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Article

Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers

by
Maryam Al-Shammari
1,
Asrar Haque
1 and
M M Hafizur Rahman
2,*
1
Department of Computer Science, CCSIT, King Faisal University, Al Ahsa 31982, Saudi Arabia
2
Department of Computer Networks & Communications, CCSIT, King Faisal University, Al Ahsa 1982, Saudi Arabia
*
Author to whom correspondence should be addressed.
Sustainability 2023, 15(8), 6766; https://doi.org/10.3390/su15086766
Submission received: 16 January 2023 / Revised: 1 April 2023 / Accepted: 4 April 2023 / Published: 17 April 2023
(This article belongs to the Special Issue To the Future: Adoption of Artificial Intelligence and Blockchain in Agriculture and Healthcare from a Sustainability Perspective)
Browse Figures
Versions Notes

Abstract

:
Many artificially intelligent systems solve complex health- and agriculture-related problems that require great computational power. Such systems are used for tracking medical records, genome sequence analysis, image-based plant disease detection, food supply chain traceability, and photosynthesis simulation. Massively parallel computers (MPCs) are among those used to solve these computation-intensive problems. MPCs comprise a million nodes; connecting such a large number of nodes is a daunting task. Therefore, hierarchical interconnection networks (HINs) have been introduced to solve this problem. A midimew-connected torus network (MTN) is a HIN that has basic modules (BM) as torus networks that are connected hierarchically by midimew links. This paper presents the performance of MTNs in terms of static topological parameters and cost-effectiveness, as measured through simulations. An MTN was compared with other networks, including mesh, torus, TESH, TTN, MMN, and TFBN. The results showed that our MTN had a low diameter with a high bisection width and arc connectivity. In addition, our MTN had a high cost–performance trade-off factor (CPTF), a high cost-effective factor (CEF), low packing density, and moderate message-traffic density with marginally higher costs, as compared to other networks, due to wire complexity. However, our MTN provided better bandwidth with higher static fault tolerance. Therefore, MTNs are suggested for further evaluation of the effective implementation of MPCs.

1. Introduction

One of the challenging issues in implementing supercomputers is the topology involved in connecting a massive number of nodes. It is necessary for the enormous computational power of these supercomputers to progress towards a more sustainable world, which will provide better healthcare, better use of natural resources, sustainable agriculture, and energy-efficient machinery and industry. To achieve these goals, it is required to advance complex modeling, comprehensive simulations, and big data analysis, which are all computation-intensive tasks. Massively parallel computers (MPCs) have been broadly utilized to perform these tasks.
The role of MPCs in building sustainable applications is of great interest, as they could lead to a next-generation agricultural revolution with the assistance of artificial intelligence (AI). MPCs utilize resources to provide important functionalities, such as image-based plant disease detection, plague control, pesticide design, understanding pesticides effects, and food supply-chain traceability [1,2]. Massive amounts of seed data have been analyzed to identify plants with the Rubisco enzyme, which aids in the conversion of plant carbon into biomass [3]. Agricultural water management (AWM) was able to integrate multidisciplinary models that used MPCs [4]. Workflows for agricultural and livestock-farming applications have been designed through hybrid data analytics [5]. Enormous computational power is required to implement deep-learning networks for the price forecasting of agricultural products, which has a significant impact on the profitability of agricultural products [6].
AI has been applied in several approaches to climate pattern prediction and weather forecasting [7,8] to increase agricultural productivity and sustainability [9]. The use of models for assessing future risks that are related to climate change and its impact on agriculture is critical [10]. The genome sequencing of bacterial species to enhance plant growth in order to overcome climate challenges and limited resources has been conducted using supercomputers [11].
The role of MPCs is crucial in providing sustainable healthcare services. In recent years, they have advanced in order to handle the unusually large and distributed medical and non-medical data related to COVID-19 [12] by enabling the tracking of infected people and effectively monitoring whole communities [13,14]. Supercomputers were used to analyze the genome sequence of the virus in order to identify an appropriate vaccine [15,16,17]. Another important domain of supercomputers in healthcare is telemedicine, which includes the diagnosis, the examination, and the medical assessment. Telemedicine relies on information systems with massive storage, and requires the precise operation of tasks by high-performance computing and communications (HPCC) strategies [18]. Researchers used next-generation AI and robot core technologies to simulate the human brain [19,20]. To control the spread of serious communicable diseases, integrating epidemiology with the rapid computing power of MPCs was proposed [21]. Deep-learning models were implemented on MPCs as accelerators to speed-up medical applications for large biomedical databases [22]. Cambridge-1 was used for building and hosting powerful AI language models in molecular chemistry [23]. Supercomputers have also contributed to everyday activities such as fraudulent insurance claim detection by analyzing massive claim data within a short period of time [24].
The performance of MPCs continues to increase in order to provide the massive computational power required for various applications. According to the TOP500 organization, the first true exascale supercomputer was the Frontier system at the Oak Ridge National Laboratory (ORNL) in the U.S. [25]. It achieved the top rank in November 2022, with a performance exceeding 1.102 Exaflop/s. MPCs can support high accuracy applications because they are driven by matrix multiplication [26]. For example, AI applications rely on supercomputers for complex modeling tasks, such as seismic imaging, climate/environmental geospatial predictions, and computational astronomy [27]. Therefore, supercomputers have been shown to be ideal for training deep neural networks (DNNs) by processing the large datasets required for fast and accurate DNN results [28,29]. Supercomputers have been linked directly to grand challenges and their solutions [30]. Therefore, many research centers are rapidly adopting supercomputer resources due to these benefits.
In this paper, we addressed a serious concern in the implementation of supercomputers, which is the topology connecting massive numbers of nodes. This topology directly impacts the performance of MPCs [31]. Therefore, hierarchical interconnection networks (HINs) have received much attention from the research community. HINs enable the connection of large numbers of nodes with reasonable delays, as compared to conventional networks [32]. Supercomputer connections have reached the capacity of one million nodes, making conventional networks infeasible [33]. In our previous work, we proposed an HIN called an MTN, which used 2D torus connected by midimew links to build higher levels in order to include greater numbers of nodes [31]. However, the current research aimed to evaluate the network topology represented by static parameters and cost effectiveness. The evaluation of topological properties is crucial for a hierarchal network to be considered for further development. MTNs provide optimal features for static and cost-effective performance. This is shown in a comparative analysis of static network performance and cost-effective parameters according to the diameter and average distance evaluated by computer simulation. MTNs are compared to six other networks, two with conventional topologies, including 2D mesh and 2D torus. The presence of conventional networks in the evaluation was because of their current usage by supercomputers [34]. In addition, an MTN is compared to other HINs, including a midimew-connected mesh network (MMN) [35], a tori-connected mesh network (TESH) [36], a tori-connected flattened butterfly network (TFBN) [37], and a tori-connected torus network (TTN) [38].
The results of the simulated MTN showed a similar diameter to MMN that was lower than the mesh and TESH but higher than the other networks. MTNs have a lower average distance than almost all networks; a node degree which is lower than TFBN but higher than conventional networks; moderate bisection width; high arc connectivity; and higher costs as a result of wire complexity. However, MTNs also have better fault tolerance and bandwidth availability. In addition, in terms of static performance, MTNs have advantages in CPTFs, moderate time–cost-effective factors (TCEFs), message traffic density (MTD), and low packing density. However, MTNs have higher costs when compared to other networks, although they are still lower than mesh networks. This is a result of their improved bandwidth availability and high fault tolerance. Therefore, practical implementations of MPCs should consider adopting the most relevant features for the intended system. For example, high fault handling and sufficient bandwidth may make an MTN ideal for certain applications; however, considering its static performance and overall costs are worthy considerations.
The remaining sections are organized as follows: Section 2 illustrates MTN architecture, followed by Section 3, which discusses the routing algorithm. Section 4 provides a comparative analysis of the static network performance. Section 5 shows the cost-effective analysis of the network. Finally, Section 6 presents the main findings and recommendations.

2. Network Architecture of MTNs

In this section, the architecture of MTNs is illustrated. With regards to our previous paper [31], we only updated the layout of the horizontal links as seen in Figure 1. This difference provides a low diameter and average distances in the network. The main goal of HINs is to achieve a low diameter and average distance values, as described in Section 4. These two properties were not evaluated in our previous paper. Furthermore, this study also evaluated a new modified network to evaluate its static performance and cost effectiveness.
The basic building block of a midimew-connected torus network is the basic module (BM), which was 2 m   ×   2 m torus, where m is a positive integer. The hierarchy connection which was implemented at higher-level connections with the 2 m   ×   2 m torus BM used midimew links. The 2D torus block represented level 1, the lowest level. Higher levels were the result of midimew connections in level 1. The packet flow in the network links was different at the lowest level (level 1), as compared to that at higher levels. All of the levels used bidirectional links for flexible connections. The following definitions illustrated the network topology:
Definition 1.
The 2D torus (level 1) was a  2 m   ×   2 m  BM containing  2 2 m  nodes created by  2 m  rows and  2 m  columns.
Definition 2.
An MTN (m, L, q) had a  2 m   ×   2 m  BM, where L is the level of hierarchy and q is the inter-level connectivity. The connections that formed the higher levels used  4   ×   2 q  of each of the BM’s free ports,  2 ( 2 q ) for the horizontal connections and  2 ( 2 q ) for the diagonal connections, where  q ∈ { 0 , 1 , … . , m } . The minimal level of inter-connectivity was when  q = 0 , while maximum was when  q = m . In this paper,  m = 2 for better granularity.
Definition 3.
The highest level that could be obtained by the MTNs ( m , L , q ) was L m a x = 2 m − q   +   1 level.
Definition 4.
The wire complexity,  W L ,  was the total number of wires used to link the network elements. Therefore, the  W L  of MTNs was found using Equation (1), where  W B M  is the number of links in a single BM.
W L = W B M × 2 m × 2 ( L − 1 ) + ∑ x = 2 L 2 2 q × 2 m × 2 ( L − 1 )
The following figures are provided to illustrate these definitions. Figure 1 represents an MTN (2,2,0). Figure 2 shows the overall network connections forming level 2. Further details of each level are shown in Table 1.
The modification to the network, as shown in Figure 1 and based on the previous version [31], did not change the total number of nodes, free ports, or wire complexity, as shown in Table 1. This indicated that both versions supported the same number of nodes for supercomputers. However, different connections were needed to achieve better values for diameter and average distance, as shown in Section 4.

3. Routing Algorithm of MTNs

This section discusses the simulation of the proposed topology. MTN was simulated using JAVA coding to generate the diameter and average distance values. The algorithm for the routing protocol is presented here, and it generated the message paths in the network with respect to its layout. The routing algorithm identified the methods by which the network chose the best path, which in this study was the shortest. This protocol, followed by the packets from their source to their destination, was critical. It’s incorrect implementation would severely impact the performance efficiency, since the main goal of the routing algorithm was to select the most cost-effective path. A supercomputer relies on the routing protocol used to connect its nodes. In HINs, these procedures are based on the network topology. For example, the main phases of specific HINs, such as STTN, FTBN, and MMN, may be adapted by applying similar routing procedures, as detailed in this section. However, due to the different layouts of the network (i.e., the topology), the result of the hop-distance, as represented by diameter and average distance, would not be the same.
The MTN routing algorithm began at the higher levels before moving to the lower levels based on the hierarchy interconnectivity. The packet dimensions were compared to determine whether its destination was on the same level. If its destination was on the same level, then the routing used an inter-BM connection. The network would follow the shortest path while transferring the packet via assigned outer nodes in each BM. In a similar manner, if the packet was traveling to another level, the routing would use the identified outer nodes to complete the packet transfer. Figure 1 shows the outer nodes for each level. For example, if a level-2 packet had a level-5 destination, then the shortest-path routing would use the bottom-right BM to exit that level. At level 5, the packet would enter through the top-left BM.
In the simulation, the MTN was programmed to generate the shortest path between any two distinct nodes of the network, so that the simulation would be able to generate all of the required paths in order to calculate the diameter and average distance. The algorithm’s implementation generated the paths with respect to a simple procedure to evaluate the proposed layout. The simulation was conducted using deterministic order routing due to its simplicity and clarity. In such protocols, the hop-distance or the shortest path was counted directly, depending on the proposed topology. The simulation followed dimensional order routing to choose paths based on the cost and dimensionality of the nodes. To move from one node to another, the simulation would first travel horizontally via the x-axis, and then, when the packet could go no further in this direction, it would switch to moving vertically along the y-axis. Therefore, the hop-distance was counted directly as the distance between the packet source and the destination to evaluate the MTN topology in terms of low hop-counts and, thus, better performance.
Figure 3 shows the steps followed to apply the MTN topology. The network used source-and-destination pairs as BMs, and it considered the levels of both BMs represented in source destination nodes. It would then choose higher or lower levels based on where each BM node was located. The MTN used an outer BM for moving from one level to another, as shown in Figure 1. Inside the level, it considered whether the BM had the destination node to route the packet to that exact node inside the BM. Routes using shortest-path followed dimensional order routing.
Figure 3 illustrates the steps which were followed to code the MTN to calculate the diameter and average distance. A supercomputer contains many interconnected nodes. Since the main goal of the routing algorithm was to simulate the MTN performance for such systems, it considered the path of all pairs of nodes in the network in order to generate diameter and average distance values. Therefore, the intent was to obtain the routing results from one node to another. The addresses and the representations of the nodes are shown in Figure 4. The layout of the BM was also presented as a reference for the numbering format used for naming, such as “level.BM.node.” From source to destination, we found the following:
Source.BM.level/destination.BM.level: This referred to the level where the source/destination BM was located. In Figure 4, the source and destination were both on level 2. Therefore, Source.BM.level and destination.BM.level = 2.
Source.BM/destination.BM: This referred to the exact source/destination BM on the same level. In Figure 4, Source.BM = (3,3), destination.BM = (2,1).
Source.BM.node/destination.BM.node: This referred to the source/destination node inside a particular BM. In Figure 4, Source.BM.node = (2,2), destination.BM.node = (2,2).
The simulation was performed by coding the BM and the higher level in order to obtain the paths between a pair of nodes in the network. The flow chart of the routing algorithm is shown in Figure 3, following the protocol and using dimensional order routing and controlled routing flow to obtain the network paths. In the next section, we evaluated the MTN via the results from this simulation for diameter and average distance, while the other parameters were derived using a graph model.

4. Comparative Static Network Performance Evaluation

To evaluate the performance of MTNs, it is beneficial to analyze the network’s static parameters. MPCs require an implementation with a low constant degree, low cost, good connectivity, high fault tolerance, and high scalability [39]. Such features could be derived from the static performance evaluation, as shown in this section. Our MTN was compared with two conventional networks, mesh and torus, since such networks are still widely used in supercomputer applications [34]. However, several proposed HINs were included in the comparison to evaluate their capabilities in connecting nodes while maintaining the required characteristics.

4.1. Node Degree

Node degree refers to the maximum number of links exiting a node. Its impact was directly linked to the network cost, where the higher the node degree, the higher the cost due to the router I/O and physical links [40]. In addition, the MPCs considered constant degrees because, when the number of nodes increased, the configuration of the router interfaces would not change. Therefore, the scalability of this network was simple and cost effective. In contrast, our MTN was more computationally demanding. However, it provided high bandwidth availability for low congestion and minimal network latency, both for a reasonable cost increase [41]. As shown in Table 2, the MTN had a moderate node degree, as compared to a high degree in the TFBN and the low values found in conventional networks. In addition, Figure 5, below, is used to show the time complexity of various networks compared to MTNs when the network size increases.

4.2. Diameter

Considering all of the shortest paths for node-to-node connections, the diameter was the maximum path. It was a main parameter that contributed to the dynamic performance of the network, as it significantly affected network latency despite regular throughput. High diameter resulted in high delays of message transmissions. In addition, higher diameters lowered the message-passing bandwidth and, thus, degraded the overall performance of the network [42]. Conventional networks, such as mesh and torus, as shown in Table 2, suffer from this specific challenge, making them infeasible for next-generation MPCs. This motivated our proposal of this novel HIN to maintain a low diameter while supporting a large number of node connections. The MTN diameter was generated according to the routing protocol described in Section 3. The MTN had a low diameter, although it was larger than that of the TFBN. However, it was considered low as compared to conventional topologies, as their diameter increased as the number of network nodes increased.

4.3. Average Distance

The average distance was the average obtained from the total value of all different pairs of nodes represented by the shortest path connection. A lower average distance indicated a lower buffer, which yielded lower contention. It was directly proportional to the network latency, as a lower average distance resulted in lower latency. Similar to the diameter, it had a direct impact on the latency of the network. However, it considered the network under no-load conditions, while the diameter considered the packet flow saturation [43]. The same as for diameter, it was generated with the help of the routing protocol described in Section 3. To promote optimal inter-node communication in terms of average message-transfer latency, the average distance needed to be low, as demonstrated by the MTN, as compared to all the other networks, with the exception of the TFBN.

4.4. Arc Connectivity

This was the minimum number of links required to divide the network into two separate parts. It was a critical value, as it indicated the fault tolerance of the network. The fault tolerance had to be prepared in advance due to the high frequency of faults [44]. Therefore, a high value in arc connectivity indicated a more robust network. The static fault tolerance of the HIN was identified by the ratio between the arc connectivity and the node degree [45]. The BMs of the MTN provided the maximum fault tolerance that a network could have due to the equality between the node degree and the arc connectivity. In addition, the arc connectivity indicated the alternative paths that could be taken in the event of a failure. The MTN had optimal arc connectivity, similar to most of the other networks. However, the BM had a maximum fault tolerance that enabled robust implementation of the underlying architecture. In addition, Figure 6, below, shows the arc connectivity of various networks compared to MTN when the network size increases.

4.5. Bisection Width

Bisection width was the minimum number of links required to ensure two equal halves of the network. In a parallel processing environment, this was vital due to the regular use of a divide-and-conquer approach, where the network is divided into two halves that are then each divided further to enable the network to perform multiple tasks simultaneously. If the value of the bisection width was low, this resulted in a lower bandwidth used to merge the two halves, leading to more congestion in the packet flow. However, a large value could not be viable without a significant increase in wiring, resulting in a more costly and complex implementation [46]. Therefore, moderate values were ideal for parallel processing, which was demonstrated in the MTN, as compared to conventional networks that were more complex and expensive. Also, Figure 7, below, shows the bisection width of the MTN in comparison with other networks as the network size increases.

4.6. Wire Complexity

Wire complexity refers to the total number of wires required to implement the network topology. It depended on the node degree, since it identified the wires exiting a node. Although wire complexity was not an actual cost of the network configuration due to physical routers and the length and wire types used, it provided a reliable indicator of resource demand. The greater the number of wires that were needed, the more cost was required for physical resources. However, costs could be justified according to their contributions to low latency and high fault tolerance, and even lower the costs of the overall network, that is, they could elicit a better performance in terms of the connectivity between nodes and hierarchal levels. The MTN had better performance on this parameter, as compared to the TFBN, due to having lower wire complexity for physical links, improving its scalability and affordability. Finally, Figure 8, below, is used to show the wire complexity of various networks in comparison with MTN when the network size increases.
This evaluation indicated the most important parameters for the implementation of a practical supercomputer. The best adaptation of the parallel processing environment would require some compromises in terms of robustness versus costs. For example, given the high probability of faults when scaling is regularly conducted, the reasonable costs of the flexible MTN makes it a better choice than the TFBN, which could become quite expensive under these circumstances. However, a TTN could provide simpler and less costly scaling, as compared to an MTN. In contrast, MTNs have better inter-node communication with respect to average message transfer latency due to the lower average distance. According to the above comparison, MTNs have better fault tolerance, enhanced cost-efficient scaling, improved inter-node communication, and fewer physical wires, as compared to other HINs with similar features.

5. Cost Effectiveness Analysis of MTNs

Static costs were used to identify the feasibility of the network prior to implementation. The hardware used, including routers, processing elements, and wiring for the topology adaptation, contributed to the actual cost. However, static costs, along with the other parameters presented in this section, were good indicators for network assessment. A cost-effective study was performed, since the overall costs were a major factor when choosing a topology for adaptation. The performance of a supercomputer was identified by the topology used to connect its nodes. Therefore, evaluating the cost of the topology was essential to clarify the characteristics of a network prior to its implementation.
The network costs of MTNs were illustrated by six parameters: cost, cost performance trade-off factor, cost-effective factor, time–cost-effective factor, packing density, and message traffic density. These factors were chosen to summarize the relationships between the costs and the most essential parameters of MPCs. The six factors which were assessed for all of the other networks and compared to MTNs were similar to the static parameters, as shown in Section 4. A tubular comparison is depicted in Table 3 for exact value comparison. The following explanation elaborates on each factor and its calculations.

5.1. Cost

Cost was determined as the product of diameter and node degree. These two main parameters represented the capability of the hardware. For example, a higher node degree resulted in higher routing expenses in either installation or maintenance. Message traffic density, the distance between nodes, network bandwidth, latency, and fault tolerance all depended on the diameter and node degree. Systems with a large diameter and node degree would be expensive, with low bandwidth and poor scalability. Cost ( C ) was calculated using Equation (2), where D is diameter and N d is node degree.
C = N d × D
Our MTN had a good cost outcome, as compared to conventional networks such as mesh, as shown in Table 3. However, it was slightly more expensive than other networks. However, this slight difference could be justified due to its better bandwidth and enhanced network latency.

5.2. Cost Performance Trade-Off Factor

The CPTF enabled the assessment of the network by providing a crucial static parameter. The CPTF summarized four of the most important parameters: diameter, node degree, wire complexity, and total number of nodes. This provided a rapid comparison regarding costs, as opposed to comparing each characteristic individually. The diameter indicated the upper latency and bandwidth. The node degree was related to the router costs and scalability. The wire complexity was directly proportional to the wire costs through the layout links. The number of nodes was the actual processing elements that were required for the execution of the programs. The C P T F was calculated using Equation (3).
C P T F = N d × G p D ,   G p = W L t o t a l   #   n o d e s
The CPTF of the MTN was higher than for the TESH, the MMN, the torus, and the mesh. It was slightly lower than the TTN, and notably different from the TFBN. Therefore, the MTN provided a good trade-off between performance and costs.

5.3. Cost-Effective Factor

Supercomputers provide fast computational power for various complex simulations. Therefore, speed, as a result of connecting large numbers of nodes, is critical. In addition, the level of efficiency while using these nodes contributed to the fast response. Therefore, speed and efficiency were indicated using CEF. In addition, the number of links used to connect the nodes depended on the topology configuration. These links had to be considered when analyzing system costs. The C E F was expressed as Equation (4), where ρ = c o s t   o f   W L c o s t   o f   n o d e s , 0 ≤ ρ ≤ 1 ; in our evaluation ρ = 0.1
C E F = 1 1 + ρ × G p
The CEF of the MTN was almost equal to that for the six compared networks. However, the MTN’s efficiency resulted in a good CEF.

5.4. Time–Cost-Effective Factor

This factor was included to show the static topology performance. MPCs conduct complex experiments that require results within a relatively short period of time. The efficiency of this process is affected by the speed of the processing nodes used, the problem-solving steps, and the routing algorithm via topology connections. However, TCEF was important for assessing the time required for large programs used by MPCs. The TCEF for the MTN was evaluated according to the final simplification using Equation (5), where p = t o t a l   n o d e s :
T C E F = 2 1 + ρ G p + 1 p
The TCEF of the MTN was similar to that of the TTN, torus, and mesh, with a slight difference when compared with TESH, MMN, and TFBN. The time comparison considered nodes and links for performing parallel tasks, resulting in a reasonable factor for use in an MPC.

5.5. Packing Density

In order to show that MTNs could be used for very large-scale integrations (VLSIs), the packing density was measured. This was the ratio between the total number of nodes and the cost, which indicated that a higher value for packing density required less area for the VLSI chip. The packing density of the MTN was evaluated using Equation (6).
p a c k i n g   d e n s i t y = t o t a l   n o d e s C
The packing density of the MTN was better, as compared to the higher values found in torus, TESH, TFBN, and MMN. The MTN was almost similar to mesh, with a slightly higher result.

5.6. Message Traffic Density

This factor was related to the alternative paths that were available from a single node to another. Multiple paths enabled the system to consider alternative choices in the event of faulty links. In other words, this factor indicated whether the traffic distribution of the network was efficient. The traffic distribution relied on the availability of multiple paths for a packet. The MTN was evaluated using Equation (7), along with other networks for comparison.
M T D = a v e r a g e   d i s t a n c e × t o t a l   n o d e s W L
The MTD of the MTN provided a better average MTD, as compared to the high TESH and TFBN results.
The MTN provided satisfactory costs with respect to various performance parameters. Therefore, using MTNs for supercomputing is feasible, with better bandwidth, latency, fault tolerance, time, and costs, as compared to conventional topologies. As compared to other hierarchal interconnection networks, there were many similarities and some differences. For supercomputing, MTNs are more expensive; however, they provide better performance with better diameter, node degree, and fault tolerance due to wrap-around links of torus BMs that are supported with hierarchal connections via midimew links.

6. Conclusions

This paper analyzed the proposed HIN for a large number of nodes for exascale and zetta-scale computational power. In this study, the static performance was completed by simulating the network. The simulation of the network-level connections resulted in low diameter and average distance values. The MTN was compared with both conventional mesh and torus networks. In addition, the comparison included hierarchal networks, such as TESH, TTN, TFBN, and MMN. We used simulations to evaluate an MTN via static parameters, including the node degree, the diameter, the average distance, the bisection width, the arc connectivity, and the wire complexity. In addition, a cost-effective analysis of the network was conducted based on the following parameters: MTD, CEF, CPTF, TCEF, packing density, and cost. Our results provide a solid foundation for further research and implementation.
This study showed that MTNs have good features with respect to network connectivity, fault tolerance, and bandwidth, as the MTN had lower parameters than conventional networks. However, it was higher than for the TFBN while not as high as the TESH. The MTN had a lower average distance than all of the other networks, except for the TFBN. In addition, in terms of static performance, the MTN had a high CPTF; moderate TCEF and MTD; and low packing density. However, the MTN had a high cost factor, as compared to other networks, although it was still lower than that of mesh. This was the result of better bandwidth and its high fault tolerance. In this case, a practical implementation of a supercomputer may consider the relative required features for specified tasks to determine an appropriate compromise.
The MTN was shown to be a promising HIN. Therefore, future work could include dynamic performance evaluations [36,38]. In addition, since this research evaluated the network with a single configuration, that is, MTNs (2,2,0), additional configurations should be assessed to determine the best layout. However, the current MTN was assessed in terms of static topological parameters and several cost factors, and these results may promote fast-emerging technological advances.

Author Contributions

Conceptualization, M.M.H.R. and A.H.; methodology, M.M.H.R. and A.H.; software, M.A.-S.; validation, M.M.H.R. and A.H.; formal analysis, M.A.-S.; investigation, M.M.H.R. and A.H.; resources, M.M.H.R. data curation, M.A.-S.; writing—original draft preparation, M.A.-S.; writing—review and editing, M.M.H.R. and A.H.; visualization, M.A.-S.; supervision, M.M.H.R. and A.H.; project administration, A.H.; funding acquisition M.A.-S. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 3214].

Institutional Review Board Statement

Not applicable.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 3214]. The authors would like to thank the anonymous reviewers for their insightful scholastic comments and suggestions, which improved the quality and clarity of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Hierarchy of MTNs (2,2,0).
Figure 1. Hierarchy of MTNs (2,2,0).
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Figure 2. Complete level 2 connection of MTNs (2,2,0).
Figure 2. Complete level 2 connection of MTNs (2,2,0).
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Figure 3. Simplified routing steps of MTN.
Figure 3. Simplified routing steps of MTN.
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Figure 4. Example of source and destination pairs in an MTN.
Figure 4. Example of source and destination pairs in an MTN.
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Figure 5. Node degree of various networks when increasing the network size.
Figure 5. Node degree of various networks when increasing the network size.
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Figure 6. Arc connectivity of various networks when increasing the network size.
Figure 6. Arc connectivity of various networks when increasing the network size.
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Figure 7. Bisection width of various networks when increasing the network size.
Figure 7. Bisection width of various networks when increasing the network size.
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Figure 8. Wire complexity of various networks when increasing the network size.
Figure 8. Wire complexity of various networks when increasing the network size.
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Table
Table 1. All levels of MTNs (2, L, 0).
Table 1. All levels of MTNs (2, L, 0).
LevelConsists ofTotal NodesFree Ports UsedWire Complexity
Level 216 of 4 × 4 torus2564544
Level 316 of 4 × 4 level 2409688736
Level 416 of 4 × 4 level 365,53612139,808
Level 516 of 4 × 4 level 41,048,576162,236,960
Table
Table 2. Comparison of static network performance parameters.
Table 2. Comparison of static network performance parameters.
DiameterAverage DistanceNode DegreeBisection WidthArc
Connectivity		Wire
Complexity
256 Node
16 × 16 Mesh3010.674162480
16 × 16 Torus1684324512
TESH (2,2,0)2110.47482416
TTN (2,2,0)157.44684544
TFBN (2,2,0)105.75884800
MMN (2,2,0)179.07482416
MTNs (2,2,0)176.93684544
Table
Table 3. Static network cost comparison.
Table 3. Static network cost comparison.
CostCost Performance Trade-Off FactorCost Effective FactorTime Cost Effective FactorPacking DensityMessage Traffic Density
256 Node
16 × 16 Mesh1200.25000.84211.678682.13335.69066
16 × 16 Torus640.50.83331.661254.00004.000
TESH (2,2,0)840.3090.86021.714663.04766.4430
TTN (2,2,0)900.8500.82471.644182.84443.5011
TFBN (2,2,0)802.50.76191.519283.20001.84
MMN (2,2,0)680.3820.86021.714663.76475.5815
MTNs (2,2,0)1020.750.82471.644182.50983.2611
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Al-Shammari, M.; Haque, A.; Rahman, M.M.H. Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers. Sustainability 2023, 15, 6766. https://doi.org/10.3390/su15086766

AMA Style

Al-Shammari M, Haque A, Rahman MMH. Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers. Sustainability. 2023; 15(8):6766. https://doi.org/10.3390/su15086766

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Al-Shammari, Maryam, Asrar Haque, and M M Hafizur Rahman. 2023. "Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers" Sustainability 15, no. 8: 6766. https://doi.org/10.3390/su15086766

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