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Critical Path Statistics of Max-Plus Linear Systems with Gaussian Noise

Part of: Special matrices Stochastic processes

Published online by Cambridge University Press:  30 January 2018

James Hook
James Hook*
Affiliation:
University of Manchester
*
∗ Postal address: Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: james.hook@manchester.ac.uk
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Abstract

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The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.

Keywords

Max-plus linear systemdiscrete event systemqueue throughput

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 15B52: Random matrices
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 654 - 670
Copyright
© Applied Probability Trust 

References

Baccelli, F. and Hong, D. (2000). Analytic expansions of max-plus Lyapunov exponents. Ann. Appl. Prob. 10, 779–827.CrossRefGoogle Scholar
Baccelli, F. L., Cohen, G., Olsder, G. J. and Quadrat, J.-P. (1992). Synchronization and Linearity. John Wiley, Chichester.Google Scholar
Cuninghame-Green, R. A. (1962). Describing industrial processes with interference and approximating their steady-state behaviour. Operat. Res. Quart. 13, 95–100.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York.Google Scholar
Goverde, R. M. P., Heidergott, B. and Merlet, G. (2011). A coupling approach to estimating the Lyapunov exponent of stochastic max-plus linear systems. Europ. J. Operat. Res. 210, 249–257.Google Scholar
Heidergott, B. F. (2007). Max-Plus Linear Stochastic Systems and Perturbation Analysis. Springer, New York.Google Scholar
Mairesse, J. (1997). Products of irreducible random matrices in the (max, +) algebra. Adv. Appl. Prob. 29, 444–477.CrossRefGoogle Scholar
Merlet, G. (2008). Cycle time of stochastic max-plus linear systems. Electron. J. Prob. 13, 322–340.CrossRefGoogle Scholar
Mittal, Y. and Ylvisaker, D. (1976). Strong laws for the maxima of stationary Gaussian processes. Ann. Prob. 4, 357–371.CrossRefGoogle Scholar
Oliveira, P. E. (2012). Asymptotics for Associated Random Variables. Springer, Heidelberg.Google Scholar