Invariance principle and McKean-Vlasov limit for randomized load balancing in heavy traffic
arxiv(2024)
摘要
We consider a load balancing model where a Poisson stream of jobs arrive at a
system of many servers whose service time distribution possesses a finite
second moment. A small fraction of arrivals pass through the so called
power-of-choice algorithm, which assigns a job to the shortest among ℓ,
ℓ≥ 2, randomly chosen queues, and the remaining jobs are assigned to
queues chosen uniformly at random. The system is analyzed at critical load in
an asymptotic regime where both the number of servers and the usual heavy
traffic parameter associated with individual queue lengths grow to infinity.
The first main result is a hydrodynamic limit, where the empirical measure of
the diffusively normalized queue lengths is shown to converge to a path in
measure space whose density is given by the unique solution of a parabolic PDE
with nonlocal coefficients.
Further, two forms of an invariance principle are proved, corresponding to
two different assumptions on the initial distribution, where individual
normalized queue lengths converge weakly to solutions of SDE. In one of these
results, the limit is given by a McKean-Vlasov SDE, and propagation of chaos
holds. The McKean-Vlasov limit is closely related to limit results for Brownian
particles on ℝ_+ interacting through their rank (with a specific
interaction). However, an entirely different set of tools is required, as the
collection of n prelimit particles does not obey a Markovian evolution on
ℝ_+^n.
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