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Issue No.04 - July-Aug. (2012 vol.38)

pp: 861-874

Mirco Tribastone , Ludwig-Maximilians-Universität, München

Jie Ding , Yangzhou University, Yangzhou

Stephen Gilmore , Edinburgh University, Edinburgh

Jane Hillston , Edinburgh University, Edinburgh

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TSE.2011.81

ABSTRACT

Reasoning about the performance of models of software systems typically entails the derivation of metrics such as throughput, utilization, and response time. If the model is a Markov chain, these are expressed as real functions of the chain, called reward models. The computational complexity of reward-based metrics is of the same order as the solution of the Markov chain, making the analysis infeasible when evaluating large-scale systems. In the context of the stochastic process algebra PEPA, the underlying continuous-time Markov chain has been shown to admit a deterministic (fluid) approximation as a solution of an ordinary differential equation, which effectively circumvents state-space explosion. This paper is concerned with approximating Markovian reward models for PEPA with fluid rewards, i.e., functions of the solution of the differential equation problem. It shows that 1) the Markovian reward models for typical metrics of performance enjoy asymptotic convergence to their fluid analogues, and that 2) via numerical tests, the approximation yields satisfactory accuracy in practice.

INDEX TERMS

Modeling and prediction, ordinary differential equations, Markov processes

CITATION

Mirco Tribastone, Jie Ding, Stephen Gilmore, Jane Hillston, "Fluid Rewards for a Stochastic Process Algebra",

*IEEE Transactions on Software Engineering*, vol.38, no. 4, pp. 861-874, July-Aug. 2012, doi:10.1109/TSE.2011.81REFERENCES

- [1] A. Clark, S. Gilmore, J. Hillston, and M. Tribastone, "Stochastic Process Algebras,"
Proc. Seventh Int'l Conf. Formal Methods for Performance Evaluation, pp. 132-179, May/June 2007.- [2] M. Tribastone and S. Gilmore, "Automatic Translation of UML Sequence Diagrams into PEPA Models,"
Proc. Fifth Int'l Conf. the Quantitative Evaluation of Systems, pp. 205-214, Sept. 2008.- [3] M. Tribastone and S. Gilmore, "Automatic Extraction of PEPA Performance Models from UML Activity Diagrams Annotated with the MARTE Profile,"
Proc. Seventh Int'l Workshop Software and Performance, June 2008.- [4] J. Hillston, "Fluid Flow Approximation of PEPA Models,"
Proc. Second Int'l Conf. Quantitative Evaluation of Systems, pp. 33-43, Sept. 2005.- [5] 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, http://doi.ieeecomputersociety.org/10.1109 TSE.2010.82, Jan./Feb. 2012.- [6] J. Hillston,
A Compositional Approach to Performance Modelling. Cambridge Univ. Press, 1996.- [7] J. Meyer, "On Evaluating the Performability of Degradable Computing Systems,"
IEEE Trans. Computers, vol. 29, no. 8, pp. 720-731, Aug. 1980.- [8] R. Smith, K. Trivedi, and A. Ramesh, "Performability Analysis: Measures, an Algorithm, and a Case Study,"
IEEE Trans. Computers, vol. 37, no. 4, pp. 406-417, Apr. 1988.- [9] K.S. Trivedi, J.K. Muppala, S.P. Woolet, and B.R. Haverkort, "Composite Performance and Dependability Analysis,"
Performance Evaluation, vol. 14, nos. 3/4, pp. 197-215, 1992.- [10] J.F. Meyer, "Performability: A Retrospective and Some Pointers to the Future,"
Performance Evaluation, vol. 14, nos. 3/4, pp. 139-156, 1992.- [11] M. Beaudry, "Performance-Related Reliability Measures for Computing Systems,"
IEEE Trans. Computers, vol. 27, no. 6, pp. 540-547, June 1978.- [12] L.T. Wu, "Operational Models for the Evaluation of Degradable Computing Systems,"
SIGMETRICS Performance Evaluation Rev., vol. 11, no. 4, pp. 179-185, 1982.- [13] W.H. Sanders and J. Meyer, "A Unified Approach for Specifying Measures of Performance, Dependability, and Performability,"
Dependable Computing for Critical Applications, pp. 215-247, Springer-Verlag 1990.- [14] B. Haverkort and K. Trivedi, "Specification Techniques for Markov Reward Models,"
Discrete Event Dynamic Systems, vol. 3, nos. 2/3, pp. 219-247, July 1993.- [15] A. Aldini and M. Bernardo, "Mixing Logics and Rewards for the Component-Oriented Specification of Performance Measures,"
Theoretical Computer Science, vol. 382, no. 1, pp. 3-23, 2007.- [16] G. Clark, S. Gilmore, and J. Hillston, "Specifying Performance Measures for PEPA,"
Proc. Fifth Int'l AMAST Workshop Formal Methods for Real-Time and Probabilistic Systems, J.-P. Katoen, ed., pp. 211-227, 1999.- [17] J.T. Bradley, R. Hayden, W.J. Knottenbelt, and T. Suto, "Extracting Response Times from Fluid Analysis of Performance Models,"
Proc. SPEC Int'l Performance Evaluation Workshop, pp. 29-43, 2008.- [18] A. Clark, A. Duguid, S. Gilmore, and M. Tribastone, "Partial Evaluation of PEPA Models for Fluid-Flow Analysis,"
Proc. Fifth European Performance Eng. Workshop Computer Performance Eng., pp. 2-16, 2008.- [19] A. Clark, A. Duguid, S. Gilmore, and J. Hillston, "Espresso, a Little Coffee,"
Proc. Seventh Workshop Process Algebra and Stochastically Timed Activities, 2008.- [20] M. Tribastone, "Scalable Analysis of Stochastic Process Algebra Models," PhD dissertation, School of Informatics, The Univ. of Edinburgh, 2010.
- [21] J. Ding, "Structural and Fluid Analysis for Large Scale PEPA Models—with Applications to Content Adaptation Systems," PhD dissertation, School of Eng., The Univ. of Edinburgh, 2010.
- [22] P. Pollet, "On a Model for Interference between Searching Insect Parasites,"
J. Australian Math. Soc. Series B, vol. 32, no. 2, pp. 133-150, 1990.- [23] P. Billingsley,
Probability and Measure, third ed. Wiley, 1995.- [24] R. Darling and J. Norris, "Differential Equation Approximations for Markov Chains,"
Probability Surveys, vol. 5, pp. 37-79, 2008.- [25] J. Kemeny and J. Snell,
Finite Markov Chains. Van Nostrand, 1960.- [26] L. Gurvits and J. Ledoux, "Markov Property for a Function of a Markov Chain: A Linear Algebra Approach,"
Linear Algebra and Its Applications, vol. 404, pp. 85-117, 2005.- [27] B.R. Haverkort,
Performance of Computer Communication Systems: A Model-Based Approach. John Wiley & Sons, 1998.- [28] C. Alexopoulos and D. Goldsman, "To Batch or Not to Batch?"
ACM Trans. Modeling and Computer Simulation, vol. 14, no. 1, pp. 76-114, 2004.- [29] U.M. Ascher and L.R. Petzold,
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, 1988.- [30] P. Fitzpatrick,
Advanced Calculus, second ed. AMS Bookstore, 2009.- [31] J. Little, "A Proof of the Queuing Formula: $L = \lambda \; W$ ,"
Operations Research, vol. 9, no. 3, pp. 383-387, 1961.- [32] M. Tribastone, A. Duguid, and S. Gilmore, "The PEPA Eclipse Plug-In,"
Performance Evaluation Rev., vol. 36, no. 4, pp. 28-33, Mar. 2009.- [33] W.J. Stewart,
Probability, Markov Chains, Queues, and Simulation. Princeton Univ. Press, 2009. |