Abstract
Computing the steady-state distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steady-state stochastic model \(\boldsymbol {W}\stackrel{d}{=}g(\boldsymbol {W},\boldsymbol {X})\) where the equality is in distribution. Given partial distributional information on the random variables X, we want to estimate information on the distribution of the steady-state vector W. Such models naturally occur in queueing systems, where the goal is to find bounds on moments of the waiting time under moment information on the service and interarrival times. In this paper, we propose an approach based on semidefinite optimization to find such bounds. We show that the classical Kingman’s and Daley’s bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach. We also report computational results in the queueing context that indicate the method is promising.
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Bertsimas, D., Natarajan, K., Teo, C.P.: Probabilistic combinatorial optimization: moments, semidefinite programming and asymptotic bounds. SIAM J. Optim. 15(1), 185–209 (2004)
Bertsimas, D., Natarajan, K., Teo, C.P.: Persistence in discrete optimization under data uncertainty. Math. Program. Ser. B 108(2), 251–274 (2006)
Bertsimas, D., Popescu, I.: On the relation between option and stock prices: a convex optimization approach. Oper. Res. 50, 358–374 (2002)
Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)
Borovkov, A.A., Foss, S.G.: Stochastically recursive sequences and their generalizations. Sib. Adv. Math. 2, 16–81 (1992)
Daley, D.J.: Inequalities for moments of tails of random variables with queueing applications. Z. Wahrscheinlichkeitstheor. Verw. Geb. 41, 139–143 (1977)
Daley, D.J.: Some results for the mean waiting-time and workload in GI/GI/k queues. In: Dshalalow, J.H. (ed.) Frontiers in Queueing, pp. 35–59. CRC Press, Boca Raton (1997)
Kiefer, J., Wolfowitz, J.: On the theory of queues with many servers. Trans. Am. Math. Soc. 78, 1–18 (1955)
Kingman, J.F.C.: Some inequalities for the GI/G/1 queue. Biometrika 49, 315–324 (1962)
Kingman, J.F.C.: Inequalities in the theory of queues. J. Roy. Stat. Soc. Ser. B 32, 102–110 (1970)
Lasserre, J.B.: Bounds on measures satisfying moment conditions. Ann. Appl. Probab. 12, 1114–1137 (2002)
Lasserre, J.B., Hernández-Lerma, O.: Markov Chains and Invariant Probabilities. Progress in Mathematics, vol. 211. Birkhäuser, Basel (2003)
Lindley, D.V.: On the theory of queues with a single server. Proc. Camb. Philos. Soc. 48, 277–289 (1952)
Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)
SeDuMi version 1.1. Available from http://sedumi.mcmaster.ca/
Wolff, R.W., Wang, C.L.: Idle period approximations and bounds for the GI/GI/1 queue. Adv. Appl. Probab. 35, 773–792 (2003)
Zuluaga, L., Pena, J.F.: A conic programming approach to generalized Tchebycheff inequalities. Math. Oper. Res. 30(2), 369–388 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of D. Bertsimas supported in part by Singapore-MIT Alliance.
Research of K. Natarajan is supported in part by NUS Academic Research Grant R146-050-070-133 and Singapore-MIT Alliance.
Rights and permissions
About this article
Cite this article
Bertsimas, D., Natarajan, K. A semidefinite optimization approach to the steady-state analysis of queueing systems. Queueing Syst 56, 27–39 (2007). https://doi.org/10.1007/s11134-007-9028-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-007-9028-7