Article

# A note on insensitivity in stochastic networks

Journal of Applied Probability (Impact Factor: 0.55). 12/2006; DOI:10.1239/jap/1175267175

Source: arXiv

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**ABSTRACT:**We address a conjecture introduced by Massouli\'e (2007), concerning the large deviations of the stationary measure of bandwidth-sharing networks functioning under the Proportional fair allocation. For Markovian networks, we prove that Proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. For monotone networks, we give a more direct proof of the same result relying on stochastic comparisons that hold for general service requirement distribution. These results comfort the intuition that Proportional fairness is 'close' to allocations of service being insensitive to the service time requirement.07/2012; -
##### Chapter: Loss Networks

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**ABSTRACT:**This chapter reviews the theory of loss networks, in which calls of various types are accepted for service provided that this can commence immediately; otherwise they are rejected. An accepted call remains in the network for some holding time, which is generally independent of the state of the network, and throughout this time requires capacity simultaneously from various network resources. Both equilibrium and dynamical behaviour are studied; for the former a new approach is taken to the theory of uncontrolled loss networks, while the latter is the key to the understanding of stability issues in such networks.11/2010: pages 701-728; - [show abstract] [hide abstract]

**ABSTRACT:**Polynomial convergence rates in total variation are established in Erlang--Sevastyanov's type problem with an infinite number of servers and a general distribution of service under assumptions on the intensity of serving.10/2013;

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