149 research outputs found
Asymptotically Optimal Load Balancing Topologies
We consider a system of servers inter-connected by some underlying graph
topology . Tasks arrive at the various servers as independent Poisson
processes of rate . Each incoming task is irrevocably assigned to
whichever server has the smallest number of tasks among the one where it
appears and its neighbors in . Tasks have unit-mean exponential service
times and leave the system upon service completion.
The above model has been extensively investigated in the case is a
clique. Since the servers are exchangeable in that case, the queue length
process is quite tractable, and it has been proved that for any ,
the fraction of servers with two or more tasks vanishes in the limit as . For an arbitrary graph , the lack of exchangeability severely
complicates the analysis, and the queue length process tends to be worse than
for a clique. Accordingly, a graph is said to be -optimal or
-optimal when the occupancy process on is equivalent to that on
a clique on an -scale or -scale, respectively.
We prove that if is an Erd\H{o}s-R\'enyi random graph with average
degree , then it is with high probability -optimal and
-optimal if and as , respectively. This demonstrates that optimality can
be maintained at -scale and -scale while reducing the number of
connections by nearly a factor and compared to a
clique, provided the topology is suitably random. It is further shown that if
contains bounded-degree nodes, then it cannot be -optimal.
In addition, we establish that an arbitrary graph is -optimal when its
minimum degree is , and may not be -optimal even when its minimum
degree is for any .Comment: A few relevant results from arXiv:1612.00723 are included for
convenienc
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