Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T20:34:19.566Z Has data issue: false hasContentIssue false
  • English
  • Français

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

Published online by Cambridge University Press:  21 July 2020

Kerry FendickOpen the ORCID record for Kerry Fendick[Opens in a new window]
Kerry Fendick*
Affiliation:
Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA E-mail: kerry.fendick@jhuapl.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

Keywords

Gaussian fluid queueGaussian Markov processmultivariate Brownian bridgemultivariate Brownian motionregulated Brownian motionstationary increments

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 1 , January 2022, pp. 144 - 168
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

References

Adler, R.J. (1981). The geometry of random fields. Chichester: Wiley.Google Scholar
Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: a close-form approximation approach. Econometria 70(1): 223–262.CrossRefGoogle Scholar
Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Annals of Statistics 36(2): 906–937.CrossRefGoogle Scholar
Arthur, W.B., Ermoliev, Y.M., & Kanjovski, Y.M. (1987). Path-dependent processes and the emergence of macro-structure. European Journal of Operational Research 30: 294–303.CrossRefGoogle Scholar
Ball, C.A. & Torous, W.N. (1983). Bond pricing dynamics and options. The Journal of Financial and Quantitative Analysis 18(4): 517–531.CrossRefGoogle Scholar
Beutler, F. (1963). Multivariate wide-sense Markov processes and prediction theory. Annals of Mathematical Statistics 34: 424–438.CrossRefGoogle Scholar
Cont, R. (2010). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1: 223–236.CrossRefGoogle Scholar
Dai, J.G. (1992). Reflected Brownian Motion in an orthant: numerical methods for steady-state analysis. Annals of Probability 2(1): 65–86.Google Scholar
Debicki, K. & Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Systems 68: 267–274.CrossRefGoogle Scholar
Debicki, K. & Rolski, T. (1995). A Gaussian fluid model. Queuing Systems 20(3–4): 443–452.CrossRefGoogle Scholar
Debicki, K. & Rolski, T. (2002). A note on transient Gaussian fluid models. Queuing Systems 41(4): 321–342.CrossRefGoogle Scholar
Debicki, K., Es-Saghouani, A., & Mandjes, M. (2009). Transient characteristics of Gaussian queues. Queuing Systems 62: 383–409.CrossRefGoogle Scholar
Debicki, K., Kosinski, K., & Mandjes, M. (2012). Gaussian queues in light and heavy traffic. Queuing Systems 71: 137–149.CrossRefGoogle Scholar
Doob, J.L. (1944). The elementary Gaussian processes. Annals of Mathematical Statistics 15(3): 229–282.CrossRefGoogle Scholar
Doob, J.L. (1953). Stochastic processes. New York: Wiley.Google Scholar
Dutilleul, P. (1999). The MLE algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation 64(2): 105–123.CrossRefGoogle Scholar
Feller, W. (1971). An introduction to probability theory and its applications, vol. II, 2nd ed. New York: Wiley.Google Scholar
Fendick, K.W. & Whitt, W. (1989). Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue. Proceedings of the IEEE 77: 171–194.CrossRefGoogle Scholar
Fendick, K.W., Saksena, V.R., & Whitt, W. (1989). Dependence in packet queues. IEEE Transactions in Communications 37: 1173–1183.CrossRefGoogle Scholar
Fendick, K.W., Saksena, V.R., & Whitt, W. (1991). Investigating dependence in packet queues with the index of dispersion for work. IEEE Transactions in Communications 39: 1231–1243.CrossRefGoogle Scholar
Fischer, J.W., Walter, W.D., & Avery, M.L. (2013). Brownian bridge movement models to characterize birds’ home ranges. The Condor 115(2): 298–305.CrossRefGoogle Scholar
Hajek, B. (1994). A queue with periodic arrivals and constant service rate. In Kelly, F.P. (ed.), Probability, statistics, and optimisation: a tribute to Peter Whittle. Chichester: Wiley, pp. 147–157.Google Scholar
Halmos, P.R. & Savage, L.J. (1949). Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Annals of Mathematical Statistics 20(2): 225–241.CrossRefGoogle Scholar
Harrison, J.M. (1985). Brownian motion and stochastic flow systems. New York: Wiley.Google Scholar
Harrison, J.M. & Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Annals of Probability 9(2): 302–308.CrossRefGoogle Scholar
Hida, T. (1960). Canonical representations of Gaussian processes and their applications. Memoirs of the College of Science, University of Kyoto, A 33: 109–155.Google Scholar
Higham, N.J. (1988). Computing a nearest symmetric positive semidefinite matrix. Linear Algebra and its Applications 103: 103–118.CrossRefGoogle Scholar
Higham, N.J. (2002). Computing the nearest correlation matrix – a problem in finance. IMA Journal of Numerical Analysis 22(3): 329–343.CrossRefGoogle Scholar
Horne, J.S. (2007). Analyzing animal movements using Brownian bridges. Ecology 88(9): 2354–2363.CrossRefGoogle ScholarPubMed
Iglehart, D.L. & Whitt, W. (1970). Multiple channel queues in heavy traffic, II: sequences, networks, and batches. Advances in Applied Probability 2: 355–369.CrossRefGoogle Scholar
Karatzas, I. & Shreve, S. (2000). Brownian motion and stochastic calculus, 2nd ed. New York: Springer.Google Scholar
Kendall, M.G. & Stuart, A. (1983). The advanced theory of statistics, vol. 2. Belmont: Wadsworth.Google Scholar
Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance 10: 603–621.CrossRefGoogle Scholar
Lipster, R.S. & Shiryayev, A.N. (1974). Statistics of random processes I: general theory. New York: Springer-Verlag.Google Scholar
Mandelbrot, B. & van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review 10(4): 422–437.CrossRefGoogle Scholar
Mandrekar, V. (1968). On multivariate wide-sense Markov processes. Nagoya Mathematical Journal 33: 7–19.CrossRefGoogle Scholar
Petersen, K.B. & Pedersen, M.S. (2012). The matrix cookbook. Kongens Lyngby: Technical University of Denmark.Google Scholar
Puntanen, S. & Styan, G.P. (2005). Schur complements in statistics and probability. In Zhang, F. (ed.), Schur complement and its applications, numerical methods and algorithms, vol. 4. Boston: Springer, pp. 163–226.Google Scholar
Rao, C. (1973). Linear statistical inference and its applications. New York: Wiley.CrossRefGoogle Scholar
Rao, C. R. & Mitra, S. K. (1972). Generalized inverse of a matrix and is applications. In Le Cam, L., Neyman, J., Scott, E. (eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1. Berkeley: University of California Press.Google Scholar
Reiman, M.I. (1984). Open queuing networks in heavy traffic. Mathematics of Operations Research 9: 441–458.CrossRefGoogle Scholar
Schäfer, J. & Strimmer, K. (2005). A shrinkage approach to large scale covariance matrix estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology 4(1), Article 32.CrossRefGoogle ScholarPubMed
Sriram, K. & Whitt, W. (1986). Characterizing superposition arrival processes in packet multiplexers for voice and data. IEEE Journal on Selected Areas of Communication 4(6): 833–846.CrossRefGoogle Scholar
Whitt, W. (1983). The queuing network analyzer. The Bell System Technical Journal 62(9): 2779–2815.CrossRefGoogle Scholar
Whitt, W. (2002). Stochastic-process limits. New York: Springer.CrossRefGoogle Scholar
Whitt, W. & You, W. (2018). Using robust queuing to expose the impact of dependence in single-server queues. Operations Research 66(1): 184–199.CrossRefGoogle Scholar
Whitt, W. & You, W. (2019). Time-varying Robust Queueing. (Operations Research) https://pubsonline.informs.org/doi/10.1287/opre.2019.1846.Google Scholar
Whittle, P. (1992). Probability via expectation, 3rd ed. New York: Springer-Verlag.CrossRefGoogle Scholar
Yaglom, A.M. (1962). Theory of stationary random functions. Englewood Cliffs: Prentice-Hall.Google Scholar