Abstract
This paper shows how to construct a Markovian arrival process of second order from information on the marginal distribution and on its autocorrelation function. More precisely, closed-form explicit expressions for the MAP(2) rate matrices are given in terms of the first three marginal moments and one parameter that characterizes the behavior of the autocorrelation function. Besides the permissible moment ranges, which were known before, also the necessary and sufficient bounds for the correlation parameter are computed and shown to depend on a free parameter related to equivalent acyclic PH(2) representations of the marginal distribution. We identify the choices for the free parameter that maximize the correlation range for both negative and positive correlation parameters.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Neuts, M.: Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press (1981)
Neuts, M.: Structured Stochastic Matrices of M/G/1-type and their Applications. Marcel Dekker, New York (1989)
Latouche, G., Ramaswami, V.: Introduction to Matrix-Analytic Methods in Stochastic Modeling. Series on statistics and applied probability. ASA-SIAM, Philadelphia (1999)
Buchholz, P.: An EM-algorithm for MAP fitting from real traffic data. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 218–236. Springer, Heidelberg (2003)
Horváth, G., Buchholz, P., Telek, M.: A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: Proc. 2nd Int. Conf. on Quantitative Evaluation of Systems, Torino, Italy, pp. 124–133 (2005)
Horváth, A., Telek, M.: Markovian modeling of real data traffic: Heuristic phase-type and MAP fitting of heavy tailed and fractal-like samples. In: Calzarossa, M.C., Tucci, S. (eds.) Performance 2002. LNCS, vol. 2459. Springer, Heidelberg (2002)
Horváth, A., Telek, M.: Fitting more than three moments with acyclic phase-type distributions (submitted, 2006)
Telek, M., Heindl, A.: Matching moments for acyclic discrete and continuous phase-type distributions of second order. Intl. Journal of Simulation 3, 47–57 (2003)
Bobbio, A., Horváth, A., Telek, M.: Matching three moments with minimal acyclic phase-type distributions. Stochastic Models 21, 303–323 (2005)
Mitchell, K., van de Liefvoort, A.: Approximation models of feed-forward G/G/1/N queueing networks with correlated arrivals. Performance Evaluation 51, 137–152 (2003)
Mitchell, K.: Constructing a correlated sequence of matrix exponentials with invariant first-order properties. Operations Research Letters 28, 27–34 (2001)
Heindl, A., Mitchell, K., van de Liefvoort, A.: The correlation region of second-order MAPs with application to queueing network decomposition. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 237–254. Springer, Heidelberg (2003)
Heindl, A.: Inverse characterization of hyperexponential MAP(2)s. In: Proc. 11th Int. Conference on Analytical and Stochastic Modelling Techniques and Applications, Magdeburg, Germany, pp. 183–189 (2004)
Heffes, H., Lucantoni, D.M.: A Markov-modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. on Selected Areas in Commun. 4, 856–868 (1986)
Ferng, H.W., Chang, J.F.: Connection-wise end-to-end performance analysis of queueing networks with MMPP inputs. Performance Evaluation 43, 39–62 (2001)
Livny, M., Melamed, B., Tsiolis, A.K.: The impact of autocorrelation on queueing systems. Management Science 39, 322–339 (1993)
Patuwo, B., Disney, R., McNickle, D.: The effect of correlated arrivals on queues. IIE Transactions 25, 105–110 (1993)
Neuts, M.: Algorithmic Probability: A Collection of Problems. Chapman and Hall, Boca Raton (1995)
van de Liefvoort, A.: The moment problem for continuous distributions. Technical Report WP-CM-1990-02, School of Computing and Engineering, University of Missouri – Kansas City, USA (1990)
Gross, K.: Analytische Konstruktion korrelierter Prozesse zur Lastmodellierung in Kommunikationssystemen. Master’s thesis, Informatik 7, Universität Erlangen-Nürnberg, Germany (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Heindl, A., Horváth, G., Gross, K. (2006). Explicit Inverse Characterizations of Acyclic MAPs of Second Order. In: Horváth, A., Telek, M. (eds) Formal Methods and Stochastic Models for Performance Evaluation. EPEW 2006. Lecture Notes in Computer Science, vol 4054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11777830_8
Download citation
DOI: https://doi.org/10.1007/11777830_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35362-1
Online ISBN: 978-3-540-35365-2
eBook Packages: Computer ScienceComputer Science (R0)