[Paper Review] On the Power of Centralization in Distributed Processing

Kuang Xu|arXiv (Cornell University)|Mar 22, 2012

Advanced Queuing Theory Analysis14 references3 citations

TL;DR

This paper proposes a multi-server queueing model that quantifies the performance gain from partial centralization in distributed systems. Using a fluid limit analysis, it demonstrates that even a small fraction of centralized resource allocation (p > 0) reduces steady-state queue length scaling from O(1/(1−λ)) to O(log(1/(1−λ))) as traffic intensity λ→1, indicating an exponential improvement over purely distributed systems.

ABSTRACT

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 $N$ approaches infinity, 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.

Motivation & Objective

• To model and analyze the performance trade-off between centralized and distributed processing in large-scale systems.
• To quantify the impact of partial centralization on steady-state queue length scaling under high load.
• To establish that even a small degree of centralization (p > 0) leads to a phase transition in delay scaling behavior.

Proposed method

• Formulates a multi-server queueing model where a fraction p of resources is centrally allocated to serve the most-loaded station, while 1−p is distributed locally.
• Applies a fluid limit approach to analyze the system in the limit as the number of stations N→∞.
• Derives a unique fluid trajectory that converges to a steady-state invariant state v^I with a closed-form expression.
• Uses coupling techniques and tightness arguments to prove convergence of finite-system queue-length processes to the fluid limit.
• Establishes uniform convergence rates of the steady-state distribution to the fluid invariant state.
• Validates results via simulation of a discrete-time Markov chain with embedded sampling for steady-state queue length estimation.

Experimental results

Research questions

• RQ1 How does the steady-state queue length scale with traffic intensity λ when a small fraction p > 0 of resources is centrally allocated?
• RQ2 What is the qualitative and quantitative impact of centralization on delay scaling in large distributed systems?
• RQ3 Does the system exhibit a phase transition in performance when p increases from 0 to any positive value?
• RQ4 Can the finite N system's steady-state distribution concentrate on the fluid limit as N→∞?
• RQ5 What is the rate of convergence of the steady-state distribution to the fluid invariant state?

Key findings

• When p > 0, the steady-state average queue length scales as log_{1/(1−p)}(1/(1−λ)) as λ→1, which is exponentially smaller than the O(1/(1−λ)) scaling when p=0.
• The fluid limit of the system converges to a unique invariant state v^I, which admits a simple closed-form expression.
• The sequence of finite-system queue-length processes converges to the fluid trajectory over any finite time horizon as N→∞.
• The steady-state distribution of the N-server system concentrates on the fluid invariant state v^I as N→∞.
• The convergence of the steady-state distribution to v^I occurs uniformly across initial conditions and is tight for all p > 0.
• Simulation results confirm the theoretical prediction: with p=0.05 and λ=0.99, the average queue length is drastically reduced compared to p=0.