157 research outputs found
Qualitative properties of -fair policies in bandwidth-sharing networks
We consider a flow-level model of a network operating under an -fair
bandwidth sharing policy (with ) proposed by Roberts and
Massouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is a
probabilistic model that captures the long-term aspects of bandwidth sharing
between users or flows in a communication network. We study the transient
properties as well as the steady-state distribution of the model. In
particular, for , we obtain bounds on the maximum number of flows
in the network over a given time horizon, by means of a maximal inequality
derived from the standard Lyapunov drift condition. As a corollary, we
establish the full state space collapse property for all . For the
steady-state distribution, we obtain explicit exponential tail bounds on the
number of flows, for any , by relying on a norm-like Lyapunov
function. As a corollary, we establish the validity of the diffusion
approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009)
1719-1780], in steady state, for the case where and under a local
traffic condition.Comment: Published in at http://dx.doi.org/10.1214/12-AAP915 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stein's method for steady-state diffusion approximations
Diffusion approximations have been a popular tool for performance analysis in
queueing theory, with the main reason being tractability and computational
efficiency. This dissertation is concerned with establishing theoretical
guarantees on the performance of steady-state diffusion approximations of
queueing systems. We develop a modular framework based on Stein's method that
allows us to establish error bounds, or convergence rates, for the
approximations. We apply this framework three queueing systems: the Erlang-C,
Erlang-A, and systems.
The former two systems are simpler and allow us to showcase the full
potential of the framework. Namely, we prove that both Wasserstein and
Kolmogorov distances between the stationary distribution of a normalized
customer count process, and that of an appropriately defined diffusion process
decrease at a rate of , where is the offered load. Futhermore,
these error bounds are \emph{universal}, valid in any load condition from
lightly loaded to heavily loaded. For the Erlang-C model, we also show that a
diffusion approximation with state-dependent diffusion coefficient can achieve
a rate of convergence of , which is an order of magnitude faster when
compared to approximations with constant diffusion coefficients.Comment: PhD Thesi
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