[Paper Review] Self-stabilizing Balls & Bins in Batches

Petra Berenbrink, Tom Friedetzky+4 more|arXiv (Cornell University)|Mar 7, 2016

Distributed systems and fault tolerance6 references4 citations

TL;DR

This paper introduces a batched, infinite balls-into-bins model where λn balls arrive each round and each non-empty bin deletes one ball, modeling dynamic request processing in distributed systems. It analyzes the Greedy[d] allocation strategy and proves that for any arrival rate λ < 1, the system is self-stabilizing: the maximum load is O(1/(1−λ) · log(n/(1−λ))) for d=1 and O(log(n/(1−λ))) for d=2, with Greedy[2] achieving exponentially better load balancing than Greedy[1] at high arrival rates.

ABSTRACT

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modeled as static balls into bins processes, where $m$ balls (tasks) are to be distributed to $n$ bins (servers). In a seminal work, Azar et al. proposed the sequential strategy \greedy{d} for $n=m$. When thrown, a ball queries the load of $d$ random bins and is allocated to a least loaded of these. Azar et al. showed that $d=2$ yields an exponential improvement compared to $d=1$. Berenbrink et al. extended this to $m\gg n$, showing that the maximal load difference is independent of $m$ for $d=2$ (in contrast to $d=1$). We propose a new variant of an \emph{infinite} balls into bins process. Each round an expected number of $\lambda n$ new balls arrive and are distributed (in parallel) to the bins. Each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server's current load but receive no information about parallel requests. We study the \greedy{d} distribution scheme in this setting and show a strong self-stabilizing property: For \emph{any} arrival rate $\lambda=\lambda(n)<1$, the system load is time-invariant. Moreover, for \emph{any} (even super-exponential) round $t$, the maximum system load is (w.h.p.) $O(\frac{1}{1-\lambda}\cdot\log\frac{n}{1-\lambda})$ for $d=1$ and $O(\log\frac{n}{1-\lambda})$ for $d=2$. In particular, \greedy{2} has an exponentially smaller system load for high arrival rates.

Motivation & Objective

• To model dynamic, distributed request processing without centralized control, where requests arrive in batches and servers delete one task per round.
• To analyze the stability and load distribution of the Greedy[d] allocation strategy in an infinite, batch-based balls-into-bins process.
• To establish that the system is self-stabilizing under any arrival rate λ < 1, even from arbitrary initial states.
• To quantify the maximum system load for both d=1 and d=2 choices, showing a fundamental advantage of the power of two choices in batched, dynamic settings.

Proposed method

• Proposes a batched, infinite balls-into-bins process where λn balls arrive per round and each non-empty bin deletes one ball.
• Introduces a potential function Ψ(X(t)) to track system load and analyze convergence, combining load imbalance and bin load distribution.
• Uses a two-part potential function Γ(X(t)) = Φ(X(t)) + Ψ(X(t)), where Φ measures load imbalance and Ψ tracks bin load distribution.
• Applies negative drift analysis and Hajek-type stability theorems to prove positive recurrence and self-stabilization.
• Employs concentration bounds and Lambert W function approximations to bound the maximum load under high arrival rates.
• Leverages the power of two choices (Greedy[2]) to achieve logarithmic load scaling, contrasting with linear-logarithmic scaling in Greedy[1].

Experimental results

Research questions

• RQ1 Can a batched, infinite balls-into-bins process with parallel arrivals and deletions be self-stabilizing under arbitrary initial loads and arrival rates λ < 1?
• RQ2 What is the maximum system load for the Greedy[1] strategy in this batched, dynamic model, and how does it scale with n and λ?
• RQ3 How does the Greedy[2] strategy compare to Greedy[1] in terms of load distribution and stability in high-arrival-rate scenarios?
• RQ4 Does the system maintain a low maximum load even when starting from an exponentially high load state, and if so, how quickly does it stabilize?
• RQ5 Can the power of two choices be preserved and leveraged in a batched, parallel setting where requests arrive simultaneously and cannot observe each other?

Key findings

• For Greedy[1], the maximum system load at any time is (w.h.p.) bounded by O(1/(1−λ) · log(n/(1−λ))), which becomes superlinear even for λ = 1 − 1/n.
• For Greedy[2], the maximum system load is (w.h.p.) bounded by O(log(n/(1−λ))), maintaining logarithmic scaling even as λ → 1.
• The system is self-stabilizing: it is positive recurrent and, at any (even super-exponentially large) time t, the load is (w.h.p.) low, implying strong stability guarantees.
• The analysis proves that the potential function Ψ(X(t)) is (w.h.p.) O(n · log n), which, combined with drift analysis, confirms the load bounds.
• The paper establishes a tight lower bound for Greedy[1] that matches the upper bound asymptotically when λ ≤ 1 − 1/poly(n), confirming the bound's tightness.
• The power of two choices yields an exponential improvement in load distribution compared to one choice, especially critical at high arrival rates (λ → 1).