Reliable networking in Ethernet ring mesh networks using

Telecommunication Systemshttps://doi.org/10.1007/s11235-019-00566-8

Reliable networking in Ethernet ring mesh networks using regulartopologies

Mohsen Jahanshahi1 · Fathollah Bistouni2

© Springer Science+Business Media, LLC, part of Springer Nature 2019

AbstractReliable networking is an important factor in Ethernet ringmesh networks (ERMs)with ITU-TG.8032Ethernet ring protectionrecommendation or with the IEC 62439-3 high-availability seamless redundancy protocol. However, there are two majorchallenges for this purpose: (1) Hitherto, irregular topologies are used in ERMs that it causes difficulty in analyzing andimproving reliability. (2) A topology is appropriate for practical implementation, if it is low cost, simple to implement, andsimple to develop for large scale-systems (i.e. scalable). However, irregular topologies are extremely difficult to implement anddevelop. To deal with these challenges, this paper introduces two regular topologies called chordal ring and k-cube networks,for the first time in the area of ERMs. In addition, the proposed topologies are carefully analyzed in terms of reliability. Theseanalyzes prove that the proposed regular topologies outperform existing irregular multiple-ring networks namely shared link,shared node, complex shared link, redundant link, and 3-connected network.

Keywords Ethernet ring mesh network · Optical networking · Reliability · Topology · Failure rate · Chordal ring

1 Introduction

Although SDH/SONET rings clearly provide fast protectionswitching, Ethernet-based equipment is cheaper and morecompliant than synchronous equipment [1]. Therefore, theenhancement of carrier-class Ethernet is rapidly growingcompared to SONET/SDH networks [2], as most of the ser-vice providers prefer using Ethernet to SONET/SDH forobtaining and gathering customer traffic [3]. Furthermore,carrier-class optical Ethernet eliminates the restrictions ofgeneric Ethernet LANs by presenting new standards for var-ious service-type supports (by the IEEE 802.1Q standards),enhanced scalability (by the IEEE 802.1Qay standards),operator manageability (by the IEEE 802.1ag standards andITU-T Y.1731 Recommendation), and reliability (by theITU-T G.8031 and G.8032 Recommendations) [2].

B Mohsen [email protected]

Fathollah [email protected]

1 Young Researchers and Elite Club, Central Tehran Branch,Islamic Azad University, Tehran, Iran

2 Department of Computer Engineering, Central TehranBranch, Islamic Azad University, Tehran, Iran

The Ethernet ring protection (ERP) presented by theITU-T G.8032 Recommendation espouses greatly reliableprotection with a rapid protection switching time of less than50 ms for Ethernet services [2], but with a restrictive failureper ring protection capability [4]. The basic idea of the ERPswitching scheme is to handle loop freeness within intercon-nected Ethernet ring meshes [5]. To prevent a loop, one linkin each ERP ring is blocked by the ERP. This ring protectionlink (RPL) is deactivated in an idle mode that means thereis no failure or active switching command in place. When afailure happens at a link, the nodes next to the failure, blockthe failed link, and the RPL is unblocked [1, 5]. In addition,the high-availability seamless redundancy (HSR) presentedby IEC 62439-3 is receiving attention due to its capabilityof providing better availability with a zero failover time [4].However, this technique increases thenumber of links/specialnodes such as HSR and quadruple boxes in the network,which increases CAPEX and OPEX [4, 6, 7]. Generally, themain principle of HSR is to duplicate frames and send themover disjointed paths at the same time [4]. The benefit of theduplicated frames is that they make sure that each node willreceive at least one copy of the sent frame, especially if a faultoccurs in one of the ring components. However, in the casethat ring is healthy, or fault-free, half of the ring bandwidthwill be lost because of the duplicated frame copies [8].

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M. Jahanshahi

In the meantime, reliability has always been considered asone of the basic parameters in Ethernet ring mesh networks(ERMs) [1–12]. However, regardless of using ERP or HSRschemes, an Ethernet single-ring mesh network cannot befault tolerant in more than one link (i.e. unreliable). To mendthis situation, a reasonable approach is to use multiple-ringmesh structures. However, in general, multiple-ring meshescan be divided into two types in terms of topology: Irreg-ular networks and regular networks. Irregular networks arethose networks, which do not have a general structure fordevelopment. Therefore, these networks are often complexto implement and difficult to improve; include the extra hard-ware costs without necessary scalability. To the best of ourknowledge, all topologies investigated in the past are ofthis kind. For example, shared link topology, redundant linktopology, and shared node topology [1, 9–11]. These topolo-gies will be discussed further in Sect. 2. In contrast, regularnetworks are those that follow a certain design map in theirstructure. Therefore, the main feature of this type of net-works is scalability and easy implementation. In addition,hardware cost control and performance optimization effortscan be better managed on the network. Also, we believe thatthese networks can provide essential parameter of reliabil-ity in a higher level of performance compared to irregularnetworks (the analyses carried out in Sect. 4 will prove thebelief). However, to the best of our knowledge, these net-works have still not been studied in the design of ERMs inany of the previous works. Therefore, the contribution of thispaper is to offer two regular topologies in this area namedchordal ring and k-cube. The structure of these networks willbe discussed in Sect. 3. In addition, the analysis carried outin Sect. 4 demonstrates that the regular networks can pro-vide higher performance compared to previous topologiesof shared link topology, redundant link topology, complexshared link topology, shared node topology, and 3-connectednetwork, in terms of reliability.

It is to be noted that two important studies have recentlybeen done in [13, 14]. However, these studies differ fromwhat is done in this paper. First of all, it should be notedthat research done in the reliability of ERM can be dividedinto two main parts: (1) reliability analysis and (2) design ofnetwork topology. The works done in the field of reliabilityanalysis improve the accuracy of reliability analysis in thisarea.On the other hand, theworks done in the design of topol-ogy improves reliability. It should be noted that although theproposed methods in the first field (i.e., reliability analysis)can be applied in the second domain (i.e., network topol-ogy design), the goals of these two areas are quite different.According to these discussions, it should be noted that thetwo studies conducted in [13, 14] are about reliability anal-ysis methods and are in the first domain. Nevertheless, thesubject of this paper is about design of network topology, andit is in the second domain. In summary, in [13], a newmethod

is proposed based on the concept of a spanning tree for reli-ability analysis. The spanning tree set method can be shownto have more accurate results for network reliability analysiscompared to earlier approaches. Furthermore, in [14], a newmethod calledR&C (remove and contraction) is proposed forexact network reliability analysis. The analysis showed thatthis method has a higher accuracy than previous methods.

It should benoted that themain contributionof this paper isto develop two regular topologies called the chordal ring andk-cube. As indicated in the later sections, these new designscan have higher reliability than previous designs.

The rest of this paper is ordered as follows: Sect. 2 isdevoted to related works. The proposed topologies are pre-sented in Sect. 3. Reliability analysis was performed inSect. 4. Section 5 presents the simulation results. Some dis-cussions have beenmade about the analysis done in this paperin Sect. 6. Finally, the conclusion is made in Sect. 7.

2 Related works

In this section, we attempt to review important recent worksin the field of ERM topologies and their reliability.

Ryoo et al. [1] has studied the concepts in ERP and dis-cussed the fundamental operating principles by which theautomatic protection switchingworks. In thiswork, two typesof irregular topologies are intended for Ethernet multiple-ring mesh networks: shared link topology and shared nodetopology. A shared link topology is made up of two rings thatone or more links is common between them. Also, A sharednode topology consists of two rings that one node is commonbetween them. These topologies are shown in Fig. 1a, b. Itshould be noted that these topologies are two basic struc-tures for designing other irregular or regular multiple-ringtopologies. Lee et al. [2] examined the designing principlesof protected Ethernet ring mesh networks and they proposedan optimized scheme for efficient ERP networking in termsof cost-effectiveness and reserved capacity. In this work, twoERM topologies are considered: shared link topology (shownin Fig. 1a) and complex shared link topologies (i.e. irregu-lar topology). Complex shared link topologies often containmultiple rings that the intersection of these rings is the com-mon links between them. In other words, these networks arebuilt from repeating the pattern of shared link topology. Anexample of this topology is shown in Fig. 1c. Lee et al. [5]addresses the problem of designing a reliable network formaximum network availability in ring mesh networks whilefollowing ERP scheme by introducing a heuristic algorithmto find a suboptimal solution. Similarly to [2], two ERMtopologies are considered in this work: shared link topology(Fig. 1a) and complex shared link topologies (i.e. irregu-lar topology (Fig. 1c). In addition, there exist some mainproblems in this work in terms of analysis approach: In this

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Reliable networking in Ethernet ring mesh networks using regular topologies

Fig. 1 a A shared link topology,b a shared node topology, c acomplex shared link topology,d a redundant link topology,e 3-connected network

work, it is supposed that the availability for each component(including nodes and links) is specified and network avail-ability is a function of its components availability. However,the failure rate and repair rate are accessible information onthe availability analysis [15]. Furthermore, to do availabilityanalysis of a system, the typical approach is to constructand solve a Markov model. In other words, the analysisintroduced in this work is a time-independent analysis of reli-ability [16, 17] instead of an actual analysis of availability.Lee et al. [6] similarly to [5] examined the problem of design-ing a reliable network for maximum network availability inring mesh networks but with regard to the HSR scheme. Interms of reliability and availability, the design of a reliableEthernet ring mesh network is independent of the use of anyof the schemes of ERP or HSR. That is because it is usuallyassumed that a ring mesh network failure occurs because offailure of the links. Therefore, in [6], the same algorithmas it was in the [5] was used. Moreover, the method of cal-culating availability in this work was similar to that of [5].

Two ERM topologies are considered: shared link topology(shown in Fig. 1a) and redundant link topology (i.e. irregularand commonly used in HSR ring networks). A Redundantlink topology is shown in Fig. 1d. Nurujjaman et al. [12]proposed an effective design approach for ERP mesh net-works, which addresses the issue of service outages becauseof double link failures. This work developed a mathematicalmodel to design ERP networks effectivelywith the balance indesign objective between minimizing capacity requirementand improving service availability. Generally, two types oftopology have been studied in this work: Complex sharedlink topologies (is shown inFig. 1c) and 3-connected network(i.e. irregular network). A 3-connected network is shown inFig. 1e. In this type of network, each node is directly con-nected to three other nodes by three separate links. Since thereis no clearmethod for the development of the network inmorenumber of nodes, the network is considered as an irregularnetwork. Assi et al. [9] considered the joint problem of RPLselection and ring hierarchy selection when routing traffic

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flows in presence of multiple ERP instances. They obtainedseveral insights and guidelines for network operators to pro-vide their customers with the required quality of servicethey demand.Generally, complex shared link topologies havebeen studied in this work. Allawi et al. [7] investigated theproblem of design optimization namely trade-off betweenredundancy investment and the realization of the networkavailability demanded by the customer in HSR-based Ether-net mesh networks. In this work, to analyze the availabilityof the single-ring networks, he employed the same methodas in [5, 6]. In other words, in this case, there are the sameproblems as in [5, 6]. Allawi et al. [7] uses an estimatedupper bound reliability equation presented in [18] for thecalculation of multiple-ring mesh availability. However, thisstrategy is suitable for calculating the reliability not for avail-ability since it can produces misleading results. Generally, inthis work, redundant link topology and complex shared linktopologies have been studied. In another effort, Allawi et al.[4] studied the problem of minimizing the cost of HSR meshnetworks given a network availability constraint. However,in this study, an inappropriate method that previously usedin [7], is used to analyze the availability and hence has thesame fundamental problems. Generally, in this work, redun-dant link topology, shared link topology, and complex sharedlink topologies have been studied. Recently, two importantresearches have been done to improve reliability analysisaccuracy in [13, 14]. Bistouni and Jahanshahi [13] developeda completely new approach based on the concept of spanningtrees for exact reliability analysis. They demonstrated thatthis method has a high precision in the calculation of relia-bility compared to the previous ones. Furthermore, Bistouniand Jahanshahi [14] proposed a new approach called R&C(remove and contraction) for network reliability analysis thatis more accurate than earlier methods. In general, in [13, 14],the following topologies have been investigated: shared linktopology, shared node topology, complex shared link topol-ogy, redundant link topology, and 3-connected network.

In general, given the above discussions, five types oftopologies can be identified based on previous works: (1)shared link topology, (2) shared node topology, (3) complexshared link topology, (4) redundant link topology, and (5) 3-connected network. In addition, since these topologies do nothave a general structure for development, all of these topolo-gies are of irregular topology. In other words, in the past,little attention has been paid to the development of regularERM topologies. However, Regular topologies are scalablein terms of development and easier to implement and perfor-mance optimization. Moreover, we believe that the regularnetworks could provide the necessary parameter of reliabil-ity, at a higher level compared to the irregular networks.Therefore, in the next section, two new regular topologiesnamed chordal ring and k-cube will be introduced.

Table 1 summarizes previous works, their goals as wellas the topologies examined in the study. Also, their short-comings which have been resolved in this research are listed.As it can be seen, none of the existence research works hasfocused on the topology design. Although considering somefactors such as RPL selection and ring hierarchy selectioncan improve reliability, these factors are a function of thegiven network topology. As a result, improving reliability bydesigning a good topology can be much more effective thanothermethods. In addition, according to Table 1, studies donein the previous works does not cover all types of topologies.Considering different types of topologies in a research canhelp to ensure the integrity of the research results. On theother hand, according to Table 1, one of the other problemsin some of the previous works is the lack of accurate methodsfor reliability analysis. Therefore, to fill the aforementionedgaps, this paper has many contributions as follows: Twotopologies of the regular type called the chordal ring and k-cube are introduced in this paper. Therefore, this paper is animportant step towards developing the methods of topologydesign in the field of Ethernet ring mesh networks. In addi-tion, all types of irregular topologies have been investigatedin this paper for comparison with proposed topologies. Asdemonstrated in the following sections, the proposed topolo-gies can achieve higher reliability than previous topologies.On the other hand, inaccurate methods have been used in pre-vious works [4–7] for reliability analysis. However, in thisresearch work, decomposition method is used as a precisemethod for determining the reliability of complex networks[19–21].

3 Proposed topologies

In this section, two new regular ERM topologies are pre-sented in the area of ERMs: Chordal ring in sub-Sect. 3.1and k-cube in sub-Sect. 3.2.

3.1 Chordal ring

Ethernet single-ringmesh networks are inexpensive and easyto implement, but they have the lowest fault tolerance. Toimprove this situation, one method is the use of additionallinks called chords that would connect the nodes of the ringin a given way. The rings with additional edges linking nodesare denoted in the literature the chordal rings (CHRs).Varioustypes ofCHRs are analyzed in different network technologiessuch as Fiber channel networks, interconnection networks,ATM (Asynchronous Transfer Mode) networks, wirelesssensor networks,Wavelength DivisionMultiplexing (WDM)networks etc. due to good properties of this topology suchas regularity, low diameter, high connectivity, and efficientrouting [22–27]. However, this topology has not been stud-

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Table 1 Related works and their deficiencies

Related works Goal Topologies Deficiencies

[1] Studying the concepts of ERP Shared link topologyShared node topology

Lack of any contributions to topologydesign

Not evaluating all irregular topologiesNot assessing any regular topology

[2] Efficient ERP networking Shared link topologyComplex shared link topologies



[5] Maximizing network availability Shared link topologyComplex shared link topologies


Not evaluating all irregular topologiesNot assessing any regular topologyInaccurate analysis of reliability

[6] Maximizing network availability Shared link topologyRedundant link topology



[12] Introducing a new approach foreffective design in ERP meshnetworks

Complex shared link topologies3-Connected network



[9] Efficient RPL selection and ringhierarchy selection

Complex shared link topologies Lack of any contributions to topologydesign


[7] Design optimization namely trade-offbetween redundancy investment andthe network availability

Redundant link topologyComplex shared link topologies



[4] Minimizing the cost of HSR meshnetworks given a network availabilityconstraint

Redundant link topologyShared link topologyComplex shared link topologies



[13] Introducing a new approach based onthe concept of spanning trees for exactreliability analysis

Shared link topologyShared node topologyComplex shared link topologyRedundant link topology3-Connected network


Not assessing any regular topology

[14] Introducing a new approach calledR&C (remove and contraction) fornetwork reliability analysis

Shared link topologyShared node topologyComplex shared link topologyRedundant link topology3-Connected network


Not assessing any regular topology

ied yet in the field of ERMs. In fact, according to previousworks, it can be argued that any regular topology not beenstudied yet in the field of ERMs. Therefore, for the first time,this paper proposes the use of CHR topologies in the area ofthe ERMs.

Generally, CHR is a ring consisting of excess chords. Thepair (p, Q) describes the CHR, in which, p represents thenumber of nodes in the ring and the set of the chords qi is Qwhere qi ∈ {2, . . . , p/2}. The nodes are at the distance qi in

the ring, are linked by each chord qi ∈ Q. The denotation ofthis structure is (p; q1, . . . , qi ), q1 < q2 < · · · < pi . In thecase, there is a chord of distance p/2, the degree of chordalring is 2i and ring’s degree is 2i − 1. Note that p is even [27,28].

The most popular type of CRHs that is used in vari-ous fields of network technology, is standard fourth degreechordal ring (CHR4). Also, this paper will focus on thisimportant type of CHRs. Generally, A structure of p nodes in

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M. Jahanshahi

Fig. 2 Example of standard graph CHR4 (25; 7)

which, each node vi is linked to four other nodes: vi−1(modp),vi+1(modp) plus to two nodes vi−q(modp), vi+q(modp), is calleda fourth degree chordal ring. Here q is the distance of excesschords [27, 28].An example of a chordal ring of fourth degreeis shown in Fig. 2.

3.2 k-Cube network

K-cubes (Hypercubes/k-dimensional hypercubes) hold lotsof fine network characteristics such as great connectivity,low density, small diameter and regular structure. The rea-son is that the binary strings are naturally encoded intotheir topology thus those are global [29–31]. k-cubes areanalyzed in different networking fields such as data cen-ter networks, interconnection networks, WDM networks etc[31–36]. However, this type of topology has not been stud-ied yet in the field of ERMs. Therefore, for the first time, thispaper studies the use of k-cube topologies in the area of theERMs.

A graph k-cube is an undirected graph consisting of 2k

vertices (nodes), where each vertex is connected to k othervertices. The vertices are labeled from 0 to 2k − 1 so thatthere is an edge between a certain pair of vertices if andonly if binary representation of them differs in one and onlyone bit. This featuremakes possible a straightforward routingmechanism. In this mechanism, the path from a source node ito destination node j can be achieved byXOR (exclusiveOR)of binary addresses of nodes i and j . If the result of the XORin a given bit equals 1, then, the dimension correspondingto the bit is the preferred route. For example, a 4-cube is

Fig. 3 The structure of a 4-cube

shown in Fig. 3. In this figure, consider the source node 0101and destination node 1011: 0101 ⊕ 1011 � 1110 and so themessages will be sent over the links corresponding to thedimensions 2, 3, and 4 (counting from right to left).

The Hamming distance between two addressesSk Sk−1 . . . S1 and DkDk−1 . . . D1 in k-cube is givenby:

H(S, D) �k∑

i�1

Si ⊕ Di (1)

In a k-cube, between any pair of nodes S and D, there areH(S, D) disjoint paths of length H(S, D) and k − H(S, D)

disjoint paths of length H(S, D) + 2. For example, consideragain the source 0101 and destination 1011 in Fig. 3: H(S, D) � 3 and therefore, three paths of length 3 and onepath of length 5 exist between the source and destination. Inaddition, if the number of faulty links and nodes is less thank in a k-cube, then there is at least one path length less thanor equal to H(S, D) + 2 between any two non-faulty nodesS and D.

One of the favorable characteristics of k-cubes is the recur-sive nature of their structure. This means that a k-cube canbe constructed by connecting nodes with the same addressin the two k − 1-cubes. For example, it should be noted thata 4-cube (shown in Fig. 3) is made up of two 3-cubes.

4 Reliability analysis

Generally, reliability is defined by the IEEE as “the abilityof a system or component to perform its required functionsunder stated conditions for a specified period of time” [37].Therefore, mathematically, reliability R(t) is the probabilitythat a system will be successful in the interval from time 0 totime t [19]. In both definitions above, there is an importantpoint. The important point is that the reliability of a system isdefined according to the required function of the system. Inthe field of networking technology, researchers have differ-

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ent interpretations of the required function, which have beenled to various metrics in the areas of reliability. In general,these metrics are three cases: terminal reliability, broadcastreliability, and network (all-terminal) reliability. In terminalreliability, required function is to connect a given source nodeto a given destination node. In broadcast reliability, requiredfunction is to connect a given source node to a group of des-tination nodes. In network (all-terminal) reliability, requiredfunction is to connect all the source nodes to all destinationnodes. According to previous works in the field of ERMs[4–7], all-terminal reliability is considered in this area. All-terminal reliability is defined as the probability of successfulconnection of each node in the network to any other node init.

Generally, in this paper, the following assumptions willbe considered for the reliability analysis:

1. It will be assumed that each link in network may fail.2. All failures are statistically independent.3. The failures are assumed to be exponentially distributed.

Therefore, if λ is the failure rate of a link, then the cor-responding reliability is given by: RL(t) � e−λt .

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

First, consider the Ethernet single-ring mesh network forall-terminal reliability analysis. In terms of all-terminal relia-bility, it should be noted that a n-node ring network is actuallya k-out-of-n system with k � n − 1. The reliability of thissystem is the probability that k or more components are suc-cessful. Binomial distribution can be used to calculate thereliability of such a system [38]. As a result, we have:

Rsingle−ring(t) �n∑

i�n−1

(ni

)(e−λt)i (1 − e−λt)n−i

�(

nn − 1

)(e−λt)n−1(

1 − e−λt)

+

(nn

)(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

� ne−(n−1)λt − (n − 1)e−nλt (2)

Next, we consider a shared link ERM (shown in Fig. 1a)that is an irregular multiple-ring topology. First, it must benoted that it is very difficult to accurately determine the relia-bility of the multiple-ring networks due to the complexity oftheir structure. Therefore, in previous works [4–7], often itis suggested to use estimation-based methods (not exact) forthis case. However, in this section, we use the decompositionmethod for accurately determining the reliability of the net-

work, which is a precise method for analyzing the reliabilityof complex networks [19–21]. The decomposition method isan application of the law of total probability. In this method,first, one of the components is selected as a key componentand then the system reliability is calculated in two modes:when the key component is activated and when the key com-ponent has been failed. Finally, these two probabilities arecombined to achieve the reliability of the whole system. Inaddition, it should be noted that with regard to the amount ofcomplexity of a system, several components of a system canbe selected as keys and sub-keys. The decompositionmethodcan be expressed by the following equation:

Rsystem(t) � Rkey(Rsystem∣∣Rkey) + Rkey(Rsystem

∣∣Rkey)(3)

Therefore, for shared link network, we have:

Rshared link(t)

� (1 − e−λt)(6e−5λt − 5e−6λt

)

+ e−λt(e−2λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt(e−λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt((

3e−2λt − 2e−3λt)(1 − e−λt)

+ e−λt(3e−2λt − 2e−3λt

))))

� 15e−5λt − 23e−6λt + 9e−7λt (4)

It should be noted here that more details about how tocompute the equations of this section is placed in Appendix.

Now, consider a shared node topology (shown in Fig. 1b).All-terminal reliability for this network is given by:

Rshared node(t)

� (1 − e−λt)(e−3λt

(4e−3λt − 3e−4λt

))

+ e−λt(e−2λt

(4e−3λt − 3e−4λt

)(1 − e−λt)

+ e−λt(e−λt

(4e−3λt − 3e−4λt

)(1 − e−λt)

+ e−λt((

4e−3λt − 3e−4λt)(1 − e−λt)


))))

� 16e−6λt − 24e−7λt + 9e−8λt (5)

Also, for the complex shared link topology in Fig. 1c, wehave:

Rcomplex shared link(t)

� (1 − e−λt)(e−2λt

(15e−5λt − 23e−6λt + 9e−7λt

))

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M. Jahanshahi

+ e−λt(e−λt

(15e−5λt − 23e−6λt + 9e−7λt

)(1 − e−λt)

+ e−λt((

15e−5λt − 23e−6λt + 9e−7λt)(1 − e−λt)

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

)

+e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)

+ e−λt(4e−2λt − 4e−3λt + e−4λt

)))

+e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)


)))))

� 65e−7λt − 155e−8λt + 125e−9λt − 34e−10λt (6)

This time, we assess the all-terminal reliability of theredundant link topology in Fig. 1d. In this case, all-terminalreliability is calculated as follows:

Rredundant link(t)

� (1 − e−λt)(15e−5λt − 23e−6λt + 9e−7λt

)

+ e−λt(e−2λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt(e−λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt((

3e−2λt − 2e−3λt)(1 − e−λt)


))))

� 24e−5λt − 50e−6λt + 36e−7λt − 9e−8λt (7)

Similarly, we analyze the all-terminal reliability of the 3-connected network (Fig. 1e). In this case, we have:

R3−connected (t)

� (1 − e−λt )((1 − e−λt )(E1) + e−λt ((1 − e−λt )(E2)

+ e−λt ((1 − e−λt )(E3) + e−λt (E4))))

+ e−λt ((1 − e−λt )((1 − e−λt )(E2) + e−λt ((1 − e−λt )(E3)

+ e−λt (E4)))

+ e−λt (E5))

(8)

In Eq. (8), equations E1 through E5 are calculated as fol-lows:

E1 � (1 − e−λt)(e−2λt

(15e−5λt − 23e−6λt + 9e−7λt

))

+ e−λt(e−λt(1 − e−λt)(15e−5λt − 23e−6λt + 9e−7λt

)

+ e−λt((1 − e−λt)(15e−5λt − 23e−6λt + 9e−7λt

)

+ e−λt(e−λt(1 − e−λt)(4e−3λt − 3e−4λt

)

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)


)))))

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

E2 � (1 − e−λt)(e−2λt (E6)

)

+ e−λt(e−λt(1 − e−λt)(E6) + e−λt((1 − e−λt)(E6)

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)

+e−λt(3e−2λt − 2e−3λt

))))(10)

E3 � (1 − e−λt)(e−2λt

(4e−3λt − 3e−4λt

)

+ e−λt(e−λt(1 − e−λt)(4e−3λt − 3e−4λt

)

+ e−λt ((1 − e−λt)(e−λt (E7)

)+ e−λt(e−λt (E7)

))))


)

+ e−λt(e−λt (E7)))

+ e−λt((1 − e−λt)((

1 − e−λt)(4e−3λt − 3e−4λt)

+ e−λt (E7))+ e−λt (E7)

))(11)

E4 � (1 − e−λt)((1 − e−λt)(E8)

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)+ e−λt (E7)

))

+ e−λt((1 − e−λt)(E9) + e−λt (E7))

(12)

E5 � (1 − e−λt )((1 − e−λt )(Rredundant link(t)) + e−λt (E4)

)

+ e−λt ((1 − e−λt )(E4) + e−λt (E10))

(13)

Also, in Eqs. (10), (11), (12), and (13) equations E6

through E10 are calculated as follows:

E6 � (1 − e−λt)(e−λt

(4e−3λt − 3e−4λt

))

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)


))

� 11e−4λt − 16e−5λt + 6e−6λt (14)

E7 � (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))

� 4e−2λt − 4e−3λt + e−4λt (15)

E8 � (1 − e−λt)(e−λt

(4e−3λt − 3e−4λt

))

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)

+ e−λt(e−λt(1 − e−λt)(1 − (

1 − e−λt)2)

123


+ e−λt(1 − (

1 − e−λt)2))))

� 12e−4λt − 18e−5λt + 7e−6λt (16)

E9 � (1 − e−λt)(4e−3λt − 3e−4λt

)


)

� 8e−3λt − 11e−4λt + 4e−5λt (17)

E10 � 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)))))

� 24e−3λt − 68e−4λt + 82e−5λt − 51e−6λt

+ 16e−7λt − 2e−8λt (18)

According to Eqs. (9)–(18), we have:

E1 � 56e−7λt − 130e−8λt + 102e−9λt − 27e−10λt (19)

E2 � 40e−6λt − 90e−7λt + 69e−8λt − 18e−9λt (20)

E3 � 32e−5λt − 68e−6λt + 49e−7λt − 12e−8λt (21)

E4 � 32e−4λt − 84e−5λt + 86e−6λt − 40e−7λt + 7e−8λt

(22)

E5 � 112e−5λt − 398e−6λt + 582e−7λt

− 434e−8λt + 164e−9λt − 25e−10λt (23)

Now, substituting Eqs. (19)–(23) into Eq. (8), we get:

R3−connected(t) � 376e−7λt − 1476e−8λt

+ 2352e−9λt − 1895e−10λt

+ 770e−11λt − 126e−12λt (24)

This time, consider the chordal ring topology. In this case,firstly, we consider a CHR4 (8; 2). We chose this type ofchordal ring because the number of its nodes fits other topolo-gies discussed in this paper. On the other hand, the numberof nodes of ERMs in real conditions is between 8 and 16nodes (up to 16 nodes) [2, 9]. However, we will also study

the topology of CHR4 (16; 2) to assess the all-terminal relia-bility in more number of nodes. To calculate the all-terminalreliability of these networks, we will consider the upper andlower bounds. Lower bound: In this case, we assume thatall the chords on the network failed, then, we will calculatereliability. In these conditions, reliability of the network isequal to the reliability of a single-ring ERM. Upper bound:In this case, it will be assumed that all chords on the networkare active and healthy. It is to be noted that the number ofchords in the network is relatively high. As a result, it is veryunlikely that all the chords in the network fail at the sametime. Therefore, it is expected that network reliability is veryclose to the upper bound.

For CHR4 (8; 2), the lower bound reliability is given by:

RCHR4(8;2)(t) � 8e−7λt − 7e−8λt (25)

Also, the upper bound reliability of CHR4 (8; 2) is givenby:

RCHR4(8;2)(t) � 1 − (1 − e−λt )8 � 8e−λt − 28e−2λt

+ 56e−3λt − 70e−4λt + 56e−5λt

− 28e−6λt + 8e−7λt − e−8λt (26)

Similarly, the lower bound for CHR4 (16; 2)‘s reliabilityis calculated as follows:

RCHR4(16;2)(t) � 16e−15λt − 15e−16λt (27)

Furthermore, the upper bound reliability of CHR4 (16; 2)is given by:

RCHR4(16; 2)(t) � 1 − (1 − e−λt )16

� 16e−λt − 120e−2λt + 560e−3λt

− 1820e−4λt + 4368e−5λt − 8008e−6λt

+ 11440e−7λt − 12870e−8λt

+ 11440e−9λt − 8008e−10λt + 4368e−11λt

− 1820e−12λt + 560e−13λt − 120e−14λt

+ 16e−15λt − e−16λt (28)

Now, consider the k-cube topology. At first, we considera 3-cube. Next, we will study the topology of 4-cube. Inaddition, we will consider the upper and lower bounds tocalculate the all-terminal reliability of 4-cube. Lower bound:In this case, it will be assumed that all links between the two3-cubes are failed, except for one of the links because work-ing at least one of them is necessary for working the entirenetwork. In this case, reliability of the network is equal to thereliability of two 3-connected networks that are connected inseries. Upper bound: In this case, it will be assumed that alllinks between the two 3-cubes are active and healthy. Here

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M. Jahanshahi

Table 2 All-terminal reliability as a function of time for λ � 10−6

Time (Hr) 1000 2000 3000 4000 5000 6000 7000 8000

8-Node single-ring 0.999972 0.999889 0.999752 0.999561 0.999317 0.999022 0.998675 0.998278

16-Node single-ring 0.999881 0.999530 0.998953 0.998158 0.997151 0.995939 0.994528 0.992926

Shared link 0.999994 0.999976 0.999946 0.999904 0.999850 0.999784 0.999706 0.999617

Shared node 0.999988 0.999952 0.999893 0.999810 0.999703 0.999574 0.999422 0.999246

Complex shared link 0.999995 0.999980 0.999955 0.999919 0.999874 0.999818 0.999752 0.999675

Redundant link 0.999994 0.999976 0.999946 0.999905 0.999851 0.999786 0.999709 0.999621

3-Connected 0.999999 0.999996 0.999991 0.999984 0.999975 0.999964 0.999951 0.999935

CHR4 (8;2)-lower bound 0.999972 0.999889 0.999752 0.999561 0.999317 0.999022 0.998675 0.998278

CHR4 (8;2)-upper bound 1 1 1 1 1 1 1 1

CHR4 (16;2)-lower bound 0.999881 0.999530 0.998953 0.998158 0.997151 0.995939 0.994528 0.992926

CHR4 (16;2)-upper bound 1 1 1 1 1 1 1 1

3-Cube 0.999999 0.999996 0.999991 0.999984 0.999975 0.999964 0.999951 0.999935

4-Cube-lower bound 0.999998 0.999992 0.999982 0.999968 0.999950 0.999928 0.999901 0.999871

4-Cube-upper bound 1 1 1 1 1 1 1 1

it should be noted that the number of links between the two3-cubes is relatively high. As a result, the probability that allthe links between the two 3-cubes fail simultaneously is verylow. Therefore, it is expected that network reliability is veryclose to the upper bound.

All-terminal reliability of 3-cube is calculated as Eq. (29)that is equal to the reliability of the 3-connected network.

R3−cube(t) � (1 − e−λt )((1 − e−λt )(E1)

+ e−λt ((1 − e−λt )(E2)

+ e−λt ((1 − e−λt )(E3) + e−λt (E4)

)))

+ e−λt ((1 − e−λt )((1 − e−λt )(E2)

+ e−λt ((1 − e−λt )(E3) + e−λt (E4)

))+ e−λt (E5)

)

� 376e−7λt − 1476e−8λt + 2352e−9λt

− 1895e−10λt + 770e−11λt − 126e−12λt (29)

Also, the lower bound and upper bound reliability for 4-cube are given by Eqs. (30) and (31), respectively:

R4−cube(t) � (R3−cube(t))(R3−cube(t)) (30)

R4−cube(t) � (1 − e−λpt

)((1 − e−λpt

)(E1)

+e−λpt((1 − e−λpt

)(E2)

+ e−λpt((1 − e−λpt

)(E3) + e−λpt (E4)

)))


)((1 − e−λpt

)(E2)


)(E3) + e−λpt (E4)

))

+ e−λpt (E5))

� 376e−7λpt − 1476e−8λpt + 2352e−9λpt

− 1895e−10λpt + 770e−11λpt − 126e−12λpt

(31)

In Eq. (31), λp is equal to:

λp � 1

1 − (1 − e−λt

)2

⎛

⎝−d(1 − (

1 − e−λt)2)

dt

⎞

⎠

�2λ

(− 1

etλ+ 1

)

− 1etλ

+ 2(32)

It should be noted here that more details about how tocompute the equations of this section is placed in Appendix.

According to the Eqs. (2)–(32), results of all-terminal reli-ability analysis as a function of time for different link failurerates (λ) 10−6, 10−5, and 10−4 per hour are summarized inTables 2, 3 and 4, respectively.

According to Tables 2, 3, and 4, single-ring ERMs are theleast reliable networks. This was expected because they canonly tolerate a single link failure. On the other hand, 8-nodesingle-ring achieves greater reliability compared to 16-nodesingle-ring. In other words, as the number of nodes increases,the reliability decreases in single-ring networks.

Among multiple-ring networks studied in the past (all ofwhich are irregular), the 3-connected network has the high-est reliability compared to shared link, shared node, complexshared link, and redundant link topologies. As a result, themain competitor of the regular networks of chordal ring andk-cube is 3-connected network in terms of all-terminal reli-ability.

First, we examine the status of the three networks inTable 2. According to Table 2, 3-connected network hasbetter all-terminal reliability compared to lower bound reli-ability of CHR4 (8; 2) and CHR4 (16; 2) chordal rings.However, the upper bound reliability of the chordal ringsis incredibly better than the 3-connected (their upper bound

123



Time (Hr) 1000 2000 3000 4000 5000 6000 7000 8000

8-Node single-ring 0.997336 0.989859 0.978279 0.963227 0.945264 0.924891 0.902548 0.878626

16-Node single-ring 0.989171 0.960856 0.920300 0.871600 0.817930 0.761721 0.704807 0.648548

Shared link 0.999401 0.997615 0.994663 0.990574 0.985385 0.979139 0.971882 0.963664

Shared node 0.998828 0.995423 0.989952 0.982577 0.973458 0.962748 0.950597 0.937149


Redundant link 0.999410 0.997679 0.994863 0.991018 0.986199 0.980456 0.973842 0.966406

3-Connected 0.999899 0.999588 0.999054 0.998277 0.997234 0.995900 0.994248 0.992247

CHR4 (8;2)-lower bound 0.997336 0.989859 0.978279 0.963227 0.945264 0.924891 0.902548 0.878626

CHR4 (8;2)-upper bound 1 1 1 1 1 1 1 1

CHR4 (16;2)-lower bound 0.989171 0.960856 0.920300 0.871600 0.817930 0.761721 0.704807 0.648548

CHR4 (16;2)-upper bound 1 1 1 1 1 1 1 1

3-Cube 0.999899 0.999588 0.999054 0.998277 0.997234 0.995900 0.994248 0.992247

4-Cube-lower bound 0.999798 0.999176 0.998108 0.996556 0.994476 0.991818 0.988529 0.984555



Time (Hr) 1000 2000 3000 4000 5000 6000 7000 8000

8-Node single-ring 0.827380 0.559500 0.344626 0.201145 0.113370 0.062356 0.033688 0.017952

16-Node single-ring 0.541635 0.185160 0.054298 0.014737 0.003817 0.000959 0.000235 0.000057

Shared link 0.944560 0.810097 0.647186 0.490807 0.357949 0.253321 0.175082 0.118731

Shared node 0.906900 0.717849 0.522290 0.358906 0.236697 0.151353 0.094492 0.057881


Redundant link 0.949264 0.829818 0.682149 0.534452 0.402952 0.294477 0.209755 0.146257

3-Connected 0.987086 0.931791 0.822039 0.673434 0.514775 0.370407 0.253185 0.165737

CHR4 (8;2)-lower bound 0.827380 0.559500 0.344626 0.201145 0.113370 0.062356 0.033688 0.017952

CHR4 (8;2)-upper bound 1 0.999999 0.999980 0.999860 0.999426 0.998283 0.995875 0.991545

CHR4 (16;2)-lower bound 0.541635 0.185160 0.054298 0.014737 0.003817 0.000959 0.000235 0.000057

CHR4 (16;2)-upper bound 1 1 1 1 1 0.999997 0.999983 0.999929

3-Cube 0.987086 0.931791 0.822039 0.673434 0.514775 0.370407 0.253185 0.165737

4-Cube-lower bound 0.974338 0.868235 0.675749 0.453513 0.264993 0.137201 0.064102 0.027469


reliability is almost equal to one). On the other hand, it shouldbe noted that for calculating the lower bound reliability ofthe chordal rings, it is assumed that all of the chord links inthese networks have failed. However, the possibility of sucha scenario is really low in chordal rings. Therefore, upperbound reliability for these networks is closer to reality. Thistime, we examine k-cube networks. Based on Table 2, 3-connected and 3-cube networks have an identical reliabilityto each other. On the other hand, lower bound reliability of4-cube is very close to the reliability of 3-connected, but 3-connected reliability is slightly better. However, the upperbound reliability of 4-cube is approximately one and muchbetter than the 3-connected reliability. In addition, in orderto calculate the lower bound reliability of the 4-cube, it isassumed that all links between the two 3-cubes are failed,

except for one of the links because working at least one ofthem is necessary for working the entire network. However,the possibility of such a scenario is low. Therefore, the upperbound reliability is more reasonable compared to the lowerbound. Now, comparing the chordal ring and k-cube wouldbe useful. For this purpose, at first, we compare the CHR4(8; 2) and 3-cube, in which the number of nodes is equal toeight. According to Table 2, 3-cube achieves higher reliabil-ity compared to CHR4 (8; 2)-Lower bound but its reliabilityis lower than CHR4 (8; 2)-Upper bound. This time, we com-pare the CHR4 (16; 2) and 4-cube that the number of nodesin both of them is equal to 16. As it is evident in Table 2, thelower bound reliability of 4-cube is higher than the CHR4(16; 2)-lower bound, but both have the same upper boundreliability.

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M. Jahanshahi

Table 3 also shows the facts similar to what was observedin Table 2. According to the table, both regular networksproposed for use in ERMs (i.e. chordal ring and k-cube)achieve a greater advantage in terms of all-terminal relia-bility than existing irregular networks. However, it is stilldifficult to choose the best network between chordal ring andk-cube. For this issue, Table 4 can be investigated. Accord-ing to this table, as the operating time and link failure rateincrease, k-cube finds a higher preference for choice in termsof reliability compared to chordal ring. Therefore, the k-cubenetwork can be selected as the best network in terms of all-terminal reliability.

In total, according to the analysis conducted in this sub-section, the proposed regular networks of chordal ring andk-cube for use in ERMs outperform irregular topologiesin terms of all-terminal reliability. Moreover, it should beconsidered that regular networks are scalable (i.e. easy todevelop, to implement, and routing).

5 Simulation results

In this section, we plan to present simulation results to showreal reliability for the CHR4 and k-cube topologies. Con-sequently, this section can help to understand more aboutthe performance of these networks in terms of reliability.To achieve this goal, we will use a minimal path set-basedapproach. A minimal path can be defined as a sequence ofnodes and edges from source to destination without a cycle.Therefore, the reliability of a system is the probability of theunion of its minimal paths. In this paper, we will use theefficient method, which first detects the minimal paths setsand then evaluates the reliability in the compact form of theSDP (Sum of disjoint products) using modified MVI (mul-tivariable inversion) algorithm [39]. The methodology canbe used to determine the time independent reliability of thenetworks [40]. The MATLAB 0.9 [41] software is used inprogramming these algorithms. This method contains:

• Finding the path set for a certain pair of nodes.• Generating the path sets of desired order of cardinality.• Obtaining the path sets in increasing order of their cardi-nality.

• The reliability evaluation of the terminal pair is done fromordered minimal path sets using the MVI algorithm.

According to the previous section, among irregular multi-ring networks, the 3-connected and the complex sharednetwork are the most reliable. Consequently, in this section,we consider these networks representing irregular networks.

First, we compare networks with a number of nodes equalto eight. Therefore, the reliability results for the 8-node

single-ring, CHR4 (8; 2), 3-cube, 3-connected network, andcomplex shared link network are shown in Fig. 4.

In Fig. 4, the reliability curve for each network is foundedon the reliability of the link. In this figure, all probabilities areconsidered for link reliability (i.e. zero to one). As a result,when the reliability of the links is low to moderate (about0–0.6), CHR4 (8; 2) can achieve the best reliability com-pared to other networks. However, with the high reliabilityof the links (more than average and about 0.6–1), 3-cube hasa higher performance than other networks. In addition, asshown in this figure, 3-connected network has a very nearperformance to 3-cube in terms of reliability. Also, 8-nodesingle-ring has the lowest performance in terms of reliabil-ity compared to other networks. In general, based on Fig. 4,CHR4 (8; 2) is a very good choice for cases where link relia-bility is low to moderate. In addition, although CHR4 (8; 2)has a good performance when link reliability is high, 3-cubeis a better choice in this case. Consequently, according tothese discussions, the two proposed regular topologies ofCHR4 and k-cube can provide higher performance in termsof reliability compared to irregular topologies.

In the next step, we compare networks with 16 nodes.Therefore, the reliability results for the 16-node single-ring,CHR4 (16; 2), and 4-cube are shown in Fig. 5. According tothis figure, when the reliability of the links is not high (about0–0.6), CHR4 (16; 2) has a better performance than 4-cubeand 16-node single-ring. However, when the reliability of thelinks in the network is higher than the mean (about 0.6–1), 4-cube has better performance than CHR4 (16; 2) and 16-nodesingle-ring. With the results of this section and the precedingsection, we found out that the proposed topologies have verygood performance and can be considered as an appropriatereplacement for irregular networks.

6 Discussion

In this section, we plan to discuss some of the issues relatedto the analysis conducted in this paper. There is a discussionof the difference between the network topologies analyzedin this paper. In this paper, these topologies were analyzed:8-node single-ring, 16-node single-ring, shared link network,Shared node network, complex shared link, redundant link, 3-connected, CHR4 (8; 2), CHR4 (16; 2), 3-cube, and 4-cube.On the other hand, a question arises here: Is it fair to comparethese topologies with different number of nodes and links?To answer this question, first youmust consider the influenceof the number of nodes and links on the network.

The number of nodes: In general, as the number ofnodes increases, network failure probability increases andnetwork reliability is reduced. For example, consider the 8-node single-ring and 16-node single-ring. As you can see inTables 2, 3 and 4, the reliability of the 16-node single-ring

123


Fig. 4 Network reliability as a function of link reliability for network size 8

is less than an 8-node single-ring. Instead, it should be notedthat the number of nodes in the proposed networks of CHR4(8; 2), CHR4 (16; 2), 3-cube, and 4-cube networks is morethan or equal to the number of nodes in other networks. As aresult, the analysis carried out in this paper has been in favorof irregular networks because the number of nodes in the pro-posed regular networks is more than or equal to the numberof nodes in the irregular networks. Therefore, the effective-ness of the proposed networks can be greater than it has beenanalyzed in this paper. In fact, the advantage of the proposedtopologies towards other topologies is related to how theyuse links (this way of using links in the network defines aparticular topology), and it is not related to the number ofnodes.

The number of links: When the number of nodes in a net-work is greater, the number of links used to connect thesenodes also increases subsequently. How to use these links toconnect nodes to each other, create a particular topology. Infact, this is what we analyzed in this paper. In this paper, wecompare the different methods of using links (which definedifferent topologies). The number of links among the pro-posed topologies is higher or equal to the number of linksused in other networks that studied in this paper. This makessense, as the number of inbound links increases as the numberof nodes increases. However, it should be noted that increas-ing the number of links in proportion to the number of nodesdoes not guarantee a high reliability for the network. Here,network performance is directly dependent on the networktopology and how to use these links. For example, consider an

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M. Jahanshahi

Fig. 5 Network reliability as a function of link reliability for network size 16

8-node single-ring and 16-node single-ring. The number oflinks in the 16-node single-ring has increased in proportion toits number of nodes in comparison to the 8-node single-ring.However, as shown in Tables 2, 3, and 4, 16-node single-ringhas a lower reliability than 8-node single-ring. Actually, thenetwork reliability is often reduced when the number of linksproportional to the number of nodes in the network increases.Of course, this depends on the network topology. In fact, thenetwork topology,whichdetermines howconnections are set,is the primary determinant of the performance of a network.

Considering the discussion in this section, it can befounded out that the evaluation of the proposed topologieswith other topologies in this paper is fair and logical becausethe number of nodes in the proposed topologies is more orequal to the number of nodes in other topologies. Also, the

number of links is reasonable in accordance with the largernumber of nodes in the proposed topologies. Furthermore,increasing the number of links in proportion to the numberof nodes does not guarantee an increase in reliability.

7 Conclusion

The issue of reliable networking in Ethernet ring meshnetworks (ERMs) recently has received a lot of attention.According to researches done in the past, irregular multiple-ring Ethernet mesh topologies are the main idea to improvereliability in this area. However, the irregular networks areoften complex to implement, difficult to improve, and are alsonon-scalable. Therefore, in this paper, we propose two regu-

123


lar network design schemes called k-cube and chordal ring,for the first time. In addition, detailed analysis of the reliabil-ity demonstrated that both regular networks proposed for usein ERMs achieve a greater advantage in terms of all-terminalreliability than single-ring networks and existing irregularmultiple-ring networks namely shared link topology, sharednode topology, complex shared link topology, redundant linktopology, and 3-connected network.

Since the regular topologies for using ERMs, is proposedfor the first time in this paper, the debate is still at the begin-ning of its way. Therefore, one of the most important tasks isfocused on the development of other regular networks in thefield of ERMs in order to provide a higher level of reliability.

Appendix

Here, how to use the decomposition method (DM) for calcu-lating the all-terminal reliability equations in Sect. 4.1 willbe described in detail.

(1) Shared link topology reliability (Eq. 4): First, we exam-ine Eq. (4), which is related to the reliability of sharedlink topology. How to choose DM key components isshown appropriately by a hierarchical graph in Fig. 6.In this figure, the key components are labeled by k1, k2,k3, and k4, in the order they were selected. Two statescorresponding to each key component (healthy or failed)is also shown in Fig. 6. For example, k1 is indicative ofthe failure of first key component.

Using Fig. 6, we can calculate the network reliabilityrelated to different network states according to different statesof key components that are shown by a sequence of key com-ponents and their states:

k1 � 6e−5λt − 5e−6λt (33)

k1k2 � e−2λt(3e−2λt − 2e−3λt

)(34)

k1k2k3 � e−λt(3e−2λt − 2e−3λt

)(35)

k1k2k3k4 � 3e−2λt − 2e−3λt (36)

k1k2k3k4 � 3e−2λt − 2e−3λt (37)

Now, according to Eqs. (33)–(37), the reliability of theentire network can be calculated as follows (i.e. equal to theEq. (4) in the previous section):

Rsharedlink(t) � (1 − e−λt)(6e−5λt − 5e−6λt

)

+ e−λt(e−2λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt(e−λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt((

3e−2λt − 2e−3λt)(1 − e−λt)


))))

� 15e−5λt − 23e−6λt + 9e−7λt (38)

(2) Shared node topology (Eq. 5): The hierarchy diagram ofthe key components selection for shared node topologyis shown in Fig. 7. Considering this figure, we have:

k1 � e−3λt(4e−3λt − 3e−4λt

)(39)

k1k2 � e−2λt(4e−3λt − 3e−4λt

)(40)


)(41)

k1k2k3k4 � 4e−3λt − 3e−4λt (42)

k1k2k3k4 � 4e−3λt − 3e−4λt (43)

As a result, according to Eqs. (39)–(43), the reliability ofthe entire network is given by (i.e. equal to the Eq. 5):

Rsharednode(t) � (1 − e−λt)(e−3λt

(4e−3λt − 3e−4λt

))

+ e−λt(e−2λt

(4e−3λt − 3e−4λt

)(1 − e−λt)

+ e−λt(e−λt

(4e−3λt − 3e−4λt

)(1 − e−λt)

+ e−λt((

4e−3λt − 3e−4λt)(1 − e−λt)


))))

� 16e−6λt − 24e−7λt + 9e−8λt (44)

(3) Complex shared link topology (Eq. 6): The hierarchydiagram of the key components selection for complexshared link topology is shown in Fig. 8. Consideringthis figure, network reliability for different modes is asfollows:

k1 � e−2λt(15e−5λt − 23e−6λt + 9e−7λt

)(45)

k1k2 � e−λt(15e−5λt − 23e−6λt + 9e−7λt

)(46)

k1k2k3 � 15e−5λt − 23e−6λt + 9e−7λt (47)

k1k2k3 � (1 − e−λt )(e−λt (1 − e−λt )(4e−3λt − 3e−4λt)

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt)

+ e−λt (4e−2λt − 4e−3λt + e−4λt)))

+ e−λt ((1 − e−λt )(4e−3λt − 3e−4λt)

123

M. Jahanshahi

Fig. 6 The hierarchy of the key components in the decomposition method for shared link topology

+ e−λt (4e−2λt − 4e−3λt + e−4λt ))

� 20e−4λt − 41e−5λt + 29e−6λt − 7e−7λt (48)

Therefore, according to the Eqs. (45)–(48), we have:

Rcomplexsharedlink(t)

� (1 − e−λt)(e−2λt

(15e−5λt − 23e−6λt + 9e−7λt

))

+ e−λt((

15e−5λt − 23e−6λt + 9e−7λt)(1 − e−λt)

+ e−λt((

15e−5λt − 23e−6λt + 9e−7λt)(1 − e−λt)


)

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)


)))

+ e−λt((1 − e−λt)(4e−3λt − 3e−4λt

)


)))))

� 65e−7λt − 155e−8λt + 125e−9λt − 34e−10λt (49)

(4) Redundant link topology (Eq. 7): Based on the hierarchyof the key components selection for redundant topologythat is shown in Fig. 9, network reliability related todifferent network states according to different states ofkey components are given by:

k1 � 15e−5λt − 23e−6λt + 9e−7λt (50)

k1k2 � e−2λt(3e−2λt − 2e−3λt

)(51)


)(52)

k1k2k3k4 � 3e−2λt − 2e−3λt (53)

k1k2k3k4 � 3e−2λt − 2e−3λt (54)

123


Fig. 7 The hierarchy of the key components in the decomposition method for shared node topology

According to the above equations, the reliability of theentire network is calculated as follows:

Rredundantlink(t)

� (1 − e−λt)(15e−5λt − 23e−6λt + 9e−7λt

)

+ e−λt(e−2λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt(e−λt

(3e−2λt − 2e−3λt

)(1 − e−λt)

+ e−λt((

3e−2λt − 2e−3λt)(1 − e−λt)


))))

� 24e−5λt − 50e−6λt + 36e−7λt − 9e−8λt (55)

(5) 3-Connected network (Eq. 8): How to choose key com-ponents in decomposition method for the 3-connected

network is shown in Fig. 10. Based on this figure, net-work reliability for different modes of key componentsis given by:

k1k2 � E1 � 56e−7λt − 130e−8λt + 102e−9λt − 27e−10λt

(56)

k1k2k3 � E2 � 40e−6λt − 90e−7λt + 69e−8λt − 18e−9λt

(57)

k1k2k3k4 � E3 � 32e−5λt − 68e−6λt + 49e−7λt − 12e−8λt

(58)

k1k2k3k4 � E4 � 32e−4λt − 84e−5λt

+ 86e−6λt − 40e−7λt + 7e−8λt (59)

k1k2k4 � E2 � 40e−6λt − 90e−7λt + 69e−8λt − 18e−9λt

(60)

k1k2k4k3 � E3 � 32e−5λt − 68e−6λt + 49e−7λt − 12e−8λt

(61)

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M. Jahanshahi

Fig. 8 The hierarchy of the key components in the decomposition method for complex shared link topology

k1k2k4k3 � E4 � 32e−4λt − 84e−5λt

+ 86e−6λt − 40e−7λt + 7e−8λt (62)

k1k2 � E5 � 112e−5λt − 398e−6λt + 582e−7λt

− 434e−8λt + 164e−9λt − 25e−10λt (63)

Considering the Eqs. (56)–(63), the reliability of the 3-connected network is calculated as Eq. (64):

R3−connected(t)

� (1 − e−λt)((1 − e−λt)(E1) + e−λt((1 − e−λt)(E2)

+ e−λt ((1 − e−λt)(E3) + e−λt (E4)

)))

+ e−λt((1 − e−λt)((1 − e−λt)(E2)

+ e−λt ((1 − e−λt)(E3) + e−λt (E4)

))+ e−λt (E5)

)

� 376e−7λt − 1476e−8λt + 2352e−9λt

− 1895e−10λt + 770e−11λt − 126e−12λt (64)

(6) Chordal rings (Eqs. (25)–(28)): To calculate the relia-bility of the chordal rings of CHR4(8; 2) and CHR4(16; 2), two cases have been considered: When all thechord links have failed (lower bound) and when all thechord links are active and healthy (upper bound). In caseof lower bound for a CHR4(p; q), network reliabilityis the reliability of a p-node single-ring network. Reli-ability block diagram (RBD) related to the lower boundis shown in Fig. 11(a). In addition, in the event of upperbound for a CHR4(p; q), network reliability is equalto the reliability of a parallel system contains p com-ponents in parallel from the standpoint of reliability.Figure 11(b) shows the RBD related to this case.

Therefore, according to Fig. 11a, the lower bound relia-bility for CHR4(8; 2) and CHR4(16; 2) is given by:

RCHR4(8;2)(t) � 8e−7λt − 7e−8λt (65)

RCHR4(16;2)(t) � 16e−15λt − 15e−16λt (66)

123


Fig. 9 The hierarchy of the key components in the decomposition method for redundant link topology

Also, according to Fig. 11b, the upper bound reliability ofCHR4(8; 2) and CHR4(16; 2) is calculated as follows:

RCHR4(8;2)(t)

� 1 − (1 − e−λt)8

� 8e−λt − 28e−2λt + 56e−3λt − 70e−4λt

+ 56e−5λt − 28e−6λt + 8e−7λt − e−8λt (67)

RCHR4(16;2)(t) � 1 − (1 − e−λt)16

� 16e−λt − 120e−2λt + 560e−3λt − 1820e−4λt

+ 4368e−5λt − 8008e−6λt

+ 11440e−7λt − 12870e−8λt + 11440e−9λt

− 8008e−10λt + 4368e−11λt − 1820e−12λt

+ 560e−13λt − 120e−14λt + 16e−15λt − e−16λt (68)

(7) k-cube networks (Eqs. 25–28): 3-cube network struc-ture is topologically equivalent to the structure of a3-connected network. Therefore, the reliability of the3-cube is equal to 3-connected reliability. In case of4-cube, two modes are intended for calculation of reli-ability: When all the links between the two 3-cubes arefailed, except for one of the links because working atleast one of them is necessary for working the entirenetwork (lower bound). In this case, reliability of thenetwork is equal to the reliability of two 3-connectednetworks that are connected in series. Second mode,when all the links between the two 3-cubes are activeand healthy (upper bound). In this case, the network canbe seen as a 3-cube network with redundant links.

123

M. Jahanshahi

Fig. 10 The hierarchy of the key components in the decomposition method for 3-connected network

Fig. 11 a Lower bound RBD of CHR4(p; q), b upper bound RBD ofCHR4(p; q)

Therefore, according to the above description, the 3-cubereliability can be calculated as follows:

R3−cube(t) � 376e−7λt − 1476e−8λt + 2352e−9λt

− 1895e−10λt + 770e−11λt − 126e−12λt (69)

Also, according to the above description, the 4-cube lowerbound reliability is given by:

R4−cube(t) � (R3−cube(t))(R3−cube(t)) (70)

Moreover, the 4-cube upper bound reliability can bedefined as the a 3-cube network reliability with the differ-ence that its link failure rate (λp) is equal to the failure rateof a parallel system includes two 3-cube links with failurerate λ:

R4−cube(t) � 376e−7λpt − 1476e−8λpt + 2352e−9λpt

− 1895e−10λpt + 770e−11λpt − 126e−12λpt

(71)

In Eq. (71), λp is equal to:

λp � 1

1 − (1 − e−λt

)2

⎛

⎝−d(1 − (

1 − e−λt)2)

dt

⎞

⎠

�2λ

(− 1

etλ+ 1

)

− 1etλ

+ 2(72)

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Publisher’s Note Note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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https://www.mathworks.com/

M. Jahanshahi

Mohsen Jahanshahi completedhis B.Sc. and M.Sc. studies inComputer Engineering in Iran,dated 2002 and 2005, respec-tively. He joined the departmentof Computer Engineering atIslamic Azad University (CentralTehran Branch) in 2005. Healso achieved his Ph.D. degreein Computer Engineering fromIslamic Azad University (TehranScience and Research branch)in 2011. He was promoted toassociate professor in 2015 andcurrently is an IEEE Senior

member. In addition, Jahanshahi has been a member of YoungResearchers and Elite club since 2012. Besides, he is a reviewer ofmany research papers submitted to Elsevier, Springer, and Wileyinternational journals, to name a few. His research interests includeperformance evaluation, multistage interconnection networks, wire-less mesh networks, wireless sensor networks, cognitive networks,learning systems, mathematical optimization, and soft computing.

Fathollah Bistouni received hisB.Sc. and M.Sc. degrees in ITEngineering from Payame NoorUniversity, Abhar, Iran in 2011and Qazvin Islamic Azad Uni-versity, Qazvin, Iran in 2014,respectively. He is currently aPh.D. candidate in ComputerEngineering in Central TehranBranch, Islamic Azad Univer-sity, Tehran, Iran. His researchinterests include Reliability Engi-neering, Performance Evaluation,Interconnection Networks, Eth-ernet Ring Mesh Networks, and

Wireless Mesh Networks.

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