Asymptotic Behavior of Total Times For Jobs That Must Start Over If a Failure Occurs

TL;DR

This paper analyzes the asymptotic behavior of total completion times in systems where tasks restart after failures, providing relations between task and total time distributions and showing heavy-tailed properties under certain conditions.

Contribution

It derives tight asymptotic relations for total times in restart scenarios, extending understanding of failure impacts on task duration distributions.

Findings

Total time distribution is heavy-tailed if task times have unbounded support.

Asymptotic expressions for tail behavior are provided under various failure scenarios.

The analysis employs Cramér–Lundberg asymptotics, Tauberian theorems, and integral asymptotics.

Abstract

Many processes must complete in the presence of failures. Different systems respond to task failure in different ways. The system may resume a failed task from the failure point (or a saved checkpoint shortly before the failure point), it may give up on the task and select a replacement task from the ready queue, or it may restart the task. The behavior of systems under the first two scenarios is well documented, but the third ({\em RESTART}) has resisted detailed analysis. In this paper we derive tight asymptotic relations between the distribution of {\em task times} without failures to the {\em total time} when including failures, for any failure distribution. In particular, we show that if the task time distribution has an unbounded support then the total time distribution $H$ is always heavy-tailed. Asymptotic expressions are given for the tail of $H$ in various scenarios. The key ingredients of the analysis are the Cram\'er--Lundberg asymptotics for geometric sums and integral asymptotics, that in some cases are obtained via Tauberian theorems and in some cases by bare-hand calculations.