${\lambda }^2-\lambda -z=0$$

(24)

having the roots

$${\lambda }_{1,2}\left(z\right)=\frac{1\pm \sqrt{1+4z}}{2}.$$

(25)

The Jacobsthal polynomials can be expressed by the following Binet-type formula [44]:

$$J_n\left(z\right)=\frac{{\left({\lambda }_1\left(z\right)\right)}^n-{\left({\lambda }_2\left(z\right)\right)}^n}{{\lambda }_1\left(z\right)-{\lambda }_2\left(z\right)},$$

(26)

hence,

$${\mathcal{L}}_n\left(z\right)=\frac{{\left({\lambda }_1\left(z\right)\right)}^{n+2}-{\left({\lambda }_2\left(z\right)\right)}^{n+2}}{{\lambda }_1\left(z\right)-{\lambda }_2\left(z\right)},$$

(27)

for every $n\ge 0$.

The Jacobsthal polynomials $J_n\left(z\right)$ are closely related to the Jacobsthal–Lucas polynomials $j_n\left(z\right)$ defined by the same recurrence relation,

$$j_n\left(z\right)=j_{n-1}\left(z\right)+z{j}_{n-2}\left(z\right),n=2,3,\dots ,$$

but with the initial terms $j_0\left(z\right)=2$, $j_0\left(z\right)=1$ [44,45]. These polynomials have the same coefficients as the well-known Lucas polynomials (defined by the same recurrence relation as the Fibonacci polynomials, but with different initial terms), being related by the formula $L_n\left(z\right)=z^n{j}_n\left(\frac{1}{z^2}\right)$ (see

Table 2. Lucas and Jacobsthal–Lucas polynomials.

	Lucas Polynomials	Jacobsthal–Lucas Polynomials
	$L_{n+2}\left(z\right)=z{L}_{n+1}\left(z\right)+L_n\left(z\right)$	$j_{n+2}\left(z\right)=j_{n+1}\left(z\right)+z{j}_n\left(z\right)$
$n=0$	$L_0\left(z\right)=2$	$j_0\left(z\right)=2$
$n=1$	$L_1\left(z\right)=z$	$j_1\left(z\right)=1$
$n=2$	$L_2\left(z\right)=z^2+2$	$j_2\left(z\right)=1+2z={\mathcal{C}}_2\left(z\right)$
$n=3$	$L_3\left(z\right)=z^3+3z$	$j_3\left(z\right)=1+3z={\mathcal{C}}_3\left(z\right)$
$n=4$	$L_4\left(z\right)=z^4+4z^2+2$	$j_4\left(z\right)=1+4z+2z^2={\mathcal{C}}_4\left(z\right)$
$n=5$	$L_5\left(z\right)=z^5+5z^3+5z$	$j_5\left(z\right)=1+5z+5z^2={\mathcal{C}}_5\left(z\right)$
$n=6$	$L_6\left(z\right)=z^6+6z^4+9z^2+2$	$j_6\left(z\right)=1+6z+9z^2+2z^3={\mathcal{C}}_6\left(z\right)$
$n=7$	$L_7\left(z\right)=z^7+7z^5+14z^3+7z$	$j_7\left(z\right)=1+7z+14z^2+7z^3={\mathcal{C}}_7\left(z\right)$
$n=8$	$L_8\left(z\right)=z^8+8z^6+20z^4+16z^2+2$	$j_8\left(z\right)=1+8z+20z^2+16z^3+2z^4={\mathcal{C}}_8\left(z\right)$

).

Now, if we consider the polynomials $\left\{{\mathcal{C}}_n\left(z\right)\right\}$ corresponding to the circular consecutive-2-out-of-n:F systems, $R_C(2,n;p)=p^n{\mathcal{C}}_n\left(z\right)$, we can see that for $k=2$, the relations (7) and (8) become:

$$\begin{array}{cc}\hfill {\mathcal{C}}_n\left(z\right)& ={\mathcal{C}}_{n-1}\left(z\right)+z{\mathcal{C}}_{n-2}\left(z\right),n=4,5,\dots \hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\mathcal{C}}_0\left(z\right)& =1,{\mathcal{C}}_1\left(z\right)=1+z,{\mathcal{C}}_2\left(z\right)=1+2z,{\mathcal{C}}_3\left(z\right)=1+3z,\hfill \end{array}$$

(29)

and we notice that ${\mathcal{C}}_n\left(z\right)=j_n\left(z\right)$, for every $n\ge 2$.

Obviously, the characteristic equation corresponding to the recurrence relation (28) is again (24), having the roots ${\lambda }_1$ and ${\lambda }_2$ expressed by (25), so the polynomial ${\mathcal{C}}_n\left(z\right)$ can be written:

$${\mathcal{C}}_n\left(z\right)={\alpha }_1{\lambda }_1{\left(z\right)}^n+{\alpha }_2{\lambda }_2{\left(z\right)}^n,n=2,3,\dots .$$

Using the initial conditions (29), it can be easily shown ${\alpha }_1={\alpha }_2=1$, so we obtain the following Binet-type formula (the same as the expression of Jacobsthal–Lucas polynomials $j_n\left(z\right)$ [44]):

$${\mathcal{C}}_n\left(z\right)={\left({\lambda }_1\left(z\right)\right)}^n+{\left({\lambda }_2\left(z\right)\right)}^n,$$

(30)

for every $n\ge 2$.

Remark 1.

From Equations (27) and (30), it follows that

$${\mathcal{L}}_{2n-2}\left(z\right)={\mathcal{L}}_{n-2}\left(z\right)·{\mathcal{C}}_n\left(z\right),$$

(31)

and multiplying by $p^{2n-2}$, we obtain the following formula:

$$R_L(2,2n-2;p)=R_L(2,n-2;p)·R_C(2,n;p).$$

(32)

Moreover, the following identity can be proved using (32) and mathematical induction on $r=1,2,\dots$:

$$R_L(2,2^{r}n-2;p)=R_L(2,n-2;p)\prod _{i=0}^{r-1}{R}_C(2,2^{i}n;p).$$

(33)

The roots of the of linear consecutive-2-out-of-n:F systems were studied in [41], and it has been proved that for any $n\ge 2$, the nonzero roots of the reliability polynomial $R_L(2,n;p)$ are real, distinct numbers. The Formula (32) allows us to state the same for circular consecutive-2-out-of-n:F systems. In particular, the next theorem establishes a stronger statement, namely that the closure of the set of roots of the reliability polynomials of circular consecutive-2-out-of-n:F systems is the same as for the linear consecutive case.

Theorem 3.

For any $n\ge 2$, the nonzero roots of the reliability polynomial $R_C(2,n;p)$ of the circular consecutive-2-out-of-n:F system are real, distinct numbers.

Let ${\mathcal{R}}_L$ and ${\mathcal{R}}_C$ denote the set of roots (for all $n\ge 2$) of the polynomials $R_L(2,n;p)$ and $R_C(2,n;p)$, respectively. Then,

$${\mathcal{R}}_C\subset {\mathcal{R}}_L$$

(34)

and

$$\overline{{\mathcal{R}}_C}=\overline{{\mathcal{R}}_L}=(-\infty ,0]\cup \left[\frac{4}{3},\infty \right).$$

(35)

Proof. 

According to (32), it follows that all the roots of the polynomial $R_C(2,n;p)$ are also roots of the polynomial $R_L(2,2n-2;p)$. Hence, ${\mathcal{R}}_C\subset {\mathcal{R}}_L$, and from ([41], Theorem 3), it readily follows that the nonzero roots of $R_C(2,n;p)$ are distinct, real numbers.

In order to prove (35), we need the explicit expression of these roots. From (5), we have that every nonzero root of $R_C(2,n;p)$ corresponds to a root $z\ne -1$ of the polynomial ${\mathcal{C}}_n\left(z\right)$. Using (30), we have:

$$\begin{array}{cc}\hfill {\mathcal{C}}_n\left(z\right)=0& \iff {\left(\frac{{\lambda }_1\left(z\right)}{{\lambda }_2\left(z\right)}\right)}^n=-1\hfill \\ \hfill & \iff \frac{1+\sqrt{1+4z}}{1-\sqrt{1+4z}}=exp\left(i\frac{(2r+1)\pi }{n}\right),r=0,1,\dots ,n-1.\hfill \end{array}$$

It follows that

$$z_{n,r}=-\frac{1}{4{cos}^2\left(\frac{(2r+1)\pi }{2n}\right)},r=0,1,\dots ,⌊\frac{n}{2}⌋-1$$

(36)

are the $⌊\frac{n}{2}⌋$ distinct roots of the polynomial ${\mathcal{C}}_n\left(z\right)$. It can be easily seen that $z_{n,r}\ne -1$ for any r; hence,

$$p_{n,r}^C=\frac{1}{1+z_{n,r}}=1+\frac{1}{2cos\left(\frac{(2r+1)\pi }{n}\right)+1},r=0,1,\dots ,⌊\frac{n}{2}⌋-1$$

(37)

are the distinct nonzero roots of the reliability polynomial $R_C(2,n;p)$. Using (37), it can be seen that $p_{n,r}^C\in (-\infty ,0)\cup \left(\frac{4}{3},\infty \right)$ for every $n\ge 2$ and $r=0,1,\dots ,⌊\frac{n}{2}⌋-1$, and the roots are dense in $(-\infty ,0]\cup \left[\frac{4}{3},\infty \right)$. □

4. Discussion

Finally, we compare the roots of circular consecutive-2-out-of-n:F systems (37) against the ones of linear consecutive-2-out-of-n:F systems (obtained in [41]):

$$p_{n,r}^L=1+\frac{1}{4{cos}^2\left(\frac{r\pi }{n+2}\right)-1}=1+\frac{1}{2cos\left(\frac{2r\pi }{n+2}\right)+1},r=1,2,\dots ,⌊\frac{n+1}{2}⌋,r\ne \frac{n+2}{3}.$$

(38)

We present in Figure 3 the roots of the reliability polynomials of consecutive-2-out-of-n systems, when $n=2,3,\dots ,40$ for both cases: linear systems (a) and circular systems (b).

It can be noticed that, as n increases, the interval between the minimum and the maximum root is growing faster for circular systems than for linear systems. Another remark is that the sequence of intervals can be split into three monotone subsequences: for $n=3t$, $n=3t+1$, and $n=3t+2$ (see Figure 4).

Let us calculate the maximum and the minimum value of $p_{n,k}^L$ and $p_{n,k}^C$, for every $n\ge 2$. The extreme values in (38) are attained when the denominator of the fraction is as close at possible (but not equal) to 0. Hence, for every $n\ge 5$, the maximum value is obtained for $r=⌊\frac{n+1}{3}⌋$, while the minimum value occurs for $r=⌊\frac{n+5}{3}⌋$:

$$1+\frac{1}{2cos\left(\frac{2⌊\frac{n+5}{3}⌋\pi }{n+2}\right)+1}\le p_{n,r}^L\le 1+\frac{1}{2cos\left(\frac{2⌊\frac{n+1}{3}⌋\pi }{n+2}\right)+1}.$$

(39)

Using the same reasoning for circular consecutive-2-out-of-n:F systems (with the remark that in (37) the denominator never vanishes), we find that the maximum root is obtained for $r=⌊\frac{2n-3}{6}⌋$, while the minimum one occurs for $r=⌊\frac{2n+3}{6}⌋$, for any $n\ge 6$:

$$1+\frac{1}{2cos\left(\frac{\left(2⌊\frac{2n+3}{6}⌋+1\right)\pi }{n}\right)+1}\le p_{n,r}^C\le 1+\frac{1}{2cos\left(\frac{\left(2⌊\frac{2n-3}{6}⌋+1\right)\pi }{n}\right)+1}.$$

(40)

5. Conclusions

This paper has investigated the roots of the reliability polynomials of circular consecutive-k-out-of-n:F systems, showing the similarities and the differences between circular and linear consecutive systems (studied in [41]). Using the Beraha–Kahane–Weiss theorem, we have proved that the roots of circular consecutive-k-out-of-n:F systems are unbounded, for any $k\ge 2$.

In the particular case when $k=2$, the reliability polynomials of linear and of circular consecutive-2-out-of-n:F systems are shown to be related to the Jacobsthal and Jacobsthal–Lucas polynomials, respectively. Thus, the relation between the two families of polynomials is revealed to be the same as the relation between Fibonacci and Lucas polynomials: both sequences of polynomials have the same recurrence relation, but different initial values.

Using the explicit expression deduced from the recurrence formula, we have shown that the roots of of circular consecutive-2-out-of-n:F systems share the same properties as the roots of linear consecutive-2-out-of-n:F systems, namely, that for every $n\ge 2$, all the nonzero roots are distinct, real numbers contained into the union of intervals $(-\infty ,0]\cup \left[\frac{4}{3},\infty \right)$.

Moreover, the set of all the roots of circular consecutive-2-out-of-n:F systems is a subset of the set of roots of the linear consecutive systems, and both sets are dense in the union of intervals above. However, the roots of circular systems exhibit a faster spreading along the real axis than the roots of linear consecutive-2-out-of-n:F systems.