On the Power of Centralization in Distributed Processing

In this thesis, 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, 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 1 1−p 1 1−λ when the traffic intensity λ goes to 1. This is exponentially smaller than the usual M/M/1-queue delay scaling of 1 1−λ , obtained when all resources are fully allocated to local stations (p = 0). This indicates a strong qualitative impact of even a small degree of centralization. We prove convergence to a fluid limit, and characterize both the transient and steady-state behavior of the finite 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 vI , for which a simple closedform expression is obtained. We also show that the steady-state distribution of the N -server system concentrates on vI as N goes to infinity. Thesis Supervisor: John N. Tsitsiklis Title: Clarence J. Lebel Professor of Electrical Engineering