On the power of (even a little) resource pooling

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 $\lambda$ 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.