10,180 research outputs found
Supermarket Model on Graphs
We consider a variation of the supermarket model in which the servers can
communicate with their neighbors and where the neighborhood relationships are
described in terms of a suitable graph. Tasks with unit-exponential service
time distributions arrive at each vertex as independent Poisson processes with
rate , and each task is irrevocably assigned to the shortest queue
among the one it first appears and its randomly selected neighbors. This
model has been extensively studied when the underlying graph is a clique in
which case it reduces to the well known power-of- scheme. In particular,
results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the
size of the clique gets large, the occupancy process associated with the
queue-lengths at the various servers converges to a deterministic limit
described by an infinite system of ordinary differential equations (ODE). In
this work, we consider settings where the underlying graph need not be a clique
and is allowed to be suitably sparse. We show that if the minimum degree
approaches infinity (however slowly) as the number of servers approaches
infinity, and the ratio between the maximum degree and the minimum degree in
each connected component approaches 1 uniformly, the occupancy process
converges to the same system of ODE as the classical supermarket model. In
particular, the asymptotic behavior of the occupancy process is insensitive to
the precise network topology. We also study the case where the graph sequence
is random, with the -th graph given as an Erd\H{o}s-R\'enyi random graph on
vertices with average degree . Annealed convergence of the occupancy
process to the same deterministic limit is established under the condition
, and under a stronger condition ,
convergence (in probability) is shown for almost every realization of the
random graph.Comment: 32 page
Randomized longest-queue-first scheduling for large-scale buffered systems
We develop diffusion approximations for parallel-queueing systems with the
randomized longest-queue-first scheduling algorithm by establishing new
mean-field limit theorems as the number of buffers . We achieve
this by allowing the number of sampled buffers to depend on the number
of buffers , which yields an asymptotic `decoupling' of the queue length
processes.
We show through simulation experiments that the resulting approximation is
accurate even for moderate values of and . To our knowledge, we are
the first to derive diffusion approximations for a queueing system in the
large-buffer mean-field regime. Another noteworthy feature of our scaling idea
is that the randomized longest-queue-first algorithm emulates the
longest-queue-first algorithm, yet is computationally more attractive. The
analysis of the system performance as a function of is facilitated by
the multi-scale nature in our limit theorems: the various processes we study
have different space scalings. This allows us to show the trade-off between
performance and complexity of the randomized longest-queue-first scheduling
algorithm
Concentration of measure and mixing for Markov chains
We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.Comment: 28 page
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