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Mean Value Analysis Approximation

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Mean Value Analysis Approximation for Multiple Server Queueing Networks Ian F. Akyildiz School of lnformation and Computer Science, Georgia Institute of Technology, Atlanta, Georgia 30332-0280, U.S.A. Gunter Bolch IMMDIV, Universitiit Erlangen-Niirnberg, Martensstrasse 1, 8520 Erlangen, Fed. Rep. Germany Received May 1986 Revised 3 June 1987 77 An approximation of mean value analysis is presented for queueing networks containing multiple server stations. The approximation is based on the estimation of the conditional marginal probabilities used by the mean residence time formula in mean value analysis. A comparison against classical mean value analysis allows us to determine the accuracy of our algorithm. In all investigated network models, the approximate results vary from the exact results by less than four percent on the average. This approximation method has all the advantages of classical mean value analysis; specifically, it is easy to implement and has a very short run time. Keywords: Queueing Network, Analytical Method, Mean Value Analysis, Conditional Marginal Probability. 1. Introduction Mean value analysis has enjoyed widespread popularity during recent years as an exact technique for providing solutions to product-form queueing networks. The basic concept of mean value analysis is the application of an iterative procedure to calculate mean residence time, system throughput, and the mean number of jobs. A number of studies about mean value analysis has been published in the last few years I.F. Akyildiz was born in 1954 in Istanbul, Turkey. He received his B.S. (1978), M.S. (1981), and Ph.D. (1984) degrees in Computer Science from the University of Edangen-Niirnberg, Fed. Rep. Germany. From 1981 through 1985 he served as a Scientific Employee in the Informatik IV (Operating Systems, chair of Prof. F. Hoffman) Department at the University of Erlangen-Niirnberg. During that time he coauthored a text book titled Analysis of Computer Systems (in German) published by Teubner Verlag in 1982. In January of 1985 he joined the faculty in the Computer Science Department at Louisiana State University as an Assistant Professor. He was also a Visiting Professor in the Computer Science Department at the University of Florida in the summer of 1985 and in the Computer Science Department of the Universidad Tecnica de Federico Santa Maria in Valparaiso, Chile in the summer of 1986. In Fall 1987 he joined the faculty in the School of Information and Computer Science at Georgia Institute of Technology as an Assistant Professor. His research interests are performance evaluation, operating systems, and communication networks. Dr. Akyildiz is a member of IEEE, ACM (SIGOPS and SIGMETRICS), GI (GeseUschaft fiir Informatik), MMB (German Interest Group in Measurement, Modeling and Evaluation of Computer Systems). Gamter Boleh received his Ph.D. in Electrical Engineering from the University of Kartsruhe in 1973, where he also worked as an Assistant Professor. In 1974 he joined the Computer Science Department of the University of Erlangen-Niirnberg. From 1977 to 1979 he was a Visiting Professor in the Computer Science Department of The Catholic University of Rio de Janeiro (PUC). Since 1982 he has been the "Akademischer Direktor" in the operating systems division of the Computer Science Department at the University of Erlangen-Niirnberg. He spent two months (April-May 1986) at the Mathematical Institute of the Academy Sciences in Minsk, U.S.S.R. and another two months (August-September 1987) at the Computer Science Department of PUC in Rio de Janeiro. Dr. Bolch is a coauthor of a text book on performance analysis and has published numerous papers. His research interest are operating systems, process control, and analytic modeling of computer systems. Dr. Bolch is a member of GI (Gesellschaft fiir Informatik) and MMB (German Interest Group in Measurement, Modeling and Evaluation of Computer Systems). North-Holland Performance Evaluation 8 (1988) 77-91 0166-5316/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)78 LF. Akyildiz, G. Bolch / Mean value analysis for multiple server queueing networks [1,3,6,7,10-15,17-21]. The principle advantage to mean value analysis lies in its ability to effectively compute the performance measures without computing the normalization constants. However, when considering systems with multiple job classes, the storage requirement of mean value analysis increases very rapidly. This problem is further magnified when one incorporated multiple server stations into mean value analysis. Therefore, Reiser and Lavenberg [13], Bard [2], and Schweitzer [15] have introduced an approximation for mean value analysis which eliminates the storage complexity problems associated with classical mean value analysis, in short form MVA. They approximate the mean number of jobs at each station for the network, and then iterate on this approximation, halting when the number of jobs at each of the stations stabilizes. However, this approach is limited to single-server models. Chandy and Neuse [4] improved the work of Bard and Schweitzer with the Core and Linearizer algorithms, but this work was still limited to single-server networks. The method involved estimating the fraction of class-r jobs in each station and the fractional change in this value when one job from class s is removed from the station. The Chandy and Neuse Core algorithm took as input these estimates for fractional changes in the number of jobs at each station with the removal of any one job. This algorithm can be shown equivalent to the method proposed by Bard and Schweitzer if one assumes this fractional change is zero. The Linearizer algorithm was an improvement over the Core algorithm as it applied to Core algorithm to each of the (K-lr) populations in order to improve their performance measures estimates. These improved performance measures were then used in calculating the necessary values for the final population K. The first significant technique introduced for the analysis of multiple server stations was done by Neuse and Chandy [10] in 1981. SCAT (for Self Correcting Approximation Technique), allowed the analysis of queueing network models with multiple job classes and multiple servers. This


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