Abstract
We show the solution to the optimal filtering problem for states of Markov jump processes by observations of multivariant point processes. A characteristic feature of observations is that their compensators are random linear functions of the system state, and the composite “state–observations” process does not possess the Markov property. The provided optimal filtering estimate is expressed via the solution of some recurrent system of linear differential equations and algebraic relations. We present examples of using theoretical results to construct typical models of real queueing networks. We establish the connections between our new optimal filtering algorithm and classical results of Kalman–Bucy and Wonham. We propose a solution for the problem of estimating the current state of a UDP connection given the observations of video stream.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gikhman, I.I. and Skorokhod, A.V., Teoriya sluchainykh protsessov, vol. 2, Moscow: Nauka, 1973. Translated into English under the title The Theory of Stochastic Processes. II, New York: Springer, 2004.
Frey, R. and Runggaldier, W., A Nonlinear Filtering Approach to Volatility Estimation with a View Towards High Frequency Data, Int. J. Theoret. Appl. Finance, 2001, vol. 4, pp. 199–210.
Cvitanic, J., Liptser, R., and Rozovskii, B., A Filtering Approach to Tracking Volatility from Prices Observed at Random Times, Ann. Appl. Probab., 2006, vol. 16, pp. 1633–1652.
Mamon, R.S. and Elliott, R.J., Hidden Markov Models in Finance, New York: Springer-Verlag, 2007.
Ceci, C. and Gerardi, A., Pricing from Geometric Marked Point Processes under Partial Information: Entropy Approach, Int. J. Theoret. Appl. Finance, 2009, vol. 12, pp. 179–207.
Miller, B.M., Avrachenkov, K.E., Stepanyan, K.V., and Miller, G.B., The Optimal Stochastic Control Problem for a Data Flow with Incomplete Information, Probl. Inf. Transmission, 2005, vol. 41, no. 2, pp. 150–170.
Rieder, U. and Winter, J., Optimal Control of Markovian Jump Processes with Partial Information and Applications to a Parallel Queueing Model, Math. Meth. Oper. Res., 2009, vol. 70, pp. 567–596.
Borisov, A.V., The Wonham Filter under Uncertainty: A Game Theoretic Approach, Automatica, 2011, vol. 47, no. 5, pp. 1015–1019.
Borisov, A.V., Miller, B.M., and Semenikhin, K.V., Filtering of the Markov Jump Process Given the Observations of Multivariate Point Process, Autom. Remote Control, 2015, vol. 76, no. 2, pp. 219–240.
Borisov, A.V., Application of Optimal Filtering Algorithms to Solving Availability Monitoring Problem for a Remote Server, Informatika Ee Primen., 2014, vol. 8, no. 3, pp. 34–50.
Wonham, W.W., Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering, SIAM J. Control, 1965, vol. 2, pp. 347–369.
Cox, D.R. and Smith, W.L., Renewal Theoryo, London: Methunen, 1962. Translated under the title Teoriya vosstanovleniya, Moscow: Sovetskoe Radi, 1967.
Wong, E. and Hajek, B., Stochastic Processes in Engineering Systems, New York: Springer–Verlag, 1985.
Elliott, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, New York: Springer, 1994.
Jacod, J. and Shiryaev, A.N., Predel’nye teoremy dlya sluchainykh protsessov, Moscow: Fizmatlit, 1994. Translated into English under the title Limit Theorems for Stochastic Processes, New York: Springer, 2013.
Stevens, U.R., TCP/IP Illustrated, vol. 1: The Protocols, Boston: Addison-Wesley, 1994. Translated under the title TCP-IP Protocols, St. Petersburg: BKhV-Peterburg, Nevskii Dialekt, 2003.
Kalman, R.E. and Bucy, R.S., New Results in Linear Filtering and Prediction Theory, Trans. ASME, J. Basic Eng., 1960, vol. 83D, pp. 95–108.
Miller, B.M. and Rubinovich, E.Ya., Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami, Moscow: Nauka, 2005. Translated into English under the title Impulsive Control in Continuous and Discrete-Continuous Systems, New York: Springer, 2012.
Liptser, R.Sh. and Shiryaev, A.N., Teoriya martingalov (Martingale Theory), Moscow: Fizmatlit, 986. Translated into English under the title Theory of Martingales, Dordrecht: Kluwer, 1986.
Elliott, R., Stochastic Calculus and Applications, New York: Springer, 1982 Translated under the title Stokhasticheskii analiz i ego prilozheniya, Moscow: Mir, 1986.
Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov, Moscow: Nauka, 1974. Translated into English under the title Statistics of Random Processes, New York: Springer, 2013.
Baccelli, F. and Bremaud, P., Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrence, Berlin: Springer, 2003.
Bremaud, P., Point Processes and Queues: Martingale Dynamics, New York: Springer, 1981.
Yushkevich, A.A., Controllable Markov Models with a Countable Set of States and Continuous Time, Theory Probab. Appl., 1978, vol. 22, no. 2, pp. 2150–235.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Borisov, 2016, published in Avtomatika i Telemekhanika, 2016, No. 2, pp. 115–141.
Rights and permissions
About this article
Cite this article
Borisov, A.V. Application of optimal filtering methods for on-line of queueing network states. Autom Remote Control 77, 277–296 (2016). https://doi.org/10.1134/S0005117916020053
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916020053