Abstract
For asymptotically valid point and confidence-interval (CI) estimation of steady-state quantiles in dependent simulation output processes, two recent output-analysis procedures assume that those processes satisfy the geometric-moment contraction (GMC) condition. Moreover, the GMC condition ensures satisfaction of most of the other assumptions underlying those procedures, which are based on the techniques of batch means and standardized time series, respectively. For performance evaluation of the associated point and CI estimators, the G/G/1 queueing system provides gold-standard test processes. We prove that the GMC condition holds for G/G/1 queue-waiting times obtained with a non-heavy-tailed service-time distribution (i.e., its moment generating function exists in a neighborhood of zero). This result complements earlier proofs that the GMC condition holds for many widely used time-series and Markov-chain processes. A robustness study illustrates empirical verification of the GMC condition for M/G/1 queue-waiting times obtained with non-heavy-tailed and heavy-tailed service-time distributions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aktaran-Kalaycı, T., Alexopoulos, C., Argon, N.T., Goldsman, D., Wilson, J.R.: Exact expected values of variance estimators in steady-state simulation. Nav. Res. Logist. 54(4), 397–410 (2007)
Alexopoulos, C., Argon, N.T., Goldsman, D., Tokol, G., Wilson, J.R.: Overlapping variance estimators for simulation. Oper. Res. 55(6), 1090–1103 (2007)
Alexopoulos, C., Boone, J.H., Goldsman, D., Lolos, A., Dingeç, K.D., Wilson, J.R.: Steady-state quantile estimation using standardized time series. In: K.H. Bae, B. Feng, S. Kim, L. Lazarova-Molnar, Z. Zheng, T. Roeder, R. Thiesing (eds.) Proceedings of the 2020 Winter Simulation Conference, pp. 289–300. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey (2020)
Alexopoulos, C., Dingeç, K.D., Goldsman, D., Lolos, A., Mokashi, A.C., Wilson, J.R.: Steady-state quantile estimation using standardized time series. Technical report (2022). https://people.engr.ncsu.edu/jwilson/files/stsms-112821.pdf. Accessed 10th Feb 2022
Alexopoulos, C., Goldsman, D., Mokashi, A.C., Tien, K.W., Wilson, J.R.: Sequest: a sequential procedure for estimating quantiles in steady-state simulations. Oper. Res. 67(4), 1162–1183 (2019). https://people.engr.ncsu.edu/jwilson/files/sequest19or.pdf. Accessed 7th Sept 2019
Alexopoulos, C., Goldsman, D., Wilson, J.R.: A new perspective on batched quantile estimation. In: C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, A.M. Uhrmacher (eds.) Proceedings of the 2012 Winter Simulation Conference, pp. 190–200. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey (2012)
Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Springer, New York (2007)
Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall/CRC, Boca Raton, Florida (1994)
Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econ. 31, 307–327 (1986)
Damerdji, H.: Strong consistency of the variance estimator in steady-state simulation output analysis. Math. Oper. Res. 19, 494–512 (1994)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Orlando, FL (1984)
Derflinger, G., Hörmann, W., Leydold, J.: Random variate generation by numerical inversion when only the density is known. ACM Trans. Model. Comput. Simul. 20(18), 1–25 (2010)
Dingeç, K., Goldsman, D., Alexopoulos, C., Lolos, A., Wilson, J.R.: Geometric-moment contraction, stationary processes, and their indicator processes. Technical report, Gebze Technical University, Georgia Institute of Technology, North Carolina State University (2022). https://people.engr.ncsu.edu/jwilson/files/gmcind-030822.pdf. Accessed 8th Feb 2022
Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4), 987–1007 (1982)
Glynn, P.W., Whitt, W.: Estimating the asymptotic variance with batch means. Oper. Res. Lett. 10, 431–435 (1991)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Keilson, J., Machihara, F.: Hyperexponential waiting time structure in hyperexponential normal upper H Subscript normal upper K Baseline divided by normal upper H Subscript normal upper L Baseline divided by 1\({{\rm H}_{\rm K}/{\rm H}_{\rm L}/1}\) systems. J. Oper. Res. Soc. Jpn. 28(3), 242–251 (1985)
Kingman, J.F.: Inequalities in the theory of queues. J. Roy. Stat. Soc. B 32(1), 102–110 (1970)
Li, W.K., Ling, S., McAleer, M.: Recent theoretical results for time series models with GARCH errors. J. Econ. Surv. 16(3), 245–269 (2002)
Mori, M.: Transient behaviour of the mean waiting time and its exact forms in M/M/1 and M/D/1. J. Oper. Res. Soc. Jpn. 19(1), 14–31 (1976)
Nicholls, D.F., Quinn, B.G.: Random Coefficient Autoregressive Models: An Introduction. Springer, New York (1982)
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, New York (1999)
Shao, X., Wu, W.B.: Asymptotic spectral theory for nonlinear time series. Ann. Stat. 35(4), 1773–1801 (2007)
Tong, H.: Nonlinear Time Series: A Dynamical System Approach. Oxford University Press, New York (1990)
Wu, W.B.: On the Bahadur representation of sample quantiles for dependent sequences. Ann. Stat. 33(4), 1934–1963 (2005)
Wu, W.B., Shao, X.: Limit theorems for iterated random functions. J. Appl. Probab. 41, 425–436 (2004)
Wu, W.B., Woodroofe, M.: A central limit theorem for iterated random functions. J. Appl. Probab. 37(3), 748–755 (2000)
Acknowledgements
We thank Pierre L’Ecuyer not only for his remarkable contributions to the theory and practice of computer simulation over the past four decades, but also for his equally remarkable contributions as an editor and reviewer in a broad diversity of scientific disciplines, where his work has been applied extensively.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Dingeç, K.D., Alexopoulos, C., Goldsman, D., Lolos, A., Wilson, J.R. (2022). Geometric-Moment Contraction of G/G/1 Waiting Times. In: Botev, Z., Keller, A., Lemieux, C., Tuffin, B. (eds) Advances in Modeling and Simulation. Springer, Cham. https://doi.org/10.1007/978-3-031-10193-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-031-10193-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-10192-2
Online ISBN: 978-3-031-10193-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)