Journal of Applied Probability

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

Sergey Foss and Masakiyo Miyazawa

Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51A (2014), 249-265.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528479

Digital Object Identifier
doi:10.1239/jap/1417528479

Mathematical Reviews number (MathSciNet)
MR3317362

Zentralblatt MATH identifier
1317.60120

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 90B15: Network models, stochastic
Secondary: 60E15: Inequalities; stochastic orderings 60G17: Sample path properties

Keywords
Fluid network Poisson and renewal arrivals heavy-tailed distribution of batch size workload process stability strong stability

Citation

Foss, Sergey; Miyazawa, Masakiyo. Two-node fluid network with a heavy-tailed random input: the strong stability case. J. Appl. Probab. 51A (2014), 249--265. doi:10.1239/jap/1417528479. https://projecteuclid.org/euclid.jap/1417528479


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