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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

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.

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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