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Large deviations for a feed-forward network

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems Miscellaneous topics in calculus of variations and optimal control

Published online by Cambridge University Press:  01 July 2016

Leila Setayeshgar  and
Hui Wang
Leila Setayeshgar*
Affiliation:
Brown University
Hui Wang*
Affiliation:
Brown University
*
∗ Postal address: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.
∗ Postal address: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.
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Abstract

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We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.

Keywords

Large deviationweak convergencediscontinuous dynamics

MSC classification

Primary: 60F10: Large deviations 65C05: Monte Carlo methods
Secondary: 49N90: Applications of optimal control and differential games
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 545 - 571
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

This research was supported in part by the National Science Foundation (NSF-DMS-0706003).

References

Alanyali, M. and Hajek, B. (1998). On large deviations of Markov processes with discontinuous statistics. Ann. Appl. Prob. 8, 45–66.Google Scholar
Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Atar, R. and Dupuis, P. (1999). Large deviations and queueing networks: methods for rate function identification. Stoch. Process Appl. 84, 255–296.Google Scholar
Borovkov, A. A. and Mogul'ski, A. A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Math. Surveys 56, 803–916.CrossRefGoogle Scholar
Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks. Springer, New York.Google Scholar
Chiang, T.-S. and Sheu, S.-J. (2000). Large deviation of diffusion processes with discontinuous drift and their occupation times. Ann. Prob. 28, 140–165.Google Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.Google Scholar
Dupuis, P., Ellis, R. S. and Weiss, A. (1991). Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Prob. 19, 1280–1297.Google Scholar
Dupuis, P., Ishii, H. and Soner, H. M. (1990). A viscosity solution approach to the asymptotic analysis of queueing systems. Ann. Prob. 18, 226–255.Google Scholar
Dupuis, P., Leder, K. and Wang, H. (2008). On the large deviations properties of the weighted-serve-the-longest-queue policy. In In and Out of Equilibrium 2, eds Sidoravicius, V. and Vares, M. E., Birkhäuser, Basel, pp. 229–256.Google Scholar
Dupuis, P., Leder, K. and Wang, H. (2009). Importance sampling for weighted-serve-the-longest-queue. Math. Operat. Res. 34, 642–660.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. John Wiley, New York.Google Scholar
Foley, R. D. and McDonald, D. R. (2001). Join the shortest queue: stability and exact asymptotics. Ann. Appl. Prob. 11, 569–607.Google Scholar
Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.Google Scholar
Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin.Google Scholar
Ignatiouk-Robert, I. (2001). Sample path large deviations and convergence parameters. Ann. Appl. Prob. 11, 1292–1329.Google Scholar
Ignatiouk-Robert, I. (2005). Large deviations for processes with discontinuous statistics. Ann. Prob. 33, 1479–1508.CrossRefGoogle Scholar
Kelly, J. L. (1951). General Topology. Springer, New York.Google Scholar
Puhalskii, A. A. and Vladimirov, A. A. (2007). A large deviation principle for Join the shortest queue. Math. Operat. Res. 32, 700–710.Google Scholar
Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis. Chapman and Hall, London.Google Scholar
Stolyar, A. L. and Ramanan, K. (2001). Largest weighted delay first scheduling: large deviations and optimality. Ann. Appl. Prob. 11, 1–48.Google Scholar