Improving the reliability of the Benes network for use in large-scale systems

Microelectronics Reliability xxx (2015) xxx–xxx

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Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel

Improving the reliability of the Benes network for use in large-scale systems Mohsen Jahanshahi a,⇑, Fathollah Bistouni b a b

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

a r t i c l e

i n f o

Article history: Received 10 September 2014 Received in revised form 11 November 2014 Accepted 12 December 2014 Available online xxxx Keywords: Parallel computers Multistage interconnection networks Rearrangeable non-blocking MINs Benes network Large-scale systems Reliability

a b s t r a c t Systems with high computational capabilities are usually made of a large number of processing elements in order to solve complex problems efficiently. To achieve this purpose, the processing elements must be able to communicate with each other, which can be provided by interconnection networks. Meanwhile, multistage interconnection networks (MINs) are often recommended for use in such systems due to the efficiency and cost-effectiveness. However, there is a fundamental problem in the general structure of these networks called blocking. An important class of MINs is rearrangeable non-blocking MINs that can be used as a cost-effective solution to cope with the blocking problem. Benes network is one of the most popular rearrangeable non-blocking MINs, which has been considered by many researchers over the years. And yet its reliability improvement is an important factor that must be considered in the review of most systems, especially large-scale ones. Based on previous works, there are three main approaches to improve the reliability of MINs: (1) Adding a number of stages to MIN. (2) Using multiple MINs in parallel. (3) Using replicated MINs. In this paper, in order to find the best solution to improve the reliability of the Benes network, all three of these methods are investigated. Reliability analyses show that the use of multiple parallel Benes networks can obtain more advantages than other methods for Benes network. The approach improves the Benes network to be used in large-scale systems in various aspects of reliability namely time-independent terminal reliability, time-independent broadcast reliability, time-dependent terminal reliability, and time-dependent broadcast reliability. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The demand for systems with high computational power has always existed for many years and has not ever stopped. A number of important problems have been identified whose solutions require a tremendous amount of computational power that should be available in the future [1–3]. Instances of those problems include long-range weather prediction, nuclear reactor design, human genome, condensed matter theory, speech recognition, natural language processing, computer vision, image processing, and automated reasoning, to name a few. In the other hand, performance of processors has been doubled in approximately every three-year [4,5]. However, high computational power cannot be achieved solely by increasing processors’ performance. That is, exploitation of other efficient means such as parallel processing techniques is very important to raise this capability. Nowadays, ⇑ Corresponding author. E-mail addresses: [email protected] (M. Jahanshahi), [email protected]. ir (F. Bistouni).

one of the main methods to improve the speed and computational power is the use of multiple processors in parallel computers. In fact, a parallel computer is a collection of processing elements that communicate and cooperate to solve large problems fast [6,7]. Therefore, parallel computers can be achieved by using parallel processing to speed up and gain more power to solve complex problems. Parallel processing is the use of multiple processors that are working simultaneously on the same problem. The hop is if a single processor can generate x floating point operations per second (FLOPS), then 100 of these may be able to produce 100x FLOPS, and 1000 processors may produce 1000x FLOPS [8]. Given the above discussion, several questions may arise. How do these processing elements communicate and cooperate? How is data transmitted among processors, what sort of interconnection is provided? In other words, a parallel computer requires some kinds of communication subsystem to interconnect processors, memories, disks, and other peripherals. The task of communicating between different nodes is the responsibility of the interconnection networks. An interconnection network is a complex connection of switching elements and links which allow communication

http://dx.doi.org/10.1016/j.microrel.2014.12.008 0026-2714/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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between processors and memories. Therefore, the design of an efficient interconnection network is very critical for the construction of efficient parallel computers [4,7]. Interconnection networks can be divided into two main classes: Static and Dynamic. In a static network the connection between input and output nodes is fixed and cannot be changed and reconfigured. The examples of these types of networks are linear array, ring, tree, star, fat tree, mesh, tours, systolic arrays, and hypercube. Dynamic interconnection networks are implemented with switched channels, which are dynamically configured to meet the communication needs of user programs. In this paper, our focus is on a particular class of multistage interconnection networks (MINs) known as Benes network, which is a dynamic network. Now, the question that arises here is that why in this paper, a dynamic network is selected for the study? We have several reasons for this choice: (1) As mentioned previously, dynamic networks are more flexible in providing connections between nodes and can be reconfigured when it is needed. On the other hand, in many modern applications, the flexible and reconfigurable connections are very important. Therefore, dynamic networks are more useful in multi-processor systems compared with static networks such that these networks are known as multi-processor interconnection networks [9]. (2) As it will be discussed below, dynamic networks (i.e. bus, crossbar, and MINs) can provide a variety of options to satisfy both cost and performance requirements. In addition, MINs achieve a reasonable balance between cost and performance compared with bus and crossbar. (3) Dynamic networks have two important solutions to deal with the blocking problem in interconnection networks: use of non-blocking networks and use of rearrangeable non-blocking networks. Also, since use of the rearrangeable non-blocking networks is a cost-effective solution (because they require the minimum number of crosspoints to connect N terminal nodes in a rearrangeably non-blocking manner), this paper is focused on the Benes network, which is a well-known rearrangeable non-blocking network [10,11]. The dynamic networks include bus, crossbar, and multistage interconnection networks (MINs), which are often used in the multiprocessor systems [11,12]. But which of these topologies can be more suitable for use in large-scale systems? In the choice of an interconnection network for use in large-scale systems, two basic parameters should be considered: cost and performance. The crossbar networks are the most efficient but also the most expensive ones. On the other hand, shared bus networks have lower costs but also lower performance. MINs provide a compromise between the above networks because they provide efficient performance using a reasonable number of switches [12,13]. Therefore, MINs are often used in the context of single-instruction multiple-data (SIMD) and multiple-instruction multiple-data (MIMD) parallel machines and are also increasingly adopted for implementing the switching fabric of high-capacity communication processors, including asynchronous transfer mode (ATM) switches, gigabit Ethernet switches, and terabit routers [14,15]. MINs can be divided into three main classes: (1) blocking MINs, (2) non-blocking MINs, and (3) rearrangeable non-blocking MINs [11,16–18]. A MIN is said to be blocking if any free output may be unavailable to any free input because existing connections prevent a path from being established across the network. The Banyan-type MINs such as shuffle-exchange network [19], Baseline [20], and Generalized Cube [21] are blocking to traffic with a random destination distribution. A MIN that is always capable of connecting a free input to a free output, but which may require existing connections to be rearranged in order to do so, is called rearrangeable non-blocking MIN. The (2n  1) stage N  N shuffle-exchange network [22] (n = log2 N) and Benes network [23,24] are examples of rearrangeable non-blocking MINs. In addition, a MIN which is always capable of connecting a free input to a free

output, regardless of the connection already established across the MIN, is said to be non-blocking MIN. The Clos network [25] is non-blocking MIN. The Clos network gets more performance compared with the other MINs [12] that this feature makes it attractive. However, these networks have two main problems that prevent them from being the practical solution. Generally, nonblocking MINs are more expensive than rearrangeable non-blocking ones and more complex to control [14,18,26,27]. It should be noted here that to estimate the cost of a MIN, one common method is to calculate the crosspoint cost. The crosspoint cost is given by the number of crosspoints within a switching element and by the number of switching elements within the network [18,27]. Therefore, since a non-blocking network requires more switching elements than a rearrangeable non-blocking network, it is not economical, especially for use in large-scale systems. Consequently, at present, the rearrangeable non-blocking MINs are commonly used in large-scale systems. Benes network is a rearrangeable non-blocking MIN that is known to many researchers in this field of research. On the other hand, one of the most important aspects of performance that should be considered in evaluation of most systems is reliability. An interconnection network should be able to deliver information reliably. Interconnection networks can be designed for continuous operation in the presence of a limited number of faults. In other words, in an interconnection network, reliability is a measure of how often the network correctly performs the task of delivering messages. In most situations, there is a need to deliver messages 100% of time without loss. For this reason, an unreliable interconnection network will have a direct negative impact on the overall performance of a parallel computer system. A parallel computer requires that its interconnection network operate without packet loss for ten thousands of hours [4,10]. Generally, reliability is the ability of a system to perform and maintain its functions in routine circumstances, as well as hostile or unexpected circumstances. Therefore, many researchers believe that it is the most important requirement for an effective MIN [12,13,19,28–33]. Hence, in this paper we are focused on improving the reliability of the Benes network. While this is the case, improving the reliability of a MIN, until now, has been presented in several approaches; (1) Adding a number of stages to a MIN or MINs. (2) Using multiple MINs in parallel. (3) Using replicated MINs. Each of these approaches is described in details in the following section. Subsequently, according to the arguments already made, our motivation in this paper is to find the best solution to improve reliability of Benes network. If we meet the vital requirement of reliability, then it can be claimed that a good progress for improving MINs’ performance in largescale systems takes place. The rest of the paper is organized as follows: A useful background will be presented in Section 2. Different reliability improvement approaches on the Benes network will be discussed in Section 3. In Section 4, reliability of these approaches will be analyzed, and finally some conclusions will be drawn in Section 5. 2. Background In Section 2.1, the general structure of MINs will be discussed. Then, the blocking problem and structure of Benes network will be described in Sections 2.2 and 2.3, respectively. Discussions about related works will be done in Section 2.4. Finally, a useful argument on the contribution of the paper will be made in Section 2.5. 2.1. General structure of MINs MINs are used for the interconnection of a set of N input terminal to N output terminals using sets of switches arranged in stages.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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Typically, an N  N MIN is constructed by (logn N) stages of n  n switching elements and there are (N/n) switches in each stage. Therefore, the network complexity of an N  N MIN (defined as the total number of switching elements) is equal to O(N logn N). As opposed to the crossbar network, which requires O(N2) switching elements, crossbar network is an irrational choice for large values of N. Fig. 1 shows the general structure of a MIN. The MINs therefore form a class of interconnection networks which includes the omega network, the flip network, the indirect binary n-cube, the baseline and the reverse baseline networks which have all been proven topologically equivalent [15,35,36]. Switching elements are connected by an interconnection pattern of links sometimes referred to as a permutation network or a shuffle. In general the shuffle between each stage of the MIN network will be different because of interconnection links pattern. 2.2. Blocking problem in MINs Routing tag consists of binary digits that control the connection through different stages of the path from input, to the output. Let the source S and destination D be represented in binary as:

S ¼ s1 . . . sðlog2 NÞ D ¼ d1 . . . dðlog2 NÞ In each switching elements of a MIN, an incident packet is prefaced by a tag indicating the required destination. The selector of the input port examines this tag and inspects the state of the arbiter of the required output port. If the selected arbiter indicates that the required output is free, the connection will be established but if it is busy, then it will be refused. All selectors may thus work concurrently and asynchronously. Therefore, at a certain time, switching element of size 2  2 can be in one of the two states, straight and exchange. These two states are shown in Fig. 2. Let the upper input and output lines be labeled i and the lower input and output lines be labeled j. (1) Straight-input i to output i, input j to output j; (2) exchange-input i to output j, input j to output i. One feature of the MINs is their self-routing. In self-routing procedure, the switches examine the destination of their input data and set themselves. The most significant bits of destination in self-routing procedure indicate the state of switch in the first stage and similarly next bits indicate the state of switch at later stages. In the MINs, if two inputs to a switch in stage x have the same value in the bit position x of destination tag, the network cannot realize this permutation. In other words, these cases require a state that is not supported by the switch (Fig. 3). This term refers to the block-

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Fig. 2. States of switching elements.

ing, which can greatly reduce network performance and consequently the performance of the whole system. Therefore, solving or moderating this problem is one of the main concerns of researchers in this field. To combat the blocking mode, so far two main solutions have been proposed: (1) use of non-blocking MIN (Clos network), (2) use of rearrangeable non-blocking MINs [11,18,37]. Previous analyses demonstrate that Clos network (non-blocking MIN) [25] is able to remove the blocking mode. However, there are several issues that compromise the efficiency of this solution. Generally, non-blocking MINs are more expensive than rearrangeable nonblocking ones and more complex to control [14,26,27]. Therefore, rearrangeable non-blocking MINs are often proposed to be used in large-scale systems. The (2n  1)-stage shuffle-exchange networks [22] (n = log2 N) and Benes network [23] are examples of rearrangeable non-blocking MINs. The Benes network is very well known and the network has been perused by many authors. Therefore, in this paper we focus on this network. 2.3. Structure of Benes network The Benes network basically has a topology that can be viewed as a connection of a Baseline network (a blocking MIN) and a reverse Baseline network (a blocking MIN) with the center stages overlapped. A Benes network of size N  N consist of (2(log2 N)  1) stages and each stage is composed of (N/2) switching elements of size 2  2. The network complexity of an N  N Benes network is N/2(2(log2 N)  1)]. A Benes network of size 8  8 is shown in Fig. 4. 2.4. Reliability and related works Reliability is one of the most important performance parameters, which is usually considered in the study of most systems. In fact, the reliability analysis is a mathematical description of a system that gives accurate information about the performance of the system. Therefore, reliability is a major concern of researchers in the field of MINs. So far, in order to improve the reliability of the

Fig. 1. General structure of MIN.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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Fig. 3. States that are leading to blocking.

MINs, considerable works have been done, which resulted in several important approaches which can be generally classified into three categories: (1) Adding a number of stages to MIN. (2) Using multiple MINs in parallel. (3) Using replicated MINs. 2.4.1. Adding a number of stages to MIN This is one of the first ideas has been used to improve the reliability of blocking MINs [19,36,38]. In this approach, a fixed number of stages are added to the network. The benefits of this approach can be outlined as follows: (1) It is proved that this approach increases reliability and fault-tolerance of blocking MINs to some extent. (2) This approach increases costs as a function of a single switching stage, therefore it slightly rises the cost (i.e. is cost effective). (3) It is easy to implement. Despite the above advantages, this approach suffers from two major defects. These imperfections can be listed as follows: (1) In previous studies on blocking MINs, it is proved that this approach is not efficient to intermittent use on a given network and cannot always increase reliability [30,31]. As a result, this method enhances the reliability of the blocking networks to a bit extent. (2) It can increase the path length between any source–destination pair. 2.4.2. Using multiple MINs in parallel In this approach, multiple MINs are in parallel alongside each other, and the sources and destinations are connected to all MINs by multiplexers and demultiplexers [13,28,29,34]. Analysis has shown that this method can provide better reliability compared to the first one in blocking MINs [13,29,34]. The advantages of this approach can be listed as follows: (1) it increases reliability and fault-tolerance at a higher level than the first method in blocking MINs. (2) There is a possibility of reducing the number of switching stages, leading to reductions of the path length between any source–destination pair. Contrary to the above merits, this approach has also some problems listed as follows: (1) Typically, the network cost increases due to the increased size of switching elements. (2) On the other hand, the increasing size of the switch may cause an increase of switch failure rate [13,28,29]. (3) In this approach, auxiliary links are commonly used to deal with some fault conditions which can cause an increase in the path length between each source–destination pair.

Here, it should be noted that although it has been proved that higher reliability can be provided using multiple MINs in parallel compared with adding a number of stages to MINs, these results are derived from the implementation of these methods on blocking MINs. On the other hand, the Benes network as a rearrangeable non-blocking MIN has different structure and behavior compared with blocking MINs. Therefore, although previous studies on improving the reliability of the networks are very useful for future studies, these results may not fully apply to the Benes network. In other words, the method of adding a number of stages to MINs cannot be excluded in the study of Benes network’s reliability improvement due to differences in the structure of the blocking and rearrangeable non-blocking MINs. As a result, to achieve a comprehensive investigation, we need to analyze all three approaches over the Benes network. 2.4.3. Using replicated MINs Generally, replicated MINs enlarge MINs by replicating them L times [18,39,60–64]. The resulting MINs are arranged in L layers. Input ports are connected to their corresponding output ports. In contrast to the MINs, replicated MINs may cause out of order packet sequences because multiple paths exist for a source–destination pair. Sending packets belonging to the same message to the same layer avoids destruction of packet order. In terms of reliability, the increase in the number of layers can cause a better improvement compared to blocking MINs. For example, consider an N  N (blocking) MIN. let r be the probability of a switch being operational. Since a typical MIN is single-path, the failure of any switch will cause network failure, so from the perspective of reliability, there are (log2 N) switching elements in series for each terminal path (path between each source–destination pair). Therefore, the terminal reliability of an N  N MIN is given by Eq. (1).

Rt ðMINÞ ¼ rlog2 N

ð1Þ

Now, consider a replicated MIN with L = 2. There are two paths between a given source and destination. From the terminal reliability point of view, this system can be represented as a parallel system, consisting of two series systems in parallel. Also, each series system is comprised of (log2 N) switching elements. Therefore, the terminal reliability of an N  N replicated MIN is given by:

Rt ðreplicated MINÞ ¼ 1  ð1  r log2 N Þ

2

ð2Þ

According to Eqs. (1) and (2), the results of terminal reliability analysis for the various switch reliability (r = 0.9, 0.95, and 0.99) are shown in Fig. 5.

Fig. 4. A Benes network of size 8  8.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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In Fig. 5, the curves corresponding to a given switch reliability are shown in different colors for more clarification. By comparing the two diagrams related to specific switch reliability, it is clear that replicated MIN achieves higher reliability compared to the MIN. In particular, as the switch reliability is reduced, the superiority of replicated MIN compared to MIN is vividly flaunted. According to the discussions made in this section, since Benes network is an important and famous MIN due to its rearrangeability feature, we have chosen it to study these reliability improvement approaches. In addition, each of these approaches has some advantages and disadvantages. For this reason, our motivation in this paper is to determine the most suitable solution to improve reliability of Benes network. To achieve a comprehensive investigation, we need to analyze all three approaches over the network. 2.5. Contribution Regarding the points mentioned in Section 2.4, it can be concluded that all three methods of reliability improvement (i.e. adding a number of stages to MIN, using multiple MINs in parallel, and using replicated MINs) are often used to improve the reliability of blocking MINs such as shuffle-exchange and Baseline networks. However, as previously mentioned, an important class of MINs is rearrangeable non-blocking MINs that can be used as a cost-effective solution to cope with the blocking problem. Also, Benes network is one of the most popular rearrangeable non-blocking MINs, which has been considered by many researchers over the years. However, to the best of our knowledge, no research has been done on improving the reliability of these types of networks. Therefore, our main contribution to the body of knowledge is to improve the reliability of the Benes network. Another point is that most previous works have focused on improving reliability using just a single method. For instance, in [19,30,31,33,36,38], improving reliability is limited only to increasing the number of stages or in [18,28,29,34], improving reliability is limited solely to using multiple MINs in parallel. However, more reasonable and more comprehensive approach is that a number of important methods to improve the reliability of the network are tested to select the best among them. Therefore, another contribution of this paper is to analyze three aforementioned popular methods to improve the reliability of Benes network that is unparalleled in comparison with previous works. However, in this paper, there are some other important contributions in terms of reliability engineering. Clearly, most of the pervious works are approximated solutions and based on reliability bounds’ estimations, especially for large network sizes due to the difficulty of reliability analysis in fault-tolerant MINs [13,19,29,32,33]. However, the methodology used in this paper is to model the networks in series–parallel or complex series–parallel

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reliability block diagrams (RBDs) instead of using the estimated upper and lower bounds RBDs. Therefore, this approach provides more accurate information about reliability of the networks compared with previous works. Moreover, the methodology is applicable for evaluating the reliability of other MINs, even with more complex structures. The next point is in relation to the time parameter in the analysis of reliability. By taking time parameter into account, most reported works in the field of MINs reliability analysis have been limited to one of these approaches: time-independent reliability analysis or time-dependent reliability analysis. For instance, in [18,30–33], reliability analysis is limited only to time-independent analysis. Or in [12,13,19,29,34], reliability analysis is limited just to time-dependent analysis. However, both time-dependent and time-independent analyses are valuable and can provide useful information. Therefore, in this paper, both types of analysis will be carried out in order to achieve perfect results. 3. Analysis of different approaches for Benes network reliability improvement In this section, three main approaches to improve the reliability of MINs (explained in the previous section) will be studied on the Benes network. The idea of using the method of increasing the number of stages on the Benes network will be discussed in Section 3.1. Discussions on the using multiple Benes networks in parallel will be presented in Section 3.2. Finally, implementation of the replicated MINs idea on the Benes network will be analyzed in Section 3.3. Each of these methods has advantages and disadvantages, but from the perspective of reliability, the one which achieves a significant improvement in reliability compared to other methods, would be the better approach. After examining these methods, their efficiency rate will be analyzed in Section 4. At the end, according to the results of the analysis, the most appropriate method for improving reliability in the Benes network will be determined. 3.1. Extra-stage Benes network In this sub-section, the approach of ‘‘adding a number of stages to MIN’’ is applied on the Benes network, and the resulted structure is called the Extra-stage Benes Network (EBN). The basic idea is to create redundancy in the number of paths between each source–destination pair and hereby reliability can be improved. An N  N EBN is an N  N Benes network with an additional stage. Fig. 6 shows an 8  8 EBN. An EBN of size N  N is comprised of (2(log2 N)) stages and each stage contains (N/2) switching elements of size 2  2. Therefore, the network complexity of an N  N EBN network is [N/2(2(log2 N))]. 3.2. Parallel Benes network

Terminal Reliability

1.2 1 0.8 0.6 r(MIN)=0.9 r(MIN)=0.95 r(MIN)=0.99 r(Replicated)=0.9 r(Replicated)=0.95 r(Replicated)=0.99

0.4 0.2 0

8

16

32

64

128

256

512

1024

2048

Network Size Fig. 5. Terminal reliability of MIN and replicated MIN vs. network size.

Another approach for improving reliability of MINs is using multiple MINs in parallel, which is studied on the Benes network in this sub-section. The new network is called Parallel Benes Network (PBN). The basic idea of this method is to create redundancy in the number of networks, which can lead to redundancy in the number of paths between each source–destination pair. A PBN network is composed of two copies of a sub network, each containing a Benes network. A PBN of size 16  16 is shown in Fig. 7. A PBN of   size N  N consists of ((2log2 N)  3) stages of N2 switching elements. All switches are of size 2  2. There is one 2  1 multiplexer for each input link of a switch in the stage 1 and one 1  2 demultiplexer for each output link of a switch in the stage ((2 log2 N)  3). Therefore, a PBN of size N  N has N multiplexers (MUXs) in the input stage and N demultiplexers (DMUXs) in the output stage.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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Fig. 6. An EBN network of size 8  8.

Fig. 7. A PBN network of size 16  16.

The network complexity of an N  N PBN is given by [N/ 2(2 + ((2 log2 N)  3))]. 3.3. Replicated Benes networks Replicated MINs enlarge MINs by replicating them L times. The resulting MINs are arranged in L layers. Sources and destinations are connected to each layer and thus the ability to communicate between each source–destination increases by increasing the number of paths. In this section, we will implement this method on the Benes network and the new network is called Replicated Benes Network (RBN). An RBN (RBN-2) network of size 8  8 is shown in Fig. 8. An RBN-L of size N  N is designed by replicating Benes network L times. There are L independent Benes networks, such

that a connection path between each source–destination pair can be established via any one of the networks. All the L sub-networks are of identical type. Each source and destination is linked to all the L sub-networks. An RBN-L consists of ((2 log2 N)  1) number of stages and [L(N2(2(log2 N)  1))] number of switching elements. The network complexity of an N  N FDOT (with L = 2) is equal to (N(2(log2 N)  1)). In this paper, we limit our discussion to RBN-2 (RBN), in which the number of layers L is equal to 2. 4. Reliability analysis of EBN, PBN, and RBN networks Reliability is the ability of a system to do and maintain its function in routine circumstances as well as hostile or unexpected ones. Therefore, one of the most important performance parameters

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

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Fig. 8. 3D view of a RBN network of size 8  8.

required for each interconnection network is reliability [13,18,19,28–33,57] and evaluation of the reliability of these networks becomes essential. In fact, reliability analysis is a mathematical description of a system and can give exact information about the performance of the system. However, the reliability analysis is very challenging problem in the case of complex networks. Complex networks are consisting 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 [40–42]. According to this definition, lifeline networks such as electrical and gas networks [40,41,43], wireless mobile ad hoc networks (MANETs) [42,44,45], wireless mesh networks [46–50], wireless sensor networks [51–53], sensors based on nano-wire networks [54], social networks [55], stochastic-flow manufacturing networks (SMNs) [56], and interconnection networks [12,13,18,19,28,31,59] 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 to 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 due to 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-tol-

erance 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. In this paper, we will examine the reliability from two important aspects, terminal and broadcast. Terminal reliability is the probability of at least one fault free path between each source–destination pair. Also, Broadcast reliability is the probability of at least one fault free path between one specific source and all network destinations. Moreover, reliability analysis can be done either time-independent or time-dependent and in the previous works usually one of them has been employed. However, in this paper we will use both of these cases to achieve a more comprehensive analysis. Another important point in improving the reliability is related to the cost. In the domain of MINs, methods to improve reliability are often accompanied by increased hardware costs. Therefore, a method that exacts a high cost to the network may not be affordable in practice. The parameter that can be used in assessment of the MINs performance in terms of cost, is the cost-effectiveness parameter (mean time to failure (MTTF)/cost ratio) that in the most reported works [13,19,28,29,60] has been emphasized. Therefore, we will discuss this parameter to attain a comprehensive review of the networks’ hardware costs. Clearly, according to the above description, in this paper we will examine the reliability in terms of the following parameters:     

Terminal reliability (time-independent reliability). Broadcast reliability (time-independent reliability). Terminal time-dependent reliability. Broadcast time-dependent reliability. Cost-effectiveness.

Thanks to precise investigation of these five parameters, it can be claimed that the reliability analysis conducted in this paper leads to integrated results. In this section, reliability block diagrams, time-independent terminal reliability, time-independent broadcast reliability, time-dependent terminal reliability, time-dependent broadcast reliability, and cost-effectiveness parameter will be discussed in Section 4.1, through 4.6, respectively. 4.1. Reliability block diagrams A system usually consists of different components. Generally, system reliability is a function of the reliabilities of the components as well as the relationships between them. Every component

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can be in one of the two conditions: working or failed. Reliability Block Diagram (RBD) is a graphical representation of how the components of a system are connected from reliability point of view. RBD is used to model the various series–parallel and complex block combinations that result in system success. Therefore, RBD helps reliability analysis using a functional diagram to portray and analyze the reliability relationship of components in a system [58,59]. Each element of a system is represented by a block that is interconnected to other blocks of the system at a desired level of assembly. The basic relationships among components are depicted as lines that may be series, parallel, or series–parallel. A series system is a configuration such that, if any one of the system components fails, the entire system fails. A graphical description of a series system is shown in Fig. 9. Assuming that Ri be the probability of component i being operational, system reliability for Fig. 9 is calculated as Eq. (3):

RðsystemÞ ¼ R1 \ R2 \ . . . \ Rn ¼ R1 ðR2 jR1 ÞðR3 jR1 R2 Þ . . . ðRn jR1 R2 . . . Rn1 Þ

ð3Þ

In the case of independent components, Eq. (3) becomes:

RðsystemÞ ¼

n Y Ri

ð4Þ

i¼1

A parallel system is a configuration such that, as long as not whole the components fail, the entire system works. A graphical description of a parallel system with n components is shown in Fig. 10. Assuming that Fi be the probability of component i failure, system unreliability is calculated as Eq. (5) for this case:

Fig. 10. Parallel RBD.

be analyzed by considering a specific source–destination pair in the network. In MINs, at least there should be a fault-free path for successful connection between each source–destination pair. 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 is equal to r. According to Benes network’s structure (shown in Fig. 4), terminal reliability RBD of 8  8 Benes network is shown in Fig. 12. Regarding to this diagram, terminal reliability of 8  8 Benes network, denoted Rt (8  8 Benes), is calculated by the Eq. (9):

h i 2 Rt ð8  8 BenesÞ ¼ r 2 1  ð1  ðr2 ð1  ð1  rÞ2 ÞÞÞ

ð9Þ

FðsystemÞ ¼ F 1 \ F 2 \ . . . \ F n ¼ F 1 ðF 2 jF 1 ÞðF 3 jF 1 F 2 Þ . . . ðF n jF 1 F 2 . . . F n1 Þ

ð5Þ

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

FðsystemÞ ¼

n Y

Fi

Also according to the structure of the Benes network, terminal reliability RBD of 16  16 Benes network is shown in Fig. 13. From this figure, the terminal reliability of an N  N Benes network is given by:

"

ð6Þ Rt ðN  N BenesÞ ¼ r

i¼1

2

  2 # N N 1  1  Rt  Benes 2 2

ð10Þ

As a result, the parallel system reliability is calculated as follows:

RðsystemÞ ¼ 1  FðsystemÞ ¼ 1 

n n Y Y F i ¼ 1  ð1  Ri Þ i¼1

ð7Þ

i¼1

Some systems are made up of combinations of several series and parallel configurations. The way to obtain system reliability in such cases is to break the total system configuration down into homogeneous subsystems. Then, consider each of these subsystems separately as a unit, and calculate their reliabilities. Finally, put these simple units back (via series or parallel recombination) into a single system and calculate its reliability. An example of series–parallel RBD is shown in Fig. 11. Reliability of the system is given by:

Taking the structure of EBN into account (shown in Fig. 6), the terminal reliability RBD of 8  8 EBN is shown in Fig. 14. According to this diagram, the terminal reliability of 8  8 EBN is calculated as follows:

  2 2 Rt ð8  8 EBNÞ ¼ r 2 1  ð1  ðr 2 ð1  ð1  rÞ2 Þ ÞÞ

ð11Þ

Similar to Benes network, we have:

"   2 # N N Rt ðN  N EBNÞ ¼ r 2 1  1  Rt  EBN 2 2

ð12Þ

RðsystemÞ ¼ R1 ½1  ð1  R2 Þð1  R3 Þ ½1  ðð1  ð1  ð1  R4 Þð1  R5 ÞÞÞ ð1  ð1  ð1  R6 Þð1  R7 ÞÞÞÞ

ð8Þ

4.2. Terminal reliability (time-independent analysis) Since terminal reliability is defined as the probability of successful communication between source–destination pair, it can

Fig. 9. Series RBD.

Fig. 11. Series–parallel RBD.

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normalized. Here, given the number of gates of each switching element, the switch reliability can be computed based on r. It is assumed that the hardware complexity of a component is directly proportional to the gate counts [13,28,29,34]. RBD of 8  8 PBN for terminal reliability is shown in Fig. 15. In Fig. 15, switching elements of size 2  2 is shown as SE2 and 2  1 multiplexers and 1  2 demultiplexers are shown as MUX2 and DEMUX2, respectively. According to the terminal reliability RBD of 16  16 PBN (shown in Fig. 15), the terminal reliability of 16  16 PBN is given by Eq. (13).

Rt ð16  16 PBNÞ ¼ 1   h i 2 2  1  r3 1  ð1  ðr 2 ð1  ð1  rÞ2 ÞÞÞ ð13Þ Fig. 12. Terminal reliability RBD of 8  8 Benes network.

Also, to N  N PBN, we have:

   2 N N Rt ðN  N PBNÞ ¼ 1  1  r Rt  Benes 2 2

ð14Þ

Considering RBN network structure (shown in Fig. 8), we can draw its terminal reliability RBD as it is depicted in Fig. 16. Reliability for this diagram is calculated as follows:

Rt ð8  8 RBNÞ ¼ 1   h i 2 2  1  r 2 1  ð1  ðr 2 ð1  ð1  rÞ2 ÞÞÞ ð15Þ Also, for RBN of size N  N, we have:

Rt ðN  N RBNÞ ¼ 1  ð1  ðRt ðN  N BenesÞÞÞ2

ð16Þ

Terminal reliability analyses results of Benes network, EBN, PBN, and RBN for different network sizes are shown in Figs. 17–19. In this study, we have considered following three values for switch reliability (r)l; 0.9 (low switch reliability), 0.95 (middle switch reliability), and 0.99 (high switch reliability).

Fig. 13. Terminal reliability RBD of 16  16 Benes network.

Fig. 14. Terminal reliability RBD of 8  8 EBN network.

In order to more accurately analyze reliability of the PBN network (shown in Fig. 7), a few points should be noted. PBN has a number of 2  1 multiplexers and 1  2 demultiplexers. These switching components have different reliability compared with 2  2 switches. Therefore, to compare different networks properly, the reliabilities of different switching components should be

Fig. 15. Terminal reliability RBD of 16  16 PBN network.

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1.005

Terminal Reliability

1 0.995 0.99 0.985 0.98 Benes network

0.975

EBN

0.97 0.965

PBN RBN

8

16

32

64

128

256

512

1024

2048

Network Size Fig. 19. Terminal reliability vs. network size in high switch reliability (r = 0.99).

Fig. 16. Terminal reliability RBD of 8  8 RBN network.

1

Terminal Reliability

0.9 0.8 0.7 0.6 0.5 0.4

Benes network

0.3

EBN

0.2

PBN

0.1

RBN

0

8

16

32

64

128

256

512

1024

2048

All three Figs. 17–19 show almost the same point and that is Benes and EBN networks have the lowest terminal reliability compared to other networks. The two networks are also very close to each other in terms of terminal reliability, but Benes network is slightly better than the EBN in some network sizes. This reflects the fact that the approach of adding stage to the Benes network has no positive effect on reliability. The main reason is the increase of network depth (number of stages) of EBN compared to Benes network, that makes the network more complex. On the other hand, previous analyses have also demonstrated that increasing the network complexity due to the increased number of stages can lead to decrease the reliability [30,31]. Moreover, these results show that the best results are owned by two networks PBN and RBN such that they can achieve much higher level of reliability than the other two networks. However, as the figures depict RBN attain higher terminal reliability compared to PBN. The main reason for this is that the PBN network has fewer stages as compared with the RBN. In other words, the number of stages in the RBN has been decreased compared with the initial state of the Benes network. Unlike EBN network, more stages in RBN rather than PBN has led to terminal reliability improvement in the RBN. Thus, according to these results, we can conclude that increasing the number of stages cannot always lead to increased reliability. Instead, there is an optimal number of stages for each network such that the network reliability will be maximized. In sum, from the above discussion it can be concluded that the best option to improve the terminal reliability of Benes network is to utilize the approach of applying Replicated Benes networks.

Network Size Fig. 17. Terminal reliability vs. network size in low switch reliability (r = 0.9).

Broadcast reliability is defined as the probability of a successful connection between a given source and all destinations in a network. Therefore, to calculate the broadcast reliability, first a given source should be taken into account, and then the connection from the source to all destinations should be considered. Similar to the previous one, in this sub-section, at first, the diagram of each network is drawn, and then using it, broadcast reliability for the network is calculated. According to Benes network’s structure (shown in Fig. 4), its broadcast reliability RBD can be plotted in Fig. 20. Referring to the diagram, the broadcast reliability of 8  8 Benes network is calculated as follows:

1

Terminal Reliability

0.98 0.96 0.94 0.92 0.9 0.88

  2  Rb ð8  8 BenesÞ ¼ r 5 1  1  ðr 3 ð1  ð1  rÞ2 ÞÞ

Benes network EBN PBN RBN

0.86 0.84

4.3. Broadcast reliability (time-independent analysis)

8

16

32

64

128

256

512

1024

2048

Network Size Fig. 18. Terminal reliability vs. network size in middle switch reliability (r = 0.95).

ð17Þ

In addition, considering the diagram depicted in Fig. 21 which is associated with the Benes network of size 16  16, we can see that the network has a behavior that its broadcast reliability for size N  N is calculated by Eq. (18).

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11

Fig. 22. Broadcast reliability RBD of 8  8 EBN network.

Fig. 20. Broadcast reliability RBD of 8  8 Benes network.

Also, for network size of N  N, we have:

Fig. 21. Broadcast reliability RBD of 16  16 Benes network.

" Rb ðN  N BenesÞ ¼ r

Nþ2 2

   2 # N N 1  1  Rb  Benes 2 2

ð18Þ

Broadcast reliability RBD for 8  8 EBN network is shown in Fig. 22. According to the diagram, broadcast reliability of 8  8 EBN is given by:

   2  2 Rb ð8  8 EBNÞ ¼ r 5 1  1  r3 ð1  ð1  rÞ2 Þ

ð19Þ

Similarly, the following equation can be used to calculate the broadcast reliability of the EBN in greater scales.

" Rb ðN  N EBNÞ ¼ r

Nþ2 2

  2 # N N 1  1  Rb  EBN 2 2 

ð20Þ

From Figs. 23 and 24, broadcast reliability of 16  16 PBN and 8  8 RBN are given by (21) and (22):

Rb ð16  16 PBNÞ ¼ 1   19 h i 2 2  1  r 2 1  ð1  ðr 3 ð1  ð1  rÞ2 ÞÞÞ ð21Þ Rb ð8  8 RBNÞ ¼ 1   h i 2 2  1  r 5 1  ð1  ðr 3 ð1  ð1  rÞ2 ÞÞÞ ð22Þ

    2 N N Nþ2 Rb ðN  N PBNÞ ¼ 1  1  r 4 Rb  Benes 2 2

ð23Þ

Rb ðN  N RBNÞ ¼ 1  ð1  ðRb ðN  N BenesÞÞÞ2

ð24Þ

Broadcast reliability analysis results for different network sizes and different switch reliabilities are shown in Figs. 25–27. As Figs. 25–27 illustrate the highest and the lowest broadcast reliability are respectively owned by PBN and EBN for different network sizes and switch reliabilities. Increasing the network complexity due to the increased number of stages can reduce broadcast reliability. That is why EBN attains the lowest reliability among the other networks. However, as the network size and switch reliability increases, the reliability of two networks Benes and EBN are more matched such that their functionality becomes very close to the switch reliability of 0.99 (shown in Fig. 27). Considering the hardware cost, there is a very undesirable situation while EBN is used. Therefore, the method of adding extra stages to improve the reliability of Benes is not preferred at all. The results demonstrate that both PBN and RBN obtain the highest broadcast reliability, respectively, as compared to other networks. In other words, the best way to improve the broadcast reliability of the Benes network is to focus on two other approaches namely replicated Benes networks and multiple Benes networks in parallel. However, according to the results it is clear that PBN is superior in comparison to RBN in terms of broadcast reliability. This implies that, unlike the previous section that less number of stages in the PBN compared to RBN led to decreased terminal reliability, this feature increases the broadcast reliability of the PBN compared with RBN. Overall, according to this section and the previous one, the following conclusions can be drawn: (1) approach of adding a number of stages, does not lead to significant reliability improvement of the Benes network. (2) Both RBN and PBN networks achieve the best results and have a close performance in terms of reliability. However, RBN is slightly superior in terms of terminal reliability compared with PBN. In addition, PBN is also slightly better in terms of broadcast reliability compared with RBN. In other words, using both replicated Benes networks and multiple parallel Benes networks are the best choices. But here a question arises, which of these methods can provide higher performance? To answer this question, the following time-dependent reliability analyses can be helpful. 4.4. Terminal reliability (time-dependent analysis) In the previous sub-sections, all analyses were based on timeindependent reliability analysis. This type of reliability analysis can provide useful information, and most of the previous works were based on this type of analysis [30–32]. However, for a more

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Fig. 23. Broadcast reliability RBD of 16  16 PBN network.

Broadcast Reliability

0.9 0.8

Benes network

0.7

EBN

0.6

PBN

0.5

RBN

0.4 0.3 0.2 0.1 0

8

16

32

64

128

256

512

1024

2048

Network Size

Broadcast Reliability

Fig. 25. Broadcast reliability vs. network size in low switch reliability (r = 0.9).

Fig. 24. Broadcast reliability RBD of 8  8 RBN network.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Benes network EBN PBN RBN

8

16

32

64

128

256

512

1024

2048

Network Size

comprehensive analysis, a time-dependent evaluation of each system can be important as well in terms of reliability. Therefore, in this paper, in addition to time-independent analysis of the networks, we will also consider the time-dependent analysis. Similar

Fig. 26. Broadcast reliability vs. network size in middle switch reliability (r = 0.95).

to the time-independent reliability analysis, time-dependent analysis can also be done from two different perspectives; terminal and

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Broadcast Reliability

1.2 Benes network

1

EBN PBN

0.8

RBN

0.6 0.4 0.2 0

8

16

32

64

128

256

512

1024

2048

Network Size Fig. 27. Broadcast reliability vs. network size in high switch reliability (r = 0.99).

broadcast. In this sub-section, we will focus on terminal reliability and broadcast reliability will be discussed in the next sub-section. In this paper, similar to some previous works, we assume that the time-to-failure of the switching components are described with an exponential distribution [28,29,34]. Therefore, according to the equations in Section 4.2, we can obtain the equations of time-dependent terminal reliability. Terminal reliability of 8  8 Benes network, denoted Rt88Benes (t), can be obtained by Eq. (25).

Rt88

Benes ðtÞ

¼ 4e5kt  2e6kt  4e8kt þ 4e9kt  e10kt

ð25Þ

Also, to calculate the terminal reliability of Benes network for larger sizes the following formula is used:

RtNN

Benes ðtÞ

   2  ¼ e2kt 1  1  RtNNBenes ðtÞ 2

ð26Þ

2

times, but Benes leads to a slightly higher terminal reliability than EBN. Therefore, the method of adding the number of stages is not suitable for improving the reliability of the Benes network. According to the results, two networks PBN and RBN are acquired the highest terminal reliability in comparison with the other two networks. Nevertheless, between these two networks, the RBN works better because compared with PBN it can obtain higher terminal reliability for different times. But as the figures show, this is not a strong excellence compared with PBN. 4.5. Broadcast reliability (time-dependent analysis) Time-dependent broadcast reliability refers to the probability of successful communication between a given source port with all destination ports as a function of time. It is clear that time has a negative impact on reliability trend, but it certainly would be useful to assess the influence. In this sub-section, like the previous one, we assume that the time-to-failure of the switching components is described with an exponential distribution and a reasonable estimate for k is about 106 per hour. The broadcast reliability of Benes network, EBN, PBN, and RBN can be obtained by following equations:

Rb88

Benes ðtÞ

¼ 4e9kt  2e10kt  4e13kt þ 4e14kt  e15kt

ð33Þ

   2  Nþ2 RbNN Benes ðtÞ ¼ e 2 kt 1  1  RbNNBenes ðtÞ 2

Rb88

EBN ðtÞ

ð34Þ

2

¼ 8e10kt  8e11kt þ 2e12kt  16e15kt þ 32e16kt  24e17kt þ 8e18kt  e19kt

Similarly, for the networks of EBN, PBN, and RBN, we have:

Rt88

EBN ðtÞ

¼ 8e6kt  8e7kt þ 2e8kt  16e10kt þ 32e11kt  24e12kt þ 8e13kt  e14kt 

RtNNEBN ðtÞ ¼ e2kt

Nþ2 kt 2



  2  1  1  RbNNEBN ðtÞ 2

ð36Þ

2

ð27Þ Rb1616

  2  1  1  RtNNEBN ðtÞ 2

RbNNEBN ðtÞ ¼ e

ð35Þ

PBN ðtÞ

ð28Þ

2

27

29

35

37

39

¼ 8e 2 kt  4e 2 kt  8e 2 kt þ 8e 2 kt  2e 2 kt  16e27kt þ 16e28kt  4e29kt þ 32e31kt  48e32kt þ 24e33kt  4e34kt  16e35kt

Rt1616

PBN ðtÞ

6kt

¼ 8e

 4e

7kt

9kt

 8e

þ 8e

10kt

11kt

 2e

þ 32e36kt  24e37kt þ 8e38kt  e39kt

 16e12kt þ 16e13kt þ 32e15kt  48e16kt

  Nþ2  2 RbNNPBN ðtÞ ¼ 1  1  e 4 kt RbNNBenes ðtÞ

þ 24e17kt  4e14kt  20e18kt þ 32e19kt

2

20kt

 24e

21kt

þ 8e

e

22kt

RtNNPBN ðtÞ ¼ 1  ð1  e Rt88

RBN ðtÞ

2

ðRtNNBenes ðtÞÞÞ 2

2

RBN ðtÞ

¼ 8e9kt  4e10kt  8e13kt þ 8e14kt  2e15kt

ð30Þ

 16e18kt þ 16e19kt  4e20kt þ 32e22kt  48e23kt þ 24e24kt  4e25kt  16e26kt

¼ 8e5kt  4e6kt  8e8kt þ 8e9kt  18e10kt

þ 32e27kt  24e28kt þ 8e29kt  e30kt

þ 16e11kt þ 32e13kt  48e14kt þ 24e15kt  4e12kt  20e16kt þ 32e17kt  24e18kt 20kt

e

RtNNRBN ðtÞ ¼ 1  ð1  ðRtNN Benes ðtÞÞÞ2

RbNNRBN ðtÞ ¼ 1  ð1  ðRbNN

Benes ðtÞÞÞ

ð39Þ

2

ð40Þ

ð31Þ 1.2

ð32Þ

According to the above equations, and with the assumption that a reasonable estimate for k is about 106 per hour [28,29,34], the time-dependent terminal reliability analysis as a function of time for the network sizes of 8, 16, and 32 are shown in Figs. 28–30, respectively. All three figures show almost the same results. According to these figures, as it is expected, the terminal reliability decreases with increasing the time, however, a better performance of a network will be obtained in case of less reduction compared with other networks. As illustrated in the charts, the weakest results are owned by two networks EBN and Benes. They operate very close to each other in terms of terminal reliability for different

Terminal Reliability

þ 8e

19kt

ð38Þ

2

ð29Þ Rb88

kt

ð37Þ

1 0.8 0.6 Benes network EBN PBN RBN

0.4 0.2 0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Time (Hr) Fig. 28. Terminal reliability as a function of time in network size 8.

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1.2

1.2

1

1

Broadcast Reliability

Terminal Reliability

14

0.8 0.6 Benes network

0.4

EBN

0.2

PBN RBN

0

10000

20000

30000

40000

50000

60000

70000

80000

0.8 0.6 Benes network

0.4

EBN PBN

0.2

RBN

90000 0

Time (Hr)

10000

20000

30000

40000

50000

60000

70000

80000

90000

Time (Hr)

Fig. 29. Terminal reliability as a function of time in network size 16.

Fig. 31. Broadcast reliability as a function of time in network size 8.

1

1.2

0.8

1

0.6 Benes network EBN PBN RBN

0.4 0.2 0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Broadcast Reliability

Terminal Reliability

1.2

0.8 0.6 Benes network

0.4

EBN PBN

0.2

RBN

Time (Hr) 0

10000

Fig. 30. Terminal reliability as a function of time in network size 32.

4.6. Cost-effectiveness During design of high-reliable interconnection networks, another concern of researchers is their hardware cost. In fact, although creating redundancy in hardware devices can improve the reliability of the network, increased cost resulting from the creation of this redundancy must be reasonable and proportionate to the extent of improvement in reliability. Therefore, the parameter that can be used in assessment of the MINs performance in terms

30000

40000

50000

60000

70000

80000

90000

Time (Hr) Fig. 32. Broadcast reliability as a function of time in network size 16.

Broadcast Reliability

1.2 1 0.8 0.6 0.4

Benes network EBN

0.2

PBN RBN

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Time (Hr) Fig. 33. Broadcast reliability as a function of time in network size 32.

Cost-Effectiveness of point-terminal

Time-dependent broadcast reliability analysis results for network sizes of 8, 16, and 32 are shown in Figs. 31–33, respectively. From the obtained results, it is clear that the highest broadcast reliability for different times is owned by PBN, among other networks. By examining each of these figures independently, it can be obtained that as time increases it becomes more advantageous to choose the PBN network over the other three ones. In addition, by putting together the results of these figures, it can be concluded that as network size increases, PBN has the most obvious advantages over other networks again. These results indicate the fact that the use of multiple parallel Benes networks is much better than other methods in terms of broadcast reliability. In contrast to the previous sub-section in which RBN will result in a weak advantage compared to the PBN in terms of terminal reliability, in this analysis, PBN leads to a clear superiority and extraordinary compared to RBN in terms of broadcast reliability. Altogether, based on the results obtained in this study and the previous one, although PBN attains slightly weaker results in terms of terminal reliability compared to RBN, however, its tangible superiority in terms of broadcast reliability makes us to choose the PBN as the best improved Benes network. Also, considering all the arguments made in the paper, it can be concluded that the approach of using multiple Benes networks in parallel compared with two other methods of adding the number of stages and using replicated Benes networks, is a better choice to improve reliability.

20000

6000 Benes network

5000

EBN PBN

4000

RBN

3000 2000 1000 0

1E-06

1E-05

0.0001

0.001

0.01

Switch Failura Rate Fig. 34. Cost-effectiveness of point-terminal as a function of switch failure rate in network size 8.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

15

M. Jahanshahi, F. Bistouni / Microelectronics Reliability xxx (2015) xxx–xxx

of cost, is the cost-effectiveness parameter that in the most reported works [13,19,28,29,60] has been emphasized. Therefore, here, this important parameter will be studied in order to deeply investigate the performance of the networks. The cost-effectiveness (CE) parameter is given by Equation (41) [13,19,28,29,60].

CE ¼

Meantime to failure ¼ Cost



0

0

RðtÞdt Cost

2

2



0

2

CET ðN  N BenesÞ ¼

0

" e





1 1 e

2kt

CET ð8  8 EBNÞ ¼

CEB ð16  16 PBNÞ !      2 2 R1 19 kt 3kt kt 2 2 dt 1  1  e 1  1  e ð1  ð1  e Þ Þ 0 ¼

224 ð54Þ R1 0

2

2

2Nð2 þ ðð2log2 NÞ  3ÞÞ ð55Þ

ð1  ð1  e

2 kt 2

2 # !

Þ Þ

dt

CEB ð8  8 RBNÞ !       2 2 R1 5kt 3kt kt 2 dt 1  1  e 1  1  e 1  ð1  e Þ 0 ¼

160 ð56Þ

    2  dt e2kt 1  1  RtNNEBN ðtÞ 2

2

CEB ðN  N RBNÞ ¼ ð45Þ

2Nð2ðlog2 NÞÞ

CET ð16  16 PBNÞ    2 ! R1 2 3kt 2kt kt 2 1  1  e 1  ð1  ðe ð1  ð1  e Þ ÞÞÞ dt 0 ¼

224 ð46Þ R1 0

   2  dt 1  1  ekt RtNNBenes ðtÞ 2

2

2Nð2 þ ðð2log2 NÞ  3ÞÞ

ð47Þ

CET ð8  8 RBNÞ !     2 2 R1 2kt 2kt kt 2 dt 1 1 e 1  1  ðe ð1  ð1  e Þ ÞÞ 0 160 R1  1  ð1  ðRtNN 0

dt Benes ðtÞÞÞ

4Nð2ðlog2 NÞ  1Þ

2

ð49Þ

CEB ð8  8 BenesÞ     2  R 1 5kt 3kt kt 2 dt e 1  1  e ð1  ð1  e Þ Þ 0 ¼ ð50Þ 80     2  R 1 Nþ2kt 2 dt e 1  1  R ðtÞ N N b 2  2 Benes 0 CEB ðN  N BenesÞ ¼ 2Nð2ðlog2 NÞ  1Þ ð51Þ

R1  1  ð1  ðRbNN 0

2 Benes ðtÞÞÞ

dt ð57Þ

4Nð2ðlog2 NÞ  1Þ

According to Eqs. (42) and (49), the results of cost-effectiveness of point-terminal analysis as a function of the switch failure rate ðkÞ for network sizes 8, 16, and 32 are shown in Figs. 34–36, respectively. These figures depict almost the same results. These results confirm that the PBN is the best choice in terms of cost-effectiveness of point-terminal compared to other networks. In other words, the PBN can achieve higher efficiency versus its increased cost. Furthermore, according to Eqs. (50) and (57), the results of costeffectiveness of point-broadcast analysis as a function of the switch failure rate ðkÞ for network sizes 8, 16, and 32 are shown in Figs. 37–39, respectively. From Figs. 37–39, it is evident that the best and weakest results in terms of cost-effectiveness of point-broadcast are owned by PBN and RBN, respectively. In other words, according to the results

Cost-Effectiveness of point-terminal

0

CET ðN  N EBNÞ ¼

CET ðN  N RBNÞ ¼

   Nþ2  2  dt 1  1  e 4 kt RbNNBenes ðtÞ

96

R1

¼

2

ð53Þ

ð44Þ

CET ðN  N PBNÞ ¼

2

2Nð2ðlog2 NÞÞ

2

2Nð2ðlog2 NÞ  1Þ

2kt

0

    2  Nþ2 dt e 2 kt 1  1  RbNNEBN ðtÞ

CEB ðN  N PBNÞ ¼

ð43Þ R1

R1

ð42Þ

    2  dt e2kt 1  1  RtNNBenes ðtÞ

dt

ð52Þ

dt

80

#! 2 2

96

CEB ðN  N EBNÞ ¼

e2kt 1  ð1  ðe2kt ð1  ð1  ekt Þ ÞÞÞ

CET ð8  8 BenesÞ ¼

   2 e5kt 1  1  e3kt 1  ð1  ekt Þ

CEB ð8  8 EBNÞ ¼

ð41Þ



R1

0

R1

Also, to estimate the cost of a MIN in Eq. (41), one common method is to calculate the crosspoint cost. The crosspoint cost is given by the number of crosspoints within a switching element and by the number of switching elements within the network [18,27]. On the other hand, similar to the reliability, cost-effectiveness parameter can be calculated from two standpoints of terminal and broadcast. Consequently, according to Eq. (41), cost-effectiveness of point-terminal, denoted CET, and cost-effectiveness of point-broadcast, denoted CEB, are given by the following equations: R1

R1

"

1800 1600

Benes network

1400

EBN

1200

PBN RBN

1000 800 600 400 200 0

1E-06

1E-05

0.0001

0.001

0.01

Switch Failure Rate Fig. 35. Cost-effectiveness of point-terminal as a function of switch failure rate in network size 16.

Please cite this article in press as: Jahanshahi M, Bistouni F. Improving the reliability of the Benes network for use in large-scale systems. Microelectron Reliab (2015), http://dx.doi.org/10.1016/j.microrel.2014.12.008

Cost-Effectiveness of point-terminal

16

M. Jahanshahi, F. Bistouni / Microelectronics Reliability xxx (2015) xxx–xxx

600 Benes network

500

EBN PBN

400

RBN

300 200

5. Conclusion

100

Multistage interconnection networks play a key role in the performance of multiprocessor systems and parallel processing. In the meantime, the Benes network is considered as one of the most important MINs for use in these systems due to its rearrangeability feature. On the other hand, multi-processor systems and supercomputers are typically made up of thousands of processor elements. Therefore, to be used in the large-scale systems, one of the requirements of the Benes network is reliability. In this paper, to find the best solution to improve the reliability of the Benes network, following three important approaches to reliability improvement of MINs was investigated; (1) Adding a number of stages to MIN. (2) Using multiple MINs in parallel. (3) Using replicated MINs. In this paper, the reliability analysis was conducted from two different perspectives; terminal and broadcast. The time-independent terminal and broadcast analyses showed that the approach of using multiple networks in parallel and using replicated Benes networks will result in the highest performance in terms of reliability. Time-independent analysis showed that the two approaches can achieve nearly the same results. However, the method of using replicated Benes networks is slightly better than using multiple Benes networks in parallel in terms of terminal reliability. Moreover, the results indicated that the approach of using multiple Benes networks in parallel, can achieve slightly better performance in terms of broadcast reliability, compared to using replicated Benes networks. For this reason, choosing the best solution from the two approaches to improve the reliability of the Benes network was difficult. However, the time-dependent analyses were instrumental in choosing the best approach. The time-dependent analysis showed that although the approach of using multiple Benes networks in parallel was slightly weaker than the using replicated Benes networks in terms of terminal reliability, but it achieved a significant broadcast reliability compared with the method of using replicated Benes networks. For this reason and due to the importance of broadcast communication in MINs, we found that the best method to improve the reliability of Benes network from the three important approaches to improve the reliability of the MINs is using multiple Benes networks in parallel. Moreover, the results of the cost-effectiveness analysis also confirmed that the using multiple Benes networks in parallel is an affordable method in comparison with other two methods.

0 1E-06

1E-05

0.0001

0.001

0.01

Switch Failure Rate

Cost-Effectiveness of point-broadcast

Fig. 36. Cost-effectiveness of point-terminal as a function of switch failure rate in network size 32.

3500 Benes network

3000

EBN

2500

PBN

2000

RBN

1500 1000 500 0

1E-06

1E-05

0.0001

0.001

0.01

Switch Failure Rate

Cost-Effectiveness of point-broadcast

Fig. 37. Cost-effectiveness of point-broadcast as a function of switch failure rate in network size 8.

700 Benes network

600

EBN

500

PBN

400

RBN

300 200 100 0

1E-06

1E-05

0.0001

0.001

0.01

Switch Failure Rate Fig. 38. Cost-effectiveness of point-broadcast as a function of switch failure rate in network size 16.

Cost-Effectiveness of point-broadcast

obtained in the previous and this sub-section, it can be concluded that the use of the several Benes networks in parallel can provide more benefits to the Benes network in terms of reliability and cost-effectiveness parameters in comparison with the idea of adding a number of stages to Benes network and using replicated Benes networks.

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140 Benes network

120

EBN

100

PBN

80

RBN

60 40 20 0

1E-06

1E-05

0.0001

0.001

0.01

Switch Failure Rate Fig. 39. Cost-effectiveness of point-broadcast as a function of switch failure rate in network size 32.

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