Self-Stabilizing Balls and Bins in Batches

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 among n bins (servers). In a seminal work, Azar et al. (SIAM J Comput 29(1):180–200, 1999. https://doi.org/10.1137/S0097539795288490) proposed the sequential strategy $\textsc {Greedy}[{d}]$ for $n=m$. Each ball queries the load of d random bins and is allocated to a least loaded of them. Azar et al. (1999) showed that $d=2$ yields an exponential improvement compared to $d=1$. Berenbrink et al. (SIAM J Comput 35(6):1350–1385, 2006. https://doi.org/10.1137/S009753970444435X) extended this to $m\gg n$, showing that for $d=2$ the maximal load difference is independent of m (in contrast to the $d=1$ case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of $\lambda n$ new balls arrive and are distributed (in parallel) to the bins. Subsequently, 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 $\textsc {Greedy}[{d}]$ distribution scheme in this setting and show a strong self-stabilizing property: for any arrival rate $\lambda =\lambda (n)<1$, the system load is time-invariant. Moreover, for any (even super-exponential) round t, the maximum system load is (w.h.p.) for $d=1$ and for $d=2$. In particular, $\textsc {Greedy}[{2}]$ has an exponentially smaller system load for high arrival rates.

Auxiliary Results

The following theorem gives necessary and sufficient conditions for a Markov Chain to be positive recurrent. Roughly speaking, we want that the Markov Chain returns to a finite number of states: Given the state x of the system at some time t, we need to show that there exists some potential $\zeta (\cdot )$ which is decreasing linearly (in $\beta (x)$, the length of a suitably chosen period) for “most states” (first condition). For a finite number of states (set C), whose size can be a function of n but not of t, the potential is not required to decrease but the potential is required to be finite after $\beta (x)$ time steps (second condition).

Theorem 6

(Fayolle et al. [12, Theorem 2.2.4]) A time-homogeneous irreducible aperiodic Markov chain $\zeta$ with a countable state space $\varOmega$ is positive recurrent if and only if there exists a positive function $\phi (x), x\in \varOmega$, a number $\eta > 0$, a positive integer-valued function $\beta (x), x \in \varOmega$, and a finite set $C \subseteq \varOmega$ such that the following inequalities hold:

• 1. , $x\not \in C$

• 2. , $x\in C$

The following theorem gives a tail bound on a potential for which the following two properties hold: In increase in the potential has a tail bound (first condition) and whenever the potential is large, in decreases in expectation (second condition).

Theorem 7

(Simplified version of Hajek [14, Theorem 2.3]) Let ${(Y(t))}_{t \ge 0}$ be a sequence of random variables on a probability space $(\varOmega , \mathcal {F}, P)$ with respect to the filtration ${(\mathcal {F}(t))}_{t\ge 0}$. Assume the following two conditions hold:

• (i) (Majorization) There exists a random variable Z and a constant $\lambda ' > 0$, such that for some finite D, and $(|Y({t+1}) - Y(t)| \big \vert \mathcal {F}(t)) \prec Z$ for all $t \ge 0$; and

• (ii) (Negative Bias) There exist $a,\varepsilon _0 > 0$, such for all t we have

  

Let . Then, for all b and t we have

$$\begin{aligned} \text {Pr}(Y(t) \ge b | \mathcal {F}(0)) \le e^{\eta (Y(0) - b)}+ \frac{2 D}{\varepsilon _0 \cdot \eta }\cdot e^{ \eta (a - b)}. \end{aligned}$$

Proof

The statement of the theorem provided in [14] requires besides (i) and (ii) to choose constants $\eta$, and $\rho$ such that $0< \rho \le \lambda '$, $\eta < \varepsilon _0/c$ and $\rho =1-\varepsilon _0\cdot \eta +c \eta ^2$ where . With these requirements it then holds that for all b and t

$$\begin{aligned} \begin{aligned} \text {Pr}(Y(t) \ge b | \mathcal {F}(0)) \le \rho ^t e^{\eta (Y(0) - b)}+ \frac{1-\rho ^t}{1-\rho }\cdot D\cdot e^{\eta (a - b)}. \end{aligned} \end{aligned}$$

(59)

In the following we bound Eq. (59) by setting . The following upper and lower bound on $\rho$ follow.

• $\rho =1-\varepsilon _0 \cdot \eta +c \eta ^2 \le 1-\varepsilon _0 \cdot \eta + \varepsilon _0 \cdot \eta \cdot c \cdot \lambda '^2/(2D) \le 1-\varepsilon _0 \cdot \eta + \varepsilon _0 \cdot \eta /2 =1-\varepsilon _0 \cdot \eta /2$, where we used $c\le D/\lambda '^2$.

• $\rho =1-\varepsilon _0 \cdot \eta +c \eta ^2 \ge 1-\varepsilon _0 /(2\varepsilon _0) \ge 0$.

We derive, from Eq. (59) using that for any $t\ge 0$ we have $0\le \rho ^t\le 1$

$$\begin{aligned} \begin{aligned} \text {Pr}(Y(t) \ge b | \mathcal {F}(0))&\le \rho ^t e^{\eta (Y(0) - b)}+ \frac{1-\rho ^t}{1-\rho }\cdot D\cdot e^{\eta (a - b)} \le e^{\eta (Y(0) - b)}\\&\quad + \frac{1}{1-\rho }\cdot D\cdot e^{\eta (a - b)}\\&\le e^{\eta (Y(0) - b)}+ \frac{2 D}{\varepsilon _0 \cdot \eta }\cdot e^{ \eta (a - b)}, \end{aligned} \end{aligned}$$

(60)

since $\frac{1}{(1-\rho )}\le \frac{2 }{\varepsilon _0 \cdot \eta }$. This yields the claim. $\square$

Theorem 8

(Raab and Steger [20, Theorem 1]) Let M be the random variable that counts the maximum number of balls in any bin, if we throw m balls independently and uniformly at random into n bins. Then if $\alpha > 1$ and if $0< \alpha < 1$, where

where $d_c$ is largest solution of $1 + x (\log c - \log x + 1) -c = 0$. We have $d_1=e$ and $d_{1.00001}= 2.7183$.