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Two-node fluid network with a heavy-tailed random input: the strong stability case

Part of: Distribution theory - Probability Stochastic processes Special processes Operations research and management science

Published online by Cambridge University Press:  30 March 2016

Sergey Foss  and
Masakiyo Miyazawa
Sergey Foss
Affiliation:
Heriot Watt University and Novosibirsk State University, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@hw.ac.uk.
Masakiyo Miyazawa
Affiliation:
Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan. Email address: miyazawa@rs.tus.ac.jp.
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Abstract

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We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Keywords

Fluid networkPoisson and renewal arrivalsheavy-tailed distribution of batch sizeworkload processstabilitystrong stability

MSC classification

Primary: 60K25: Queueing theory 90B15: Network models, stochastic
Secondary: 60E15: Inequalities; stochastic orderings 60G17: Sample path properties
Type
Part 6. Heavy tails
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 249 - 265
Copyright
Copyright © Applied Probability Trust 2014 

References

Baccelli, F., and Foss, S. (2004). Moments and tails in monotone-separable stochastic networks. Ann. Appl. Prob. 14, 612–650.CrossRefGoogle Scholar
Baccelli, F., Foss, S., and Lelarge, M. (2005). Tails in generalized Jackson networks with subexponential service-time distributions. J. Appl. Prob. 42, 513–530.CrossRefGoogle Scholar
Boxma, O., and Ivanovs, J. (2013). Two coupled Lévy queues with independent input. Stoch. Systems 3, 574–590.Google Scholar
Dai, J. G., and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Systems 1, 146–208.CrossRefGoogle Scholar
Foss, S., and Korshunov, D. (2006). Heavy tails in multi-server queue. Queueing Systems 52, 31–48.Google Scholar
Foss, S., and Miyazawa, M. (2014). Two-node fluid network with a heavy-tailed random input: the weak stability case. Working paper.Google Scholar
Foss, S., Korshunov, D., and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York.Google Scholar
Kella, O. (1996). Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Prob. 6, 186–199.CrossRefGoogle Scholar
Kobayashi, M., and Miyazawa, M. (2014). Tail asymptotics of the stationary distribution of a two-dimensional reflecting random walk with unbounded upward jumps. Adv. Appl. Prob. 46, 365–399.CrossRefGoogle Scholar
Lelarge, M. (2005). Fluid limit of generalized Jackson queueing networks with stationary and ergodic arrivals and service times. J. Appl. Prob. 42, 491–512.Google Scholar
Lieshout, P., and Mandjes, P. (2008). Asymptotic analysis of Lévy-driven tandem queues. Queueing Systems 60, 203–226.CrossRefGoogle Scholar
Miyazawa, M. (2011). Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP 19, 233–299.CrossRefGoogle Scholar
Zwart, B., Borst, S., and Mandjes, M. (2004). Exact asymptotics for fluid queues fed by multiple heavy-tailed on-off sources. Ann. Appl. Prob. 14, 903–957.Google Scholar