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Issue No.06 - June (2013 vol.39)
pp: 744-756
M. Tribastone , Dept. for Inf., Ludwig-Maximilians Univ. of Munich, Munich, Germany
ABSTRACT
Layered queueing networks are a useful tool for the performance modeling and prediction of software systems that exhibit complex characteristics such as multiple tiers of service, fork/join interactions, and asynchronous communication. These features generally result in nonproduct form behavior for which particularly efficient approximations based on mean value analysis (MVA) have been devised. This paper reconsiders the accuracy of such techniques by providing an interpretation of layered queueing networks as fluid models. Mediated by an automatic translation into a stochastic process algebra, PEPA, a network is associated with a set of ordinary differential equations (ODEs) whose size is insensitive to the population levels in the system under consideration. A substantial numerical assessment demonstrates that this approach significantly improves the quality of the approximation for typical performance indices such as utilization, throughput, and response time. Furthermore, backed by established theoretical results of asymptotic convergence, the error trend shows monotonic decrease with larger population sizes-a behavior which is found to be in sharp contrast with that of approximate mean value analysis, which instead tends to increase.
INDEX TERMS
Approximation methods, Unified modeling language, Stochastic processes, Sociology, Statistics, Servers, Accuracy,mean value analysis, Modeling and prediction, Markov processes, PEPA, ordinary differential equations, queueing networks
CITATION
M. Tribastone, "A fluid model for layered queueing networks", IEEE Transactions on Software Engineering, vol.39, no. 6, pp. 744-756, June 2013, doi:10.1109/TSE.2012.66
REFERENCES
[1] P.A. Jacobson and E.D. Lazowska, "The Method of Surrogate Delays: Simultaneous Resource Possession in Analytic Models of Computer Systems," SIGMETRICS Performance Evaluation Rev., vol. 10, pp. 165-174, Sept. 1981.
[2] C.M. Woodside, "Throughput Calculation for Basic Stochastic Rendezvous Networks," Performance Evaluation, vol. 9, no. 2, pp. 143-160, 1989.
[3] J.A. Rolia and K.C. Sevcik, "The Method of Layers," IEEE Trans. Software Eng., vol. 21, no. 8, pp. 689-700, Aug. 1995.
[4] G. Franks, T. Omari, C.M. Woodside, O. Das, and S. Derisavi, "Enhanced Modeling and Solution of Layered Queueing Networks," IEEE Trans. Software Eng., vol. 35, no. 2, pp. 148-161, Mar. 2009.
[5] M. Reiser and S.S. Lavenberg, "Mean-Value Analysis of Closed Multichain Queuing Networks," J. ACM, vol. 27, no. 2, pp. 313-322, 1980.
[6] Y. Bard, "Some Extensions to Multiclass Queueing Network Analysis," Proc. Third Int'l Symp. Modelling and Performance Evaluation of Computer Systems, pp. 51-62, 1979.
[7] K.M. Chandy and D. Neuse, "Linearizer: A Heuristic Algorithm for Queueing Network Models of Computing Systems," Comm. ACM, vol. 25, no. 2, pp. 126-134, 1982.
[8] P. Schweitzer, "Approximate Analysis of Multiclass Closed Networks of Queues," Proc. Int'l Conf. Stochastic Control and Optimization, pp. 25-29, June 1979.
[9] K.R. Pattipati, M.M. Kostreva, and J.L. Teele, "Approximate Mean Value Analysis Algorithms for Queuing Networks: Existence, Uniqueness, and Convergence Results," J. ACM, vol. 37, no. 3, pp. 643-673, 1990.
[10] M. Woodside, J. Neilson, D. Petriu, and S. Majumdar, "The Stochastic Rendezvous Network Model for Performance of Synchronous Client-Server-Like Distributed Software," IEEE Trans. Computers, vol. 44, no. 1, pp. 20-34, Jan. 1995.
[11] S. Ramesh and H. Perros, "A Multilayer Client-Server Queueing Network Model with Synchronous and Asynchronous Messages," IEEE Trans. Software Eng., vol. 26, no. 11, pp. 1086-1100, Nov. 2000.
[12] D.A. Menascé, "Simple Analytic Modeling of Software Contention," SIGMETRICS Performance Evaluation Rev., vol. 29, no. 4, pp. 24-30, Mar. 2002.
[13] J. Hillston, A Compositional Approach to Performance Modelling. Cambridge Univ. Press, 1996.
[14] M. Tribastone, S. Gilmore, and J. Hillston, "Scalable Differential Analysis of Process Algebra Models," IEEE Trans. Software Eng., vol. 38, no. 1, pp. 205-219, Jan./Feb. 2012.
[15] R.A. Hayden and J.T. Bradley, "A Fluid Analysis Framework for a Markovian Process Algebra," Theoretical Computer Science, vol. 411, nos. 22-24, pp. 2260-2297, 2010.
[16] T.G. Kurtz, "Solutions of Ordinary Differential Equations as Limits of Pure Markov Processes," J. Applied Probability, vol. 7, no. 1, pp. 49-58, Apr. 1970.
[17] M. Tribastone, J. Ding, S. Gilmore, and J. Hillston, "Fluid Rewards for a Stochastic Process Algebra," IEEE Trans. Software Eng., vol. 38, no. 4, pp. 861-874, July/Aug. 2012.
[18] J. Hillston, "Exploiting Structure in Solution: Decomposing Compositional Models," Proc. Sixth Int'l Workshop Process Algebra and Performance Modelling, pp. 1-15, 1998.
[19] J. Hillston and N. Thomas, "Product form Solution for a Class of PEPA Models," Performance Evaluation, vol. 35, nos. 3/4, pp. 171-192, 1999.
[20] P.G. Harrison, "Turning Back Time in Markovian Process Algebra," Theoretical Computer Science, vol. 290, no. 3, pp. 1947-1986, 2003.
[21] N. Thomas and Y. Zhao, "Mean Value Analysis for a Class of PEPA Models," Computer J., 2010.
[22] M. Tribastone, "Relating Layered Queueing Networks and Process Algebra Models," Proc. First Joint WOSP/SIPEW Int'l Conf. Performance Eng., pp. 183-194, 2010.
[23] M. Tribastone, "Scalable Analysis of Stochastic Process Algebra Models," PhD dissertation, School of Informatics, The Univ. of Edinburgh, 2010.
[24] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, 1996.
[25] MATLAB, Version 7.10.0 (R2010a), The MathWorks Inc., 2010.
[26] J. Hillston and V. Mertsiotakis, "A Simple Time Scale Decomposition Technique for Stochastic Process Algebras," Computer J., vol. 38, no. 7, pp. 566-577, 1995.
[27] Real-Time and Distributed Systems Group, Dept. of Systems and Computer Eng., Univ. of Carleton, "LQNS Software Package," http://www.sce.carleton.ca/radslqns, 2012.
[28] M. Tribastone, A. Duguid, and S. Gilmore, "The PEPA Eclipse Plug-In," SIGMETRICS Performance Evaluation Rev., vol. 36, no. 4, pp. 28-33, Mar. 2009.
[29] U. Herzog and J.A. Rolia, "Performance Validation Tools for Software/Hardware Systems," Performance Evaluation, vol. 45, nos. 2/3, pp. 125-146, 2001.
[30] M. Tribastone, "Approximate Mean Value Analysis of Process Algebra Models," Proc. 19th IEEE Int'l Symp. Modelling, Analysis and Simulation of Computer and Telecomm. Systems, pp. 369-378, July 2011.
[31] R. Darling and J. Norris, "Differential Equation Approximations for Markov Chains," Probability Surveys, vol. 5, pp. 37-79, 2008.
[32] J. Meyer, "On Evaluating the Performability of Degradable Computing Systems," IEEE Trans. Computers, vol. 29, no. 8, pp. 720-731, Aug. 1980.
[33] R.A. Hayden, A. Stefanek, and J.T. Bradley, "Fluid Computation of Passage-Time Distributions in Large Markov Models," Theoretical Computer Science, vol. 413, no. 1, pp. 106-141, 2012.