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


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 $\lambda$ 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 $N\rightarrow \infty$), and that this fluid trajectory converges to a unique invariant state $\mathbf{v}^{I}$, for which a simple closed-form expression is obtained. We also show that the steady-state distribution of the $N$-server system concentrates on $\mathbf{v}^{I}$ as $N$ goes to infinity.


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John N. Tsitsiklis. Kuang Xu. "On the power of (even a little) resource pooling." Stoch. Syst. 2 (1) 1 - 66, 2012.


Published: 2012
First available in Project Euclid: 24 February 2014

zbMATH: 1296.60253
MathSciNet: MR2960735
Digital Object Identifier: 10.1214/11-SSY033

Primary: 60K25
Secondary: 37C10, 60F17, 60K30, 90B15, 90B22

Rights: Copyright © 2012 INFORMS Applied Probability Society


Vol.2 • No. 1 • 2012
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