Discrete-event stochastic systems with correlated inputs: Modeling and performance evaluation

Abstract

In the majority of the previous works on discrete-event stochastic systems, they have been assumed to have independent input processes. However, in many applications, these input processes can be highly correlated. Furthermore, the performance measures of the systems with correlated inputs can be significantly different from those with independent inputs. In this paper, we provide an overview on some commonly used methods for modeling correlated input processes, and we discuss the difficulties and possible future research topics in the study of discrete-event stochastic systems with correlated inputs.

Keywords

discrete-event stochastic system / correlated input / performance evaluation

References

[1] Altiok,T Melamed,B 2001. The case for modeling correlation in manufacturing systems. IIE Transactions, 33( 9): 779– 791

[2] Bardsley,E 2017. A finite mixture approach to univariate data simulation with moment matching. Environmental Modelling & Software, 90: 27– 33

[3] Biller,B 2009. Copula-based multivariate input models for stochastic simulation. Operations Research, 57( 4): 878– 892

[4] Biller,B Gunes Corlu,C 2012. Copula-based multivariate input modeling. Surveys in Operations Research and Management Science, 17( 2): 69– 84

[5] Biller,B Nelson,B L 2003. Modeling and generating multivariate time-series input processes using a vector autoregressive technique. ACM Transactions on Modeling and Computer Simulation, 13( 3): 211– 237

[6] Biller,B Nelson,B L 2008. Evaluation of the ARTAFIT method for fitting time-series input processes for simulation. INFORMS Journal on Computing, 20( 3): 485– 498

[7] BoxG E JenkinsG M ReinselG C LjungG M (1970). Time Series Analysis: Forecasting and Control. Hoboken, NJ: John Wiley & Sons

[8] Cario,M C Nelson,B L 1998. Numerical methods for fitting and simulating autoregressive-to-anything processes. INFORMS Journal on Computing, 10( 1): 72– 81

[9] Carrizosa,E Olivares-Nadal,A V Ramírez-Cobo,P 2016. Robust newsvendor problem with autoregressive demand. Computers & Operations Research, 68: 123– 133

[10] CesarJ V R (2015). Markovian Arrival Processes: The Identifiability Issue and Other Applied Aspects. Dissertation for the Doctoral Degree. Leganés: Universidad Carlos III de Madrid

[11] Choi,D I Kim,T S Lee,S 2008. Analysis of an MMPP/G/1/K queue with queue length dependent arrival rates, and its application to preventive congestion control in telecommunication networks. European Journal of Operational Research, 187( 2): 652– 659

[12] Darsow,W F Nguyen,B Olsen,E T 1992. Copulas and Markov processes. Illinois Journal of Mathematics, 36( 4): 600– 642

[13] Diaz,R Bailey,M P Kumar,S 2016. Analyzing a lost-sale stochastic inventory model with Markov-modulated demands: A simulation-based optimization study. Journal of Manufacturing Systems, 38: 1– 12

[14] Embrechts,P 2009. Copulas: A personal view. Journal of Risk and Insurance, 76( 3): 639– 650

[15] Fischer,W Meier-Hellstern,K 1993. The Markov-modulated Poisson process (MMPP) cookbook. Performance Evaluation, 18( 2): 149– 171

[16] Frey,R McNeil,A 2003. Dependent defaults in models of portfolio credit risk. Journal of Risk, 6( 1): 59– 92

[17] Gaur,V Giloni,A Seshadri,S 2005. Information sharing in a supply chain under ARMA demand. Management Science, 51( 6): 961– 969

[18] Giloni,A Hurvich,C Seshadri,S 2014. Forecasting and information sharing in supply chains under ARMA demand. IIE Transactions, 46( 1): 35– 54

[19] Girish M K, Hu J Q (1999). Modeling of correlated arrival processes in the Internet. In: Proceedings of the 38th IEEE Conference on Decision and Control. Phoenix, AZ, 4454–4459

[20] Gong,W B Hu,J Q 1992. The MacLaurin series for the GI/G/1 queue. Journal of Applied Probability, 29( 1): 176– 184

[21] Hu,J Q 1996. The departure process of the GI/G/1 queue and its MacLaurin series. Operations Research, 44( 5): 810– 815

[22] Hu,J Q Zhang,C Zhu,C B 2016. (s, S) inventory systems with correlated demands. INFORMS Journal on Computing, 28( 4): 603– 611

[23] Ibragimov,R 2009. Copula-based characterizations for higher order Markov processes. Econometric Theory, 25( 3): 819– 846

[24] Jagerman,D L Melamed,B 1992. The transition and autocorrelation structure of TES processes. Communications in Statistics: Stochastic Models, 8( 2): 193– 219

[25] Jaoua,A L’Ecuyer,P Delorme,L 2013. Call-type dependence in multiskill call centers. Simulation, 89( 6): 722– 734

[26] Kugiumtzis,D Bora-Senta,E 2014. Simulation of multivariate non-Gaussian autoregressive time series with given autocovariance and marginals. Simulation Modelling Practice and Theory, 44: 42– 53

[27] Kuhl,M E Ivy,J S Lada,E K Steiger,N M Wagner,M A Wilson,J R 2010. Univariate input models for stochastic simulation. Journal of Simulation, 4( 2): 81– 97

[28] Lee,H L Padmanabhan,V Whang,S 1997. Information distortion in a supply chain: The bullwhip effect. Management Science, 43( 4): 546– 558

[29] Lee,H L So,K C Tang,C S 2000. The value of information sharing in a two-level supply chain. Management Science, 46( 5): 626– 643

[30] Lei,L Hu,J Q Zhu,C B 2022. Discrete-event stochastic systems with copula correlated input processes. IISE Transactions, 54: 321– 331

[31] Lim,S Y Hur,S Noh,S J 2006. Departure process of a single server queuing system with Markov renewal input and general service time distribution. Computers & Industrial Engineering, 51( 3): 519– 525

[32] Livny,M Melamed,B Tsiolis,A K 1993. The impact of autocorrelation on queuing systems. Management Science, 39( 3): 322– 339

[33] Lucantoni,D M 1991. New results on the single server queue with a batch Markovian arrival process. Communications in Statistics:Stochastic Models, 7( 1): 1– 46

[34] Marshall,A W Olkin,I 1988. Families of multivariate distributions. Journal of the American Statistical Association, 83( 403): 834– 841

[35] Melamed,B 1991. TES: A class of methods for generating autocorrelated uniform variates. ORSA Journal on Computing, 3( 4): 317– 329

[36] MelamedB (1993). An overview of TES processes and modeling methodology. In: Donatiello L, Nelson R, eds. Performance Evaluation of Computer and Communication Systems. Berlin, Heidelberg: Springer, 359– 393

[37] Melamed,B Hill,J R 1995. A survey of TES modeling applications. Simulation, 64( 6): 353– 370

[38] NelsenR B (2006). An Introduction to Copulas. 2nd ed. New York, NY: Springer Science & Business Media

[39] Neuts,M F 1979. A versatile Markovian point process. Journal of Applied Probability, 16( 4): 764– 779

[40] Patuwo,B E Disney,R L McNickle,D C 1993. The effect of correlated arrivals on queues. IIE Transactions, 25( 3): 105– 110

[41] Pereira D C, del Rio Vilas D, Monteil N R, Prado R R, del Valle A G (2012). Autocorrelation effects in manufacturing systems performance: A simulation analysis. In: Proceedings of the Winter Simulation Conference (WSC). Berlin: IEEE, 1–12

[42] Runnenburg,J T 1962. Some numerical results on waiting-time distributions for dependent arrival-intervals. Statistica Neerlandica, 16( 4): 337– 347

[43] Shang,K H 2012. Single-stage approximations for optimal policies in serial inventory systems with nonstationary demand. Manufacturing & Service Operations Management, 14( 3): 414– 422

[44] Szekli,R Disney,R L Hur,S 1994. MR/GI/1 queues by positively correlated arrival stream. Journal of Applied Probability, 31( 2): 497– 514

[45] Zhang,X 2004. Technical note: Evolution of ARMA demand in supply chains. Manufacturing & Service Operations Management, 6( 2): 195– 198

[46] Zhu,Y Li,H 1993. The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services. Queuing Systems, 14( 1–2): 125– 134

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