Advertisement

Introduction

  • Peter Buchholz
  • Jan Kriege
  • Iryna Felko
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Quantitative analysis of man-made systems like computer systems, communication networks, manufacturing plants, logistics networks, to mention only few examples, is often done by means of discrete event models that are analyzed numerically [152] or by simulation [105]. One key issue in these models is the adequate modeling of the load which describes the occurrence of events, let it be customer arrivals in queueing networks, failure times in reliability models or packet lengths in simulation models of computer networks. In more abstract terms one can think of arrival, service or failure times that are part of a model. We will use the term inter-event times to capture the different quantities in a model. Inter-event times are characterized by random variables or stochastic processes generating non-negative numbers.

References

  1. 1.
    Alfa, A.S., Neuts, M.F.: Modelling vehicular traffic using the discrete time Markovian arrival process. Transport. Sci. 29(2), 109–117 (1995)CrossRefzbMATHGoogle Scholar
  2. 5.
    Asmussen, S., Bladt, M.: Point processes with finite-dimensional conditional probabilities. Stoch. Process. Their Appl. 82, 127–142 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 11.
    Bause, F., Buchholz, P., Kriege, J.: A comparison of Markovian arrival processes and ARMA/ARTA processes for the modelling of correlated input processes. In: Proceedings of the Winter Simulation Conference (2009)Google Scholar
  4. 14.
    Biller, B., Gunes, C.: Introduction to simulation input modeling. In: Johansson, B., Jain, S., Montoya-Torres, J., Hugan, J., Yücesan, E. (eds.) Proceedings of the Winter Simulation Conference (WSC), pp. 49–58 (2010)Google Scholar
  5. 20.
    Bobbio, A., Horváth, A., Scarpa, M., Telek, M.: Acyclic discrete phase type distributions: properties and a parameter estimation algorithm. Perform. Eval. 54(1), 1–32 (2003)CrossRefGoogle Scholar
  6. 25.
    Breuer, L.: An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Ann. OR 112(1–4), 123–138 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 27.
    Brickner, C., Indrawan, D., Williams, D., Chakravarthy, S.R.: Simulation of a stochastic model for a service system. In: Johansson, B., Jain, S., Montoya-Torres, J., Hugan, J., Yücesan, E. (eds.) Proceedings of the Winter Simulation Conference (WSC), pp. 1636–1647 (2010)Google Scholar
  8. 31.
    Buchholz, P.: An EM-algorithm for MAP fitting from real traffic data. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation/TOOLS. Lecture Notes in Computer Science, vol. 2794, pp. 218–236. Springer, New York (2003)Google Scholar
  9. 32.
    Buchholz, P., Kriege, J.: A heuristic approach for fitting MAPs to moments and joint moments. In: Proceedings of the 6th International Conference on the Quantitative Evaluation of Systems (QEST), pp. 53–62. IEEE Computer Society, Budapest (2009)Google Scholar
  10. 37.
    Buchholz, P., Telek, M.: Rational arrival processes associated to labelled Markov processes. J. Appl. Probab. 49(1), 40–59 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 38.
    Buchholz, P., Telek, M.: On minimal representations of rational arrival processes. Ann. Oper. Res. 202(1), 35–58 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 41.
    Casale, G., Zhang, E.Z., Smirni, E.: KPC-toolbox: simple yet effective trace fitting using Markovian arrival processes. In: Proceedings of the 5th International Conference on the Quantitative Evaluation of Systems (QEST), pp. 83–92. IEEE Computer Society, St. Malo (2008)Google Scholar
  13. 48.
    Dayar, T.: On moments of discrete phase-type distributions. In: Bravetti, M., Kloul, L., Zavattaro, G. (eds.) Proceedings of the EPEW/WS-FM. Lecture Notes in Computer Science, vol. 3670, pp. 51–63. Springer, New York (2005)Google Scholar
  14. 58.
    Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Perform. Eval. 31, 245–279 (1998)CrossRefGoogle Scholar
  15. 63.
    Goseva-Popstojanova, K., Trivedi, K.S.: Effects of failure correlation on software in operation. In: Proceedings of the 2000 Pacific Rim International Symposium on Dependable Computing (PRDC), pp. 69–76. IEEE Computer Society, Los Angeles (2000)Google Scholar
  16. 68.
    Heckmüller, S., Wolfinger, B.E.: Using load transformations for the specification of arrival processes in simulation and analysis. Simulation 85(8), 485–496 (2009)CrossRefGoogle Scholar
  17. 76.
    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.) Proceedings of the Performance 2002. Lecture Notes in Computer Science, vol. 2459, pp. 405–434. Springer, Berlin (2002)Google Scholar
  18. 82.
    Horváth, G., Telek, M., Buchholz, P.: A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag-correlation. In: Proceedings of the 2nd International Conference on the Quantitative Evaluation of Systems (QEST), pp. 124–133. IEEE CS Press, Torino (2005)Google Scholar
  19. 92.
    Kelton, W.D., Sadowski, R.P., Sadowski, D.A.: Simulation with Arena, 4th edn. McGraw-Hill, New York (2007)Google Scholar
  20. 94.
    Khayari, R.E.A., Sadre, R., Haverkort, B.: Fitting world-wide web request traces with the EM-algorithm. Perform. Eval. 52, 175–191 (2003)CrossRefGoogle Scholar
  21. 97.
    Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. Perform. Eval. 54(2), 149–173 (2003)CrossRefGoogle Scholar
  22. 104.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  23. 105.
    Law, A.M., Kelton, W.D.: Simulation Modeling and Analysis, 3rd edn. McGraw-Hill, Boston (2000). ISBN 0-07-059292-6Google Scholar
  24. 106.
    Law, A.M., McComas, M.G.: ExpertFit distribution-fitting software: how the ExpertFit distribution-fitting software can make your simulation models more valid. In: Chick, S.E., Sanchez, P.J., Ferrin, D.M., Morrice, D.J. (eds.) Proceedings of the Winter Simulation Conference, pp. 169–174. ACM, Berlin (2003)Google Scholar
  25. 109.
    Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Netw. 2(1), 1–15 (1994)CrossRefGoogle Scholar
  26. 111.
    Lipsky, L.: Queueing Theory: A Linear Algebraic Approach. Springer, New York (2008)Google Scholar
  27. 124.
    Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16, 764–779 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 125.
    Neuts, M.F.: Matrix-geometric solutions in stochastic models. Johns Hopkins University Press, Baltimore (1981)zbMATHGoogle Scholar
  29. 128.
    Nightingale, E.B., Douceur, J.R., Orgovan, V.: Cycles, cells and platters: an empirical analysis of hardware failures on a million consumer PCs. In: Kirsch, C.M., Heiser, G. (eds.) Proceedings of the EuroSys, pp. 343–356. ACM, Salzburg (2011)Google Scholar
  30. 132.
    O’Cinneide, C.A.: Phase-type distributions: open problems and a few properties. Stoch. Model. 15(4), 731–757 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 133.
    Okamura, H., Dohi, T., Trivedi, K.S.: Markovian arrival process parameter estimation with group data. IEEE/ACM Trans. Netw. 17(4), 1326–1339 (2009)CrossRefGoogle Scholar
  32. 140.
    Paxson, V., Floyd, S.: Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Netw. 3(3), 226–244 (1995)CrossRefGoogle Scholar
  33. 141.
    Rahnamay-Naeini, M., Pezoa, J.E., Azar, G., Ghani, N., Hayat, M.M.: Modeling stochastic correlated failures and their effects on network reliability. In: Proceedings of 20th International Conference on Computer Communications and Networks (ICCCN), pp. 1–6 (2011)Google Scholar
  34. 147.
    Ruiz-Castro, J.E., Fernández-Villodre, G., Pérez-Ocón, R.: Discrete repairable systems with external and internal failures under phase-type distributions. IEEE Trans. Reliab. 58(1), 41–52 (2009)CrossRefGoogle Scholar
  35. 148.
    Sauer, C.H., Chandy, K.M.: Computer Systems Performance Modeling. Prentice Hall, Englewood Cliffs (1981)Google Scholar
  36. 152.
    Stewart, W.J.: Probability, Markov Chains, Queues, and Simulation. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  37. 155.
    Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9–12), 1153–1168 (2007)CrossRefGoogle Scholar
  38. 158.
    Van Houdt, B., Lenin, R.B., Blondia, C.: Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times. Queueing Syst. 45(1), 59–73 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Peter Buchholz, Jan Kriege, Iryna Felko 2014

Authors and Affiliations

  • Peter Buchholz
    • 1
  • Jan Kriege
    • 1
  • Iryna Felko
    • 1
  1. 1.Department of Computer ScienceTechnical University of DortmundDortmundGermany

Personalised recommendations