On a new approach to estimating response time quantiles of a fork-join queueing system

The article proposes a new approach to estimating the quantiles of the response time distribution of the fork-join queueing system. We consider a classic version of this system with a Poisson input flow and exponential service times on homogeneous servers. Upon receipt of tasks into the system, they are instantly divided into a fixed number of subtasks and sent for service to the appropriate subsystem with an unlimited capacity storage device and one server. The task is considered served after all its components have been serviced. This system allows you to simulate many real processes, which, in order to increase efficiency, are characterized by dividing large tasks into smaller components, for example, parallel or distributed computing systems. The difficulty of analyzing systems lies in the presence of a dependence between the sojourn times of subtasks, which significantly complicates the analysis of all performance characteristics of such systems. The main contribution of the article is the approach to determining the quantiles of the response time distribution, the assessment of which is no less valuable than the assessment of the mean response time. At the same time, a much larger number of works in this area are devoted to calculating the mean, which is explained, among other things, by the complexity of carrying out such an analysis even for a given characteristic, and estimating quantiles seems to be an even more laborious task.