Reliability Engineering and System Safety 167 (2017) 362–375

Contents lists available at ScienceDirect

Reliability Engineering and System Safety

journal homepage: www.elsevier.com/locate/ress

Remove and contraction: A novel method for calculating the reliability of

Ethernet ring mesh networks

Fathollah Bistouni a , Mohsen Jahanshahi b , ∗

a Department of Computer Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran b Young Researchers and Elite Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

a r t i c l e i n f o

Keywords:

Ethernet ring mesh networks

Reliability

Topology

Remove and contraction

Failure rate

a b s t r a c t

Reliability is a key parameter in Ethernet ring mesh networks. Therefore, an accurate analysis of reliability in

Ethernet ring mesh including single or multiple rings is a main requirement. Since the calculation of reliability

in such networks is an NP-hard problem, nearly most of the previous methods in this regard, were based on

approximation. In this article, a new method namely, Remove and Contraction (R&C) is presented which is

applicable to Ethernet ring mesh networks for precise calculation of reliability. In addition, the proposed method

is a general method, which can be used for reliability analysis in other network systems as well. The analyses

show the superiority of the R&C method to previous methods in terms of accuracy.

© 2017 Elsevier Ltd. All rights reserved.

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. Introduction

The cost-effectiveness and compliancy of Ethernet-based equipment

ompared to synchronous equipment [1] , makes Ethernet a better op-

ion for service providers to obtain and gather customer traffic [2] ,

s a result carrier-class Ethernet is progressively growing compared to

ONET/SDH networks [3] .

A set of ring nodes constituting a closed loop is called an Ethernet

ing mesh (ERM) by which each node is linked to two immediate nodes

ia duplex ring link. The topology can be a single ring or collection of

nterconnected rings [1,3] . A fundamental necessity for an Ethernet net-

ork is to prevent from any loops. Each loop leads to endless circulation

f data frames, hence flooding the network. The lack of supporting fault

olerance is the main reason for developing and standardizing different

thernet protocols rather than standard Ethernet, among which Ethernet

ing protection (ERP) [3,4] and high-availability seamless redundancy

HSR) [5-9] are the most important ones.

Anyway, regardless of the type of used protocol (ERP or HSR), there

s an inherent and fundamental problem in these networks. This prob-

em is that an Ethernet single-ring mesh network is not fault tolerant

n more than one link. To overcome this problem, multiple-ring mesh

tructures are used. However, based on the placement of rings to each

ther and the number of associated shared links/nodes, different ERM

opologies can be designed. Consequently, network reliability is differ-

nt for each network design. Therefore, in recent years, good endeavors

∗ Corresponding author.

E-mail addresses: fat.bistouni.eng@iauctb.ac.ir (F. Bistouni), mohsenjahanshahi@gmail.com

ttp://dx.doi.org/10.1016/j.ress.2017.06.016

eceived 30 June 2016; Received in revised form 29 May 2017; Accepted 11 June 2017

vailable online 17 June 2017

951-8320/© 2017 Elsevier Ltd. All rights reserved.

ere made to designate the optimum design of ERMs in terms of some

erformance factors, especially, reliability and availability. Significant

fforts undertaken in this subject are elaborated in Section 2 .

As will be discussed in Section 2 , previous works indicate that de-

igning ERMs with high reliability is one of the major concerns of re-

earchers in this field. On the other hand, it demands the correct analy-

is of reliability and availability. However, there exist some remarkable

roblems in the analyses performed in some recent works in terms of re-

iability engineering. Moreover, this challenge cannot be ignored since

ncorrect analysis of reliability has a direct negative impact on the re-

ults obtained in the field of network design optimization. Therefore, the

ontribution of this paper is to introduce a novel method called Remove

nd Contraction (R&C) that analyzes the reliability of the different types

f ring mesh networks exactly, including the single-ring and multiple-

ing. This method will be discussed in Section 3 . Moreover, as will be

xplained in Section 4 , the values obtained for reliability are more accu-

ate than those of in previous works. Another important issue is about

he prevalence of this method so that it can be expanded to other engi-

eering areas. This method can be used as a general method for use in

ny complex network in graph theory to determine reliability.

The rest of this paper is organized as follows: Related works will be

iscussed in Section 2 . The proposed R&C method will be presented in

ection 3 . Reliability analyses will be carried out in Section 4 . Runtime

ill be analyzed in Section 5 . Finally, Section 6 concludes the article.

, mjahanshahi@iauctb.ac.ir (M. Jahanshahi).

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

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. Related works

In this section, we will discuss two types of related works: (1) The

elated works done in the field of Ethernet ring mesh networks. Here,

e are going to explain the advantages of the proposed method in this

mportant area. (2) The related works done in the field of other network

echnologies. Here, we are going to explain some of the other areas of

etwork engineering that the proposed method can improve reliability

nalysis in which. This is because the proposed method can be used as

general method in other engineering fields.

.1. Ethernet ring mesh networks

Lee et al. [4] introduces the issue of designing a reliable network for

aximum network availability in ring mesh networks while following

RP scheme by presenting a heuristic algorithm to find a suboptimal

olution. However, some major problems exist in this research: (i) to

dentify the best topology in terms of availability, the first task is to cal-

ulate the availability of single-ring networks. In this work, it is assumed

hat the availability for each component (including nodes and links)

s identified and network availability is a function of its components

vailability. However, the failure rate and repair rate are attainable in-

ormation on the availability analysis [10] . Furthermore, to analyze the

vailability of a system, constructing and solving Markov model is the

onventional approach. In other words, the analysis performed in this

ork is a time-independent reliability analysis that is different to actual

vailability analysis [11,12] . (ii) The next problem is about the reliabil-

ty of multiple-ring meshes. In [4] , the assumption is that the rings in

multiple-ring network are in series to each other from the reliability

iewpoint. However, often several shared links exist between the rings

13,14] . Hence, the calculation of the reliability of the whole network

s very difficult and it cannot be assumed that these rings are in series

o each other.

Lee et al. [7] similarly to [4] studied the issue of designing a reliable

etwork for maximum network availability in ring mesh networks but

ith regard to the HSR scheme. In terms of reliability and availability,

he design of a reliable Ethernet ring mesh network is independent of

he use of any of the schemes of ERP or HSR. That is why it is usually

ssumed that a ring mesh network failure occurs because of failure of

he links. Therefore, in [7] , the same algorithm as it was in the [4] was

sed. Moreover, the method of calculating availability in this work was

imilar to that of [4] . As a result, it has the same problems in terms of

eliability engineering.

Allawi et al. [8] investigate the problem of design optimization

amely trade-off between redundancy investment and the realization of

he network availability demanded by the customer in HSR-based Eth-

rnet mesh networks. To do so, one of the requirements is the correct

eliability/availability analysis of these networks. In this work, to ana-

yze the availability of the single-ring networks, they employed the same

ethod as in [4] and [7] . In other words, in this case, there are the same

roblems as in [4] and [7] : In fact, the analysis in this work is a time-

ndependent reliability analysis rather than a true availability analysis.

he next point is the reliability analysis for multiple-ring meshes. In this

ase, Allawi et al. [8] uses an estimated upper bound reliability equation

resented in [15] for the calculation of network availability. However,

his strategy also has two problems: Firstly, this equation is suitable for

alculating the reliability not for calculating availability since it can lead

o misleading results. It is an estimation method, and hence is not accu-

ate enough. In another effort, Allawi et al. [6] studied the problem of

inimizing the cost of HSR mesh networks given a network availability

onstraint. However, in this study, an inappropriate method that previ-

usly used in [8] , is used to analyze the availability and hence has the

ame fundamental problems.

According to the above discussions in this section, the analysis car-

ied out in some recent works; especially in [4,6-8] , have some impor-

ant problems in terms of reliability engineering. Clearly, we refer to the

363

ethods used in [4] and [6,8] , which derive lower and upper bounds for

eliability, respectively (i.e. inexact). Moreover, since incorrect analysis

f reliability has a direct negative impact on the results achieved in the

eld of network design optimization, this challenge is important and is

on-negligible. Therefore, the contribution of this paper is to provide a

ovel method named R&C for correct reliability analysis of the different

ypes of ring mesh networks, including the single-ring and multiple-ring

eshes. In addition, we will try to verify the proposed method by us-

ng some well-known reliability analysis techniques. Furthermore, the

roposed technique will be compared with the methods used in [4,6-8] .

s the analyses will show, the proposed method greatly improves the

ccuracy of reliability analysis.

.2. Other network technologies

Another important contribution of this paper is that the proposed

ethod of R&C can also be used to analyze the reliability of other net-

ork systems. In this section, some important related works done in

hese areas will be discussed.

In [32] , an optimal design model is developed using an optimization

ethod called harmony search (HS) to maximize seismic reliability with

imited budgets. The model is applied to actual water supply systems to

etermine pipe diameters that can maximize seismic reliability. Water

upply systems often have a complex structure topology consisting of

anks, pumps, pipelines, and valves. Therefore, the seismic reliability of

water supply network usually analyzed as estimated.

Other research in [33] provides an applied framework to derive the

onnectivity reliability and vulnerability of interurban transportation

ystems under network disruptions. The proposed model integrates sta-

istical reliability analysis to find the reliability and vulnerability of

ransportation networks. This work generally uses a mixture of statisti-

al analysis, probabilistic modeling, and simulation. However, transport

etworks are often very complex networks that require a careful analysis

f the network topology. In such an analysis, reliability equations can be

alculated so that each specific topology has your own exact equations.

sing these equations, a variety of aspects of network performance can

e evaluated.

In [34] , a Monte Carlo simulation approach is proposed for the eval-

ation of network reliability of MANET using the propagation-based link

eliability model and imperfect nodes. Under the assumption that at any

nstant the network topology will be momentarily fixed, the reliability of

he network at that particular instant can be computed. However, Monte

arlo simulation is the most complex, challenging, and time-consuming

imulation to run and is intensive, even for modern computing platforms

10] . On the other hand, in case of more complex structures, the situa-

ion could escalate. In addition, a simulation-based approach does not

uarantee exact reliability values [35] .

In Ref. [35] , the social network reliability under the various in-

uence levels is calculated by exhaustive enumeration for small-size

etworks and Monte Carlo simulation for larger networks. However,

s the network size increases, complete enumeration or minimum

ut/path sets computation is computationally inefficient. In addition,

simulation-based approach like Monte Carlo simulation does not pro-

ide exact reliability values. Also, Monte Carlo simulation is complex

nd intensive to execute even for modern computing platforms [10] .

Ref. [36] includes reliability analysis of hypercube interconnection

etwork and assesses its fault tolerance. The research showed the dif-

erent methods that can be used to enhance the reliability and fault tol-

rance aspects of existing hypercube networks. However, the reliability

nalysis carried out in it, is not general and cannot be used in other

ypes of static interconnection networks such as ring, tree, star, mesh,

nd tours.

According to the discussions in this section, it can be argued that

here is a need for careful reliability analysis in various network sys-

ems. For this purpose, it is necessary to develop general methods that

an be used for exact reliability analysis of network systems. The pro-

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

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osed method in this paper can be utilized in all types of networks that

re definable in graph theory. On the other hand, most of the network

ystems are applicable in graph theory. Therefore, another contribution

f this paper is to provide a new method named R&C that can be used

or exact reliability analysis in a variety of network technologies.

. The proposed method: remove and contraction (R&C)

This section is divided into two sections: Section 3.1 is dedicated to

he expression of the proposed method. Section 3.2 provides a reliabil-

ty analysis of the Ethernet single-ring mesh networks by the proposed

ethod as an example.

.1. Remove and contraction (R&C) method

In this section, the proposed method of R&C will be explained in

etail. This method has two main advantages: (1) R&C can be used for

ccurately determining the reliability of simple as well as complex struc-

ures. (2) R&C has a logical procedure that is easy to use and is based

n two main functions: remove and contraction.

First, it is better to give some assumption of reliability in ERMs that

s common among researchers in this field:

(1) Here, reliability is defined as the probability of successful con-

nection of each node in the network to any other node in it.

(2) It will be assumed that each link in network may fail.

(3) All failures are statistically independent.

(4) The network links have two states: working or failing.

A network topology has two main components: nodes and links. The

asic philosophy of the R&C method is that most of network topologies

re complex for accurate reliability analysis. However, if one or more

inks to be removed from the network, then we can achieve a topology

hat is easy in terms of reliability engineering. This idea has led to an

nnovative method for the exact analysis of complex ring meshes that

ill be explained in more detail below.

The main steps of the R&C are as follows:

Step 1 : If the network structure is easy to analyze, network reliability

s calculated.

Here, it is better that we describe our purpose about a structure that

s easy to analyze. In this paper, it is assumed that a network structure is

asy to analyze, if do not need complex calculations to analyze the reli-

bility. In the other words, the network structure should be in the form

f one of the following simple structures: (a) Open ring: this structure is

n the form of a usual single-ring structure with the exception that one

f links between two specific nodes of the network does not exist. In this

ase, reliability is calculated by considering the fact that the links in the

etwork are in the form of a series system. (b) Usual single-ring: In this

ase, as will be proved in this section, reliability can be calculated ac-

ording to the Eq. (7) . (c) Shared node structure: This is a multiple-ring

tructure that only the common point between the rings that are con-

ected to each other is a shared node. Therefore, this structure can be

een as a series system of several single-rings in terms of reliability. (e)

inally, there are some cases where the reliability of the sub-structure

eeded to calculate the entire network structure is computed in some

revious calculations. As a result, these ready calculations can be used

n analyzing the larger structures.

Step 2 : Otherwise, if the structure is difficult to determine the exact

etwork reliability, follow the steps below:

1. Choose one of the links and remove it from the network (Here, it is

assumed that the link has failed). For this sub-structure, go to Step

1. Then, after calculating the reliability for the sub-structure, go to

stage 2 below:

2. Selected link is contracted (Here, it is assumed that the link is

healthy), so that two nodes related to the link are put on each other

and are converted to one node. For this sub-structure, go to Step 1.

364

Step 3 : Summation of the reliabilities obtained from the above steps

s equal to the entire network reliability.

In fact, network reliability using two functions of deletion and con-

raction is calculated as follows:

𝑅𝑒𝑚𝑜𝑣𝑒 =

(1 − 𝑅 𝑙𝑖𝑛𝑘

)(𝑅 𝑠𝑢𝑏 − 𝑛𝑒𝑡𝑤𝑜𝑟𝑘

)(1)

𝐶𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =

(𝑅 𝑙𝑖𝑛𝑘

)(𝑅 𝑠𝑢𝑏 − 𝑛𝑒𝑡𝑤𝑜𝑟𝑘

)(2)

𝑁𝑒𝑡𝑤𝑜𝑟𝑘 =

(1 − 𝑅 𝑙𝑖𝑛𝑘

)(𝑅 𝑠𝑢𝑏 − 𝑛𝑒𝑡𝑤𝑜𝑟𝑘

)+

(𝑅 𝑙𝑖𝑛𝑘

)(𝑅 𝑠𝑢𝑏 − 𝑛𝑒𝑡𝑤𝑜𝑟𝑘

)(3)

As can be seen, the R&C method can systematically divide the reli-

bility problems into small problems and calculate the reliability of the

ntire network based on them.

For a better understanding of the R&C method, Fig. 1 can be consid-

red. In this figure that is in the form of a tree, the logic of divide and

onquer for the R&C method is quite evident. Summation of reliabilities

btained in the leaves of this tree is the whole network reliability.

In addition, the algorithmic shape of R&C method is shown in Fig. 2 .

.2. Reliability analysis of Ethernet single-ring mesh networks

Now, we use the R&C method for analyzing reliability in Ethernet

ingle-ring mesh networks that can help to better understand the R&C

ethod. Also, we can examine the authenticity of the R&C method by

his analysis.

A single-ring network is a collection of Ethernet nodes that form a

losed circle. In this network, each node is connected to two adjacent

ing nodes through full duplex communication links. Therefore, failure

n more than one link can lead to the failure of the entire network.

First, consider a two-node ring network. For more accurate tracking,

ree of remove and contraction of the R&C method for this network is

hown in Fig. 3 . Before entering into analysis, it should be noted that

eliability analysis can be done in two forms considering operating time:

ime-independent analysis and time-dependent analysis. Because the re-

iability of a product can be affected over time by such random factors

s inherent defects, loss of precision, accidental over-load, environmen-

al corrosion, etc. Therefore, at different times, the system may have

different probability of successfully performing the required function

nder stated conditions [23] .

In this paper, in order to have a more comprehensive analysis, we

ave used both types of analysis. Here, it should be noted that for the

ime-dependent analysis, the component failure characteristics can be

escribed by life distributions. In electronic components (such as Ether-

et components), after the system has been in operation for some time,

any of the parts can be failed and replaced. The system becomes a

ixture of old and new parts, and the failure rate approaches a con-

tant [24] . Therefore, most electronic devices and telecommunication

ystems demonstrate such a constant failure rate during their useful

ifetime [10,25] . For this case, the exponential distribution can be used

26,27] .

Based on the above discussion and given that the Ethernet equip-

ents are electronic components, let us assume that each link in network

ay fail and the failures are assumed to be exponentially distributed.

herefore, if 𝜆 be as the failure rate of a link, then the corresponding

eliability is given by: 𝑅 𝐿 ( 𝑡 ) = 𝑒 − 𝜆𝑡 .

Thus, according to Fig. 3 and Eq. (3) , we have:

𝑡𝑤𝑜 − 𝑛𝑜𝑑𝑒 ( 𝑡 ) = 𝑒 − 𝜆𝑡 (1 − 𝑒 − 𝜆𝑡

)+ 𝑒 − 𝜆𝑡 =

2 𝒆 𝜆𝑡

−

1 𝒆 2 𝜆𝑡 (4)

Now, it is better to investigate the correctness of the Eq. (4) and thus

he authenticity of the R&C method. It should be noted that in two-node

ingle ring, the two links are in parallel in terms of reliability. Therefore,

he reliability of the network is given by:

𝑡𝑤𝑜 − 𝑛𝑜𝑑𝑒 ( 𝑡 ) = 1 −

(1 − 𝑒 − 𝜆𝑡

)2 =

2 𝒆 𝜆𝑡

−

1 𝒆 2 𝜆𝑡 (5)

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 1. Tree of remove and contraction in the R&C method.

Fig. 2. The R&C algorithm.

Fig. 3. Tree of remove and contraction for two-node ring mesh.

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According to Eq. (5) , it is clear that the R&C method achieves accu-

ate results for reliability.

365

This time, consider a three-node ring network. Tree of remove and

ontraction for this network is shown in Fig. 4 . According to this figure,

e have:

𝑡ℎ𝑟𝑒𝑒 − 𝑛𝑜𝑑𝑒 ( 𝑡 ) = 𝑒 −2 𝜆𝑡 (1 − 𝑒 − 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

(𝑒 − 𝜆𝑡

(1 − 𝑒 − 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

)=

3 𝒆 2 𝜆𝑡 −

2 𝒆 3 𝜆𝑡

(6)

According to the Eqs. (4) and (6) , it can be deduced that reliability

or an n -node ring network is calculated as follows:

𝑛 − 𝑛𝑜𝑑𝑒 ( 𝑡 ) =

𝑛

𝒆 ( 𝑛 −1 ) 𝜆𝑡

−

𝑛 − 1 𝒆 𝑛𝜆𝑡

(7)

Moreover, to prove the authenticity of the above equation and thus

&C method, we can use the concept of k -out-of- n system. A k -out-of-

system is successful if any k out of the n components is successful.

herefore, the reliability of this system is the probability that k or more

omponents are successful. Binomial distribution can be used to calcu-

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 4. Tree of remove and contraction for three-node ring mesh.

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ate the reliability of such a system [18] :

𝑘 − 𝑜𝑢𝑡 − 𝑜𝑓− 𝑛 ( 𝑡 ) =

𝑛 ∑𝑖 = 𝑘

(

𝑛

𝑖

) (𝑒 − 𝜆𝑡

)𝑖 (1 − 𝑒 − 𝜆𝑡 )𝑛 − 𝑖

(8)

Now, it should be noted that a ring network is actually a k -out-of- n

ystem with 𝑘 = 𝑛 − 1 . As a result, according to Eq. (5) , we have:

𝑟𝑖𝑛𝑔 𝑛𝑒𝑡𝑤𝑜𝑟𝑘 ( 𝑡 ) =

𝑛 ∑𝑖 = 𝑛 −1

(

𝑛

𝑖

) (𝑒 − 𝜆𝑡

)𝑖 (1 − 𝑒 − 𝜆𝑡 )𝑛 − 𝑖

=

(

𝑛

𝑛 − 1

) (𝑒 − 𝜆𝑡

)𝑛 −1 (1 − 𝑒 − 𝜆𝑡 )+

(

𝑛

𝑛

) (𝑒 − 𝜆𝑡

)𝑛 (1 − 𝑒 − 𝜆𝑡 )𝑛 − 𝑛

= 𝑛 (𝑒 − 𝜆𝑡

)𝑛 −1 (1 − 𝑒 − 𝜆𝑡 )+

(𝑒 − 𝜆𝑡

)𝑛 = 𝑛

(𝑒 − 𝜆𝑡

)𝑛 −1 − 𝑛 (𝑒 − 𝜆𝑡

)𝑛 +

(𝑒 − 𝜆𝑡

)𝑛 =

𝑛

𝒆 ( 𝑛 −1 ) 𝜆𝑡

−

𝑛 − 1 𝒆 𝑛𝜆𝑡

(9)

According to Eqs. (7) and (9) , the accuracy of R&C method is proved.

Here, it should be noted that single-ring network that was analyzed

n this section is simplest structure in ring networks. However, as we

hall see in the next section, the R&C method is also suitable for ana-

yzing the multiple-ring networks that are complex in terms of reliabil-

ty analysis. In the next section, it will be demonstrated that the R&C

ethod achieves exact results and has high accuracy compared to meth-

ds used in the previous works.

. Reliability analysis

In this section, at first (in Section 4.1 ), we will analyze some sam-

les of multiple-ring meshes by R&C method. Then, the results will be

ompared with results obtained in previous works. In Section 4.2 , we

ill compare the different types of multiple-ring topologies in terms of

eliability, by R&C method. This analysis can help to determine the best

opologies for Ethernet ring meshes in terms of reliability. Finally, in

ection 4.3 , we analyze the most well-known complex network refer-

nce model namely ARPA2, as a practical instance.

.1. Comparison with previous methods

In this section, we will compare the proposed method with the meth-

ds used in previous works of [4,6-8] . An exact reliability analysis of

ultiple-ring networks is difficult. To fix the problem, previous works

4,6-8] used an estimation method. First, it is better that we do a tech-

ical review of previous works:

.1.1. Previous research conducted in [4]

Lee et al. [4] offered the following equation to calculate the avail-

bility:

𝐸𝑅𝑃 =

∏𝑅 ∈𝑀

𝐴 𝑟𝑖𝑛𝑔 𝑅 (10)

366

In [4] , it is supposed that the availability for each component (links)

s specified and network availability is a function of its components

vailability. However, the failure rate and repair rate are accessible in-

ormation on the availability analysis [10] . Furthermore, to do availabil-

ty analysis of a system, the typical approach is to construct and solve

Markov model. In other words, the analysis introduced in this work

s a time-independent analysis of reliability [11,12] instead of an actual

nalysis of availability.

In the Eq. (10) , R is a set of links that forms a ring. Also, R is a

ubset of the total links. For different rings i and j , 𝑅 𝑖 ∩ 𝑅 𝑗 = ∅. M is a

et of rings that forms an Ethernet ring mesh that includes all nodes, R

M. A ring R is the availability of a single Ethernet network. Eq. (10) is a

ethod to estimate the reliability. Because it assumes a multiple-ring as

network of independent rings that have no common link among each

ther.

.1.2. Previous research conducted in [ 6,8 ]

In another instance, [6] and [8] used the following equation to cal-

ulate the upper bound of availability, which is actually a method to

stimate the reliability of multiple-ring networks [15] :

𝐻𝑆𝑅 ≤ 1 −

⎡ ⎢ ⎢ ⎢ ⎣ 𝑁 ∑𝑖 =1

⎛ ⎜ ⎜ ⎜ ⎝ ∏𝑙 𝑘,𝑖 ∈𝐸 𝑖

(1 − 𝐴 𝑙 𝑘,𝑖

) 𝑖 −1 ∏𝑗=1

⎛ ⎜ ⎜ ⎜ ⎝ 1 −

∏𝑙 𝑘,𝑗 ∈𝐸 𝑗

(1 − 𝐴 𝑙 𝑘,𝑗

)(1 − 𝐴 𝑙 𝑖,𝑗

) ⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ (11)

In Eq. (10) , l k, j is an undirected link connecting nodes k and i, E i is

he set of links incident to node i , and 𝐴 𝑙 𝑘,𝑖 is the availability of link l k, j .

n this case, an estimated upper bound reliability equation presented in

15] is used for the calculation of network availability. However, this

trategy has two problems: Firstly, this equation is suitable for calculat-

ng the reliability. On the other hand, it is an estimation method, and

ence is not accurate enough.

.1.3. New analyses by the R&C method

In order to compare the R&C method with the above methods, let

s consider the multiple-ring mesh topologies (1) and (2) in Fig. 5 (b)

elated to physical network topology in Fig. 5 (a).

First, consider the topology (1) . For this topology, tree of remove

nd contraction in the R&C method is shown in Fig. 6 . According to this

gure, we have:

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 1 ) ( 𝑡 ) =

(−

1 𝒆 𝑡𝜆

+ 1 )(

8 𝒆 7 𝑡𝜆 −

7 𝒆 8 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

( (

3 𝒆 2 𝑡𝜆 −

2 𝒆 3 𝑡𝜆

) (

5 𝒆 4 𝑡𝜆 −

4 𝒆 5 𝑡𝜆

) )

=

15 𝒆 9 𝑡𝜆 +

23 𝒆 7 𝑡𝜆 −

37 𝒆 8 𝑡𝜆 (12)

This time, consider the topology (2) . Remove and contraction dia-

ram for this topology is shown in Fig. 7 . According to this figure, the

eliability of topology (2) is calculated as follows:

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 2 ) ( 𝑡 ) =

(−

1 𝒆 𝑡𝜆

+ 1 )(

15 𝒆 9 𝑡𝜆 +

23 𝒆 7 𝑡𝜆 −

37 𝒆 8 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

( (−

1 𝒆 𝑡𝜆

+ 1 )(

3 𝒆 2 𝑡𝜆 −

2 𝒆 3 𝑡𝜆

) (

5 𝒆 4 𝑡𝜆 −

4 𝒆 5 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

(

3 𝒆 2 𝑡𝜆 −

2 𝒆 3 𝑡𝜆

) 2 (

2 𝒆 𝑡𝜆

−

1 𝒆 2 𝑡𝜆

)

)

=

56 𝒆 7 𝑡𝜆 +

102 𝒆 9 𝑡𝜆 −

27 𝒆 10 𝑡𝜆 −

130 𝒆 8 𝑡𝜆 (13)

Similarly to pervious works [4,6,8] , let us assume that 𝑅 𝑙𝑖𝑛𝑘 ( 𝑡 ) = . 999 . Therefore, according to Eqs. (12) and (13) , results of the relia-

ility analysis are summarized in Table 1 . This table compares the re-

ults obtained by the proposed R&C method with the results obtained

y previous estimation-based methods in [4,6,8] .

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 5. (a) Physical topology, (b) Ring network examples.

Fig. 6. Remove and contraction diagram for topology (1) .

Fig. 7. Remove and contraction diagram for topology (2) .

Table 1

Comparison of various methods for reliability analysis.

Topology (1) (2)

No. of links 9 10

No. of nodes 8 8

No. of rings 2 3

Reliability using R&C method 0.999987 0.999993

Reliability using Eq. (9) [ 4 ] 0.999984 0.999988

Reliability using Eq. (10) [ 6 , 8 ] 0.999992 0.999998

v

t

i

a

q

b

[

w

s

i

r

s

i

s

o

h

i

p

n

According to the Table 1 , results by method in [4] are far from the

alues obtained by the exact R&C method in this paper. This is because

he method presented in [4] actually offers a lower bound of reliabil-

ty, which is always less than the actual value of network reliability. In

ddition, the reliability values obtained by method in [6] and [8] ac-

367

uire the values far from the reality as this method provides an upper

ound for network reliability [15] . In other words, both approaches in

4,6] , and [8] provide a non-exact value for the reliability of these net-

orks, which is far from the reality. Therefore, these methods are not

uitable for reliability-based design optimization problems. Generally,

n network design optimization, the aim is to improve reliability to a

equired level that can be achieved by fault tolerance or fault avoidance

trategies. Another point is that effective and precise methods for analyz-

ng reliability are needed to better understand the reliability of network

ystems under faulty situations to give suitable emergency management

perations. That is why researchers in the field of reliability engineering

ave always been looking for accurate methods for determining reliabil-

ty [11,15-19] . Therefore, it is evident that R&C method proposed in this

aper is more preferable to the pervious methods in [4,6] , and [8] for

etwork design optimization.

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 8. (a) A shared link topology, (b) A shared node topology, (c) A complex shared link

topology, (d) A redundant link topology, (e) 3-connected network.

Fig. 9. Remove and contraction diagram for topology (a).

4

n

b

t

o

p

c

a

𝑅

n

r

𝑅

l

𝑅

l

𝑅

c

l

𝑅

E

𝑅

E

f

s

e

o

o

m

b

w

m

p

b

b

a

l

r

c

u

s

i

f

b

i

.2. New reliability analyses of different topologies

In this section, the different types of topologies in Ethernet ring mesh

etworks will be analyzed in terms of reliability by R&C method. To the

est of our knowledge, this is the first exact analysis of the reliability in

hese networks.

In general, five types of topologies can be identified based on previ-

us works: (1) shared link topology, (2) shared link topology, (3) com-

lex shared link topology, (4) redundant link topology, and (5) 3-

onnected network. These topologies are shown in Fig. 8 .

For topology (a) (shared link topology), remove and contraction di-

gram is shown in Fig. 9 . Based on this diagram, we have:

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 𝑎 ) ( 𝑡 ) =

(−

1 𝒆 𝑡𝜆

+ 1 )(

6 𝒆 5 𝑡𝜆 −

5 𝒆 6 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

(

3 𝒆 2 𝑡𝜆 −

2 𝒆 3 𝑡𝜆

) 2

=

9 𝒆 7 𝑡𝜆 +

15 𝒆 5 𝑡𝜆 −

23 𝒆 6 𝑡𝜆 (14)

Now, consider the topology (b), shared node topology. It should be

oted that this topology can be seen as a series system of two four-node

ing. Therefore, according to Eq. (7) , network reliability is given by:

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 𝑏 ) ( 𝑡 ) =

(

4 𝒆 3 𝑡𝜆 −

3 𝒆 4 𝑡𝜆

) 2 =

16 𝒆 6 𝑡𝜆 +

9 𝒆 8 𝑡𝜆 −

24 𝒆 7 𝑡𝜆 (15)

Remove and contraction diagram for topology (c) (complex shared

ink topology), is shown in Fig. 10 . Based on this diagram, we have:

368

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 𝑐 ) ( 𝑡 ) =

(−

1 𝒆 𝑡𝜆

+ 1 )(

1 𝒆 2 𝑡𝜆

(

16 𝒆 6 𝑡𝜆 +

9 𝒆 8 𝑡𝜆 −

24 𝒆 7 𝑡𝜆

) )

+

1 𝒆 𝑡𝜆

( (−

1 𝒆 𝑡𝜆

+ 1 )(

1 𝒆 𝑡𝜆

(

16 𝒆 6 𝑡𝜆 +

9 𝒆 8 𝑡𝜆 −

24 𝒆 7 𝑡𝜆

) )

+

1 𝒆 𝑡𝜆

( (−

1 𝒆 𝑡𝜆

+ 1 )(

16 𝒆 6 𝑡𝜆 +

9 𝒆 8 𝑡𝜆 −

24 𝒆 7 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

( (−

1 𝒆 𝑡𝜆

+ 1 )( (

2 𝒆 𝑡𝜆

−

1 𝒆 2 𝑡𝜆

) (

4 𝒆 3 𝑡𝜆 −

3 𝒆 4 𝑡𝜆

) )

+

1 𝒆 𝑡𝜆

( (−

1 𝒆 𝑡𝜆

+ 1 )( (

−

1 𝒆 𝑡𝜆

+ 1 )(

4 𝒆 3 𝑡𝜆 −

3 𝒆 4 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

(

2 𝒆 𝑡𝜆

−

1 𝒆 2 𝑡𝜆

) 2 )

+

1 𝒆 𝑡𝜆

(

2 𝒆 𝑡𝜆

−

1 𝒆 2 𝑡𝜆

) 2 ) ) ) )

=

7 𝒆 8 𝑡𝜆 +

20 𝒆 7 𝑡𝜆 +

92 𝒆 10 𝑡𝜆 −

27 𝒆 11 𝑡𝜆 −

91 𝒆 9 𝑡𝜆 (16)

Also, according to the Fig. 11 , reliability of topology (d) (redundant

ink topology) is calculated by R&C as Eq. (17) .

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 ( 𝑑 ) ( 𝑡 ) =

(−

1 𝒆 𝑡𝜆

+ 1 )(

9 𝒆 7 𝑡𝜆 +

15 𝒆 5 𝑡𝜆 −

23 𝒆 6 𝑡𝜆

)

+

1 𝒆 𝑡𝜆

(

3 𝒆 2 𝑡𝜆 −

2 𝒆 3 𝑡𝜆

) 2 =

24 𝒆 5 𝑡𝜆 +

36 𝒆 7 𝑡𝜆 −

9 𝒆 8 𝑡𝜆 −

50 𝒆 6 𝑡𝜆 (17)

Finally, the remove and contraction diagram of topology (e) (3-

onnected network) is shown in Fig. 12 . Based on this figure, the re-

iability of 3-connected network is given by:

3− 𝑐 𝑜𝑛𝑛𝑒𝑐 𝑡𝑒𝑑 ( 𝑡 ) = ( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 1 ) + 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 2 )

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 3 ) + 𝑒 − 𝜆𝑡 ( 𝐸 4 ))))

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 2 ) + 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 3 )

+ 𝑒 − 𝜆𝑡 ( 𝐸 4 ))) + 𝑒 − 𝜆𝑡 ( 𝐸 5 )) (18)

In Eq. (18) , each of the equations E 1 through E 5 is calculated as

qs. (24) through (28) in the Appendix.

Now, substituting Eqs. (24) through (28) into Eq. (18) , we get:

3− 𝑐 𝑜𝑛𝑛𝑒𝑐 𝑡𝑒𝑑 ( 𝑡 ) = 376 𝑒 −7 𝜆𝑡 − 1476 𝑒 −8 𝜆𝑡 + 2352 𝑒 −9 𝜆𝑡 − 1895 𝑒 −10 𝜆𝑡

+ 770 𝑒 −11 𝜆𝑡 − 126 𝑒 −12 𝜆𝑡 (19)

Let us assume that 𝜆 is 10 −6 per hour. In this case, according to

qs. (14) , (15) , (16) , (17) , and (19) , the results of reliability analysis

or five topologies (a), (b), (c), (d), and (e), as a function of time, are

hown in Fig. 13 .

According to Fig. 13 , it is evident that the highest reliability and low-

st reliability belong to topology (e) (3-connected network) and topol-

gy (b) (shared node topology), respectively. In addition, three networks

f shared link, redundant link, and complex shared link have a perfor-

ance close to each other in terms of reliability. However, the relia-

ility of complex shared link topology is better than the other two net-

orks. These results reflect the fact that the sharing link in Ethernet ring

esh networks is not a good idea for the efficient design from the stand-

oint of reliability. As a result, as much as possible, such cases should

e avoided in the design. In addition, according to the results obtained

y the topologies of 3-connected and complex shared link, it is infer-

ble that the designing networks by numerous and sophisticated shared

inks can be a good idea in terms of reliability. Also, in this case, more

egular structures (e.g. 3-connected network) can have a higher priority

ompared to irregular structures (e.g. complex shared link topology).

In addition, time-independent analysis of reliability can also provide

seful information about the reliability of a system. For this case, Fig. 14

hows reliability ( R ) of various networks as a function of the link reliabil-

ty ( r ). In this figure, network reliability is shown for all possible values

or the link reliability ( r ∈ [0, 1]). As the figure shows, when link relia-

ility is low (between zero to 0.25), every reliability of all five network

s very close together. In other words, weak link reliability can affect

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 10. Remove and contraction diagram for topology (c).

Fig. 11. Remove and contraction diagram for topology (d).

t

t

r

h

4

i

R

a

c

i

e

i

b

i

a

𝜆

he reliability of each network topology. However, the differences be-

ween network reliabilities become more apparent at the time that link

eliability increases. As can be seen, 3-connected network can achieve

igher reliability compared to other networks.

Fig. 12. Remove and contraction

369

.3. Practical instance

In this section, as practical instance, we examine the exact reliabil-

ty of well-known complex network reference model namely ARPA2 by

&C method. To the best of our knowledge, this is the first exact reli-

bility analysis of the ARPA2 network, capable of leading to the exact

alculation of reliability equations.

ARPA2 network originally forms a ring mesh topology. The network

s shown in Fig. 15 .

To facilitate the calculations, links that are in tandem can be consid-

red as one link. In this case, failure rate for the new link (which this link

s representative of the several links in tandem) ( 𝜆NL ) can be calculated

y Eq. (20) [20-22] . This equation is calculated based on failure rates of

ndividual links ( 𝜆) and considering the fact that the consecutive links

re as a series system in the viewpoint of reliability.

𝑁𝐿 =

(

−

1 𝑅 ( 𝑡 )

)

𝑑

𝑑𝑡

(𝑅 𝑙𝑖𝑛𝑘𝑠 ( 𝑡 )

)(20)

𝑙𝑖𝑛𝑘𝑠

diagram for topology (e).

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 13. Reliability as a function of time.

Fig. 14. Reliability as a function of link reliability.

S

𝜆

o

𝑅

E

w

In the above equation, the R links ( t ) is the reliability of the serial links.

o, if we label the number of links with L , then we have:

𝑁𝐿 =

(−

1 𝑒 − 𝐿𝜆𝑡

)𝑑

𝑑𝑡

(𝑒 − 𝐿𝜆𝑡

)= 𝐿𝜆 (21)

Based on the above discussions, the remove and contraction diagram

f ARPA2 network is shown in Fig. 16 .

According to Fig. 16 , the reliability of ARPA2 is given by:

𝐴𝑅𝑃𝐴 2 ( 𝑡 ) = ( 1 − 𝑒 −3 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 1 )

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 2 ) + 𝑒 − 𝜆𝑡 ( 𝐸 3 )))

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 −2 𝜆𝑡 )( ( 1 − 𝑒 −2 𝜆𝑡 )( 𝐸 ) + 𝑒 −2 𝜆𝑡 ( 𝐸 ) )

4 5

370

+ 𝑒 −2 𝜆𝑡 ( ( 1 − 𝑒 −2 𝜆𝑡 )( 𝐸 6 ) + 𝑒 −2 𝜆𝑡 ( 𝐸 7 ) )))

+ 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 8 ) + ( 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )( 𝐸 9 )

+ 𝑒 −2 𝜆𝑡 ( 𝐸 10 ))) + 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 11 ) + 𝑒 − 𝜆𝑡 ( 𝐸 12 ) )))

+ 𝑒 −3 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( ( 1 − 𝑒 −2 𝜆𝑡 )( 𝐸 13 ) + 𝑒 −2 𝜆𝑡 ( 𝐸 14 ) )

+ 𝑒 −2 𝜆𝑡 ( ( 1 − 𝑒 −2 𝜆𝑡 )( ( 1 − 𝑒 −2 𝜆𝑡 )( 𝐸 15 ) + 𝑒 −2 𝜆𝑡 ( 𝐸 16 ) ) + 𝑒 −2 𝜆𝑡 ( 𝐸 17 ) ))

(22)

In Eq. (22) , each of the equations E 1 through E 17 is calculated as

qs. (39) through (55) in the Appendix.

Therefore, by replacing the Eqs. (39) through (55) in the Eq. (22) ,

e have:

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 15. Ethernet ring network example of well-known network reference model: ARPA2.

𝑅

+

3𝒆 17

𝒆 2 𝑡

a

r

i

o

v

i

r

r

t

f

fi

r

0

b

t

i

l

i

F

n

5

c

a

a

r

r

o

a

t

b

t

t

c

c

b

r

a

i

p

d

c

l

p

a

O

h

o

6

t

T

𝐴𝑅𝑃𝐴 2 ( 𝑡 ) =

(

−

1 𝒆 3 𝑡𝜆 + 1

)

⎛ ⎜ ⎜ ⎜ ⎝ (−

1 𝒆 𝑡𝜆+ 1

)(( 𝐴 ) − ( 𝐵 ) +

1 𝒆 7 𝑡𝜆 +

1 𝒆 8 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 +

2 𝒆 11 𝑡𝜆

+

( 𝐸 ) (−

1 𝒆 𝑡𝜆+ 1

)+ ( 𝐹 )

𝒆 5 𝑡𝜆

In Eq. (23) , each of the equations A, B, C, D, E, and F is calculated

s Eqs. (56) through (61) in the Appendix.

Therefore, according to the Eqs. (23) through (61) , the results of

eliability for different failure rates ( 𝜆) as a function of time, are shown

n Fig. 17 . As the figure shows, as operating time increase, the reliability

f ARPA2 network decreases. Moreover, the failure rate of links is a

ery influential factor on network reliability. According to the figure,

ncreasing link failure rate can lead to a significant reduction in network

eliability. Therefore, one of the fault-avoidance strategies to improve

eliability is to increase the quality of the links in the network in order

o reduce their failure rate.

In addition, in order to analyze the reliability of the network as a

unction of the reliability of the link, Fig. 18 can be considered. In this

gure, network reliability is shown for all possible values for the link

eliability ( r ∈ [0, 1]). This shows that for low link reliability (between

and 0.5), network reliability is very low and close to zero. This could

e because there are many links in series between neighbor nodes, in

he ARPA2 network. On the other hand, the increasing number of links

n series, increase the probability of failure and reduces the network re-

iability. Therefore, a method to improve the reliability of the network

s trying to reduce the number of links or increase link reliability. As

ig. 18 shows, improving the reliability of the link could lead to a sig-

ificant improvement in the reliability of the entire network.

Fig. 16. Remove and contraction d

371

𝑡𝜆+

7 𝒆 18 𝑡𝜆 −

1 𝒆 13 𝑡𝜆 −

2 𝒆 16 𝑡𝜆 −

3 𝒆 14 𝑡𝜆 −

3 𝒆 20 𝑡𝜆 −

5 𝒆 15 𝑡𝜆

)+ ( 𝐶 )

𝜆+

(

−

1 𝒆 2 𝑡𝜆 + 1

)

( 𝐷 ) ⎞ ⎟ ⎟ ⎟ ⎠

(23)

. Runtime analysis

One other concern in the reliability analysis of complex networks

an be about time required for analysis. Therefore, in this section, we

nalyze runtime of the proposed R&C method. In this analysis, we will

ssume that runtime is due to the following three items: (1) The time

equired to enumerate samples. (2) The time needed to calculate the

eliability of samples. (3) The time required to calculate the reliability

f the entire system.

In addition, the MATLAB 9.0 platform is used for programming these

lgorithms. The results of this analysis are summarized in Table 2 . In this

able, a variety of topologies discussed in the previous sections have

een investigated for three different methods: R&C method proposed in

his paper, the method used in [4] , and the method used in [6,8] .

As Table 2 shows, the proposed R&C method can operate better than

he other two methods in terms of runtime. The reason for this event

an be searched in the first step of the proposed method Section 3 ). As

an be seen in the first step of the proposed method, four items have

een included to reduce the enumerated samples as follows: )a) Open

ing (b) Usual single-ring (c) Shared node structure (e) Finally, there

re some cases where the reliability of the sub-structure is computed

n some previous calculations. In the event of any of these four items,

rocess of remove and contraction stops and thus runtime extraordinary

ecreases. For example, consider the topology ( (2) of Fig. 5 (b). In this

ase, the number of cases investigated by method presented in [4] at

east will be equal to seven (shown in Fig. 19 ). However, the method

roposed in this paper examines only four cases (as shown in Fig. 7 ). In

ddition, the computational complexity of the method in [6] is equal to

( N

3 ), where N is the number of nodes in the network. Therefore, the

igh complexity leads to increased runtime, especially when the number

f nodes increases in the network.

. Conclusion and future works

This paper provides a new method named R&C (remove and contrac-

ion) for the exact analysis of reliability in Ethernet ring mesh networks.

his method is suitable for reliability analysis in a variety of topologies,

iagram for ARPA2 network.

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Fig. 17. Time-dependent reliability for ARPA2 network.

Fig. 18. Time-independent reliability for ARPA2 network.

e

R

fi

t

s

S

a

A

s

s

c

b

t

s

r

s

s

a

p

c

I

ven for complex topologies. In addition, the analyses demonstrated that

&C method achieves exact results in comparison with previous works.

An important debate is about computational complexity: ( 1 ) In the

rst step of the proposed method, in case of four types of network struc-

ures, process of remove and contraction stopped and reliability of the

tructure directly be calculated: (a) Open ring (b) Usual single-ring (c)

hared node structure (e) Finally, there are some cases where the reli-

bility of the sub-structure is computed in some previous calculations.

s a result, these ready calculations can be used in analyzing the larger

tructures. These four items can extremely reduce the number of counted

amples. Since all four conditions often occur in all structures, the worst

372

ase will not happen. (2) In previous works, a case that prevented from

eing a precise method for calculating the reliability of network sys-

ems was the high complexity of the calculations, even for small network

izes [4,6,8,11,28-31] . However, this paper proposes a method that can

educe the complexity of network structure by converting it to simple

tructures that are easier for exact analysis of reliability. From this per-

pective, the proposed method reduces the computational complexity by

systematic procedure that is simple. (3) The next argument is that the

roposed method can perform better than previous methods in terms of

omputational complexity. First, consider the method proposed in [4] .

n this work, to calculate the reliability of a network, we need to iden-

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

Table 2

Comparison of various methods in terms of runtime (s).

Topology (1) (2) (a) (b) (c) (d) (e)

Method R&C 0 .01053 0 .02142 0 .01032 0 .0081 0 .03427 0 .01264 0 .05997

[ 4 ] 0 .01474 0 .02613 0 .01238 0 .00972 0 .04249 0 .01643 0 .08996

[ 6 ] 0 .02211 0 .02653 0 .01609 0 .01069 0 .03714 0 .01807 0 .06314

Fig. 19. The process of enumerating the sub rings by the method in [5] for topology (2) .

t

m

i

r

f

t

w

a

[

t

t

i

a

g

r

i

t

s

A

𝐸

𝐸

𝐸

𝐸 )

)

𝐸

c

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

ify the major rings and sub rings. On the other hand, choosing different

odes of the major rings and sub rings will lead to different reliabil-

ties. Therefore, in [4] , a case is considered which could bring greater

eliability to the network. As a result, this method should enumerate dif-

erent modes for network reliability analysis. For example, consider the

opology (2) of Fig. 5 (b). In this case, the number of cases investigated

ill be equal to seven. However, the method proposed in this paper ex-

mines only four cases (shown in Fig. 7 ). In addition, the method in

6] also has a high computational complexity. This complexity is equal

o O( N

3 ), where N is the number of nodes on the network. Therefore, as

he number of nodes increases, the computational complexity can also

ncrease.

Some of the issues can be considered as future works: a debate is

bout the formulation of reliability equations by R&C in certain topolo-

ies that grow uniformly. Another issue is to use R&C to analyze the

eliability of the other types of networks and graphs in other engineer-

ng fields. In addition, some future works could be focused to improve

he R&C method. In this regard, some ideas can be considered to find

trategies that help to increase the efficiency of this method.

ppendix

In Eq. (18) , equations E 1 through E 5 are calculated as follows:

1 =

(1 − 𝑒 − 𝜆𝑡

)(𝑒 −2 𝜆𝑡

(15 𝑒 −5 𝜆𝑡 − 23 𝑒 −6 𝜆𝑡 + 9 𝑒 −7 𝜆𝑡

))+ 𝑒 − 𝜆𝑡

(𝑒 − 𝜆𝑡

(1 − 𝑒 − 𝜆𝑡

)(15 𝑒 −5 𝜆𝑡 − 23 𝑒 −6 𝜆𝑡 + 9 𝑒 −7 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

((1 − 𝑒 − 𝜆𝑡

)(15 𝑒 −5 𝜆𝑡 − 23 𝑒 −6 𝜆𝑡 + 9 𝑒 −7 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

(𝑒 − 𝜆𝑡

(1 − 𝑒 − 𝜆𝑡

)(4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

((1 − 𝑒 − 𝜆𝑡

)(4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

(3 𝑒 −2 𝜆𝑡 − 2 𝑒 −3 𝜆𝑡

)))))= 56 𝑒 −7 𝜆𝑡 − 130 𝑒 −8 𝜆𝑡 + 102 𝑒 −9 𝜆𝑡 − 27 𝑒 −10 𝜆𝑡

(24)

2 =

(1 − 𝑒 − 𝜆𝑡

)(𝑒 −2 𝜆𝑡

(𝐸 6

))+ 𝑒 − 𝜆𝑡

(𝑒 − 𝜆𝑡

(1 − 𝑒 − 𝜆𝑡

)(𝐸 6

)+ 𝑒 − 𝜆𝑡

((1 − 𝑒 − 𝜆𝑡

)(𝐸 6

)+ 𝑒 − 𝜆𝑡

((1 − 𝑒 − 𝜆𝑡

)(4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

(3 𝑒 −2 𝜆𝑡 − 2 𝑒 −3 𝜆𝑡

))))(25)

3 = ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 −2 𝜆𝑡 ( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) + 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 )

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 𝐸 7 ) ) + 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 𝐸 7 ) ))))

+ 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) + 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 𝐸 7 ) ) )

+ 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( ( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) + 𝑒 − 𝜆𝑡 ( 𝐸 7 ) ) + 𝑒 − 𝜆𝑡 ( 𝐸 7 ) ) )

(26)

4 = ( 1 − 𝑒 − 𝜆𝑡 )( ( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 8 ) + 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) + 𝑒 − 𝜆𝑡 ( 𝐸 7 ) )

+ 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 9 ) + 𝑒 − 𝜆𝑡 ( 𝐸 7 )) (27

373

5 = ( 1 − 𝑒 − 𝜆𝑡 )( ( 1 − 𝑒 − 𝜆𝑡 )( 𝑅 𝑟𝑒𝑑 𝑢𝑛𝑑 𝑎𝑛𝑡 𝑙𝑖𝑛𝑘 ( 𝑡 ) ) + 𝑒 − 𝜆𝑡 ( 𝐸 4 ) )

+ 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( 𝐸 4 ) + 𝑒 − 𝜆𝑡 ( 𝐸 10 ) ) (28)

Also, in Eqs. (25) , (26) , (27) , and (28) , equations E 6 through E 10 are

alculated as follows:

6 = ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) ) + 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 )

+ 𝑒 − 𝜆𝑡 ( 3 𝑒 −2 𝜆𝑡 − 2 𝑒 −3 𝜆𝑡 ))

= 11 𝑒 −4 𝜆𝑡 − 16 𝑒 −5 𝜆𝑡 + 6 𝑒 −6 𝜆𝑡 (29)

7 = ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) ) + 𝑒 − 𝜆𝑡 (( 1 − 𝑒 − 𝜆𝑡 )( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 )

+ 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 )) = 4 𝑒 −2 𝜆𝑡 − 4 𝑒 −3 𝜆𝑡 + 𝑒 −4 𝜆𝑡 (30)

8 = ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 ) )

+ 𝑒 − 𝜆𝑡 ( ( 1 − 𝑒 − 𝜆𝑡 )( 4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡 )

+ 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 1 − 𝑒 − 𝜆𝑡 )( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) + 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) ) )

= 12 𝑒 −4 𝜆𝑡 − 18 𝑒 −5 𝜆𝑡 + 7 𝑒 −6 𝜆𝑡 (31)

9 =

(1 − 𝑒 − 𝜆𝑡

)(4 𝑒 −3 𝜆𝑡 − 3 𝑒 −4 𝜆𝑡

)+ 𝑒 − 𝜆𝑡

(4 𝑒 −2 𝜆𝑡 − 4 𝑒 −3 𝜆𝑡 + 𝑒 −4 𝜆𝑡

)= 8 𝑒 −3 𝜆𝑡 − 11 𝑒 −4 𝜆𝑡 + 4 𝑒 −5 𝜆𝑡 (32)

10 = 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 4 ) + ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 4 )

+ ( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) ) + 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) )))

+ ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 4 ) + ( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 4 )

+ ( 1 − 𝑒 − 𝜆𝑡 )(( 1 − 𝑒 − 𝜆𝑡 )( 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 ) ) + 𝑒 − 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 ) 2 )))))

= 24 𝑒 −3 𝜆𝑡 − 68 𝑒 −4 𝜆𝑡 + 82 𝑒 −5 𝜆𝑡 − 51 𝑒 −6 𝜆𝑡 + 16 𝑒 −7 𝜆𝑡 − 2 𝑒 −8 𝜆𝑡 (33)

According to Eqs. (24) through (33) , we have:

1 = 56 𝑒 −7 𝜆𝑡 − 130 𝑒 −8 𝜆𝑡 + 102 𝑒 −9 𝜆𝑡 − 27 𝑒 −10 𝜆𝑡 (34)

2 = 40 𝑒 −6 𝜆𝑡 − 90 𝑒 −7 𝜆𝑡 + 69 𝑒 −8 𝜆𝑡 − 18 𝑒 −9 𝜆𝑡 (35)

3 = 32 𝑒 −5 𝜆𝑡 − 68 𝑒 −6 𝜆𝑡 + 49 𝑒 −7 𝜆𝑡 − 12 𝑒 −8 𝜆𝑡 (36)

4 = 32 𝑒 −4 𝜆𝑡 − 84 𝑒 −5 𝜆𝑡 + 86 𝑒 −6 𝜆𝑡 − 40 𝑒 −7 𝜆𝑡 + 7 𝑒 −8 𝜆𝑡 (37)

5 = 112 𝑒 −5 𝜆𝑡 − 398 𝑒 −6 𝜆𝑡 + 582 𝑒 −7 𝜆𝑡 − 434 𝑒 −8 𝜆𝑡 + 164 𝑒 −9 𝜆𝑡 − 25 𝑒 −10 𝜆𝑡 (38)

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐴

𝐵

𝐶

𝐷

In Eq. (22) , equations E 1 through E 17 are calculated as follows:

1 = 𝑒 −6 𝜆𝑡 ((1 − 𝑒 −6 𝜆𝑡

)(𝑒 −6 𝜆𝑡

)+ 𝑒 −6 𝜆𝑡

(3 𝑒 −4 𝜆𝑡 − 2 𝑒 −6 𝜆𝑡

))=

1 𝒆 6 𝑡𝜆 +

3 𝒆 10 𝑡𝜆 −

3 𝒆 12 𝑡𝜆

𝒆 6 𝑡𝜆 (39)

2 = 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −8 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 (( 1 − 𝑒 −4 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −4 𝜆𝑡 ( 2 𝑒 −2 𝜆𝑡 − 𝑒 −4 𝜆𝑡 )))

=

1 𝒆 8 𝑡𝜆 +

1 𝒆 10 𝑡𝜆 +

2 𝒆 12 𝑡𝜆 −

3 𝒆 14 𝑡𝜆

𝒆 2 𝑡𝜆 (40)

3 = 𝑒 −2 𝜆𝑡 ( ( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 ) ) )

=

1 𝒆 5 𝑡𝜆 +

1 𝒆 8 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 −

2 𝒆 11 𝑡𝜆

𝒆 2 𝑡𝜆 (41)

4 = ( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −8 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 ( ( 1 − 𝑒 −4 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 ) + 𝑒 −4 𝜆𝑡 ( 2 𝑒 −2 𝜆𝑡 − 𝑒 −4 𝜆𝑡 ) )

=

1 𝒆 8 𝑡𝜆 +

1 𝒆 10 𝑡𝜆 +

2 𝒆 12 𝑡𝜆 −

3 𝒆 14 𝑡𝜆 (42)

5 = ( 1 − 𝑒 −4 𝜆𝑡 )( 𝑒 −6 𝜆𝑡 ) + 𝑒 −4 𝜆𝑡 (( 1 − 𝑒 −3 𝜆𝑡 )( 𝑒 −3 𝜆𝑡 )

+ 𝑒 −3 𝜆𝑡 ( 1 − ( 1 − 𝑒 − 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) ))

=

1 𝒆 6 𝑡𝜆 +

1 𝒆 7 𝑡𝜆 +

1 𝒆 8 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 −

3 𝒆 10 𝑡𝜆 (43)

6 = ( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 ) )

=

1 𝒆 5 𝑡𝜆 +

1 𝒆 8 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 −

2 𝒆 11 𝑡𝜆 (44)

7 = ( 1 − ( 1 − 𝑒 − 𝜆𝑡 )( 1 − 𝑒 −4 𝜆𝑡 ) )( 1 − ( 1 − 𝑒 −3 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) )

=

(

1 𝒆 2 𝑡𝜆 +

1 𝒆 3 𝑡𝜆 −

1 𝒆 5 𝑡𝜆

) (

1 𝒆 𝑡𝜆

+

1 𝒆 4 𝑡𝜆 −

1 𝒆 5 𝑡𝜆

)

(45)

8 = 𝑒 −4 𝜆𝑡 (( 1 − 𝑒 −3 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )( 𝑒 −10 𝜆𝑡 ) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 ))) + 𝑒 −3 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 )))

=

(−

1 𝒆 3 𝑡𝜆 + 1

)(1

𝒆 6 𝑡𝜆 +

3 𝒆 10 𝑡𝜆 −

3 𝒆 12 𝑡𝜆

)+

1 𝒆 4 𝑡𝜆 +

2 𝒆 8 𝑡𝜆 −

2 𝒆 10 𝑡𝜆

𝒆 3 𝑡𝜆

𝒆 4 𝑡𝜆 (46)

9 = ( 1 − 𝑒 −3 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )( 𝑒 −12 𝜆𝑡 ) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −6 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −4 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) ))) + 𝑒 −3 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −6 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −4 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) ))

=

1 𝒆 8 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 +

1 𝒆 10 𝑡𝜆 +

2 𝒆 12 𝑡𝜆 +

3 𝒆 17 𝑡𝜆 −

3 𝒆 14 𝑡𝜆 −

4 𝒆 15 𝑡𝜆 (47)

10 = ( 1 − 𝑒 −2 𝜆𝑡 )( ( 1 − 𝑒 −3 𝜆𝑡 )( 𝑒 −8 𝜆𝑡 ) + 𝑒 −3 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −6 𝜆𝑡 ) ) )

+ 𝑒 −2 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −6 𝜆𝑡 ) )

=

1 𝒆 4 𝑡𝜆 +

1 𝒆 5 𝑡𝜆 +

1 𝒆 9 𝑡𝜆 +

2 𝒆 8 𝑡𝜆 +

2 𝒆 13 𝑡𝜆 −

1 𝒆 7 𝑡𝜆 −

2 𝒆 10 𝑡𝜆 −

3 𝒆 11 𝑡𝜆 (48)

11 = 𝑒 −4 𝜆𝑡 ( ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 )( ( 1 − 𝑒 −6 𝜆𝑡 )( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 ) ) + 𝑒 −6 𝜆𝑡 ) )

(49)

374

12 =

((1 − 𝑒 −4 𝜆𝑡

)(1 −

(1 − 𝑒 −2 𝜆𝑡

)2 ) + 𝑒 −4 𝜆𝑡 )

×((1 − 𝑒 −6 𝜆𝑡

)(1 −

(1 − 𝑒 −2 𝜆𝑡

)(1 − 𝑒 −3 𝜆𝑡

))+ 𝑒 −6 𝜆𝑡

)=

(

1 𝒆 8 𝑡𝜆 +

2 𝒆 2 𝑡𝜆 −

2 𝒆 6 𝑡𝜆

) ( (

−

1 𝒆 6 𝑡𝜆 + 1

) (

1 𝒆 2 𝑡𝜆 +

1 𝒆 3 𝑡𝜆 −

1 𝒆 5 𝑡𝜆

)

+

1 𝒆 6 𝑡𝜆

)

(50)

13 = 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −2 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )( 𝑒 −11 𝜆𝑡 ) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 (1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 )))) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 (1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 ))))

=

2 𝒆 7 𝑡𝜆 +

2 𝒆 10 𝑡𝜆 +

3 𝒆 11 𝑡𝜆 +

3 𝒆 15 𝑡𝜆 −

1 𝒆 9 𝑡𝜆 −

1 𝒆 12 𝑡𝜆 −

7 𝒆 13 𝑡𝜆

𝒆 2 𝑡𝜆 (51)

14 = ( 1 − 𝑒 −2 𝜆𝑡 )(( 1 − 𝑒 −2 𝜆𝑡 )(( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −7 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 (( 1 − 𝑒 −3 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −3 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 ))) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 ))) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −2 𝜆𝑡 )(( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −3 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) )) + 𝑒 −2 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −6 𝜆𝑡 ) ))

=

2 𝒆 6 𝑡𝜆 +

2 𝒆 7 𝑡𝜆 +

3 𝒆 14 𝑡𝜆 +

4 𝒆 11 𝑡𝜆 +

5 𝒆 10 𝑡𝜆 +

10 𝒆 15 𝑡𝜆 −

1 𝒆 8 𝑡𝜆 −

3 𝒆 9 𝑡𝜆 −

3 𝒆 17 𝑡𝜆

−

8 𝒆 12 𝑡𝜆 −

10 𝒆 13 𝑡𝜆 (52)

15 = ( 1 − 𝑒 −2 𝜆𝑡 )(( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −7 𝜆𝑡 ) + 𝑒 −6 𝜆𝑡 (( 1 − 𝑒 −3 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 )

+ 𝑒 −3 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 ))) + 𝑒 −2 𝜆𝑡 (( 1 − 𝑒 −6 𝜆𝑡 )( 𝑒 −5 𝜆𝑡 )

+ 𝑒 −6 𝜆𝑡 ( 1 − ( 1 − 𝑒 −3 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) ))

=

2 𝒆 7 𝑡𝜆 +

2 𝒆 10 𝑡𝜆 +

3 𝒆 11 𝑡𝜆 +

3 𝒆 15 𝑡𝜆 −

1 𝒆 9 𝑡𝜆 −

1 𝒆 12 𝑡𝜆 −

7 𝒆 13 𝑡𝜆 (53)

16 = ( 1 − 𝑒 −2 𝜆𝑡 )( ( 1 − 𝑒 −3 𝜆𝑡 )( 𝑒 −4 𝜆𝑡 ) + 𝑒 −3 𝜆𝑡 ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 2 ) )

+ 𝑒 −2 𝜆𝑡 ( 1 − ( 1 − 𝑒 −3 𝜆𝑡 )( 1 − 𝑒 −2 𝜆𝑡 ) )

=

2 𝒆 4 𝑡𝜆 +

2 𝒆 9 𝑡𝜆 +

3 𝒆 5 𝑡𝜆 −

1 𝒆 6 𝑡𝜆 −

5 𝒆 7 𝑡𝜆 (54)

17 = ( 1 − ( 1 − 𝑒 −2 𝜆𝑡 ) 3 )( 1 − ( 1 − 𝑒 −2 𝜆𝑡 )( 1 − 𝑒 −3 𝜆𝑡 )( 1 − 𝑒 −6 𝜆𝑡 ) ) (55)

In Eq. (23) , equations A, B, C, D, and E are calculated as follows:

=

(−

1 𝒆 3 𝑡𝜆 + 1

)(1

𝒆 6 𝑡𝜆 +

3 𝒆 10 𝑡𝜆 −

3 𝒆 12 𝑡𝜆

)+

1 𝒆 4 𝑡𝜆 +

2 𝒆 8 𝑡𝜆 −

2 𝒆 10 𝑡𝜆

𝒆 3 𝑡𝜆

𝒆 4 𝑡𝜆 (56)

=

(−

1 𝒆 3 𝑡𝜆 + 1

)(1

𝒆 6 𝑡𝜆 +

3 𝒆 10 𝑡𝜆 −

3 𝒆 12 𝑡𝜆

)+

1 𝒆 4 𝑡𝜆 +

2 𝒆 8 𝑡𝜆 −

2 𝒆 10 𝑡𝜆

𝒆 3 𝑡𝜆

𝒆 5 𝑡𝜆 (57)

= 1

𝒆 17 𝑡𝜆 +

2 𝒆 5 𝑡𝜆 +

2 𝒆 6 𝑡𝜆 +

2 𝒆 20 𝑡𝜆 +

4 𝒆 15 𝑡𝜆 +

5 𝒆 16 𝑡𝜆 +

6 𝒆 12 𝑡𝜆 −

1 𝒆 19 𝑡𝜆 −

3 𝒆 11 𝑡𝜆 −

3 𝒆 13 𝑡𝜆 −

3 𝒆 14 𝑡𝜆 −

5 𝒆 10 𝑡𝜆 −

6 𝒆 18 𝑡𝜆

𝒆 𝑡𝜆

(58)

=

(

1 𝒆 11 𝑡𝜆 +

1 𝒆 12 𝑡𝜆 +

1 𝒆 17 𝑡𝜆 +

2 𝒆 8 𝑡𝜆 +

3 𝒆 16 𝑡𝜆 +

4 𝒆 9 𝑡𝜆 +

6 𝒆 18 𝑡𝜆 −

1 𝒆 10 𝑡𝜆

−

1 𝒆 15 𝑡𝜆 −

3 𝒆 20 𝑡𝜆 −

5 𝒆 13 𝑡𝜆 −

7 𝒆 14 𝑡𝜆

)

(59)

F. Bistouni, M. Jahanshahi Reliability Engineering and System Safety 167 (2017) 362–375

𝐸

𝐹

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

=

(

2 𝒆 6 𝑡𝜆 +

4 𝒆 7 𝑡𝜆 +

4 𝒆 14 𝑡𝜆 +

7 𝒆 10 𝑡𝜆 +

8 𝒆 11 𝑡𝜆 +

20 𝒆 15 𝑡𝜆 −

1 𝒆 8 𝑡𝜆 −

6 𝒆 9 𝑡𝜆 −

6 𝒆 17 𝑡𝜆

−

11 𝒆 12 𝑡𝜆 −

20 𝒆 13 𝑡𝜆

)

(60)

=

4 𝒆 19 𝑡𝜆 +

5 𝒆 6 𝑡𝜆 +

7 𝒆 10 𝑡𝜆 +

8 𝒆 7 𝑡𝜆 +

8 𝒆 14 𝑡𝜆 +

18 𝒆 11 𝑡𝜆 +

26 𝒆 15 𝑡𝜆 −

2 𝒆 16 𝑡𝜆 −

6 𝒆 8 𝑡𝜆

−

11 𝒆 12 𝑡𝜆 −

17 𝒆 17 𝑡𝜆 −

19 𝒆 9 𝑡𝜆 −

20 𝒆 13 𝑡𝜆 (61)

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