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Semi-Markov processes can be considered as a generalization of both Markov and renewal processes. One of the principal characteristics of these processes is that in opposition to Markov models, they represent systems whose evolution is dependent not only on their last visited state but on the elapsed time since this state. Semi-Markov processes are replacing the exponential distribution of time intervals with an optional distribution. In this paper, we give a statistical approach to test the semi-Markov hypothesis. Moreover, we describe a Monte Carlo algorithm able to simulate the trajectories of the semi-Markov chain. This simulation method is used to test the semi-Markov model by comparing and analyzing the results with empirical data. We introduce the database of Network traffic which is employed for applying the Monte Carlo algorithm. The statistical characteristics of real and synthetic data from the models are compared. The comparison between the semi-Markov and the Markov models is done by computing the Autocorrelation functions and the probability density functions of the Network traffic real and simulated data as well. All the comparisons admit that the Markovian hypothesis is rejected in favor of the more general semi Markov one. Finally, the interval transition probabilities which show the future predictions of the Network traffic are given.


Semi-Markov processes Monte Carlo simulation Synthetic time series Hypothesis test Network Traffic

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How to Cite
Kordnoori, S., Mostafaei, H., Kordnoori, S., & Ostadrahimi, M. (2020). Testing the Semi Markov Model Using Monte Carlo Simulation Method for Predicting the Network Traffic . Pakistan Journal of Statistics and Operation Research, 16(4), 713-720.


  1. Adeyemi,O.J., Popoola,S.I., Atayero, A.A., Afolayan,D.G., Ariyo, M., Adetiba,E. (2018). Exploration of daily Internet data traffic generated in a smart university campus. Data in brief, 20, 30-52.
  2. Dainotti,A.,Pescape,A., Rossi,P.S.,Palmieri,F. ,Ventre,G.l. (2008). Internet traffic modeling by means of Hidden Markov Models. Computer Networks, 52(14), 2645-2662.
  3. D'Amico, G. P. (2018). Copula based multivariate semi-Markov models with applications in high-frequency finance. European Journal of Operational Research, 267(2), 765-777.
  4. D'Amico,G. ,Manca, R. ,Corini, C., Petroni, F.,Prattico, F. (2016). Tornadoes and related damage costs: statistical modelling with a semi-Markov approach. Geomatics, Natural Hazards and Risk, 7(5), 1600-1609.
  5. D'Amico,G. ,Petroni,F. (2011). A semi-Markov model with memory for price changes. Journal of statistical mechanics: Theory and experiment, 12, 12009.
  6. Devolder, P.,Janssen,J. and R. Manca. (2015). Basic stochastic processes. London: Mathematics and Statistics Series, ISTE, John Wiley and Sons, Inc.
  7. Georgiadis, S. L. (2015). Nonparametric estimation of the stationary distribution of a discrete-time semi-Markov process. Communication in Statistics-Theory and Methods, 44(7), 1319-1337.
  8. Guoa, Z., Zhaob, W., Luc, H.,Wang, J. (2012). Multi-step forecasting for wind speed using a modified EMD-based artificial neural network mode. Renewal Energy, 37, 241-249.
  9. Hocaoglua,F., Gerekb, O., Kurbanb,M. (2010). Wind Engineering and Industrial Aerodynamics.
  10. Katris, C. (2015). Comparing forecasting approaches for Internet traffic. Expert systems with applications, 42(21), 8172-8183.
  11. Kavasseri, R. (2009). Day-ahead wind speed forecasting using f-ARIMA models. Renewal Energy, 34, 1388-1393.
  12. Levy, P. (1954). Processus semi-markoviens. Proceedings from the international congress of Mathematics, 3, 416-426.
  13. Malefaki, S. (2014). Reliability of maintained systems under a semi-Markov setting. Reliability Engineering& system safety, 131, 282-290.
  14. Marnerides, A.K.,Pezaros, D.P., Hutchison,D. (2018). Internet traffic characterization: Third-order statistics & higher-order spectra for precise traffic modelling. Computer Networks, 134, 193-201.
  15. Masala, G. (2014). Wind Time Series Simulation with Underlying Semi-Markov Model: An Application to Weather Derivatives. Journal of Statistics and Management Systems, 17(3), 285-300.
  16. Nfaoui,H.,Essiarab, H., Sayigh,A.A.M. (2004). A stochastic Markov chain model for simulating wind speed time series at Tangiers, Morocco. Renewal Energy, 29(8), 1407-1418.
  17. Pertsinidou, C. E. (2017). Study of the seismic activity in central Ionian Islands via semi-Markov modelling. Acta Geophysica, 65(3), 533-541.
  18. Peschansky, A. K. (2015). Semi-Markov Model of a Single-Server Queue with Losses and Maintenance of an Unreliable Server. Cybernetics and Systems Analysis, 51(4), 632-642.
  19. Silvestrov, D. (2004). Limit Theorems for randomly stopped stochastic processes . Springer-Verlag .
  20. Smith, W. L. (1955). Regenerative stochastic processes. Proceedings of the royal society, Ser. A., 6, 232.
  21. Stenberg,F., Manca, R. and Silvestrov, D. (2006). Semi markov reward models for disability insurance. Theory of Stochastic Processes,12(28), no. 3–4, pp.239-254., 12(28)(3-4), 239-254.
  22. Vishnevski, V. A. (2017). Estimating the throughput of wireless hybrid systems operating in a semi-Markov stochastic environment. Automation and remote control, 78(12), 2154-2164