Random Permutations and Queues

TL;DR

This paper investigates the long-term behavior of fixed points in random permutations generated by growth rules, revealing differences between discrete and continuous-time models and linking permutation cycles to queueing theory.

Contribution

It demonstrates that the continuous-time embedding of the Chinese Restaurant Process exhibits limiting proportions of fixed points, unlike the discrete-time version, and connects cycle structures to infinite-server queues.

Findings

Discrete-time CRP does not have a fixed point limit.

Continuous-time CRP has well-defined fixed point proportions.

Cycle structures can be modeled as tandem queues.

Abstract

Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-server queues. We use this connection to show how results from the queuing theory can be interpreted in terms of the evolution of the cycle counts of permutations.