**Author(s):**John N. Tsitsiklis | Kuang Xu

**Journal:**Stochastic Systems

ISSN 1946-5238

**Volume:**2;

**Issue:**1;

**Start page:**1;

**Date:**2012;

Original page

**Keywords:**Queueing | service flexibility | resource pooling | asymptotics | fluid approximation

**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 insteady 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

**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

**v**

*I*as

*N*goes to infinity.