Fundamental Limits of Throughput and Availability: Applications to prophet inequalities & transaction fee mechanism design

TL;DR

The paper studies fundamental limits of availability and throughput for independent, heterogeneous demands under capacity constraints, introducing a concentration-inequality generator that bounds feasible availability-througput pairs via a binomial surrogate $X \sim \text{Binomial}(n, \kappa T/n)$ and a ReLU-based function $f$; the framework extends to up-to-unit demands and informs multi-unit prophet inequalities. It shows that tighter bounds on throughput for a given availability (and vice versa) translate into stronger welfare guarantees for posted-price mechanisms, with explicit results for multi-unit settings and novel invertible tail bounds that relate unavailability to throughput. The analysis yields tractable, closed-form comparisons against Poisson-worst-case benchmarks and yields practical guidance for transaction-fee design in blockchains, where high availability reduces costly emergency mechanisms. Overall, the work provides stronger, more tractable tools than classical Chernoff bounds to reason about availability-throughput tradeoffs in economic and computational systems, enabling sharper mechanism design and planning under capacity constraints.

Abstract

This paper studies the fundamental limits of availability and throughput for independent and heterogeneous demands of a limited resource. Availability is the probability that the demands are below the capacity of the resource. Throughput is the expected fraction of the resource that is utilized by the demands. We offer a concentration inequality generator that gives lower bounds on feasible availability and throughput pairs with a given capacity and independent but not necessarily identical distributions of up-to-unit demands. We show that availability and throughput cannot both be poor. These bounds are analogous to tail inequalities on sums of independent random variables, but hold throughout the support of the demand distribution. This analysis gives analytically tractable bounds supporting the unit-demand characterization of Chawla, Devanur, and Lykouris (2023) and generalizes to up-to-unit demands. Our bounds also provide an approach towards improved multi-unit prophet inequalities (Hajiaghayi, Kleinberg, and Sandholm, 2007). They have applications to transaction fee mechanism design (for blockchains) where high availability limits the probability of profitable user-miner coalitions (Chung and Shi, 2023).

Figures (7)

• Figure 1: The solid blue line represents the true bound on performant (availability, throughput) pairs, while the dashed green line is a lower bound known to the auditor. The soup kitchen achieving $(\availability_{3}, \throughput_{3})$ above the solid blue line performs well, and the soup kitchens achieving $(\availability_{1}, \throughput_{1})$ and $(\availability_{2}, \throughput_{2})$ below the solid blue line are underperforming. However, the auditor can only detect that $(\availability_{1}, \throughput_{1})$ is underperforming, since it does not know the true location of the solid blue line.
• Figure 2: The solid red line is an example $\frac{f(x)}{f(\threshold)}$, and the dotted blue line is $\frac{\max(x - \reluparam, 0)}{\max(\threshold - \reluparam, 0)}$.
• Figure 3: Step 1 moves between different $(\mu^{\uparrow}, \mu^{\downarrow})$ pairs. At all points along this path in step 1, the RHS stays constant and thus above $\mathbb{P}(D \geq \threshold)$. Then, since the RHS is increasing along the $\mu^{\downarrow}$ axis, moving from $b$ to $c$ in step 2 raises the RHS even further above $\mathbb{P}(D \geq \threshold)$.
• Figure 4: Graph of minimum throughput $\throughput = \mathbb{E}[\min(D / \threshold, 1)]$ as a function of availability $\availability = \mathbb{P}(D < \threshold)$ when $\threshold = 100$. Dashed orange line is produced when $f$ is a near-optimal ReLU function, the red line is produced when $f(x) = \exp(\lambda x) - 1$ for a carefully chosen $\lambda$, and the dotted green line is the conventional Chernoff-style bound. Compare to solid blue line, which represents the true $(\availability, \throughput)$ pairs for a Poisson distribution.
• Figure 5: Zoomed in version of \ref{['fig:chernoff-comparison']} on the point (availability, throughput) = (1, 0), with $\threshold = 5$. Observe that all bounds save for the dotted green conventional Chernoff-style bound very steeply approach (1, 0) as $\mathbb{P}(D < \threshold) \to 1$, whereas the Chernoff-style bound becomes negative.

Theorems & Definitions (11)

• proof
• theorem 1
• proof
• lemma 1
• proof
• proof
• proof
• proof : Proof of \ref{['premarkov_tail_bound']}
• proof : Proof sketch of \ref{['main_theorem']}
• proof