Abstract
Quality of performance measure bounds is crucial for an accurate dimensioning of computer network resources. We study stochastic comparisons of multidimensional Markov processes for which quantitative analysis could be intractable if there is no specific solution form. On partially ordered state space, different stochastic orderings can be defined as the strong or the less constrained weak ordering. The goal of the present paper is to compare these two orderings with respect the quality of derived bounds. We propose to study a system similar to a Jackson network except that queues have finite capacity. Different bounding systems are built either in the sense of the strong ordering with hard constraints, or in the sense of the weak ordering with less ones. The proofs of the stochastic comparisons are done using the coupling and the increasing set methods, with an intuitive event based formalism. The qualities of bounding systems are compared regarding to blocking probabilities.
Keywords
Partially supported by french research project ANR-SETI06-02.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ridder, A.: Weak stochastic ordering for multidimensionnal Markov chains. Oper. Research Letters 18, 121–126 (1995)
Castel-Taleb, H., Mokdad, L.: Performance measure bounds in wireless networks by state space reduction. In: 13th Annual Meeting of the IEEE International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS 2005), Atlanta Georgia, September 27-29 (2005)
Castel-Taleb, H., Mokdad, L., Pekergin, N.: Loss rates bounds for IP switches in MPLS networks. In: ACS/IEEE International Conference on Computer Systems and Applications, AICCSA 2006, Dubai/Sharjah, UAE, March 8-11 (2006)
Castel-Taleb, H., Mokdad, L., Pekergin, N.: Aggregated bounding Markov processes applied to the analysis of tandem queues. In: Second International Conference on Performance Evaluation Methodologies and Tools, ACM Sigmetrics, ValueTools 2007, Nantes, France, October 23-25 (2007)
Castel-Taleb, H., Pekergin, N.: Weak stochastic comparisons for performability verification. In: Al-Begain, K., Fiems, D., Horváth, G. (eds.) ASMTA 2009. LNCS, vol. 5513, pp. 294–308. Springer, Heidelberg (2009)
Castel-Taleb, H., Pekergin, N.: Stochastic monotonicity in queueing networks. In: Bradley, J.T. (ed.) EPEW 2009. LNCS, vol. 5652, pp. 116–130. Springer, Heidelberg (2009)
Castel-Taleb, H., Pekergin, N.: Stochastic monotonicity for the comparison of multidimensionnal Markov chains (submitted)
Doisy, M.: Comparaison de processus Markoviens. PHD thesis, Univ. de Pau et des pays de l’Adour 92
Economous, A.: Necessary and sufficient condition for the stochastic comparison of Jackson Networks. Prob. in the Engineering and Informational Sciences 17, 143–151 (2003)
Fourneau, J.M., Pekergin, N.: An algorithmic approach to stochastic bounds. In: Calzarossa, M.C., Tucci, S. (eds.) Performance 2002. LNCS, vol. 2459, p. 64. Springer, Heidelberg (2002)
Lindvall, T.: Lectures on the coupling method. Wiley series in Probability and Mathematical statistics (1992)
Lindvall, T.: Stochastic monotonicities in Jackson queueing networks. Prob. in the Engineering and Informational Sciences 11, 1–9 (1997)
Lopez, F.J., Martinez, S., Sanz, G.: Stochastic domination and Markovian couplings. Adv. Appl. Prob. 32, 1064–1076 (2000)
Massey, W.: An Operator-analytic approach to the Jackson network. J. Appl. Prob. 21, 379–393 (1984)
Massey, W.: Stochastic orderings for Markov processes on partially ordered spaces. Mathematics of Operations Research 12(2) (May 1987)
Massey, W.: A family of bounds for the transient behavior of a Jackson network. J. Appl. Prob., 543–549 (1986)
Mokdad, L., Castel-Taleb, H.: Stochastic comparisons: a methodology for the performance evaluation of fixed and mobile networks. Computer Communications 31(17) (November 2008)
Shaked, M., Shantikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, Boston (1994)
Stoyan, D.: Comparison methods for queues and other stochastics models. J. Wiley and Son, Chichester (1976)
Van Dijk, N.M.: Queueing networks and product forms, a system approach. John Wiley and Sons, Chichester (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Castel-Taleb, H., Pekergin, N. (2010). Accuracy of Strong and Weak Comparisons for Network of Queues. In: Müller-Clostermann, B., Echtle, K., Rathgeb, E.P. (eds) Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance. MMB&DFT 2010. Lecture Notes in Computer Science, vol 5987. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12104-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-12104-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12103-6
Online ISBN: 978-3-642-12104-3
eBook Packages: Computer ScienceComputer Science (R0)