2,042 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
Asymptotic optimality of maximum pressure policies in stochastic processing networks
We consider a class of stochastic processing networks. Assume that the
networks satisfy a complete resource pooling condition. We prove that each
maximum pressure policy asymptotically minimizes the workload process in a
stochastic processing network in heavy traffic. We also show that, under each
quadratic holding cost structure, there is a maximum pressure policy that
asymptotically minimizes the holding cost. A key to the optimality proofs is to
prove a state space collapse result and a heavy traffic limit theorem for the
network processes under a maximum pressure policy. We extend a framework of
Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams
[Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing
network setting to the stochastic processing network setting to prove the state
space collapse result and the heavy traffic limit theorem. The extension can be
adapted to other studies of stochastic processing networks.Comment: Published in at http://dx.doi.org/10.1214/08-AAP522 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotically optimal load balancing in large-scale heterogeneous systems with multiple dispatchers
We consider the load balancing problem in large-scale heterogeneous systems with multiple dispatchers. We introduce a general framework called Local-Estimation-Driven (LED). Under this framework, each dispatcher keeps local (possibly outdated) estimates of the queue lengths for all the servers, and the dispatching decision is made purely based on these local estimates. The local estimates are updated via infrequent communications between dispatchers and servers. We derive sufficient conditions for LED policies to achieve throughput optimality and delay optimality in heavy-traffic, respectively. These conditions directly imply delay optimality for many previous local-memory based policies in heavy traffic. Moreover, the results enable us to design new delay optimal policies for heterogeneous systems with multiple dispatchers. Finally, the heavy-traffic delay optimality of the LED framework also sheds light on a recent open question on how to design optimal load balancing schemes using delayed information
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