1

Abstract

Multistage interconnection networks (MINs) play a key role in the performance of parallel computers and

multiprocessor systems. A non-negligible demand on today's modern systems is to deliver multicast traffic.

Therefore, design of efficient MINs that meets the routing requirement is vital. One of the main ideas to cope with

this problem is the use of replicated MINs. However, one of the major concerns about these networks is the problem

of unnecessary layer replication in the first stages, which recently proposes a new idea called multilayer MINs.

Previous analyzes demonstrated that this new idea could lead to cost-effective topologies that had a very close

performance to the replicated MINs in terms of throughput. Also, these analyzes indicate that these networks

outperform replicated MINs in terms of delay. However, another critical parameter to prove the performance of

most systems is reliability. Therefore, in this paper, we will focus on two essential parameters of cost and reliability

to achieve both objectives; first, evaluating the performance of multilayer MINs in terms of reliability, second, to

find the best topology among the multilayer MINs introduced in previous works in terms of cost-effectiveness (mean

time to failure/cost ratio).

Keywords: Parallel computers, Multilayer multistage interconnection network, Reliability block diagrams, Cost-

effectiveness, Mean time to failure

1. Introduction

A parallel computer requires some kinds of communication subsystems to interconnect processors, memories, and

other peripherals [1, 2]. In addition, most of the proposals of switching fabric architectures for ATM (Asynchronous

Transfer Mode) networks are based on self-routing multistage interconnection networks usually known as banyan

networks [3, 4]. Furthermore, different proposed interconnection topologies for parallel computing have been

studied and adapted for networks-on-chip (NoCs) [5, 6]. Therefore, design of an efficient interconnection network is

very crucial for construction of efficient parallel computers and systems-on-chip (SoCs) [2, 5-8, 14].

Dynamic interconnection networks include bus, crossbar, and multistage interconnection networks (MINs), which

are often used in the multiprocessor systems [15-17]. The crossbar networks are the most efficient but the most

expensive ones [9, 10]. On the other hand, shared bus networks have lower costs but lower performance. MINs

provide a compromise between the above networks because they provide efficient performance using a reasonable

number of switches [11, 12, 17-21].

Typically, an N×N MIN is constructed by stages of c×c switching elements and there are

switches

per stage. Therefore, while the crossbar network needs switching elements, the network complexity of an N×N MIN (defined as the total number of switching elements) is equal to . Improving the capacity of fault-tolerance and reliability is an essential for these networks. [13, 20-24]. Fig. 1 shows the general structure of a

MIN [25-28].

Fig. 1. General structure of MINs.

On the other hand , in case of applying MINs in multicore processors or parallel computers, multicast traffic will

be likely raised [29, 34]. Multicasting can be performed by copying the packets across c×c switching elements. This

Reliability Analysis of Multilayer Multistage Interconnection Networks

Fathollah Bistouni1 and Mohsen Jahanshahi

2

1Young Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran 2Young Researchers and Elite Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

1f_bistouni@qiau.ac.ir, 2mjahanshahi@iauctb.ac.ir

mailto:f_bistouni@qiau.ac.ir

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scheme is called message replication while routing [29]. In this scheme, the packet is waiting to reach the middle

stage then it is copied. Next, the copies of the packet will continue their routes through the rest of stages. Therefore,

because of packet replication, the number of packets will be increased from stage to stage. Compared to the first

stages of a network, more switching power is needed in the last stages [29]. According to these discussions, we

deduce that MINs require three important factors for using in modern systems: (1) High reliability. (2) Efficient

multicasting. (3) Cost-effectiveness.

One of the main ideas that adapt the MINs to provide better reliability and multicasting is using the replicated

MINs. Replicated MINs enlarge the typical MINs by replicating those L times. The resulting MINs are arranged in

L layers. Fig. 2 shows the architecture of such an 8×8 replicated MIN consisting of two layers (L=2) in a three

dimensional view. A single L-layer replicated MIN of size N×N consists of stages of

switching

elements. Typically, all switching elements are of size 2×2. In addition, L-layer replicated MINs of size N×N have

demultiplexers of size 1×L in the input stage and multiplexers of size L×1 in the output stage. The network

complexity of an N×N L-layer replicated MIN is

.

In contrast to the conventional MINs, replicated MINs offer better performance in terms of reliability and

multicasting because multiple paths exist for a source-destination pair. However, in replicated MINs, the whole

network is replicated. This can lead to unnecessary layer replications in the first stages, whereas, we need more layer

replications for last stages to supply the required switching power. Consequently, in the replicated MINs, although a

large number of switching elements are used, but we cannot achieve the maximum performance for the cost spent.

Fig. 2. Three-dimensional perspective of an 8×8 two-layer replicated MIN.

In order to supply the switching power requirements more efficiently, a new topology named multilayer MINs is

proposed in [29]. In this type of MINs, the main idea is that the layer replication is defined for each stage instead of

the entire network. Therefore, the replicating the number of layers in each stage avoids unnecessary layer

replications in the first stages as in case of the replicated MINs.

In addition, in multilayer MINs, to restrict the number of layers and the amount of required chip space, two

choices are considered: beginning the replication in a more rear stage and/or stopping further layer replication if a

given number of layers are reached. In order to further reduction of network complexity, the combination of both

choices can be used. Such a network is described using (start of replication) (growth factor) and (layer limit). The structure of the multilayer MINs will be discussed in details in Section 2.

In [29], performance of the multilayer MINs and replicated MINs were analyzed in terms of three parameters:

throughput, delay, and cost. In this study, initially, the 16×16 networks consisting of 2×2 switching elements were

evaluated. In [29], it was demonstrated that the idea of the multilayer MINs could lead to cost-effective topologies

that can achieve higher performance in terms of throughput and delay parameters compared to replicated MINs.

Therefore, it is desirable to do more and more research on this type of MINs to investigate them from different

aspects. On the other hand, many researchers concluded that one of the essential requirements for the most systems

is reliability. That is the reason, until now, many researchers have emphasized that interconnection networks should

also be evaluated in terms of this important parameter [17, 18, 21, 28, 30-33]. However, to the best of our

knowledge, reliability of the multilayer MINs has been fully considered in none of the reported works. Therefore, in

this paper, our goal is to evaluate the reliability of these networks in order to analyze the behavior of multilayer

MINs with regard to many important dimensions. Moreover, in this paper, the cost of these networks will be

examined to verify their cost-effectiveness.

Overall, the contribution of this paper is twofold: First, performance evaluation of multilayer MINs in terms of

reliability and cost-effectiveness compared to the replicated MINs. Second, finding the best topology from the

multilayer MINs introduced in [29] in terms of cost-effectiveness (i.e., mean time to failure / cost ratio).

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The rest of the paper is organized as follow: Section 2 will provide a useful background. Three reliability

parameters, terminal, broadcast, and network will be analyzed in section 3. Finally, some conclusions will be

derived in section 4.

2. Background

In subsection 2.1, first, the structure of the multilayer MINs will be explained in more detail. Then, in subsection

2.2, related works within the context of MINs reliability is presented.

2.1. Structure of multilayer MINs

Following three main factors are defined to describe the structure of multilayer MINs [29, 34]: (1) Start

replication factor (GS): denotes the stage number at which the replication starts. (2) Growth factor (GF): denotes the

number of layers which can be developed in a stage by each switching element. (3) Layer limit factor (GL): denotes

the maximum number of layer replication.

For instance, Fig. 3 shows an 8×8 multilayer MIN consisting of 2×2 switching elements with GS=2, GF=2, and

GL= no limit, in lateral view. In addition, to better appreciate this, an 8×8 multilayer MIN consisting of 2×2

switching elements with GS=2, GF=2, and GL=2 is depicted in Fig. 4.

Fig. 3. Lateral view of an 8×8 multilayer MIN with GS=2, GF=2, and GL=no limit.

Fig. 4. (a) Three dimensional view of 8×8 multilayer MIN with GS=2, GF=2, and GL=2, (b) Lateral view.

Given the above discussion, it can be inferred that typical MINs and replicated MINs can be considered as special

cases of multilayer MINs. The typical MINs are equivalent to multilayer MINs with GF=1. In this case, GS and GL

have no effect. Also, replicated MINs are equivalent to multilayer MINs with GS=1, GF=L, and GL=L.

Overall, a multilayer MIN of size N×N with GS=S, GF=F, and GL=L consists of switching stages. The number of switching elements (NOSE) in the multilayer MIN is given by the Eq. (1):

(1)

Also, the size of the switching elements (SOSE) for both cases of S=1 and S>1 is given by Eqs. (2) and (3):

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(2)

(3)

In the multilayer MINs, there are N multiplexers of size L×1 for output links of switches in stage . Also, the presence or absence of demultiplexers for any input before stage 1 of the network depends on the parameter S. if

S=1, then there are N demultiplexers of size 1×L for sources before stage 1. Otherwise, none of the demultiplexers

will be used in the network. Assuming S=1, then according to Eqs. (1), (2), and (3), the network complexity (NC) of

an N×N multilayer MIN is given by:

(4)

Now, assuming S > 1, then according to Eqs. (1), (2), and (3), the network complexity (NC) of an N×N multilayer

MIN is given by Eq. (5):

(5)

2.2. State-of-the-art

Reliability analysis of multilayer MINs has not been investigated so far and hence it is the main objective of this

paper. Moreover, in this paper, further studies will be discussed below in terms of reliability analysis:

Reliability analysis of MINs can be carried out from three different perspectives of terminal, broadcast, and

network. In the study of MINs, individual investigation of these parameters is less valuable. In the previous works,

usually only one of these parameters was considered for analysis. For instance, in [18, 28, 33, 36, 37], network

reliability has been examined solely or in [32, 35], just terminal reliability has been analyzed. However, in this paper,

we will analyze all aspects of the reliability (i.e. terminal, broadcast, and network) to achieve a comprehensive

analysis.

From another point of view, reliability analysis can be done either time-independent or time-dependent. In time

independent analysis, each component of the system has a probability of success and it is assumed that this

probability is constant for all times. In contrast, time-dependent analysis looks at reliability as a function of time.

That is, a known failure distribution is assigned to each component. According to [18, 28, 32, 33, 37], the time-

dependent analysis is usually performed to obtain the mean time to failure (MTTF), which is one of the important

parameters to evaluate the performance of the most systems. However, in the most of reported works, just one of

these analyses (time-dependent or time-independent) has been studied. For instance, in [17, 18, 33], the reliability

analysis just focuses on the time-dependent analysis, or in [21, 30, 35, 36, 38], the reliability analysis only focuses

on time-independent analysis. However, this paper will examine both types of analysis to fully investigate the

behavior of MINs.

In total, given the above discussions, it can be claimed that reliability analysis are worthwhile in this paper.

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3. Reliability and Cost

Reliability is defined by the IEEE as “the ability of a system or component to perform its required functions under

stated conditions for a specified period of time” [39]. Therefore, in the domain of interconnection networks, many

researchers have been convinced that it is the most immediate parameter for each efficient network [16, 21, 23, 32-

37].

Complex networks consist of multiple source and destination nodes, complex topology, interdependencies at the

component and system levels, and uncertainties in actual conditions of network components and deterioration

models [46-48]. According to this definition, lifeline networks such as railway and water distribution networks [46,

47, 49], wireless mobile ad hoc networks (MANETs) [48, 50, 51], wireless mesh networks [52-56], wireless sensor

networks [57-60], sensors based on nano-wire networks [61], social networks [62], stochastic-flow manufacturing

networks (SMNs) [63], and interconnection networks [16, 21, 23, 32-38, 64-74] are known as complex network

systems from the viewpoint of reliability.

With regard to the reported researches, reliability investigation of the complex networks can be accomplished by

simulation or analytical models. Although simulation-based approaches are easily implemented, there are some

restrictions on their effectiveness. For instance, the statistical nature of simulation models may need a large number

of samples to achieve an acceptable level of convergence for very small or large probability estimation. In addition,

their computational efficiency depends on the number of nodes and links in a network as they use a path searching

algorithm to check the connectivity between the terminal nodes for each sample. Also, the random number

generation process is computationally demanding especially when the components are statistically correlated as the

process includes a matrix decomposition process such as the Cholesky decomposition. Furthermore, simulation

presents a small range of results compared to the analytical methods. However, since the simulation based methods

just require the generation of random samples of hazard intensity measures and the corresponding uncertain status of

network components with no need to identify complex connection system events, analytical methods have been

avoided because of their complexity in favor of the simplicity of using simulation. Using reliability equations,

analytical methods have been developed to present an exact solution for computing the reliability of a system.

Therefore, the time-consuming calculations and the non-repeatability issue of the simulation methodology should be

eliminated. Given the reliability equation for a system, further analyses on the system such as computing exact

values of the reliability, failure rate at specific points in time, computation of the system MTTF (mean time to

failure) can be performed. In addition, reliability optimization or fault-tolerance techniques can be utilized to

promote design improvement efforts. Therefore, in this paper, we will use the reliability block diagram (RBD)

method as an accurate analytical method for evaluating the reliability of complex systems.

This section is divided into five subsections: subsection 3.1. “Reliability block diagrams”, subsection 3.2.

“Terminal reliability”, subsection 3.3. “Broadcast reliability”, subsection 3.4. “Network reliability”, and subsection

3.5. “Discussion”. Initially, in subsection 3.1, an explanation will be given on how to perform the reliability

analysis. Then, in subsections 3.2 through 3.4, both time-dependent and time-independent analysis will be carried

out. In addition, a detailed analysis of the cost will be conducted in each subsection to evaluate the cost-

effectiveness.

It should be noted that investigated MINs in each subsection are exactly as it is in [29]. Therefore, initially,

reliability of 16×16 networks consisting of 2×2 switching elements will be analyzed. Then, reliability of 64×64

networks consisting of 4×4 switching elements is investigated. Also, in this paper similar to [29], the networks will

be shown using the number of layers in each stage. For example, network 8888 representing a four-stage replicated

MIN with eight layers, or network 1248 representing a four-stage multilayer MIN with one layer at stage 1, two

layers at stage 2, four layers at stage 3, and eight layers at stage 4.

3.1. Reliability block diagrams

Reliability of a system often depends on the reliability of its components. Therefore, the reliability of a system

cannot be properly measured, regardless of the role of individual system components and how they impact on

system performance. As a result, we need to have precise information about the components of a system and the

relationships among them from the reliability standpoint. A reliability block diagram (RBD) is a visualization that

illustrates the simple and redundant arrangement of critical modules that are required to be operational to deliver

service [39-41]. This simple visualization is surprisingly powerful because it enables one to gradually expose the

complexity of a system to analysis. Therefore, the RBDs can be very useful for the analysis of MINs reliability and

assist us in a better understanding of them [18, 28, 33, 35] and, we will take advantage of these diagrams.

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The premise is that for a service to be up (available) there must be at least one path across the diagram through

modules that are all up. Therefore, generally, the components of a system can be placed in three different statuses:

series, parallel, and series-parallel. In series, the success of a system depends upon the success of individual

components. For example, a series RBD is shown in Fig. 5.

Fig. 5. Series RBD.

Assuming that is the probability of the component i being operational, system reliability is given by Eq. (6) for the scenario depicted in Fig. 5.

(6)

In case of independent components, Eq. (6) becomes:

(7)

In parallel, the success of a single component can cause the entire system to work successfully. For example, a

parallel RBD is shown in Fig. 6. Assuming that is the failure probability of component i, system unreliability is calculated using Eq. (8) for Fig. 6.

(8)

Again, in case of independent components, Eq. (8) becomes:

(9)

As a result, the parallel system reliability is calculated by the following equation:

(10)

Fig. 6. Parallel RBD.

Components of a system can be a combination of series and parallel modes that known as the series-parallel. An

example of this is shown in Fig. 7. In this type, the system reliability is calculated according to Eqs. (7) and (10).

The way to obtain system reliability in such cases is to break the total system configuration down into homogeneous

subsystems. Then, considering each of these subsystems separately as a unit, and calculating their reliabilities.

Finally, putting these simple units back (via series or parallel recombination) into a single system and calculating its

reliability.

Fig. 7. A series-parallel RBD.

3.2. Terminal reliability

Terminal reliability is defined as the probability of a successful communication between a source-destination pair.

Therefore, it can be calculated by considering a source-destination pair in the network. Consequently, with regard to

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this issue, we will model the networks in a RBD. Then, according to the RBDs, the reliability of the networks will

be calculated. In this paper like [29], first, the 16×16 networks consisting of 2×2 switching elements will be

examined. In addition, networks studied in this paper are as those are in [29]. Therefore, initially, networks 1248,

1888, 1188, and 8888 will be analyzed, which all have eight layers in last stage. The networks 1248, 1888, and 1188

are multilayer-type MINs and network 8888 is a replicated-type MIN. The terminal reliability RBDs of the four

networks are shown in Fig. 8. In these RBDs, blocks corresponding to switching elements of size c1×c2 are shown

with SEc1×c2 and blocks corresponding to multiplexers and demultiplexers are shown with MUX and DMUX,

respectively.

Fig. 8. Terminal reliability RBD of (a) network 1248, (b) network 1888, (c) network 1188, and (d) network 8888.

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In the field of MINs, the researchers believe that the switching elements are significantly more unreliable than

links [35, 36]. Therefore, the switch fault model is used in virtually all previous works [9, 18, 21, 28, 30, 32, 33, 35-

38, 42]. For this reason, in this paper, we will use the switch fault model for reliability analysis of MINs. Therefore,

it will be assumed that each switching component (i.e. switching elements, multiplexers, and demultiplexers) may

fail. In addition, we will assume that reliability of a 2×2 switch (SE2) is equal to r. Therefore, given the number of

gates per switching element, the probability of it being operational can be computed based on r [18]. It is assumed

that the hardware complexity of a component is directly proportional to the number of gates [32, 33]. Given the

above discussions, the reliability of each switching components is calculated as follow:

(11)

(12)

Therefore, according to Fig. 8, the terminal reliability of each N×N network is calculated as follow:

(13)

(14)

(15)

(16)

Considering the Eqs. (13) through (16), terminal reliability for network size 16×16 is given by:

(17)

(18)

(19)

(20)

Another important parameter that is used to study the reliability of the most systems is mean time to failure

(MTTF). This parameter is considered as one of the performance metrics within interconnection networks [18, 28,

32, 33, 37, 42], which can be obtained by time-dependent reliability analysis. For this reason, we will also analyze

the networks by the key parameter to achieve a comprehensive assessment. In this paper, like [32, 33, 42], we

assume that the times-to-failure of the switching elements is described with an exponential distribution. The MTTF

is calculated using Eq. (21).

(21)

According to Eq. (21), we have:

(22)

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(23)

(24)

(25)

The next parameter that should be discussed is the cost. This parameter is a major issue in MIN design and if such

high performance comes at the expense of high cost, it may have less avail in practice. To estimate the cost of a

MIN, one common method in [21, 29, 32, 33, 42-45], is to calculate the switch complexity with an assumption that

the cost of a switch is proportional to the number of gates involved, which is roughly proportional to the number of

cross-points within a switch. For example, a 2×2 switch has four units of hardware cost, whereas a 3×3 switch has

nine units. For the multiplexers and demultiplexers, we roughly assume that each of m×1 multiplexers or 1× m

demultiplexers have m units of cost.

According to the above description, the cost of multilayer MINs is given by the Eqs. (26) and (27). Suppose that

S=1, then, we have:

(26)

And, assuming S>1, we have:

(27)

According to Eqs. (17) through (20), the results of the terminal reliability analysis for the networks 1248, 1888,

1188, and 8888, as a function of switch reliability, are summarized in Table 1. Also, according to Eqs. (22) through

(25) the results of the MTTF analysis as a function of the failure rate (per hour) for the switching elements (λ) are

given in Table 2. It should be noted that a reasonable estimate for λ is about 10-6

per hour [32, 33]. However, in this

paper, λ will be considered in the range from 10-8

to 10-2

. Moreover, considering the Eqs. (26) and (27), the results

of the cost analysis are given in Table 3.

Table 1. Terminal reliability as a function of switch reliability.

Switch reliability (r) Rt(network 1248) Rt(network 1888) Rt(network 1188) Rt(network 8888)

0.9 0.623 827 0.348 668 0.313 81 0.655 972

0.91 0.658 779 0.389 411 0.354 368 0.685 684

0.92 0.694 453 0.434 386 0.399 637 0.716 363

0.93 0.730 794 0.483 981 0.450 103 0.748 04

0.94 0.767 749 0.538 615 0.506 298 0.780 745

0.95 0.805 271 0.598 737 0.568 8 0.814 505

0.96 0.843 314 0.664 833 0.638 239 0.849 346

0.97 0.881 837 0.737 424 0.715 301 0.885 293

0.98 0.920 807 0.817 073 0.800 731 0.922 368

0.99 0.960 2 0.904 382 0.895 338 0.960 596

Table 2. MTTF as a function of switch failure rate.

Switch failure rate (λ) MTTFt(network 1248) MTTFt(network 1888) MTTFt(network 1188) MTTFt(network 8888)

10-8 1.8646324099615258×107 9958760.604221504 9086339.474278633 2.2222222222222224×107

10-6 186463.2409961526 99587.60604221505 90863.39474278632 222222.2222222222

10-4 1864.632409961526 995.8760604221505 908.6339474278632 2222.222222222222

10-2 18.64632409961526 9.958760604221505 9.086339474278631 22.22222222222222

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Table 3. The cost for each of the networks 1248, 1888, 1188, and 8888.

Network Parameter

Cost GS GF GL

1248 2 2 - 832

1888 2 8 8 1152

1188 3 8 8 928

8888 1 8 8 1280

According to Table 1, it is evident that the best and the worst results in terms of terminal reliability are owned by

networks 8888 and 1188, respectively. Also, the performance of networks 1248 and 8888 is much better than the

networks 1888 and 1188 in terms of terminal reliability. However, according to Table 3, the network 1248 has a

lower cost compared to the networks 1888, 1188, and 8888. As a result, the main competition is between two

networks 8888 and 1248 for the best performance. Although network 8888 achieves slightly higher terminal

reliability compared to network 1248 (according to Table 3), it imposes much higher costs than the network 1248.

Therefore, the network 1248 is the best network in terms of both cost and reliability among four networks. Because,

it provides an acceptable terminal reliability with low cost compared to the other three networks.

Similarly to Table 1, Table 2 shows that the best and the worst results in terms of MTTF are owned by the

networks 8888 and 1188, respectively. Also, the MTTF of network 1248 is close to network 8888 and MTTF of

these networks has a significant advantage in comparison with both networks 1888 and 1188. Therefore, although

the network 1248 is close to network 8888 in terms of MTTF, it has the lowest cost among the other networks.

Consequently, it is reasonable that network 1248 is selected as the most efficient network in terms of reliability and

cost compared to the other three networks. However, according to [18, 32, 33, 37, 42] Eq. (28) is used to

demonstrate the cost-effectiveness (CE) of network 1248 compared to three other networks:

(28)

According to Eq. (28), the results of cost-effectiveness analysis are summarized in Table 4. As it is expected, the

results in Table 4 clearly confirm that network 1248 is cost-effectiveness compared to the other three networks.

Indeed, the results prove that the multilayer MINs have better performance than replicated MINs, in terms of cost

and terminal reliability, which are two crucial parameters in the field of interconnection networks.

Table 4. Cost-effectiveness for each of the networks 1248, 1888, 1188, and 8888.

Switch failure rate (λ) CEt(network 1248) CEt(network 1888) CEt(network 1188) CEt(network 8888)

10-8 22411.447235 8644.757469 9791.314089 17361.111111

10-6 224.114472 86.447575 97.913141 173.611111

10-4 2.241145 0.864476 0.979131 1.736111

10-2 0.022411 0.008645 0.009791 0.017361

This time, the performance of 64×64 networks consisting of 4×4 switching elements is examined in terms of

terminal reliability, MTTF, and cost. Therefore, similarly to [29], the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-

7, and 8-8-8 will be considered for the analysis (again, the digits of the legend refer to the number of layers at stage

1, stage 2, etc). Care should be taken that networks 7-7-7 and 8-8-8 are kinds of replicated MINs and the rest of

them are multilayer MINs. In a manner similar to the previously studied networks (1248, 1888, 1188, and 8888), the

terminal reliability of these networks of size N×N is given by:

(29)

(30)

(31)

(32)

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(33)

(34)

Now, according to Eqs. (29) through (34), in case of network size 64×64, we have:

(35)

(36)

(37)

(38)

(39)

(40)

Also, the MTTF of these networks is calculated as follow:

(41)

(42)

(43)

(44)

(45)

(46)

According to Eqs. (35) through (46), the results of terminal reliability and MTTF analysis of the networks 1-4-16,

1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8 are summarized in Tables 5 and 6, respectively. Also, the results of the cost

analysis for these networks are given in Table 7. According to Table 5, networks 1-4-16, 1-116, and 1-16-16 cannot

compete with the networks 1-2-4, 7-7-7, and 8-8-8 in terms of terminal reliability. Therefore, the main competition

on efficiency is among three networks 1-2-4, 7-7-7, and 8-8-8. The networks 7-7-7 and 8-8-8 have a very close

performance in terms of terminal reliability. Also, two networks are slightly better than the network 1-2-4, especially

in low switch reliabilities. However, as the switch reliability increases, reliability of these three networks is very

close to each other. To select the most efficient one from these three networks, the cost should be considered as an

important parameter. According to Table 7, the network 1-2-4 is the least-cost network among other networks,

which cannot be ignored. In addition, like Table 5, Table 6 shows that the networks 1-2-4, 7-7-7, and 8-8-8 are the

top three networks in terms of MTTF. However, as Table 7 demonstrates, the networks 7-7-7 and 8-8-8 have much

higher cost than the network 1-2-4. According to the arguments made, it can be concluded that the network 1-2-4 is

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a better solution to satisfy both parameters of cost and terminal reliability; it is able to obtain an acceptable

performance in terms of reliability and MTTF without imposing a high cost on the network.

The results of cost-effectiveness analysis of network 1-2-4 are summarized in Table 8. As it can be observed, the

network 1-2-4 is the most cost-effective network among other networks. In fact, these results are indicative of the

fact that multilayer MINs have great potential to achieve high terminal reliability at a low cost compared to

replicated MINs.

Table 5. Terminal reliability as a function of switch reliability.

Switch reliability (r) Rt( 1-4-16) Rt(1-1-16) Rt(1-16-16) Rt(1-2-4) Rt(7-7-7) Rt( 8-8-8)

0.9 0.067 332 0.000 508 0.000 773 0.238 283 0.623 841 0.609 98

0.91 0.095 22 0.001 125 0.001 64 0.285 861 0.671 75 0.655 301

0.92 0.132 626 0.002 47 0.003 448 0.340 5 0.716 7 0.698 082

0.93 0.181 717 0.005 38 0.007 192 0.402 529 0.758 274 0.738 285

0.94 0.244 613 0.011 62 0.014 883 0.472 047 0.796 537 0.776 306

0.95 0.323 131 0.024 894 0.030 564 0.548 838 0.832 044 0.812 884

0.96 0.418 565 0.052 909 0.062 294 0.632 273 0.865 728 0.848 917

0.97 0.531 741 0.111 574 0.126 031 0.721 206 0.898 652 0.885 224

0.98 0.663 721 0.233 495 0.253 147 0.813 881 0.931 713 0.922 364

0.99 0.817 509 0.484 991 0.504 886 0.907 861 0.965 435 0.960 596

Table 6. MTTF as a function of switch failure rate.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

MTTFt(1-4-16) 4256068.230337063 42560.68230337063 425.6068230337063 4.256068230337063

MTTFt(1-1-16) 1388888.88825871 13888.8888825871 138.888888825871 1.38888888825871

MTTFt(1-16-16) 1470587.046191497 14705.87046191498 147.0587046191498 1.470587046191498

MTTFt(1-2-4) 7298193.346973835 72981.93346973835 729.8193346973835 7.298193346973835

MTTFt(7-7-7) 1.4386283952890933×107 143862.8395289093 1438.628395289093 14.38628395289093

MTTFt(8-8-8) 1.413341921236658×107 141334.1921236658 1413.341921236658 14.13341921236658

Table 7. The cost for each of the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8.

Network Parameter

Cost GS GF GL

1-4-16 2 4 - 10240

1-1-16 3 16 - 9472

1-16-16 2 16 16 13312

1-2-4 2 2 - 2816

7-7-7 1 7 7 6272

8-8-8 1 8 8 7168

Table 8. Cost-effectiveness of point-terminal for each of the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

CEt(1-4-16) 415.631663 4.156317 0.041563 0.000416

CEt(1-1-16) 146.631006 1.46631 0.014663 0.000147

CEt(1-16-16) 110.470782 1.104708 0.011047 0.00011

CEt(1-2-4) 2591.687978 25.91688 0.259169 0.002592

CEt(7-7-7) 2293.731498 22.937315 0.229373 0.002294

CEt(8-8-8) 2253.415053 22.534151 0.225342 0.002253

In sum, in analysis of 16×16 networks consisting of 2×2 switching elements, it was concluded that the multilayer

network 1-2-4-8 is a good choice in terms of both terminal reliability and cost parameters, which is affordable. Also,

in the analysis of 64×64 networks consisting of 4×4 switching elements, it was concluded that the multilayer

network 1-2-4 is a reasonable choice in terms of both terminal reliability and cost parameters.

3.3. Broadcast reliability

Broadcast reliability is defined as the possibility of a connection establishment between one source to all

destinations in the network. According to the description of broadcast reliability, similarly to subsection 3.2, first,

the networks will be modeled as a RBD. Then, according to the RBD, the broadcast reliability for each network will

be calculated. In addition, in this subsection, similarly to subsection 3.2, first, the 16×16 networks consisting of 2×2

switching elements (networks 1248, 1888, 1188, and 8888) will be analyzed. The broadcast reliability RBDs related

to each of these networks is shown in Fig. 9.

13

Fig. 9. Broadcast reliability RBD of (a) network 1248, (b) network 1888, (c) network 1188, and (d) network 8888.

According to Fig. 9, the broadcast reliability for each N×N network is calculated by the following Eqs.:

(47)

(48)

(49)

14

(50)

As a result, for the size 16×16, we have:

(51)

(52)

(53)

(54)

Also, the MTTF of point-broadcast is given by the following Eqs.:

(55)

(56)

(57)

(58)

According to the Eqs. (51) through (58), the results of the reliability and MTTF analyzes is given in Tables 9 and

10, respectively. According to Table 9, it is clear that the networks 1888 and 1188 have less broadcast reliability

compared to the networks 1248 and 8888. Although these results show that the network 8888 is slightly better than

the network 1248 in terms of broadcast reliability, both of them achieve a very close results. On the other hand,

according to Table 3, it was demonstrated that the network 1248 had been the least-cost network among four

networks. Therefore, similarly to subsection 3.2, it is rational to choose the network 1248 as the most efficient

network compared to the other three networks considering both reliability and cost parameters.

Table 10 also indicates the fact that the networks 1248 and 8888 have the best performance in terms of MTTF.

However, according to Table 3, the network 1248 is a better choice due to its cost-effectiveness. In order to fully

prove the cost-effectiveness of network 1248 compared to three other networks, the results of cost-effectiveness of

point-broadcast analysis are summarized in Table 11. As Table 11 shows, the highest value in terms of cost-

effectiveness is owned by the network 1248. Therefore, it can be concluded that the idea of multilayer MINs can

lead to cost-effective topologies, which have also high broad cast reliability.

Table 9. Broadcast reliability as a function of switch reliability.

Switch reliability (r) Rb(network 1248) Rb(network 1888) Rb(network 1188) Rb(network 8888)

0.9 0.015 183 0.012 931 0.005 324 0.023 415

0.91 0.025 203 0.021 081 0.009 404 0.036 129

0.92 0.041 036 0.033 801 0.016 382 0.054 767

0.93 0.065 479 0.053 362 0.028 181 0.081 627

0.94 0.102 315 0.083 092 0.047 95 0.119 815

0.95 0.158 52 0.127 902 0.080 833 0.173 621

0.96 0.234 514 0.195 114 0.135 229 0.249 104

0.97 0.344 587 0.295 65 0.224 792 0.354 892

0.98 0.497 724 0.445 694 0.371 6 0.503 126

0.99 0.709 051 0.668 972 0.611 117 0.710 553

15

Table 10. MTTF of point-broadcast as a function of switch failure rate.

Switch failure rate (λ) MTTFb(network 1248) MTTFb(network 1888) MTTFb(network 1188) MTTFb(network 8888)

10-8 2753202.762817228 2482012.9467939287 2037027.54425225 2896068.1174419126

10-6 27532.027640461245 24820.12946997553 20370.275451832702 28960.68117057186

10-4 275.320276485077 248.20129477386715 203.70275458758874 289.60681178778754

10-2 2.753202574996651 2.4820129499749735 2.0370275173078976 2.8960681536519672

Table 11. Cost-effectiveness of point-broadcast for each of the networks 1248, 1888, 1188, and 8888.

Switch failure rate (λ) CEb(network 1248) CEb(network 1888) CEb(network 1188) CEb(network 8888)

10-8 3309.137936 2154.525127 2195.072785 2262.553217

10-6 33.091379 21.545251 21.950728 22.625532

10-4 0.330914 0.215453 0.219507 0.226255

10-2 0.003309 0.002155 0.002195 0.002263

In the next analysis, the performance of 64×64 networks consisting of 4×4 switching elements, the networks 1-4-

16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8 will be examined. Similarly to the previous analysis, the broadcast

reliability of these networks for size N×N is calculated by the following Eqs.:

(59)

(60)

(61)

(62)

(63)

(64)

According to Eqs. (59) through (64), the broadcast reliability for the networks of size 64×64 is obtained by the

following equations:

(65)

(66)

(67)

(68)

(69)

(70)

Moreover, the MTTF of point-broadcast for each network is calculated as follow:

16

(71)

(72)

(73)

(74)

(75)

(76)

The results of broadcast reliability and MTTF of point-broadcast analysis are shown in Tables 12 and 13,

respectively. As the tables show, the networks 1-4-16, 1-1-16, and 1-16-16 because of poor results cannot compete

with the networks 1-2-4, 7-7-7, and 8-8-8 in terms of broadcast reliability and MTTF. On the other hand, the results

demonstrate that the network 1-2-4 has the best performance compared to the networks 7-7-7 and 8-8-8 in terms of

broadcast reliability and MTTF of point-broadcast. In addition, according to Table 7, it was proven that the network

1-2-4 had been the least-cost among six networks. Therefore, network 1-2-4 is the best choice for both the broadcast

reliability and cost.

To complete the discussion, the results of cost-effectiveness analysis are shown in Table 14. Table 14 shows that

network 1-2-4 is the most cost-effective network among other networks. These results reflect the fact that the

multilayer MINs have a greater potential for providing two important parameters of broadcast reliability and cost,

compared to the replicated MINs.

Table 12. Broadcast reliability as a function of switch reliability.

r Rb( 1-4-16) Rb(1-1-16) Rb(1-16-16) Rb(1-2-4) Rb(7-7-7) Rb( 8-8-8)

0.9 7.94972316428074E-18 2.79497219821819E-17 7.95082663481238E-18 8.21349333703834E-8 6.2570198007455E-9 1.29053305237407E-9

0.91 6.59331623870778E-16 2.01561737445077E-15 6.59105985523186E-16 0.000 001 5.55982242745457E-8 1.37213476579393E-8

0.92 5.20061032235253E-14 1.3746414047872E-13 5.1895114085152E-14 0.000 003 4.81900979661001E-7 1.42005404644458E-7

0.93 3.89757067153724E-12 8.79970157429262E-12 3.86830576980396E-12 0.000 02 0.000 004 0.000 001

0.94 2.76632768287561E-10 5.20888504572868E-10 2.70608270184503E-10 0.000 121 0.000 033 0.000 014

0.95 1.84320918103762E-8 2.77352110809128E-8 1.74019479557306E-8 0.000 705 0.000 265 0.000 13

0.96 0.000 001 0.000 001 0.000 001 0.003 972 0.001 983 0.001 149

0.97 0.000 06 0.000 047 0.000 045 0.021 155 0.013 485 0.009 06

0.98 0.002 411 0.001 428 0.001 512 0.101 488 0.076 094 0.058 013

0.99 0.060 884 0.038 53 0.040 108 0.391 871 0.312 549 0.267 733

Table 13. MTTF of point-broadcast as a function of switch failure rate.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

MTTFb(1-4-16) 361765.1356538159 3617.6513568313558 36.176513603775355 0.36176517436326266

MTTFb(1-1-16) 308628.0419673688 3086.280419678657 30.86280420107203 0.3086279867852753

MTTFb(1-16-16) 312435.5005125005 3124.355005322764 31.24355007240392 0.3124353405682797

MTTFb(1-2-4) 968142.0545482 9681.420546090934 96.8142055063278 0.9681419825186068

MTTFb(7-7-7) 817466.3271115157 8174.663271349142 81.74663278633666 0.8174661522739078

MTTFb(8-8-8) 735503.484195854 7355.034842345751 73.55034851023994 0.7355032076284417

Table 14. Cost-effectiveness of point-broadcast for each of the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

CEb(1-4-16) 35.328 627 0.353 286 0.003 533 0.000 035

CEb(1-1-16) 32.583 197 0.325 832 0.003 258 0.000 033

CEb(1-16-16) 23.470 215 0.234 702 0.002 347 0.000 023

CEb(1-2-4) 343.800 446 3.438 004 0.034 38 0.000 344

CEb(7-7-7) 130.335 83 1.303 358 0.013 034 0.000 13

CEb(8-8-8) 102.609 303 1.026 093 0.010 261 0.000 103

17

3.4. Network reliability

Network reliability is defined as the probability of successful communication between all sources and all

destinations in the network. According to this description, at first, the networks will be modeled as a RBD, and then

the network reliability will be calculated according to the RBDs.

Like the previous subsections, first, the 16×16 networks consisting of 2×2 switching elements (networks 1248,

1888, 1188, and 8888) will be analyzed and then, network reliability RBDs associated with each one is shown in

Fig. 10.

Fig. 10. Network reliability RBD of (a) network 1248, (b) network 1888, (c) network 1188, and (d) network 8888.

According to the RBDs in Fig .10, the network reliability for each of these networks is given by:

18

(77)

(78)

(79)

(80)

According to Eqs. (77) through (80), the equations of network reliability to network size 16×16 can be calculated

as follow:

(81)

(82)

(83)

(84)

Also, the MTTF of point-network is given by the following equations:

(85)

(86)

(87)

(88)

According to Eqs. (81) through (88), the results of network reliability and MTTF analysis are summarized in

Tables 15 and 16, respectively. As Table 15 shows, the highest and lowest network reliability is owned by the

networks 1248 and 1188, respectively. On the other hand, Table 16 also shows that network 1248 has the best

performance among the other networks in terms of MTTF of point-network. In addition, according to Table 3, it was

demonstrated that the network 1248 is the least-cost network compared to the other three networks. Consequently, it

is reasonable to choose network 1248 as the best network among four networks in terms of cost and network

reliability. Moreover, the results of cost-effectiveness analysis in Table 17, confirm that the network 1248 is the

most affordable network compared to the other three networks. Therefore, it can be concluded that the multilayer

MINs have a higher efficiency in comparison with the replicated MINs in terms of reliability, cost, and cost-

effectiveness.

19

Table 15. Network reliability as a function of switch reliability.

Switch reliability (r) Rn(network 1248) Rn(network 1888) Rn(network 1188) Rn(network 8888)

0.9 0.000 541 0.000 02 0.000 014 0.000 288

0.91 0.001 355 0.000 068 0.000 048 0.000 79

0.92 0.003 315 0.000 229 0.000 157 0.002 105

0.93 0.007 896 0.000 741 0.000 501 0.005 403

0.94 0.018 238 0.002 295 0.001 565 0.013 255

0.95 0.040 629 0.006 81 0.004 776 0.030 822

0.96 0.086 705 0.019 403 0.014 289 0.067 481

0.97 0.175 828 0.053 433 0.042 077 0.139 143

0.98 0.335 852 0.143 713 0.122 32 0.273 729

0.99 0.599 735 0.381 045 0.351 609 0.525 579

Table 16. MTTF of point-network as a function of switch failure rate.

Switch failure rate (λ) MTTFn(network 1248) MTTFn(network 1888) MTTFn(network 1188) MTTFn(network 8888)

10-8 1757724.670750557 1039562.2892624302 961321.798678224 1527777.777776152

10-6 17577.24670733699 10395.622895841287 9613.21798702277 15277.777777983547

10-4 175.77246714080468 103.95622902062402 96.13217992615348 152.7777778594781

10-2 1.7577246549291525 1.0395623416674948 0.9613216801096037 1.5277777086322266

Table 17. Cost-effectiveness of point-network for each of the networks 1248, 1888, 1188, and 8888.

Switch failure rate (λ) CEn(network 1248) CEn(network 1888) CEn(network 1188) CEn(network 8888)

10-8 2 112.649 845 902.397 821 1 035.907 111 1 193.576 389

10-6 21.126 498 9.023 978 10.359 071 11.935 764

10-4 0.211 265 0.090 24 0.103 591 0.119 358

10-2 0.002 113 0.000 902 0.001 036 0.001 194

Here, the performance of 64×64 networks consisting of 4×4 switching elements (the networks 1-4-16, 1-1-16, 1-

16-16, 1-2-4, 7-7-7, and 8-8-8) will be examined. Similarly to the previous analysis, the network reliability of these

networks of size N×N is calculated by the following equations:

(89)

(90)

(91)

(92)

(93)

(94)

Therefore, considering the Eqs. (89) through (94), reliability equations for size 64×64 are given by:

(95)

(96)

(97)

20

(98)

(99)

(100)

Also, the MTTF of point-network for these networks is calculated as follow:

(95)

(96)

(97)

(98)

(99)

(100)

The results of network reliability and MTTF analysis for the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and

8-8-8 are shown in Table 18 and 19, respectively.

Table 18. Network reliability as a function of switch reliability.

R Rn( 1-4-16) Rn(1-1-16) Rn(1-16-16) Rn(1-2-4) Rn(7-7-7) Rn( 8-8-8)

0.9 1.34421108731531E-

37

5.93932540845249E-

64

5.99195779222718E-

64

1.07381009322923E-

17 6.4562729739825E-19

2.53357714859519E-

20

0.91 1.33477314790267E-

33

3.36223651787142E-

57

3.42280173775187E-

57

7.47196858374132E-

16

6.40202273809694E-

17

3.57794542444517E-

18

0.92 1.1815386913263E-29 1.59137786628338E-

50

1.64932192369777E-

50

4.96065750380977E-

14 6.0371792581161E-15

4.78670286118193E-

16

0.93 9.45508649324501E-

26

6.26722763368008E-

44

6.72683081880809E-

44

3.14354109680796E-

12

5.42005234453769E-

13

6.07364757845657E-

14

0.94 6.82427693563417E-

22

2.02840860768679E-

37

2.32784805567728E-

37

1.90105002401822E-

10

4.63741660094131E-

11

7.31741810087039E-

12

0.95 4.42222804563885E-

18

5.26149146311902E-

31

6.82929401508471E-

31 1.09566107630846E-8 3.78485286649694E-9

8.37885827743867E-

10

0.96 2.54526506119947E-

14 1.0479891596599E-24 1.6771884378234E-24 0.000 001 2.94725013041205E-7 9.12016624282917E-8

0.97 1.27179528668147E-

10

1.52163234210995E-

18

3.26560547201867E-

18 0.000 031 0.000 022 0.000 009

0.98 0.000 001 1.60426844462569E-

12

4.20039881125473E-

12 0.001 449 0.001 473 0.000 873

0.99 0.001 513 0.000 001 0.000 003 0.055 513 0.070 162 0.054 561

Table 19. MTTF of point-network as a function of switch failure rate.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

MTTFn(1-4-16) 182171.3259755653 1821.713259755653 81.21713259755654 0.1821713259755653

MTTFn(1-1-16) 74404.76189898324 744.0476189898325 7.440476189898324 0.07440476189898323

MTTFn(1-16-16) 78124.98529199969 781.2498529799969 7.81249852919997 0.0781249852919997

MTTFn(1-2-4) 383748.1962481962 3837.481962481962 38.37481962481962 0.3837481962481962

MTTFn(7-7-7) 404222.3614398143 4042.223614398143 40.42223614398143 0.4042223614398143

MTTFn(8-8-8) 366369.2393133183 3663.692393133183 36.63692393133183 0.3663692393133183

21

According to tables 18 and 19, it is clear that the networks 1-4-16, 1-1-16, and 1-16-16 are not able to compete

with the networks 1-2-4, 7-7-7, and 8-8-8 in terms of network reliability and MTTF. Therefore, the main

competition is among three networks 1-2-4, 7-7-7, and 8-8-8. As Tables 18 and 19 shows, the three networks have a

very close performance in terms of network reliability and MTTF. In these circumstances, which network is a more

appropriate choice? To answer this question, the cost as another important parameter should be considered.

Considering Table 7, it is clear that the network 1-2-4 has the least-cost in comparison with other five networks.

Consequently, not only the network 1-2-4 provides an adequate reliability, but also is the least-cost network

compared to other networks. Therefore, it is a more sensible choice compared to the other networks. To better

understand this issue, the results of cost-effectiveness analysis are given in Table 20. These results prove that the

network 1-2-4 is the most cost-effective network compared to the networks 1-4-16, 1-1-16, 1-16-16, 7-7-7, and 8-8-

8.

Table 20. Cost-effectiveness of point-network for each of the networks 1-4-16, 1-1-16, 1-16-16, 1-2-4, 7-7-7, and 8-8-8.

Switch failure rate (λ) 10-8 10-6 10-4 10-2

CEn(1-4-16) 17.790 169 0.177 902 0.007 931 0.000 018

CEn(1-1-16) 7.855 232 0.078 552 0.000 786 0.000 008

CEn(1-16-16) 5.868 764 0.058 688 0.000 587 0.000 006

CEn(1-2-4) 136.274 217 1.362 742 0.013 627 0.000 136

CEn(7-7-7) 64.448 718 0.644 487 0.006 445 0.000 064

CEn(8-8-8) 51.111 78 0.511 118 0.005 111 0.000 051

In sum, it can be concluded from the analyzes carried out in section 3 that the multilayer MIN 1248 of size 16×16

and the multilayer MIN 1-2-4 of size 64×64 achieve better reliability and cost from different aspects of the terminal,

broadcast, and network compared to other networks studied in this paper. This means that although these networks

have a high reliability, they do not impose high cost on the network which makes them affordable. Indeed, the

results of these analyzes are indicative of the fact that multilayer MINs have a higher potential to meet reliability

requirements compared to replicated MINs. For a more detailed discussion in this case, the percentage of

improvement in various parameters for these networks is analyzed in the next sub-section.

3.5. Discussion

In this sub-section, for a more detailed analysis, the percentage of improvement in various parameters in

multilayer MINs compared to replicated MINs will be quantified. The first issue is about cost-effectiveness of point-

terminal. The results of the analyses show that the multilayer network of 1248 can be achieved an improvement of

29% in terms of cost-effectiveness of point-terminal compared to the replicated network of 8888. Also, multilayer

network of 1-2-4 achieved an improvement of 13% and 15% in terms of cost-effectiveness of point-terminal

compared to the replicated networks of 7-7-7 and 8-8-8, respectively.

The next issue is related to the parameters of broadcast reliability, MTTF of point-broadcast, and cost-

effectiveness of point-broadcast. Broadcast reliability in moderate switch reliability (r = 0.95): the results indicate

the fact that improvements achieved by the network 1-2-4 compared to networks of 7-7-7 and 8-8-8 is 166% and

442%, respectively. MTTF of point-broadcast: improvements achieved by the network 1-2-4 compared to networks

of 7-7-7 and 8-8-8 is 18% and 32%, respectively. Cost-effectiveness of point-broadcast: improvements achieved by

the network 1-2-4 compared to networks of 7-7-7 and 8-8-8 is 164% and 235%, respectively, and network of 1248

can be achieved an improvement of 46% compared to the replicated network of 8888.

The next issue is about the parameters of network reliability, MTTF of point-network, and cost-effectiveness of

point-network. Network reliability: network 1248 attains an improvement of 31% compared to network 8888. Also,

network 1-2-4 attains an improvement of 189% compared to network 7-7-7. MTTF of point-network: multilayer

networks of 1248 and 1-2-4 attains an improvement of 15% and 5% compared to replicated networks of 8888 and 8-

8-8, respectively. Cost-effectiveness of point-network: the primacy of network 1248 compared to 8888 is 77%. Also,

the primacy of network 1-2-4 compared to 7-7-7 and 8-8-8 is 111% and 166%, respectively.

The last parameter to be checked is the cost. The hardware cost of network 1248 has improved 53% compared to

network 8888. In addition, the hardware cost of network 1-2-4 has improved 123% and 154% compared to networks

of 7-7-7 and 8-8-8, respectively.

Finally, it should be noted that based on the analyzes conducted in this paper, replicated networks have some

advantages in some cases and in some parameters such as terminal reliability. However, it should be noted that this

22

advantage exists in a small number of parameters, Secondly, as discussed above, replicated MINs have much higher

costs in comparison to multilayer MINs.

4. Conclusion and Future Works

Reliability is one of the major concerns for the most of systems especially multiprocessor systems. On the other

hand, design of an efficient interconnection network has a great impact on the performance of these systems. MINs

have a great potential in terms of cost and efficiency for use in large-scale systems. That's why, so far, great efforts

have been made to improve the reliability of these networks. However, the cost is a fundamental issue during

reliability improvement. The cost is a very important parameter such that applying any proposed scheme is not

reasonable in practice due to their high cost. Therefore, in this paper, our aim was to analyze the reliability and cost

of multilayer MIN which is the latest idea in design of high-performance MINs. In this paper, we have studied all three aspects of reliability in MINs: terminal, broadcast, and network. Also, unlike previous works that were mostly

focused only on the time-dependent or time-independent analysis, in this paper, both types of these analyzes were

performed in order to achieve a comprehensive analysis. All the results of these analyze signify that the idea of the

multilayer MINs can lead to low-cost networks but high reliability. In other words, it was proven that the multilayer

MINs could provide high reliability in a more efficient manner than the replicated MINs. Overall, the analyzes

undertaken in this paper demonstrate that the multilayer MINs have a great potential to provide better performance

in terms of various important parameters. Therefore, this is reflecting the fact that these networks are qualified for

future studies.

Some issues should be considered in the future works; with punctilious investigation in the conducted analyzes, it

can be seen that the proper choice of parameters of GS, GF, and GL are very influential in the performance of

multilayer MINs such that the incorrect choice of these parameters leads to reliability reduction. Therefore, one of

the important directions in the future works can be focused on computing the optimal values for these parameters in

terms of reliability and cost. The next issue that can be considered as future works, is implementing the idea of

multilayer MINs on the other MINs to achieve higher performance. In this paper and [29], the idea of multilayer

MINs has been discussed on banyan-like (typical) MINs. However, there are many other advanced MINs that can be

evaluated with the idea of multilayer MINs.

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