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On the analysis of randomized load balancing schemes

It is well known that simple randomized load balancing schemes can
balance load effectively while incurring only a small overhead, making
such schemes appealing for practical systems. In this paper we provide
new analyses for several such dynamic randomized load balancing
schemes.
Our work extends a previous analysis of the supermarket model, a model
that abstracts a simple, efficient load balancing scheme in the
setting where jobs arrive at a large system of parallel processors. In
this model, customers arrive at a system of n servers as a Poisson
stream of rate u n , u < 1 , with service requirements exponentially
distributed with mean 1. Each customer chooses d servers independently
and uniformly at random from the n servers, and is served according to
the First In First Out (FIFO) protocol at the choice with the fewest
customers. For the supermarket model, it has been shown that using d=2
choices yields an exponential improvement in the expected time a
customer spends in the system over d=1 choice (simple random
selection) in equilibrium. Here we examine several variations,
including constant service times and threshold models, where a
customer makes up to d successive choices until finding one below a
set threshold.

Our approach involves studying limiting, deterministic models
representing the behavior of these systems as the number of servers n
goes to infinity. Results of our work include useful general theorems
for showing that these deterministic systems are stable or converge
exponentially to fixed points. We also demonstrate that allowing
customers two choices instead of just one leads to exponential
improvements in the expected time a customer spends in the system in
several of the related models we study, reinforcing the concept that
just two choices yields significant power in load balancing.