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 Stochastic Systems > Vol. 2 (2012) open journal systems 


On the power of (even a little) resource pooling

John N. Tsitsiklis, Massachusetts Institute of Technology
Kuang Xu, Massachusetts Institute of Technology


Abstract
We propose and analyze a multi-server model that captures a performance trade-off between centralized and distributed processing. In our model, a fraction p of an available resource is deployed in a centralized manner (e.g., to serve a most-loaded station) while the remaining fraction 1 ‒ p is allocated to local servers that can only serve requests addressed specifically to their respective stations.
Using a fluid model approach, we demonstrate a surprising phase transition in the steady-state delay scaling, as p changes: in the limit of a large number of stations, and when any amount of centralization is available (p > 0), the average queue length in steady state scales as $\log_{\frac{1}{1-p}}{\frac{1}{1-\lambda}}$ when the traffic intensity λ goes to 1. This is exponentially smaller than the usual M/M/1-queue delay scaling of $\frac{1}{1-\lambda}$, obtained when all resources are fully allocated to local stations (p = 0). This indicates a strong qualitative impact of even a small degree of resource pooling.
We prove convergence to a fluid limit, and characterize both the transient and steady-state behavior of the actual system, in the limit as the number of stations N goes to infinity. We show that the sequence of queue-length processes converges to a unique fluid trajectory (over any finite time interval, as N → ∞), and that this fluid trajectory converges to a unique invariant state vI, for which a simple closed-form expression is obtained. We also show that the steady-state distribution of the N-server system concentrates on vI as N goes to infinity.

AMS 2000 subject classifications: Primary 60K25; secondary 60K30, 60F17, 90B15, 90B22, 37C10.

Keywords: Queueing, service flexibility, resource pooling, asymptotics, fluid approximation.

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Tsitsiklis, John N., Xu, Kuang, On the power of (even a little) resource pooling, Stochastic Systems, 2, (2012), 1-66 (electronic). DOI: 10.1214/11-SSY033.

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