Quantitative Estimates of Stability in Controlled GI | D | 1 | #INF# Queueing Systems and in Control of Water Release in a Simple Dam Model

Evgueni Gordienko, Juan Ruiz de Chávez

2012

Abstract

We consider two applied discrete-time Markov control models: waiting times process in $\mbox{GI}\mid \mbox{D}\mid 1 \mid\infty$ queues with a controlled service rate and water release control in a simple dam model with independent water inflows. The stochastic dynamics of both models is determinated by a sequence of independent and identically distributed random variables with a distribution function $F$. In the situation when an available approximation $\tilde{F}$ is used in place of the unknown ${F}$, we estimate the deterioration of performance of control policies optimal with respect to the total discounted cost and the average cost per unit of time. For this purpose we introduce a stability index and find uppers bounds for this index expressed in terms of the Prokhorov distance between the distributions functions $F$ and $\tilde{F}$. When $\tilde{F} \equiv \tilde{F_{m}}$ is the empirical distribution function obtained from a sample of size m in average the stability index is less than a constant times $m^{\frac{-1}{3}}$.

References

  1. Abramov, V. (2008). Continuity theorems for the M/M/1/n queueing system. Queueing Syst., 59(1):68-86.
  2. Anderson, D., Sweeney, D., Thomas, A., Williams, T., Camm, J., and Martin, K. (2012). Introduction to Management Science: Quantitative Approaches to Decision Making, Revised. South-Western Cengage Learning, USA, 13-th edition.
  3. Asmussen, S. (1987). Applied Probability and Queues. John Willey Chichester.
  4. Bae, J., Kim, S., and Lee, E. (2003). Average cost under the PlM;t policy in a finite dam with compound Poisson inputs. J. Appl. Probab., 40:519-526.
  5. Barbu, V. and Sritharan, S. (1998). H-infinity control theory of fluid dynamics. Proceedings of the Royal Society of London, Ser. A., 545:3009-3033.
  6. Dynkin, E. and Yushkevich, A. (1979). Controlled Markov Processes. Springer, New York.
  7. Gordienko, E., Lemus-Rodriguez, E., and Montes-de Oca, R. (2009). Average cost Markov control processes: stability with respect to the Kantorovich metric. Math. Meth. Oper. Res., 70:13-33.
  8. Gordienko, E. and Ruiz de Chavez, J. (1998). New estimates of continuity in M=GI=1=¥ queues. Queueing Syst., 29:175-188.
  9. Gordienko, E. and Salem, F. (2000). Estimates of stability of Markov control processes with unbounded cost. Kybernetika, 36:195-210.
  10. Hernández-Lerma, O. (1989). Adaptive Markov Control Processes. Springer, New York.
  11. Hernández-Lerma, O. and Lasserre, J. (1996). Discretetime Markov Control Processes, Basic Optimization Criteria. Springer, New York.
  12. Kalashnikov, V. (1983). The analysis of continuity of queueing systems. In Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Math.(1021) Springer.
  13. Kitaev, M. and Rykov, V. (1995). Controlled Queueing Systems. CRC Press, Boca Raton FL.
  14. Litrico, X. and Georges, D. (2001). Robust LQG control of single input multiple outputs dam-river systems. International Journal of Systems Science, 32:795-805.
  15. MacPhee, I. and Müller, L. (2006). Stability criteria for controlled queueing systems. Queueing Syst., 52(3):215- 229.
  16. Meyn, S. and Tweedie, R. (2009). Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge, 2nd edition.
  17. Van Dijk, M. (1988). Perturbation theory for unbounded Markov reward processes with applications to queueing. Adv. Appl. Probab., 20:91-111.
  18. Van Dijk, M. and Sladky, K. (1999). Error bounds for nonnegative dynamic models. J. Optim. Theory Appl., 101:449-474.
  19. Zolotarev, V. (1976). On stochastic continuity of queuing systems of type G jGj 1. Theor. Probab. Appl., 21:250- 269.
Download


Paper Citation


in Harvard Style

Gordienko E. and Ruiz de Chávez J. (2012). Quantitative Estimates of Stability in Controlled GI | D | 1 | #INF# Queueing Systems and in Control of Water Release in a Simple Dam Model . In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-8565-21-1, pages 309-312. DOI: 10.5220/0004012603090312


in Bibtex Style

@conference{icinco12,
author={Evgueni Gordienko and Juan Ruiz de Chávez},
title={Quantitative Estimates of Stability in Controlled GI | D | 1 | #INF# Queueing Systems and in Control of Water Release in a Simple Dam Model},
booktitle={Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2012},
pages={309-312},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004012603090312},
isbn={978-989-8565-21-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Quantitative Estimates of Stability in Controlled GI | D | 1 | #INF# Queueing Systems and in Control of Water Release in a Simple Dam Model
SN - 978-989-8565-21-1
AU - Gordienko E.
AU - Ruiz de Chávez J.
PY - 2012
SP - 309
EP - 312
DO - 10.5220/0004012603090312