## Stochastic Systems

- Stoch. Syst.
- Volume 2, Number 1 (2012), 1-66.

### On the power of (even a little) resource pooling

John N. Tsitsiklis and Kuang Xu

#### 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.

#### Article information

**Source**

Stoch. Syst., Volume 2, Number 1 (2012), 1-66.

**Dates**

First available in Project Euclid: 24 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ssy/1393252040

**Digital Object Identifier**

doi:10.1214/11-SSY033

**Mathematical Reviews number (MathSciNet)**

MR2960735

**Zentralblatt MATH identifier**

1296.60253

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60F17: Functional limit theorems; invariance principles 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20] 37C10: Vector fields, flows, ordinary differential equations

**Keywords**

Queueing service flexibility resource pooling asymptotics fluid approximation

#### Citation

Tsitsiklis, John N.; Xu, Kuang. On the power of (even a little) resource pooling. Stoch. Syst. 2 (2012), no. 1, 1--66. doi:10.1214/11-SSY033. https://projecteuclid.org/euclid.ssy/1393252040