Reliable Gamma-Interconnection Network for Data Analysis in Sensor Networks: Design and Performance Evaluation

In today’s era of high speed 5G internet all electronic sensor networks are connected through IoT. Bank transactions are digitized, people can access any data through their mobile phones, organizations and companies handle their projects through online meetings etc. Military and medical surveillance, navy navigation, weapon controlling, weather forecasting etc. involve big data analysis collected from sensors, that too at a very high speed with reliable results. This requires large number of parallel processors connected with huge Bank of memory modules to store big data. Reliable interconnection network is needed to connect these large number of parallel processors and memory modules efficiently hence Multistage Interconnection Networks (MINs) come into play, as they provide highly reliable communication for big data transfer between processors and memory modules whenever required. In this manuscript a new network named Reliable Gamma-interconnection Network (RGN) is introduced which possesses multiple paths between processors and memory modules with two totally disjoint path availability. It provides high reliability and minimum path distance between source node to destination node than other gamma networks known, with the minimum hardware complexity. Reliability estimation and evaluation of RGN has been presented in this paper and comparison of results achieved with other gamma networks has been done for validation purpose.

High speed internet accessibility at sensor nodes required big data to be analyzed efficiently at server end to acknowledge large number of requests per second. [1][2][3] This application requires multiple processors to be connected in parallel with large memory banks to store big data collected through distributed sensors using IoT and cloud computing. The applications of parallel processors connected in conjunction with large number of memory modules are not only restricted to the use of internet but also involve super computers used for large number of applications such as navigation, 1-3 weather forecasting, banking and financial assessment, controlling nuclear reactions, medical, security, spacecraft engineering, and many more where huge number of sensor are deployed in the field. In many or all of these applications distributed sensing system has been employed where multiple sensors are installed to collect data from real world and send it back to the suitable node connected to the appropriate processing unit for further processing. Hence the area of parallel computing is emerging and critical which requires efficient computing through reliable communication of data from memory to processor and vice versa. 1 Min provides a reliable communication network, which communicate data between sensing node to processors and then to memory modules at reasonably high speed at modest cost. 4 This approach can help improve the scalability and performance of distributed sensing systems by providing high-speed data transfer between the sensors and the computing system. Much of research has been done to increase the speed of data communication using MIN to cope up with increasing demands of high speed internet connectivity.  Gamma MIN is a member of PM2 i class, which is known to be reliable and redundant class of networks and hence find application in high speed ATM switches and broadband communication. [4][5][6][7][8] Although, Gamma MIN is a high speed packet switch network which possesses multiple paths between source-to-destination node (SD node) but it behaves as a single path MIN when source address and destination address is same (tag value "T" = 0). in this case gamma MIN possesses no fault tolerance. This in turn limits its application and hence improvement in the structure is required.
Much of research work has been reported from early 2000 to till date to improve the fault tolerance capability of gamma interconnection network. The modifications/improvements suggested are summarized as follow: 1. Additional stages have been added to the basic structure of MIN to provide multiple path between each SD node pair including tag value "T" = 0. The networks in which this technique has been used are: extra stage gamma, 9 incomplete gamma, incomplete cyclic gamma, 19 balanced gamma, 7 enhanced improved augmented data manipulator 14 etc. 2. Bigger sized switching elements have been used to add redundancy in the network such as in fault tolerant interconnection network (FTIN), 25 3-disjoint gamma interconnection network (3-DGIN), 14 4-disjoint gamma interconnection network (4-DGIN) 21 etc. 3. Intra-stage changing has been used to improve fault tolerance such as in partially chained gamma interconnection network (PCGIN), 12 fully chained gamma interconnection network (FCGIN) 12 etc. 4. Reshuffling of connection pattern used in gamma MIN, for example, shuffling pattern used in mono gamma MIN, 9 cyclic gamma MIN 8 etc.
Although, these modifications done in gamma network increases its fault tolerance capability and reliability but its hardware complexity also increases. [18][19][20][21] This increment in hardware in turn increases the overall cost and transmission latency of the network. Moreover, most of these technique based on provision of providing additional redundancy in the network improves fault tolerance at intermediate stages only. 22 Whereas, there is no redundancy provided at input or output stages. It has been assumed in the available literature that component used at input and output stages are critical and must not fail. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] This belief appears to be unfounded because, regardless of the stage they are used in, all electronics switches are produced using the same design technology. No component may therefore be expected to be failure-free, and if a component installed at intermediate stage is prone to failures, then those faults/failures are likely to occur with the component installed input or output stages as well. Consequently, redundancy must also be offered at all stages. Some of the methods described in previous articles involve altering the network topology using intra-stage chaining between SE, 12,17,45 which results in the formation of nonuniform alternate paths from the source node to the destination nodes. Due to these uneven paths available for data transmission, it is challenging to attach packet headers to identify alternate routes in the occurrence of failures. All these gaps provide motivation to propose new reliable network belong to Gamma family which z E-mail: shilpa1_goyal@rediffmail.com provide minimum number of stages used with less number SEs at each stage. This will lead to cost reduction. To bridge the gap of providing reliable MIN with reduced overall cost a new network has been presented in this manuscript which is named as Reliable Gamma-interconnection Network (RGN). The structural to topology and reliability issues of proposed network has been discuss in the preceding sections.

Related Literature
Gamma interconnection network is a redundant multistage network which provide full connectivity between its input and output nodes. It belongs to PM 2 i class of MIN, which uses ±2 i connection pattern between its adjacent stages. 4 Gamma network is known to be highly reliable MIN and is employed in high performance computing system. The fault tolerance capability of Gamma MIN is due to the presence of multiple paths among its network nodes. Due to which it can tolerate more than one failure in runtime without affecting its overall performance. It is also known as fast speed packets switch network due to the fact that it uses virtual cut through routing. It implies that in this topology, data packets are only forwarded once the packet header has been received, which cuts down on transmission time. hence improving the overall network performance and providing maximum time to re-allocate the output node to data packet in case of node failure. The additional feature of the gamma interconnection network is the inclusion of error correcting code in the packet header of the data to be transmitted. This feature enables the error to be fixed if it only happened during transmission. This helps to increase the accuracy of the data obtained through the gamma interconnection network. A uniform number of switching elements (SEs) is present in each stage of the architectural topology of the gamma interconnection. For a gamma network of size N, N input and output nodes are connected via "n" stages total, where n is equal to log 2 n, and N input and output nodes are connected through these n stages. Except for the input and output stages, where SEs of sizes 1 × 3 and 3 × 1 respectively have been employed, SEs of configuration 3 × 3 are used. In the gamma network, the pattern for connecting the SEs at stage i (named from "0 to n") is based on the j ± 2 i pattern, wherein the SE at stage "i" is connected to three SEs at stage j with the connection patterns "(j − 2 i ) mod N," "j" and "(j + 2 i ) mod N." Figure 1 depicts the 3 × 3 SE's configuration. Three paths-upper, straight, and lower connection-are to be followed with respect to "(j − 2 i ) mod N," "j," and "(j + 2 i ) mod N," as illustrated in Fig. 1.
In order to connect inter-stage SEs for parallel computing, this connection pattern offers a scalable approach. It helps in effectively delivering the packet between stages while lowering latency and congestion. Every data packet that needs to be sent includes a packet header that lists the source and destination addresses from which a tag value must be determined. To route the data packet to its destination, a series of SEs are chosen at each stage based on the tag value. The difference between the source and destination addresses is to be used to calculate the tag value "T". Three possible values for "T" are "−1", "0", and "1". Data would be sent via an upper link for "T = −1", a straight link for "T = 0", and a lower link for "T = 1". This connection system is called as Binary Redundant Number System (BRNS). It is well known that BRNS is a fault tolerant system that, by isolating individual component failure and upholding failure-free communication, improves the performance of the communication system as a whole. Architecture of gamma interconnection network of size "8" is shown in Fig. 2. As illustrated in Fig. 2 there are multiple paths available between each source node to every destination node. Hence, gamma network is a redundant network except in case when source and destination are having same address and tag value is 0. In this case gamma network behaves as unique path MIN and have no fault tolerance.
In distributed computing, critical data must be gathered and processed through sensors used in the outside environment. For instance, failures are unacceptable in military security navigation, remote medical surveillance, in case of natural calamities, drone surveillance in conflict zones, unmanned weapon controlling for attacks, etc. Even a single failure in these crucial applications might pose a serious risk to national security or harm those serving in combat or other crucial areas. In these circumstances, parallel computing and distributed sensor data mining must function well and without errors. As a result, each node in the network, including the input and output terminals, needs to be made more redundant. Therefore, it is essential to get beyond the gamma network's restriction. In spite of this constraints, the gamma network still has a number of other prohibitions that limit its use in critical situations. These are discussed as follows: • Scalability for larger systems • Highest transmission latency due to the presence of multiple hops between nodes • Limited fault tolerance • Complex routing algorithms and • Higher cost of hardware Although it has several drawbacks, is frequently used for highspeed packet switching networks, and in recent years, a number of improvements have also been proposed in the literature to address its scalability and fault tolerance capabilities.  The enhancements recommended in the existing research are: 1. Redundancy: It entails adding extra nodes or linkages across stages to offer an alternative route for coping with errors at run time. 2. To ensure error-free connection, a Quality of Service (QoS) system is created to prioritize the data received from the sensors to the most important applications, such as medical applications. 3. The network's dynamic re-configurability during operation in the event of errors. To do this, whenever a problem arises in the following step, the preceding information of faults is gathered and stored in the little internal memory of each SE to avoid the faulty SE during transmission by re-configuration.  In order to solve the challenges outlined above, numerous novel network topologies have been proposed in the literature. 20,[26][27][28]30,31 Table I provides a comparison of currently available gamma networks along with concerns that have been addressed and suggestions for improvement. Even if the researchers have made enormous efforts to get to a stage where there is little chance that the system will fail due to a malfunction, there are still a number of problems that need to be resolved. Which are: 1. Expensive hardware requirements: The implementation of these suggested topologies may necessitate substantial hardware resources, making gamma topology an expensive undertaking. 2. Limited fault tolerance: Although these upgrades make the network resistant to node failures, multiple fault tolerance is challenging, particularly at the input and output stages. 37,41 3. Significant Latency: Because the data packet must make several hops before reaching its destination, the latency of the data packet received at the output terminal is significant. 42,43 4. Limited connectivity: Due to the limited connectivity between nodes, many of the proposed topologies for the gamma network have a limited number of potential pathways.
In this research, a new gamma network has been proposed to enhance fault tolerance and reliability. This network offers the shortest distance between input and output ports at the lowest possible cost in order to overcome the issues with the majority of gamma networks. Of all known gamma networks, the suggested network topology has the least hardware due to its simpler connections. Because of the reduction in the number of stages, the packet header needed in the proposed topology to transfer data packets from the input section to the output port has also been reduced. New gamma interconnection network has high reliability and can sustain several failures at each level. It also has excellent fault tolerance. A thorough comparison of the proposed architecture with the existing one has been offered in the following sections to support this claim.

Proposed Topology of Reliable Gamma-interconnection Network (RGN)
For big data processing in sensor networks, the proposed Reliable Gamma-interconnection Network (RGN) seeks to provide improved performance. To facilitate seamless data transfer and processing, this proposed network architecture is created to offer reliable and effective communication among interlinked nodes. RGN leverages the unique properties of the gamma interconnection network to ensure robustness, fault tolerance, and scalability in handling large volumes of data recovered from different sensor nodes. This design can improve performance in terms of decreased latency and increased reliability in multiprocessing systems used to analyze data collected from sensor networks. 44 The RGN architecture shows considerable promise for enhancing sensor network capabilities and enabling more efficient use of big data in a variety of fields, including military, navy, healthcare, agriculture, space and environmental monitoring.
The design topology of newly proposed RGN is shown in Fig. 3. RGN consist of log 2 N-1 total stages which is a considerable reduction in number of stages from the networks belong to gamma class. In gamma interconnection network, number of stages utilized is log 2 N + 1. Most of the network do possess same number of stages. Though, some of the architectures used lesser number of stages, but that too is limited to one stage reduction only. In RGN the reduction of two stage per architecture is achieved while maintaining high reliability and availability of totally disjoint paths (node and link both) between each source to every destination node. The topological architecture of RGN is as follow: (i) It consist of N number of input and output nodes.
(ii) "N/2" number of SEs (2 × 3 at input 3 × 3 at intermediate and 3 × 2 at output) are associated in each stage with "N" numbers of 2 × 1 MUX and 1 × 2 DEMUX. (iii) The total number of SE stages utilized is log 2 N-1. (iv) The connection pattern is same as that of gamma network.
RGN reduces the amount of hardware compared to other gamma networks by using fewer SEs at each stage. For example, while other gamma networks require N SEs in each stage, RGN only uses N/2 SEs. In addition, RGN uses log 2 N-1 SE stages overall, which is also fewer (two less than others) than other gamma networks which use log 2 N + 1 SE stages. The RGN's connection pattern is identical to the gamma network's, which furthers the hardware savings. The input and output terminals of RGN SE stages are also connected to 2 × 1 MUX and 1 × 2 DEMUX, which offer two distinct paths. This RGN feature enhances the network's reliability by enabling redundancy and fault tolerance. The other path can still be used for communication if the first path has a malfunction. Additionally, the effective routing of data packets made possible by the usage of MUX and DEMUX in RGN adds to the network's reliability. As a result, despite having less hardware, RGN's reliability is increased because to its distinctive architecture and features.
The two completely distinct pathways for each source-destination node combination in RGN, including the input and output nodes, is shown in Fig. 4 with red line. The routing of data packet from input node "4" to output node "7;" is shown with the red lines in Fig. 4. These pathways shown share no parts or links; hence, they offer enhanced fault tolerance. Additionally, fewer SEs are used in the specified path to route the data from source to destination in the suggested RGN. As a result, the path becomes more reliable; hence, the likelihood of SE failure decreases with decreasing SE numbers in the path. Additionally, splitting the data into many packets and delivering them across these disconnected paths to the destination can increase the efficiency of data transmission when routing through disjoint paths. This method can be used to prevent congestion in the network when large data is to be transmitted and computed for sensor networks. Additionally, it can lessen the possibility of packet loss caused by SE failures along the indicated paths. The likelihood of network/SE failures is greater in large-scale networks that are deployed with numerous sensors. The possibility of data loss in such an application could endanger human life or national security. This makes it necessary to increase the number of distinct pathways in the high-speed switch fabric used by parallel processors equipped with various sensors. By making it more difficult for attackers to intercept or manipulate the data, routing over disjoint paths also increases the security of the data while it is being transmitted. The suggested network, despite being the smallest gamma network, has all these benefits. RGN offers excellent fault tolerance, reliability, and compute efficiency through distinct pathways, which improves performance in applications that need low latency and high-speed data transmission. The proceeding part displays the suggested RGN's reliability and performance evaluation.

Performance Evaluation of RGN
Performance evaluation of RGN can be conducted by analyzing reliability, fault tolerance and associated cost for different network sizes. Reliability can be referred as the ability of a network or its components to perform their intended task without failure in the given circumstances. Reliability analysis for multistage interconnection network is very important to determine its performance in critical conditions. Three reliability measure are available to compute the performance of any network architecture, these are defined as below:    . Routing of data from input node "4 = 100" to output node "7 = 111" in RGN for size "8".
establishment of at least one fault free path between minimum number of network component which are required to communicate data from all source node to all destination node. It is an important parameter for ensuring that all of the network ports can be provided with its intended services without interruption or degradation.
For networks with a size of 16 or above, the Reliability Block Diagram (RBD) approach can be used to evaluate each of these reliability criteria. However, the inclusion-exclusion (IE) method applied to the design RBD can be used to conduct the analysis for networks up to size "8". To evaluate the performance of RGN with other network topologies, the RBD of RGN is shown in Fig. 5 for  Reliability block diagram (RBD).-MIN may be inferred from their equivalent RBD, which combines a series-parallel connection of SE. The reliability equation can then be inferred by applying probability principles to both the series model and the parallel model.
Series configuration model.-Because the failure of one component will trigger the failure of the entire system, this configuration type is not fault tolerant. A series model's reliability is always worse than the reliability of the system's least reliable part. The reliability equation of the series model can be expressed as shown by Eq. 1 if the reliability of each SE connected in series (all SE are assumed to be identical) is denoted by "r". Where R series (t) is the reliability of a series system.
Parallel configuration model.-All SE are connected in parallel in this establishment, which offers excellent reliability because to its ability to withstand "(n-1)" number failures for "n" parallel pathways. Reliability equation of a parallel model can be written as demonstrated by Eq. 2 if reliability of each SE connected in parallel is denoted by "r". Where R parallel (t) is the reliability of a parallel system. Some fundamental presumptions have been employed in order to assess the proposed network's reliability using RBD. These assumptions are: a. There are just two possible states for SE-failed or workingand none of them are repairable. b. Links are more trustworthy than system components (SEs). c. The reliability of a 3 × 3 SE is defined as "r" or "r 9/9 " (for 3 × 3 = 9 cross-points), and the reliability of other SEs can be determined by dividing the number of cross points by "9".
Where "r" could be interpreted as: Determining a trade-off between performance in terms of reliability and fault tolerance employing redundancy and increased expense for a particular application is therefore crucial.
Cost analysis.-Gamma networks are a popular option for largescale parallel computing systems used with sensor networks because they are recognized to be reliable, cost-effective multipath networks. Numerous studies have been done to increase the reliability of Gamma networks while lowering or maintaining the cost of the hardware. The cost of gamma networks might vary depending on the overall topology used in the design and the desired level of performance. The best cost-effective method for building a reliable network for a given application must therefore be determined after doing a detailed study of cost with relation to performance in terms of reliability of all Gamma networks. The most popular method found in the body of existing literature is utilized to determine the cost of the proposed gamma network (RGN). According to this method, the price of any system can be determined by adding up the prices of all the components that were used to make it, whereas the price of any component can be determined by adding up the total cross-points that were used to make it. The overall components and their costs can be estimated as follows to assess the proposed RGN's cost: At input and output stage "N" number of 2 × 1 MUXs and 1 × 2 DEMUXs are used. 2 × 1 MUXs and 1 × 2 DEMUXs has 2 × 1 = 2 Cross-Point each.
Similarly, for input and output switching stages there are N/2 SEs of size 2 × 3 and 3 × 2 respectively. So, the cost can be calculated as 12 N/2 = 6 N.
References 31, 46 contains the computed values of the cost functions of additional gamma networks that were formulated using the mention approach. According to literature, ST reliability is assumed to be "r" = 0.9 for network size and is equal to 16 and 64 in order to evaluate the performance of the analyzed gamma network in relation to cost function shown by Eq. 6.

Results and Discussion
Regardless of alternative connecting networks that employ N SEs in each stage, the presented RGN structure uses a minimum amount of hardware that is N/2 SEs per stage. The minimal path distance between each input and output node is another feature of it. As a result, it becomes a high-speed low-cost network inside its own family. The computed values of all reliability measures, including ST reliability, SAT reliability, and ASAT reliability values, are displayed in Tables II to IV. These values are calculated using the equations given in (2)    Although RGN's reliability improvement is commendable in compared to previous gamma networks, but it is not as good as in case of SEGIN-Minus through the understandings drawn from the results achieved. However, RGN has a shorter path length than SEGIN-Minus, making it a high-speed interconnection network. With regard to path length, SEGIN-Minus has a path length of log 2 N stages, whereas RGN has a path length of log 2 N-1 stages. The ST path length describes the number of network stages necessary for a signal to move from its source to its destination. In other words, it is the quantity of steps a signal takes to get to its final location. The difference in their connecting patterns also count for the causes of RGN to have a shorter ST path length than SEGIN-Minus. In order to provide the shortest possible link between any two network nodes, RGN employs the gamma network connection design. The butterfly network connection pattern, on the other hand, is used by SEGIN-Minus and is renowned for its capacity to offer a balanced and faulttolerant network. To attain the same level of interconnection as the gamma network connection pattern, the butterfly network connection pattern needs more stages.
Thus, despite the fact that SEGIN-Minus and RGN are both intended to provide effective and fault-tolerant network connectivity, RGN does so due to its special connection pattern with fewer stages and a shorter ST path length. As a result, RGN's communication speed is slower than SEGIN-Minus MIN's, while RGN's reliability is higher. Also, The path length between network nodes increases as the network size does. This is due to the fact that larger networks necessitate more connections between nodes, which lengthens the distance between nodes and make signal or information transmission more challenging. It should be noted that the path length in the case of RGN is already short in comparison to SEGIN-Minus and other networks. This implies that RGN is built to reduce path length and enhance signal or information transmission across the network. And Thus, it is a likely candidate for a high-speed packet switch network, where a large number of sensor networks are connected for data processing using parallel and distributed computing techniques. The MIN performance and cost may benefit from the hardware reduction made possible by RGN. Compared to other studied interconnection networks, RGN can achieve faster communication and reduced latency since there are fewer SEs in each stage and fewer SE stages. RGN is a more effective and affordable option for network connectivity because of the decreased hardware, which can also result in lower power consumption and production costs.
While RGN is intended to offer fault tolerance for a single defect at any stage, it might not be able to do so for numerous faults or system failures. Depending on the particular implementation and hardware utilized, the network may become less reliable. Therefore, while assessing the fault tolerance and reliability of RGN or any other network design, it is crucial to carefully analyze the unique application and implementation specifics in case if the network is having high levels of congestion.
But, overall, RGN is less expensive and simpler to install than other network designs because to its reduced hardware needs, which may be especially advantageous for big data processing applications that demand large-scale networks. Additionally, for massive data processing applications such as space exploration, missile targeting, rocket launching, bomb detonation and security navigation that need quick and effective communication between nodes, the low path distance of RGN may reduce latency and enhance overall performance.

Conclusions
In MINs achieving high reliability and fault tolerance at low cost is always been challenging as these two statements conflict each other. Redundancy is frequently added to MIN structures to increase reliability, which raises the network's used cost. On the other hand, a network's reliability declines when its cost is reduced. The suggested design for the RGN not only improves the reliability of the network but also offers a practical means of lowering the overall cost of the network. However, because the number of ST pathways at intermediate stages is lower than some of the recently suggested Gamma IN, such as FTGM and SEGIN-Minus, redundancy and fault tolerance of the proposed RGN are slightly degraded at intermediate stages. Nevertheless, compared to these recently proposed topologies, reliability per unit of hardware incorporated in the newly proposed RGN is lower. As a result, RGN's overall effectiveness is improving noticeably in terms of reliability attained, overall cost employed, and reliability-cost ratio as demonstrated in the results. Table V displays these improvements in percentage terms as well. For several tested Gamma networks, the suggested RGN demonstrates improvements in RCR ranging from 2%-42% for networks of size 16 to 1%-87% for networks of size 64, which is quite fair and has the lowest cost in its class.
Although the cost of the proposed RGN network and SEGIN-Minus is the same, and although the proposed RGN network demonstrates minimal reliability gains when compared with SEGIN-Minus, the proposed RGN network's ST path length is one less than SEGIN-Minus. As a result, suggested RGN is a better contender for applications where speed of operation is crucial, including high security navigation. The suggested way of improving gamma MIN reliability is an attractive choice for the future and can be used to improve the topological structures of other regular MINs as SEN, Cube, Omega, etc. in order to improve reliability at the lowest possible cost.