Abstract
We consider the well-studied MMPP/PH/1 queue and illustrate a method to find an almost equivalent model, the MTCP/PH/1. MTCP stands for Markovian Transition Counting Process. It is a counting process that has similar characteristics to MMPP (Markov Modulated Poisson Process). We prove that for a class of MMPPs there is an equivalent class of MTCPs. We then use this property to suggest an approximation for MMPP/PH/1 in terms of the first two moments. We numerically show that the steady state characteristics of MMPP/PH/1 are well approximated by the associated MTCP/PH/1 queue. Our numerical analysis leaves some open problems on bounds of the approximations. Of independent interest, this paper also contains a lemma on the workload expression of MAP/PH/1 queues which to the best of our knowledge has not appeared elsewhere.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Asmussen, S.: Phase-type representations in random walk and queueing problems. Ann. of Prob., 772–789 (1992)
Asmussen, S.: Matrix-analytic models and their analysis. Scand. Jour. of Stat. 27(2), 193–226 (2000)
Asmussen, S.: Applied probability and queues. Stochastic Modelling and Applied Probability, vol. 51. Springer (2003)
Bini, D.A., Meini, B., Steffé, S., Van Houdt, B.: Structured Markov chains solver: software tools. In: Proc. 2006 Workshop on Tools for Solving Structured Markov Chains, Article No. 14. ACM (2006)
Fischer, W., Meier-Hellstern, K.: The Markov-modulated Poisson process (MMPP) cookbook. Perf. Eval. 18(2), 149–171 (1993)
Fomundam, S., Herrmann, J.W.: A survey of queuing theory applications in healthcare (2007)
Gun, L.: An Algorithmic Analysis of the MMPP/G/1 Queue (No. ISR-TR-88-40). Maryland Univ. College Park Inst. For Systems Research (1988)
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, pp. 405–434. Springer, Heidelberg (2002)
Latouche G., Ramaswami, V.: Introduction to matrix analytic methods in stochastic modeling, vol. 5. SIAM (1999)
Lucantoni, D.M.: The BMAP/G/1 queue: a tutorial. In: Donatiello, L., Nelson, R. (eds.) SIGMETRICS 1993 and Performance 1993. LNCS, vol. 729, pp. 330–358. Springer, Heidelberg (1993)
Nazarathy, Y., Weiss, G.: The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Sys. 59(2), 135–156 (2008)
Rudemo, M.: Point processes generated by transitions of Markov chains. Adv. Appl. Prob., 262–286 (1973)
Ramesh, N.I.: Statistical analysis on Markov-modulated Poisson processes. Environmetrics 6(2), 165–179 (1995)
Riska, A., Squillante, M., Yu, S.Z., Liu, Z., Zhang, L.: Matrix-analytic analysis of a MAP/PH/1 queue fitted to web server data. Matrix-Analytic Methods; Theory and Applications, 333–356 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Asanjarani, A., Nazarathy, Y. (2016). A Queueing Approximation of MMPP/PH/1. In: van Do, T., Takahashi, Y., Yue, W., Nguyen, VH. (eds) Queueing Theory and Network Applications. QTNA 2015. Advances in Intelligent Systems and Computing, vol 383. Springer, Cham. https://doi.org/10.1007/978-3-319-22267-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-22267-7_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22266-0
Online ISBN: 978-3-319-22267-7
eBook Packages: EngineeringEngineering (R0)