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UID:www.tcs.tifr.res.in/event/919
DTSTAMP:20230914T125943Z
SUMMARY:Uniform Geometric and Subgeometric Ergodicity for Multiclass Many S
erver Queues in Heavy Traffic
DESCRIPTION:Speaker: Aristotle Arapostathis (University of Texas\nDepartmen
t of Electrical and\nComputer Engineering\nAustin\, Texas\, United States\
n )\n\nAbstract: \nAbstract: We study ergodic properties of multiclass mu
lti-server queues\, which are uniform over scheduling policies\, as well a
s the size n of the system. The system is heavily loaded in the Halfin-W
hitt regime\, and the scheduling policies are work-conserving and preempti
ve. We provide a unified approach via ‘matching’ Foster-Lyapunov equat
ions for both the limiting diffusion and the prelimit diffusion-scaled que
ueing processes simultaneously. We first study the limiting controlled di
ffusion\, and we show that if the spare capacity (safety staffing) paramet
er is positive\, then the diffusion is exponentially ergodic uniformly ove
r all stationary Markov controls\, and the invariant probability measures
have sub-exponential tails. This result is sharp\, since when there is no
abandonment and the spare capacity parameter is negative\, then the contro
lled diffusion is transient under any Markov control. In addition\, we sho
w that if all the abandonment rates are positive\, the invariant probabili
ty measures have sub-Gaussian tails\, regardless whether the spare capacit
y parameter is positive or negative.\nUsing the above results\, we proceed
to establish the corresponding ergodic properties for the diffusion-scale
d queueing processes. In addition to providing a simpler proof of the resu
lts in Gamarnik and Stolyar [Queueing Syst. (2012) 71:25--51]\, we extend
these results to GI/M/n+M queues with renewal arrival processes\, albeit u
nder the assumption that the mean residual life functions are bounded. F
or the Markovian model with Poisson arrivals\, we obtain stronger results
and show that the convergence to the stationary distribution is at a geome
tric rate uniformly over all work-conserving stationary Markov scheduling
policies.\nWe then turn to the case when arrivals are heavy-tailed\, or th
e system suffers from asymptotically negligible service interruptions. I
n these models\, the Itô equations are driven by either (1) a Brownian mo
tion and a pure-jump Levy process\, or (2) an anisotropic Levy process wit
h independent one-dimensional symmetric alpha-stable components\, or (3) a
n anisotropic Lévy process and a pure-jump Lévy process. We identify con
ditions on the parameters in the drift\, the Lévy measure and/or covarian
ce function which result in subexponential and/or exponential ergodicity.
We show that these assumptions are sharp. In addition\, we show that for t
he queueing models described above with no abandonment\, the rate of conve
rgence is polynomial\, and we provide a sharp quantitative characterizatio
n of this rate via matching upper and lower bounds (joint work with Hassan
Hmedi\, Guodong Pang and Nikola Sandric).\n\nBio: Ari Arapostathis is a
professor at the Department of Electrical and Computer Engineering at the
University of Texas at Austin. His research interests include analysis a
nd estimation techniques for stochastic systems\, stability properties of
large-scale interconnected power systems\, and stochastic and adaptive con
trol theory. His main technical contributions are in the areas of adapti
ve control and estimation of stochastic systems with partial observations\
, controlled diffusions\, adaptive control of nonlinear systems\, geometri
c nonlinear theory\, and stability of large scale interconnected power sys
tems. His research is currently funded by the National Science Foundatio
n (Division of Mathematical Sciences)\, Army Research Office (Applied Prob
ability)\, and the Office of Naval Research. He is a Fellow of IEEE.\n
URL:https://www.tcs.tifr.res.in/web/events/919
DTSTART;TZID=Asia/Kolkata:20181122T160000
DTEND;TZID=Asia/Kolkata:20181122T170000
LOCATION:A-201 (STCS Seminar Room)
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