# A Bitcoin-inspired infinite-server model with a random fluid limit

## TL;DR

This paper introduces a novel infinite-server queue model inspired by Bitcoin's synchronization process, revealing unique properties like a busy period distribution independent of arrival rate and a stochastic fluid limit under scaling.

## Contribution

It presents a new Bitcoin-inspired queue model with closed-form characteristics and a novel random fluid limit, expanding understanding of such systems with interacting customers.

## Key findings

- Busy period distribution is independent of arrival rate.
- Derived a closed-form stationary distribution.
- Established a stochastic fluid limit with growth-collapse behavior.

## Abstract

The synchronization process inherent to the Bitcoin network gives rise to an infinite-server model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type).