Sharding is used to improve the scalability and performance of blockchain systems. We investigate the stability of blockchain sharding, where transactions are continuously generated by an adversarial model. The system consists of $n$ processing nodes that are divided into $s$ shards. Following the paradigm of classical adversarial queuing theory, transactions are continuously received at injection rate $\rho \leq 1$ and burstiness $b > 0$. We give an absolute upper bound $\max\{ \frac{2}{k+1}, \frac{2}{ \left\lfloor\sqrt{2s}\right\rfloor}\}$ on the maximum injection rate for which any scheduler could guarantee bounded queues and latency of transactions, where $k$ is the number of shards that each transaction accesses. We next give a basic distributed scheduling algorithm for uniform systems where shards are equally close to each other. To guarantee stability, the injection rate is limited to $\rho \leq \max\{ \frac{1}{18k}, \frac{1}{\lceil 18 \sqrt{s} \rceil} \}$. We then provide a fully distributed scheduling algorithm for non-uniform systems where shards are arbitrarily far from each other. By using a hierarchical clustering of the shards, stability is guaranteed with injection rate $\rho \leq \frac{1}{c_1d \log^2 s} \cdot \max\{ \frac{1}{k}, \frac{1}{\sqrt{s}} \}$, where $d$ is the worst distance of any transaction to the shards it will access, and $c_1$ is some positive constant. We also conduct simulations to evaluate the algorithms and measure the average queue sizes and latency throughout the system. To our knowledge, this is the first adversarial stability analysis of sharded blockchain systems.

翻译：分片技术用于提升区块链系统的可扩展性与性能。本文研究在对抗模型持续生成交易场景下区块链分片的稳定性问题。系统由$n$个处理节点组成，这些节点被划分为$s$个分片。遵循经典对抗排队理论范式，交易以注入率$\rho \leq 1$和突发性$b > 0$持续到达。我们给出了任意调度器能够保证交易有界队列和延迟的最大注入率的绝对上界$\max\{ \frac{2}{k+1}, \frac{2}{ \left\lfloor\sqrt{2s}\right\rfloor}\}$，其中$k$为每笔交易访问的分片数量。随后针对分片间距相等的均匀系统提出基础分布式调度算法。为保证稳定性，注入率被限制为$\rho \leq \max\{ \frac{1}{18k}, \frac{1}{\lceil 18 \sqrt{s} \rceil} \}$。针对分片间距任意不等的非均匀系统，我们进一步提出全分布式调度算法。通过分片层次聚类，当注入率满足$\rho \leq \frac{1}{c_1d \log^2 s} \cdot \max\{ \frac{1}{k}, \frac{1}{\sqrt{s}} \}$时可保证系统稳定性，其中$d$为任意交易到其访问分片的最远距离，$c_1$为正常数。我们还通过仿真评估了算法性能，测量了系统全局平均队列长度和延迟。据我们所知，这是首项关于分片区块链系统的对抗稳定性分析工作。