2,041 research outputs found

    Asymptotically Optimal Load Balancing Topologies

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    We consider a system of NN servers inter-connected by some underlying graph topology GNG_N. Tasks arrive at the various servers as independent Poisson processes of rate Ξ»\lambda. 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 GNG_N. Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case GNG_N 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 Ξ»<1\lambda < 1, the fraction of servers with two or more tasks vanishes in the limit as Nβ†’βˆžN \to \infty. For an arbitrary graph GNG_N, the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph GNG_N is said to be NN-optimal or N\sqrt{N}-optimal when the occupancy process on GNG_N is equivalent to that on a clique on an NN-scale or N\sqrt{N}-scale, respectively. We prove that if GNG_N is an Erd\H{o}s-R\'enyi random graph with average degree d(N)d(N), then it is with high probability NN-optimal and N\sqrt{N}-optimal if d(N)β†’βˆžd(N) \to \infty and d(N)/(Nlog⁑(N))β†’βˆžd(N) / (\sqrt{N} \log(N)) \to \infty as Nβ†’βˆžN \to \infty, respectively. This demonstrates that optimality can be maintained at NN-scale and N\sqrt{N}-scale while reducing the number of connections by nearly a factor NN and N/log⁑(N)\sqrt{N} / \log(N) compared to a clique, provided the topology is suitably random. It is further shown that if GNG_N contains Θ(N)\Theta(N) bounded-degree nodes, then it cannot be NN-optimal. In addition, we establish that an arbitrary graph GNG_N is NN-optimal when its minimum degree is Nβˆ’o(N)N - o(N), and may not be NN-optimal even when its minimum degree is cN+o(N)c N + o(N) for any 0<c<1/20 < c < 1/2.Comment: A few relevant results from arXiv:1612.00723 are included for convenienc

    Asymptotic optimality of maximum pressure policies in stochastic processing networks

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    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

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    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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