Beyond series-parallel concurrent systems: the case of arch processes

TL;DR

This paper introduces arch processes, a simplified model of concurrent systems with futures, providing enumeration formulas, asymptotic analysis, and efficient algorithms for random sampling and enumeration.

Contribution

It develops a new simplified model called arch processes, derives enumeration formulas, and presents algorithms for sampling and enumerating process runs.

Findings

Exact enumeration formula for arch process runs

Asymptotic behavior of run counts

Efficient algorithms for sampling and enumeration

Abstract

In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process with futures is more or less a directed acyclic graph. Since the quantitative study of such increasingly labeled graphs (directly related to processes) seems out of reach, we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of an uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.