Abstract
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.
Highlights
Semi-Markov processes (SMPs) are a broad class of stochastic processes that overgeneralize the Markov and renewal processes at the same time
The Network traffic was forecasted by different mathematical models such as follows. (Dainotti,A.,Pescape,A., Rossi,P.S.,Palmieri,F. ,Ventre,G.l, 2008) suggested a Hidden Markov Model for Internet traffic sources at the packet level, jointly analyzing Inter Packet time and size. (Marnerides, A.K.,Pezaros, D.P., Hutchison,D., 2018) focused on the validation of the statistical characteristics of stationarity, Gaussianity and linearity of the packet interarrival time processes by 3rd order statistics. (Katris, 2015) modeled the Network traffic by the time series and Neural Network approaches. (Adeyemi,O.J., Popoola,S.I., Atayero, A.A., Afolayan,D.G., Ariyo, M., Adetiba,E., 2018) investigated the Internet traffic data based on regression Analysis and the Analysis of Variance(ANOVA)
We introduced the statistics for testing the hypothesis of being the semiMarkovian
Summary
Semi-Markov processes (SMPs) are a broad class of stochastic processes that overgeneralize the Markov and renewal processes at the same time. We apply the (SMP) to the Network traffic time series. We applied the semi-Markov chain which can perform better than the Markov models by generating synthetic time series data more precisely. The Monte Carlo method is a stochastic simulation technique based on pseudo-random numbers which allow us to forecast the system's behavior. The combination of this method with the semi-Markov model can yield precise results.
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