Reliability Analysis of Gamma Interconnection Network Systems

Transcription

1 International Journal of Performability Engineering, Vol. 5, No. 5, October 9, pp RAMS Consultants Printed in India Reliability Analysis of Gamma Interconnection Network Systems 1. Introduction INDRA GUNAWAN 1* and GAN MING LEE 1 Department of Mechanical and Manufacturing Engineering Auckland University of Technology, New Zealand Department of Systems Engineering and Management Malaysia University of Science and Technology, Malaysia (Received on September 8, 8 and finally revised on May, 9) Abstract: Gamma Interconnection Network (GIN) is a type of Multistage Interconnection Network (MIN) that consists of layers of switching elements connected together in a predefined topology providing the connectivity between the set of processors and the set of memory modules. GIN is constructed by 1x, x and x1 switching elements. In this paper, network reliability of four other different types of Gamma hybrid networks namely Monogamma Interconnection Network (MGIN), Reliable Gamma Interconnection Network (REGIN), Cyclic Gamma Interconnection Network () and Extra Stage Gamma Network (ESGN) are analyzed. The results from the proposed simulation method show that the ESGN, and REGIN exhibit high network reliability performance. This is subsequently followed by the MGIN, with a marginal network reliability improvement compared to the GIN. It is also shown that the reliability model used for simulation closely approximates the exact network reliability of MGIN,, and REGIN. Keywords: Interconnection, multistage, network, reliability, stages and switching element. Interconnection network is used for inter-processor or processor-memory communication. The purpose of integrating interconnection network into a system is to enable it to perform parallel processing. Computational power and speed is greatly increased with the help of parallel processing. Among some applications that require parallel processing are real-time simulations, image processing, weather forecasting, air-traffic control, and speech recognition. SIMD (single-instruction stream-multiple-data stream) machine is an example of where the interconnection network is incorporated into a system to perform parallel processing. The SIMD machines could be either in the Processing-Element-to-Processing-Element or Processor-to-Memory configuration. Other examples of parallel-machines models are the MIMD (multiple-instruction stream-multiple-data stream) and the multiple-simd [1]. *Corresponding author s indra.gunawan@aut.ac.nz 485

2 486 I. Gunawan and M. L. Gan Since the interconnection network plays an important role by providing communication between multiple processors or between processors and memory, failure in the interconnection network may cause the whole system to fail. Thus the reliability of the interconnection network becomes a concern [6-1]. There are three different types of reliability analysis for interconnection network which are terminal to terminal (between one switch at input stage and one switch at output stage, also known as terminal reliability), terminal to all (among one switch at input stage and all switches at output stage, also known as broadcast reliability) and all to all (among all switches at input stage and all switches at output stage, also known as network reliability) [1]. Reliability of the network is usually measured by the number of Switching Element (SE) failures tolerated. The network is considered failed once the connection between an input and output is broken. This paper analyzes the network reliability of the Gamma Interconnection Network (GIN) and its hybrid networks. The paper is organized as follows: Section gives a brief overview about the Gamma Interconnection Network (GIN) and its hybrids. Section discusses how the networks are modeled in the proposed simulation method. Section 4 evaluates the accuracy of the simulated data and extends the analysis on other network variants and sizes Finally, Section 5 concludes the paper.. Gamma Interconnection Network (GIN) and Its Hybrid Networks The Gamma Interconnection Network (GIN) [1-4, 9, 11-1, 14] is a type of Multistage Interconnection Network (MIN). Figure 1 shows an 8x8 Gamma Network. Each Switching Element is represented by a block. Generally, the GIN is made up of 1x SEs at the input stage, x SEs at the intermediate stages and x1 SEs at the output stage. Figure 1: 8 x 8 Gamma Network (GIN)

3 Reliability Analysis of Gamma Interconnection Network 487 Tag Value (x) has positive and negative values. Both values are essential to generate the Routing Paths. The initial Tag Value is defined as the difference between the Output Number (O), and the Input Number (I). x = O I (1) where, x = Tag Value (could be either positive or negative value) O = Output Number I = Input Number The relationship between the positive and negative Tag Value is: where, x = x + N ve ve () N = Total number of Inputs or Outputs. However, when the Tag Value equals zero, x =, there exists only one Routing Path in the GIN. Tag Value (x) equals zero means the Input Number is equals to the Output Number. This characteristic makes the GIN not fault tolerant. Any SE failure will cause the Routing Path with Tag Value equals zero to be broken. Thus network failure occurs. Various GIN hybrids have been proposed to increase the number of paths for connections when Tag Value equals zero. The Extra Stage Gamma Network (ESGN) [11] adds a redundant Link Stage to the GIN. Three other GIN hybrids namely the Monogamma Interconnection Network (MGIN) [4], Reliable Gamma Interconnection Network (REGIN) [14] and Cyclic Gamma Interconnection Network () [4] introduce some modifications to the existing GIN interconnection pattern.. Network Reliability Modeling Although modifications made to the gamma network increases the number of possible paths for all Tag Values, these paths are not necessarily disjointed. Disjoint paths are paths that do not share the same SEs in the intermediate stages. Thus when one of the SE in the intermediate stage fails, it may break one or more paths which could result in network failure. Reliability analysis is needed to determine the fault tolerant level of the interconnection network. Several assumptions are made for the analysis: all the links in the network are considered to be more reliable than the switches, all switches are assumed to have the same reliability (r) or unreliability (q) and fail independently, switches are either completely working or completely fail, the first and last stages of switches are assumed to be always operational. The reliability of GIN and its hybrids are analyzed by simulation using Crystal Ball [5]. Crystal Ball is an Add-Ins for Microsoft Excel. Before running the simulation, the network must first be modeled into a Microsoft Excel spreadsheet. Figure explains the Flowchart for modeling and analyzing the network reliability of GIN and its hybrid networks. Analysis is done on the following networks: 8x8 GIN, 8x8 ESGN, 8x8, 8x8 REGIN, 16x16 GIN, 16x16 ESGN, 16x16 MGIN 4, 16x16 MGIN, 8x8 4 and REGIN 4. The expression that precedes the network abbreviation represents the 16x16 number of inputs and outputs in the network. For example, 8x8 means the network has eight

4 488 I. Gunawan and M. L. Gan inputs and eight outputs. The abbreviation for MGIN, and REGIN each has a subscript and superscript term. The subscript term for the MGIN, and REGIN refers to the total number of Link Stage in the network. The MGIN abbreviation can also be written without the superscript term [4]. The superscript specifies the connection pattern of the first and last Link Stage [4] while the REGIN superscript indicates the SE Stage number that the modification of the connection pattern begins [14]. Figure : Flowchart for Modeling and Simulating the Gamma Network and Its Hybrids The following steps describe in detail the flowchart above: Step : Begin Network Modeling Step 1: Set Input M as Step : Check whether M is greater than (Total Input, N)-1. If M is greater than (N-1) proceed to Step 8 or else proceed to Step Step : Set Tag Value x as Step 4: Check whether Tag Value x is greater than (Total Input, N)-1. If x is greater than N-1 proceed to Step 7 or else proceed to Step 5

5 Reliability Analysis of Gamma Interconnection Network 489 Step 5: Program all possible paths from Input M to Output associated with Tag Value x Step 6: Increment current Tag Value x by 1 and proceed to Step 4 Step 7: Increment current value of M by 1 and proceed to Step Step 8: Step 9: Step 1: Step 11: End Network Modeling. Set simulation parameter Run simulation. (While simulation is running, the number of trial success and failure will be recorded by Crystal Ball and the average network reliability computed) Simulation ends. Obtain final average network reliability. End Crystal Ball utilizes Monte Carlo simulation method to dynamically produce alternative situations of the spreadsheet model [5]. Hundreds or thousands of trials can be simulated by using Crystal Ball. 4. Network Reliability Analysis Various schemes have been suggested to increase the fault-tolerant capability of the GIN. This, as explained earlier, is through introducing modifications on the existing GIN design topology that result in an increase of connection paths for a given Tag Value. But, there is very little numerical analysis carried out that actually calculates the reliability value of the GIN and its hybrid. As the network size increases, the process to obtain the exact reliability through conventional method such as the matrix enumeration [1] becomes very tedious. Even on a network size of 16x16, it is virtually unfeasible to analyze all possible permutations. Nevertheless, with the help of Crystal Ball, this complexity is reduced considerably. The following section compares some results produced from Crystal Ball with the actual exact value. Subsequently, it discusses the network reliability of the GIN, MGIN, REGIN, and ESGN of size 8x8 and 16x16 that are modeled using Crystal Ball. The simulation is set to run a maximum of 1, trials with the SE reliability, r, varying from.99 to Comparison of Results from Simulation and Exact Network Reliability Values By using the matrix enumeration method [1], the exact derived network reliability of the 8x8 MGIN, 8x8 and 8x8 REGIN are as follows; 8x8 8x8 8x8 MGIN : R MGIN = r 16 (11r 14 (1-r) + 8r 15 (1-r) + r 16 ) () : R = r 16 (4r 1 (1-r) r 1 (1-r) + 48r 14 (1-r) + 16r 15 (1-r) + r 16 ) (4) REGIN : R REGIN = r 16 (4r 1 (1-r) + 4r 14 (1-r) + 16r 15 (1-r) + r 16 ) (5) where, R = Network Reliability and r = SE reliability. Figure, 4 and 5 compare the network reliability of the 8x8 MGIN, 8x8 and 8x8 REGIN obtained through Crystal Ball simulation with the exact derived reliability values.

6 49 I. Gunawan and M. L. Gan Network Reliability (R) 8x8 MGIN (,) Network Reliability Crystal Ball Values Exact Derived Values Network Reliability (r) x8 (,) Network Reliability Crystal Ball Values Exact Derived Values Figure : 8x8 MGIN Network Reliability Figure 4: 8x8 Network Reliability 8x8 REGIN (,) Network Reliability Network Reliability (R) Crystal Ball Values Exact Derived Values Figure 5: REGIN Network Reliability From the graph shown in Figure, 4 and 5, the network reliability values obtained through Crystal Ball simulation closely approximate the exact derived network values. Thus it can be concluded that the values produced via Crystal Ball simulation is fairly accurate. This

7 Reliability Analysis of Gamma Interconnection Network 491 provides the basis to further extend the analysis on other network variants as well as networks of larger sizes. 4. GIN, MGIN, REGIN, and ESGN Network Reliability Figures 6 and 7 show the network reliability of GIN, MGIN, REGIN, and ESGN. The sizes of the network simulated are 8x8 and 16x16. Network Reliability (R) Crystal Ball Simulated Network Reliability (size 8x8) MGIN (,) (,) GIN REGIN (,) ESGN Network Reliability (R) Crystal Ball Simulated Network Reliability (size 16x16) GIN REGIN (,4) (,4) MGIN (,4) ESGN Figure 6: 8x8 GIN,, MGIN, REGIN and ESGN Network Reliability Figure 7: 16x16 GIN,, MGIN, REGIN and ESGN Network Reliability From the simulation results of these five gamma networks in Figures 6 and 7, the ESGN, REGIN and can be considered to be the most reliable in terms of network reliability. The GIN has the lowest network reliability. With a slight modification on the GIN to form the MGIN, the network reliability increases slightly compared to the GIN. These characteristic remains consistent as the size of the network increases from 8x8 to 16x Conclusion This paper introduces a simulation method to model and analyze the network reliability of the Gamma Interconnection Network and its hybrid networks. The implementation of Crystal Ball significantly reduces the network complexity. As shown earlier, the approach is able to produce an estimated result which closely approximates the exact network reliability. Apart from the network reliability, this proposed methodology of modeling the GIN, MGIN,, REGIN and ESGN could be applied to estimate the terminal and broadcast reliability as well. By generalizing the modeling approach, it can be used on almost any type of interconnection network systems.

8 49 I. Gunawan and M. L. Gan Acknowledgement: The authors would like to thank anonymous referees and the editor for improving the manuscript to the present form. References [1] Chen, C. W., N. P. Lu, T. F. Chen, and C. P. Chung, Fault Tolerant Gamma Interconnection Network by Chaining, IEE Proceedings Comput. Digit. Tech.; 147 (): 75-81,. [] Chen, C. W. and C. P. Chung, Fault Tolerant Gamma Interconnection Network Without Backtracking. The Journal of Systems and Software; 58(1): -1, 15 Aug. 1. [] Chen, C. W., N. P. Lu, and C. P. Chung, -Disjoint Gamma Interconnection Network. The Journal of Systems and Software; 66 ():19-14, May. [4] Chuang, P., : A Fault Tolerant Modified Gamma Interconnection Network. IEEE Transactions on Parallel and Distributed Systems, 7 (1), 11-16, [5] Decisioneering, Inc. Crystal Ball Tutorial. Retrieved 5 th May 6 from [6] Fard, N. S. and I. Gunawan, Terminal Reliability Improvement of Shuffle-Exchange Network Systems. International Journal of Reliability, Quality and Safety Engineering; 1 (1): [7] Gunawan, I., Multistage Interconnection Networks Reliability Evaluation Based on Stratified Sampling Monte Carlo Method. International Journal of Modelling and Simulation; 8 (): 9-14, 8. [8] Gunawan, I., Performance Analysis of a Multistage Interconnection Network System Based on a Minimum Cut Set Method. International Journal of Performability Engineering; 4 (): 111-1, 8. [9] Gunawan, I., Redundant Paths and Reliability Bounds in Gamma Networks. Applied Mathematical Modelling. (4): , 8. [1] Gunawan, I., Reliability Analysis of Shuffle-Exchange Network Systems. Reliability Engineering and System Safety; 9 (): 71-76, 8. [11] Lee, K.Y., and W. Hegazy, The Extra Stage Gamma Network. IEEE Transactions on Computers; 7 (11): , [1] Parker, D. S., and C. S. Raghavendra, The Gamma Network: A Multiprocessor Interconnection Network with Redundant Paths. ACM SIGARCH Computer Architecture News; 1 (): 7-8, 198. [1] Siegel, H. J., Interconnection Networks for Large Scale Parallel Processing ( nd ed.). McGraw-Hill Publishing Company, New York, 199. [14] Tzeng, N. F., P. J. Chuang, and C. H. Wu, Creating Disjoint Paths in Gamma Interconnection Networks. IEEE Transactions on Computers; 4 (1):146-15, 199. Indra Gunawan is a Senior Lecturer in the Department of Mechanical and Manufacturing Engineering at Auckland University of Technology, New Zealand. He completed his Ph.D. degree in Industrial Engineering from Northeastern University, USA. His main areas of research are reliability engineering, production and operations management, application of operations research, applied statistics, probability modeling, and project management. Besides IJPE, he has published papers in IJRQSE, RESS, IJMS, AMM, and other journals. Gan Ming Lee received his M. Sc. in Systems Engineering and Management from Malaysia University of Science and Technology (MUST). His research interest is in the area of reliability modeling of interconnection network systems.