Copulas and Quantiles in Fork-Join Queueing Systems

Abstract

The article considers a classic fork-join queueing system with a Poisson input flow and exponential service times on homogeneous servers. When entering the system, tasks are divided into smaller components (subtasks), the number of which is equal to the number of subsystems. Then the subtasks are sent for service to the corresponding subsystems, consisting of a storage device of unlimited capacity and one server. The described functioning mechanism allows, using fork-join systems, to simulate processes occurring in many different real physical systems where tasks are parallelized. The article studies the dependence between the sojourn times of subtasks in subsystems, which is at the same time the main reason for the complexity of analyzing such systems. An approach is proposed for determining the quantiles of the response time distribution, while most works in the field of analysis of fork-join systems are concentrated on obtaining approximations only for the mathematical expectation of the response time. In addition, the approximation of the copula of sojourn times of subtasks (parts of one task) by the Gumbel copula and a new Kendall’s correlation coefficient estimation are obtained.