A Fast Cross-Entropy Method for Estimating Buffer Overflows in Queueing Networks

In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for theM/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[3]  C. Morris.  Natural Exponential Families with Quadratic Variance Functions , 1982 .

[4]  Jean Walrand,et al.  A quick simulation method for excessive backlogs in networks of queues , 1989 .

[5]  J. Sadowsky.  Large deviations theory and efficient simulation of excessive backlogs in a GI/GI/m queue , 1991 .

[6]  Michael R. Frater,et al.  Optimally efficient estimation of the statistics of rare events in queueing networks , 1991 .

[7]  J. N. Kapur,et al.  Entropy Optimization Principles and Their Applications , 1992 .

[8]  J. N. Kapur,et al.  Entropy optimization principles with applications , 1992 .

[9]  Michael Devetsikiotis,et al.  An algorithmic approach to the optimization of importance sampling parameters in digital communication system simulation , 1993, IEEE Trans. Commun..

[10]  Michael Devetsikiotis,et al.  Statistical optimization of dynamic importance sampling parameters for efficient simulation of communication networks , 1993, TNET.

[11]  R. Rubinstein,et al.  Regenerative rare events simulation via likelihood ratios , 1994, Journal of Applied Probability.

[12]  Jason H. Goodfriend,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1995 .

[13]  Michael Devetsikiotis,et al.  Stochastic gradient optimization of importance sampling for the efficient simulation of digital communication systems , 1995, IEEE Trans. Commun..

[14]  Philip Heidelberger,et al.  Fast simulation of rare events in queueing and reliability models , 1993, TOMC.

[15]  Paul Glasserman,et al.  Analysis of an importance sampling estimator for tandem queues , 1995, TOMC.

[16]  J. Dshalalow.  Advances in Queueing Theory, Methods, and Open Problems , 1995 .

[17]  R. Rubinstein,et al.  Quick estimation of rare events in stochastic networks , 1997 .

[18]  Reuven Y. Rubinstein,et al.  Optimization of computer simulation models with rare events , 1997 .

[19]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[20]  R. Rubinstein,et al.  Polynomial time algorithms for estimation of rare events in queueing models , 1998 .

[21]  Reuven Y. Rubinstein,et al.  Modern simulation and modeling , 1998 .

[22]  Kluwer Academic Publishers.  Methodology and computing in applied probability , 1999 .

[23]  R. Rubinstein.  The Cross-Entropy Method for Combinatorial and Continuous Optimization , 1999 .

[24]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[25]  Pieter Tjerk de Boer,et al.  Analysis and efficient simulation of queueing models of telecommunications systems , 2000 .

[26]  V.F. Nicola,et al.  Adaptive importance sampling simulation of queueing networks , 2000, 2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165).

[27]  R. Rubinstein.  Combinatorial Optimization, Cross-Entropy, Ants and Rare Events , 2001 .

[28]  Pieter-Tjerk de Boer,et al.  Adaptive state- dependent importance sampling simulation of markovian queueing networks , 2002, Eur. Trans. Telecommun..

[29]  Reuven Y. Rubinstein,et al.  Cross-entropy and rare events for maximal cut and partition problems , 2002, TOMC.

[30]  Dirk P. Kroese,et al.  Combinatorial Optimization via Cross-Entropy , 2004 .

[31]  Dirk P. Kroese,et al.  Application of the Cross-Entropy Method to the Buffer Allocation Problem in a Simulation-Based Environment , 2005, Ann. Oper. Res..