Correlated phase-type distributed random numbers as input models for simulations

The adequate modeling of correlated input processes is an important step in building simulation models. Modeling independent identically distributed data is well established in simulation whereas the integration of correlation is still a challenge. In this paper, ARTA processes which have been used several times for describing correlated input processes in simulation are extended by using ARMA instead of AR processes to realize the correlation and Acyclic Phase Type distributions to model the marginal distribution. For this new process type a fitting algorithm is presented. By means of some real network traces it is shown that the extended model allows a better fitting of the marginal distribution as well as the correlation structure and results in a compact process description that can be used in simulation models.

[1]  Sally Floyd,et al.  Wide-area traffic: the failure of Poisson modeling , 1994 .

[2]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[3]  Peter Buchholz,et al.  ProFiDo - The Processes Fitting Toolkit Dortmund , 2010, 2010 Seventh International Conference on the Quantitative Evaluation of Systems.

[4]  A. Cumani.  On the canonical representation of homogeneous markov processes modelling failure - time distributions , 1982 .

[5]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[6]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[7]  Robert E. Wheeler,et al.  Quantile estimators of Johnson curve parameters , 1980 .

[8]  Miron Livny,et al.  The Impact of Autocorrelation on Queuing Systems , 1993 .

[9]  Eric R. Ziegel,et al.  Probability and Statistics for Engineering and the Sciences , 2004, Technometrics.

[10]  Alma Riska,et al.  Long-Range Dependence at the Disk Drive Level , 2006, Third International Conference on the Quantitative Evaluation of Systems - (QEST'06).

[11]  D. Pfeffermann,et al.  Small area estimation , 2011 .

[12]  Bahar Biller,et al.  Copula-Based Multivariate Input Models for Stochastic Simulation , 2009, Oper. Res..

[13]  Y. L. Tong.  The multivariate normal distribution , 1989 .

[14]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[15]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[16]  Kishor S. Trivedi,et al.  Effects of failure correlation on software in operation , 2000, Proceedings. 2000 Pacific Rim International Symposium on Dependable Computing.

[17]  Barry L. Nelson,et al.  Autoregressive to anything: Time-series input processes for simulation , 1996, Oper. Res. Lett..

[18]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[19]  Barry L. Nelson,et al.  Evaluation of the ARTAFIT Method for Fitting Time-Series Input Processes for Simulation , 2008, INFORMS J. Comput..

[20]  Evgenia Smirni,et al.  Trace data characterization and fitting for Markov modeling , 2010, Performance evaluation (Print).

[21]  Barry L. Nelson,et al.  Modeling and generating multivariate time-series input processes using a vector autoregressive technique , 2003, TOMC.

[22]  M. Neuts.  A Versatile Markovian Point Process , 1979 .

[23]  J. Nash.  Compact Numerical Methods for Computers , 2018 .

[24]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[25]  Barry L. Nelson,et al.  Numerical Methods for Fitting and Simulating Autoregressive-to-Anything Processes , 1998, INFORMS J. Comput..

[26]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[27]  Peter Buchholz,et al.  A Heuristic Approach for Fitting MAPs to Moments and Joint Moments , 2009, 2009 Sixth International Conference on the Quantitative Evaluation of Systems.

[28]  P. Young,et al.  Time series analysis, forecasting and control , 1972, IEEE Transactions on Automatic Control.

[29]  G. O. Wesolowsky,et al.  On the computation of the bivariate normal integral , 1990 .

[30]  Peter Buchholz,et al.  An Empirical Comparison of MAP Fitting Algorithms , 2010, MMB/DFT.

[31]  Peter Buchholz,et al.  A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation , 2005, Second International Conference on the Quantitative Evaluation of Systems (QEST'05).

[32]  Jay L. Devore,et al.  Probability and Statistics for Engineering and the Sciences, Eighth Edition , 2012 .

[33]  Peter Buchholz,et al.  An EM-Algorithm for MAP Fitting from Real Traffic Data , 2003, Computer Performance Evaluation / TOOLS.

[34]  Evgenia Smirni,et al.  Bound analysis of closed queueing networks with workload burstiness , 2008, SIGMETRICS '08.

[35]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[36]  Peter Buchholz,et al.  A Novel Approach for Phase-Type Fitting with the EM Algorithm , 2006, IEEE Transactions on Dependable and Secure Computing.

[37]  Walter Willinger,et al.  On the Self-Similar Nature of Ethernet Traffic ( extended version ) , 1995 .

[38]  Barry L. Nelson,et al.  Fitting Time-Series Input Processes for Simulation , 2005, Oper. Res..

[39]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .